1. Introduction
Fractional Orlicz–Sobolev spaces (see, for instance, [Reference Bahrouni, Ounaies and Tavares8, Reference Bonder and Salort11]) constitute a non-homogeneous extension of the classical fractional Sobolev spaces (for an introduction, see [Reference Caffarelli12, Reference Di Nezza, Palatucci and Valdinoci20]). In recent years, these spaces have attracted growing attention due to their connection with non-local nonlinear operators of non-standard growth, in particular the fractional 
$\Phi$-Laplacian (see, e.g., [Reference Alberico, Cianchi, Pick and Slavíková3, Reference Azroul, Benkirane and Srati6, Reference Bahrouni, Missaoui and Ounaies7] and the references therein).
Simultaneously, the study of singular local and non-local homogeneous problems has evolved into a vast area of research (see [Reference Canino, Montoro, Sciunzi and Squassina13, Reference Coclite and Palmieri17, Reference Fan23, Reference Fulks and Maybee26, Reference Giacomoni, Schindler and Takac28, Reference Vázquez47–Reference Yijing and Shujie50] and references therein). Nevertheless, to the best of our knowledge, the investigation of singular problems involving the fractional 
$\Phi$-Laplacian within the framework of non-homogeneous and non-reflexive fractional Sobolev spaces remains largely unexplored. So far, only one paper has addressed this topic, focusing on the reflexive case (see [Reference Sbai, Hadfi and Ouardy44]).
Moreover, a quantitative analysis of fractional equations in the fractional Orlicz framework is still lacking in the literature. This deficiency can be mainly attributed to the intrinsic non-homogeneity of the 
$\Phi$-Laplacian operator. The purpose of this paper is to contribute to closing this gap by proving the existence of multiple solutions to a singular problem with a superlinear term in the setting of non-reflexive fractional Orlicz spaces. Our approach relies on the Nehari manifold method in combination with a nonlinear Rayleigh quotient.
More precisely, we establish existence, multiplicity, and regularity results for non-local quasilinear elliptic problems driven by the fractional 
$\Phi$-Laplacian. Namely, we consider the following non-local quasilinear elliptic problem:
\begin{equation}
\begin{array}{rl}
\displaystyle \left\{\begin{array}{lcr}
\displaystyle ( - \Delta_{\Phi})^s u = \lambda u^{-\gamma} + u^{p - 1}, & \text{in} \ \Omega \\
u \gt 0, & \text{in} \ \Omega \\
u = 0, & \text{in} \ \mathbb{R}^N \backslash \Omega
\end{array}\right.
\end{array}
\end{equation}where 
$\Omega\subset \mathbb{R}^N$ is a bounded domain with 
$N \geq 1$, 
$0 \lt s, \gamma \lt 1$ and 
$(- \Delta_{\Phi})^s$ denotes the nonlinear fractional 
$\Phi$-Laplacian operator which is defined by
\begin{eqnarray*}
(- \Delta_{\Phi})^s u(x)& = & p.v \int_{\mathbb{R}^N} \phi \left(\frac{|u(x) - u (y)|}{|x - y|^{s}} \right) \frac{(u(x) - u (y))}{|x - y|^{s}} \frac{dy}{|x - y|^{N}}, \\
&=& \lim_{\epsilon \rightarrow 0} \int_{\mathbb{R}^N \backslash B_{\epsilon}(0)} \phi \left(\frac{|u(x) - u (y)|}{|x - y|^{s}} \right) \frac{(u(x) - u (y))}{|x - y|^{s}} \frac{dy}{|x - y|^{N}}.
\end{eqnarray*} Throughout this work, we shall consider the following 
$N$-function
\begin{equation*}\Phi(t):=\int_0^{|t|}\phi(s)s\ ds, t \in \mathbb{R}\end{equation*}where 
$\phi: (0, \infty) \rightarrow (0, \infty)$ is in 
$C^1$ class. Later on, we shall consider the assumptions on 
$\phi$.
Recently, non-local problems and operators, particularly in the case 
$\Phi(t)=t^{2}$, have been extensively investigated in the literature and have attracted considerable attention from mathematicians across various research areas. Such non-local operators naturally arise in the modelling of a wide range of phenomena in the applied sciences, including optimization, finance, phase transitions, stratified materials, anomalous diffusion, crystal dislocation, and the ultra-relativistic limits of quantum mechanics (see [Reference Di Nezza, Palatucci and Valdinoci20]).
It is worth emphasizing that in the quasilinear setting, especially for 
$\Phi(t)=t^{p}$, the fractional 
$p$-Laplacian operator has been widely recognized as a suitable model for quasilinear reaction–diffusion equations, which have received significant attention in recent years. The main motivation for studying such problems lies in combining nonlinear and quasilinear non-local terms in order to capture the behaviour of nonlinear diffusion processes. For further references on this subject, we refer the reader to [Reference Vázquez46–Reference Vázquez48], where many of these nonlinear non-local diffusion problems are investigated.
The study of nonlinear elliptic equations involving quasilinear homogeneous operators is typically carried out within the framework of Sobolev spaces, which provide the natural setting for establishing weak solutions. In the case of non-homogeneous differential operators, this role is played by Orlicz–Sobolev spaces. For further details on the theory of Orlicz and Orlicz–Sobolev spaces, we refer to [Reference Mihăilescu and Rădulescu36, Reference Rao and Ren39] and the references therein. It is, therefore, natural to ask what kind of results can be obtained when the local 
$\Phi$-Laplace operator is replaced by its non-local counterpart, namely, the fractional 
$\Phi$-Laplacian.
It is worth mentioning that there exist some contributions concerning equations involving the fractional 
$\Phi$-Laplacian operator and fractional Orlicz–Sobolev spaces. In fact, results in this direction are still quite scarce. In 2019, J. Fernández Bonder first introduced results in this context (see [Reference Bonder and Salort11]), where the 
$\Phi$-Laplacian operator was generalized to the fractional setting. In the same work, the authors also introduced a suitable functional framework to study equations involving the fractional 
$\Phi$-Laplacian operator. Subsequently, in 2020, several authors, including S. Bahrouni and A. M. Salort, further advanced this line of research (see [Reference Bahrouni, Ounaies and Tavares8, Reference Salort40, Reference Salort41]), establishing fundamental results such as embedding theorems and other essential properties. We also wish to highlight the important contributions of Andrea Cianchi and collaborators, who obtained optimal results in the context of fractional Orlicz spaces, including continuous and compact embedding theorems, see [Reference Alberico, Cianchi, Pick and Slavíková2, Reference Alberico, Cianchi, Pick and Slavíková3].
As highlighted above, the literature on problems involving the fractional 
$\Phi$-Laplacian remains rather limited, as this field is still in active development. To the best of our knowledge, no paper has yet addressed the existence of multiple solutions for singular equations driven by the fractional 
$\Phi$-Laplacian.
It is important to emphasize that singular elliptic problems have been widely studied in recent years (see [Reference Coclite and Palmieri17, Reference Fulks and Maybee26, Reference Yijing, Shaoping and Yiming49, Reference Yijing and Shujie50]). In [Reference Yijing, Shaoping and Yiming49], the authors considered a class of singular elliptic problems with Dirichlet boundary conditions, where the nonlinearity involves both a superlinear term and a singular term. This framework allows one to establish the existence and multiplicity of solutions by considering a positive parameter 
$\lambda \gt 0$ in front of the singular term. Specifically, they studied the following singular elliptic problem:
\begin{equation}
\left\{
\begin{array}{ll}
- \Delta u = \lambda u^{\beta} + p(x) u^{-\gamma}, & u \in \Omega,\\[1mm]
u \gt 0, & \text{in } \Omega,\\
u = 0, & \text{on } \partial\Omega,
\end{array}
\right.
\end{equation}where 
$\Omega \subset \mathbb{R}^N$ is a bounded domain, 
$p:\Omega \to \mathbb{R}$ is a non-negative, non-trivial function in 
$L^2(\Omega)$, 
$1 \lt \beta \lt 2^* - 1$, 
$0 \lt \gamma \lt 1$, 
$2^* = 2N/(N-2)$ is the usual critical Sobolev exponent, 
$N \geq 3$, and 
$\lambda \gt 0$ is a parameter.
In [Reference Yijing and Shujie50], the authors considered another singular elliptic problem of the form
\begin{equation}
\left\{
\begin{array}{ll}
- \Delta u = \lambda a(x) u^{-\gamma} + b(x) u^p, & u \in \Omega,\\[1mm]
u \gt 0, & \text{in } \Omega,\\
u = 0, & \text{on } \partial\Omega,
\end{array}
\right.
\end{equation}where 
$a, b \in C(\overline{\Omega})$, 
$a \gt 0$, and 
$b$ can be a sign-changing function. They proved that there exists 
$\lambda^* \gt 0$ such that Problem (Q2) admits at least two non-negative solutions for each 
$\lambda \in (0, \lambda^*)$.
Many researchers have been attracted due to the pioneering on singular elliptic problems introduced in [Reference Fulks and Maybee26]. For example, in [Reference Coclite and Palmieri17, Reference Yijing, Shaoping and Yiming49], the authors proved that there exists 
$\Lambda^* \in (0, \infty)$ such that problem (Q1) has at least one solution for each 
$\lambda \in (0,\Lambda^*)$ and does not exist any solution for 
$\lambda \gt \Lambda^* $ whenever 
$p \equiv 1$ in 
$\overline{\Omega}$. The authors showed that there is a unique solution under extra assumptions on 
$p$.
For further studies on singular elliptic equations involving the classical Laplacian operator in bounded domains, we refer the interested reader to [Reference Crandall, Rabinowitz and Tatar18, Reference Lazer and Mckenna35, Reference Papageorgiou, Rãdulescu and Repovs38, Reference Yijing, Shaoping and Yiming49], where various techniques and types of nonlinearities are explored. For results on unbounded domains, we refer to [Reference Santos, Alves and Silva43] and references therein.
In the case of the local 
$p$-Laplacian, that is, when 
$\Phi(t) = t^p$ and 
$s = 1$, few results are known regarding (Pλ). In [Reference Giacomoni, Schindler and Takac28], the problem (Pλ) was studied with a superlinear term, and global multiplicity of solutions with respect to the parameter 
$\lambda$ was established by combining a Brezis–Nirenberg type result with sub- and supersolution methods; see also [Reference Bal and Garain9].
In the case of the non-local fractional 
$p$-Laplacian, that is, when 
$\Phi(t) = t^p$ and 
$0 \lt s \lt 1$, the problem (Pλ) has been studied by Fang [Reference Fan23] in the semilinear case 
$p=2$ without a perturbation term, and in [Reference Canino, Montoro, Sciunzi and Squassina13, Reference Garain and Mukherjee27] for the quasilinear case 
$p \gt 2$. The perturbed singular case has been investigated in [Reference Barrios, De Bonis, Medina and Peral10, Reference Mukherjee and Sreenadh37].
As mentioned at the beginning of the introduction, research on fractional Orlicz–Sobolev spaces and the associated nonlinear equations is still ongoing, and only a few results are available in this direction. To the best of our knowledge, there is only one paper that investigates equation (Pλ) without a perturbation term (see [Reference Sbai, Hadfi and Ouardy44]). In that work, the authors established the existence and uniqueness of solutions by employing critical point theory in reflexive fractional Orlicz spaces.
To the best of our knowledge, no results are available for problem (Pλ) in the non-reflexive case that prove the existence of multiple solutions. The main aim of this paper is to fill this gap by establishing the existence of at least two distinct solutions to problem (Pλ) in the non-reflexive setting by combining the Nehari manifold method with a nonlinear Rayleigh quotient.
In particular, the main contribution of the present work is to study non-local quasilinear elliptic problems driven by the 
$\Phi$-Laplacian operator and to establish the existence of a positive constant 
$\lambda^* \gt 0$ such that problem (Pλ) admits at least two positive weak solutions for each 
$\lambda \in (0, \lambda^*)$. In fact, 
$\lambda^*$ is the smallest positive number for which the associated fibering map admits inflection points. For further results on this subject, we refer the reader to [Reference Carvalho, Silva and Goulart16, Reference Il’yasov31, Reference Il’yasov32].
Moreover, we also address the singular case in the nonlinearity, where the corresponding energy functional 
$I_\lambda$ belongs only to the 
$C^0$ class.
Another contribution of this paper lies in the analysis of the non-reflexive case, where the weak
$^*$ topology is required. Problems of this type have been extensively studied in recent years using different methods; see, for instance, [Reference Alves, Silva and Pimenta4, Reference Gossez29, Reference Gossez30, Reference Silva, Gonçalves and Silva45]. The main difficulty here is to ensure the strong convergence of minimizing sequences restricted to the Nehari set. By establishing this strong convergence, we are able to prove the existence and multiplicity of solutions for a broad class of non-local elliptic problems.
1.1. Statements of the main results
As mentioned earlier, we shall investigate the existence and multiplicity of solutions for problem (Pλ). The main idea is to combine the Nehari method with the nonlinear Rayleigh quotient. To this end, we prove the existence of a critical value 
$\lambda^* \gt 0$ such that the Nehari method can be applied for every 
$\lambda \in (0,\lambda^*)$.
We now state the assumptions on the function 
$\phi$, which is required to satisfy the following conditions:
•
$(\phi_1)$
$t \mapsto t\phi(t)$ is increasing;•
$(\phi_2)$
$\displaystyle\lim_{t \rightarrow 0} t\phi(t)=0$ and
$\displaystyle\lim_{t \rightarrow \infty} t\phi(t)=\infty$;•
$(\phi_3)$
$\displaystyle\int_{0}^{1} \left(\frac{\tau}{\Phi(\tau)}\right)^{\frac{s}{N-s}} d\tau \lt \infty$ and
$\displaystyle\int_{1}^{\infty} \left(\frac{\tau}{\Phi(\tau)}\right)^{\frac{s}{N-s}} d\tau = \infty$;•
$(\phi_4)$
$-1 \leq l-2= \inf \frac{\phi^{'}(t)t}{\phi(t)} \leq m-2= \sup \frac{\phi^{'}(t)t}{\phi(t)} \lt \infty$.
Furthermore, we assume that:
•
$(H_{1})$
$m \lt p \lt l_{s}^*$,
$l_{s}^* = \frac{Nl}{N-ls}$;•
$(H_{2})$ The function
is strictly decreasing on
\begin{equation*}t \longmapsto\frac{(1+\gamma)\phi(t) + \phi^{'} (t)t}{t^{p-2}}\end{equation*}
$(0,+\infty)$.
Due to the nonlinear nature of the fractional 
$\Phi$-Laplacian operator, the appropriate function space for solving Problem (Pλ) is the fractional Orlicz–Sobolev space 
$W^{s,\Phi}_0(\Omega)$, see § 2.2. This space is defined as the weak
$^*$ closure of 
$C_0^\infty(\Omega)$ in the space
\begin{equation*}
W^{s,\Phi}(\Omega) := \left\{u \in L^{\Phi}(\Omega) : \frac{|u(x)-u(y)|}{|x-y|^s} \in L^{\Phi}(\mathbb{R}^N \times \mathbb{R}^N, d\mu) \right\},
\end{equation*}endowed with the norm
\begin{equation*}
\|u\|_{s,\Phi} := \|u\|_{\Phi} + [u]_{s,\Phi},
\end{equation*}where the Orlicz space 
$L^\Phi(\Omega)$ is defined as
\begin{equation*}
L^{\Phi}(\Omega) = \left\{u : \Omega \to \mathbb{R}\,\text{measurable} : \int_{\Omega} \Phi\left(\frac{|u|}{\lambda}\right) \, dx \lt +\infty\ \text{for some}\ \lambda \gt 0 \right\}.
\end{equation*}Recall also that the Luxemburg norm is given by
\begin{equation*}
\|u\|_{\Phi} := \inf \left\{\lambda \gt 0 : \int_{\Omega} \Phi\left(\frac{|u(x)|}{\lambda}\right) \, dx \leq 1 \right\}.
\end{equation*} Furthermore, 
$[u]_{s,\Phi}$ is the 
$(s,\Phi)$-Gagliardo seminorm which is given by
\begin{equation*}
[u]_{s,\Phi} := \inf \left\{\lambda \gt 0 : \iint_{\mathbb{R}^N \times \mathbb{R}^N} \Phi\left(\frac{|D_s u|}{\lambda}\right) \, d\mu \leq 1 \right\}.
\end{equation*}Notice also that reflexive fractional Orlicz spaces were introduced by Bonder and Salort in [Reference Bonder and Salort11] as a natural generalization of the well-known fractional Sobolev spaces, see for instance [Reference Di Nezza, Palatucci and Valdinoci20]. After that, Alberico, Cianchi, and Slavíková, in [Reference Alberico, Cianchi, Pick and Slavíková2], considered a systematic study of embeddings from the fractional Sobolev–Orlicz spaces into the Lebesgue spaces. The main difficulty in that work was to consider non-reflexive Sobolev–Orlicz spaces and non-reflexive Orlicz spaces where the standard embedding results cannot be verified.
Now, we consider the following definition:
Definition 1.1. A positive function 
$u \in W_{0}^{s,\Phi}(\Omega)$ is said to be a weak solution of problem (Pλ) if 
$u^{-\gamma}v\in L^1(\Omega)$ for all 
$v\in W_{0}^{s,\Phi}(\Omega)$ and
\begin{equation*}\frac{1}{2}\iint\limits_{\mathbb{R}^N \times \mathbb{R}^N} \phi \left(|D_su| \right) D_suD_sv~ d\mu = \int_{\Omega} (\lambda u^{-\gamma} + u^{p} )v \,dx,~\forall \in W_{0}^{s,\Phi}(\Omega),\end{equation*}where 
$D_{s}u =\frac{u(x) - u (y)}{(x - y)^{s}}~\text{and}~d\mu:=\frac{dxdy}{|x - y|^{N}}$.
From the standard variational point of view, finding weak solutions for Problem (Pλ) is equivalent to finding critical points for the associated energy functional 
$I_{\lambda}: W_{0}^{s,\Phi}(\Omega) \rightarrow \mathbb{R}$ which is defined by
\begin{equation*}
I_{\lambda}(u)=\iint\limits_{\mathbb{R}^N \times \mathbb{R}^N} \Phi(|D_{s}u|) d\mu - \frac{\lambda}{1-\gamma} \|u\|_{1-\gamma}^{1-\gamma} - \frac{1}{p}\int_{\Omega}|u|^{p} dx,
\end{equation*}where 
$\displaystyle\|u\|_{1-\gamma}^{1-\gamma}:=\int_{\Omega}|u|^{1 - \gamma} dx$.
It is worth noting that the functional 
$I_\lambda$ belongs only to the class 
$C^{0}$. Nevertheless, we observe that
\begin{equation*}
\lim_{t \to 0} \frac{I_\lambda((1+t)u) - I_\lambda(u)}{t}
\end{equation*}exists for every 
$u \in W^{s,\Phi}_0(\Omega)$. We denote this limit by 
$I_\lambda'(u)u$. In other words, the mapping 
$u \mapsto I_\lambda(u)$ admits a Gâteaux derivative at 
$u$ in the direction of 
$u$, for all 
$u \in W^{s,\Phi}_0(\Omega)$.
Under this condition, we adopt the following notation:
\begin{equation*}
I_\lambda^{''}(u)(u,u) := \lim_{t \to 0} \frac{I_\lambda^{'}((1+t)u)(1+t)u - I_\lambda^{'}(u)u}{t}.
\end{equation*}It is not hard to prove that
\begin{equation*}
I_{\lambda}^{'}(u)u = \displaystyle\iint\limits_{\mathbb{R}^N \times \mathbb{R}^N} \phi(|D_{s}u|)|D_{s}u|^2 d\mu - \lambda \|u\|_{1-\gamma}^{1-\gamma} - \int_{\Omega}|u|^{p} dx
\end{equation*}and
\begin{align*}
I_{\lambda}^{''}(u)(u,u) & = \displaystyle\iint\limits_{\mathbb{R}^N \times \mathbb{R}^N}\phi'(|D_{s}u|)(D_{s}u)^3+ \phi(|D_{s}u|)(D_{s}u)^2 ) d\mu + \lambda\gamma\|u\|_{1-\gamma}^{1-\gamma}\\ &\quad - (p-1)\int_{\Omega} |u|^{p}dx.
\end{align*}Throughout this work, we shall consider the Nehari set as follows
\begin{equation*}\mathcal{N}_{\lambda} = \left\{u \in W_{0}^{s,\Phi} (\Omega)\setminus \{0\} ; I_\lambda^{'}(u)u=0 \right\}.\end{equation*} The Nehari set 
$\mathcal{N}_{\lambda}$ can be split into the following sets:
\begin{equation*}\mathcal{N}_{\lambda}^{+} := \{u \in \mathcal{N}_{\lambda}; I_\lambda''(u)(u,u) \gt 0 \};\end{equation*}
\begin{equation*}\mathcal{N}_{\lambda}^{-} := \{u \in \mathcal{N}_{\lambda}; I_\lambda''(u)(u,u) \lt 0 \};\end{equation*}
\begin{equation*}\mathcal{N}_{\lambda}^{0} := \{u \in \mathcal{N}_{\lambda}; I_\lambda''(u)(u,u) = 0 \}.\end{equation*}Under our assumptions, we need to find solutions for the following minimization problems:
\begin{equation}
\mathcal{C}_{\mathcal{N}_{\lambda}^{+}}:=\inf\{I_{\lambda}(u): u \in \mathcal{N}_{\lambda}^{+} \}
\end{equation}and
\begin{equation}
\mathcal{C}_{\mathcal{N}_{\lambda}^{-}}:=\inf\{I_{\lambda}(u): u \in \mathcal{N}_{\lambda}^{-} \}.
\end{equation} The main goal is to prove that 
$\mathcal{C}_{\mathcal{N}_{\lambda}^{+}}$ and 
$\mathcal{C}_{\mathcal{N}_{\lambda}^{-}}$ are attained.
Now, we also need to consider some auxiliary sets in order to apply the nonlinear Rayleigh quotient. In fact, we consider
\begin{equation*}\mathcal{E}_\lambda=\{u\in W_0^{s,\Phi}(\Omega):I_\lambda(u)=0\}.\end{equation*} At this stage, we are in a position to define the nonlinear generalized Rayleigh quotients 
$R_n, R_e: W_0^{s,\Phi}(\Omega)\setminus\{0\}\to \mathbb{R}$ in the following form:
\begin{align}
R_n(u) & = \frac{\displaystyle\iint\limits_{\mathbb{R}^N \times \mathbb{R}^N} \phi(|D_{s}u|)(D_{s}u)^2 d\mu - ||u||^{p}_{p}}{||u||^{1-\gamma}_{1 - \gamma}}\qquad\mbox{and}\qquad\nonumber\\ R_e(u) & = \frac{(1-\gamma)\displaystyle\iint\limits_{\mathbb{R}^N \times \mathbb{R}^N} \Phi(|D_{s}u|) d\mu - \frac{1-\gamma}{p}||u||_{p}^{p}}{||u||_{1-\gamma}^{1-\gamma}}.
\end{align} Note that 
$u\in\mathcal{N}_\lambda$ if, and only if, 
$R_n(u)=\lambda$. Furthermore, 
$u\in\mathcal{E}_\lambda$ if, and only if, 
$R_e(u)=\lambda$. In order to state our main results, we define the following extremes:
\begin{equation*}\lambda_*=\inf_{u\in W_0^{s,\Phi}\setminus \{0\}}\max_{t \gt 0}R_e(tu)\qquad\mbox{and}\quad \lambda^*=\inf_{u\in W_0^{s,\Phi}\setminus \{0\}}\max_{t \gt 0}R_n(tu).\end{equation*}Under these conditions, we shall prove the following result:
Theorem 1.2. Suppose that 
$(\phi_1)$–
$(\phi_4)$ and 
$(H_1)$–
$(H_2)$ are satisfied. Then 
$0 \lt \lambda_{*} \lt \lambda^{*} \lt \infty$. Furthermore, for each 
$\lambda \in (0, \lambda^{*})$, Problem (Pλ) admits at least one weak positive solution 
$u\in W_{0}^{s,\Phi}(\Omega)$ that satisfies the following statements:
(i)
$I_{\lambda}^{''}(u)(u,u) \gt 0$ and
$I_{\lambda}(u) \lt 0$;(ii)
$[u]_{s,\Phi} \rightarrow 0$ as
$\lambda \rightarrow 0$.
Now, we shall state our second result as follows:
Theorem 1.3. Suppose that 
$(\phi_1)$–
$(\phi_4)$ and 
$(H_1)$–
$(H_2)$ are satisfied. Then, for each 
$\lambda \in (0, \lambda^{*})$, Problem (Pλ) admits at least one weak positive solution 
$u\in W_{0}^{s,\Phi}(\Omega)$ that verifies the following assertions:
(i)
$I_{\lambda}^{''}(u)(u,u) \lt 0$;(ii)
$I_{\lambda}(u) \gt 0$ for each
$\lambda \in (0, \lambda_{*})$;(iii)
$I_{\lambda}(u)= 0$ for
$\lambda = \lambda_{*}$;(iv)
$I_{\lambda}(u) \lt 0$ for each
$\lambda \in (\lambda_{*}, \lambda^{*})$.
As a consequence, we consider the following result:
Corollary 1.4. Suppose that 
$(\phi_1)$–
$(\phi_4)$ and 
$(H_1)$–
$(H_2)$ hold. Then, for each 
$\lambda \in (0, \lambda^{*})$, Problem (Pλ) has at least two weak positive solutions.
The next result establishes the existence of a solution to Problem (Pλ) at level 
$\lambda=\lambda^*$.
Theorem 1.5. Assume that 
$(\phi_{1})$–
$(\phi_{4})$, 
$(H_1)$, and 
$(H_2)$ hold. Assume that 
$\lambda = \lambda^*$. Then Problem (Pλ) admits at least one non-trivial solution in 
$\mathcal{N}_{\lambda}^{+} \cup \mathcal{N}_{\lambda}^{0}$.
Now, we shall prove a regularity result for our main problem. Firstly, we require the following definition:
Definition 1.6. We say that an 
$N$-function 
$\Phi$ satisfies the 
$\Delta'$-condition (
$\Phi \in \Delta'$) if there exists 
$C \gt 0$ such that
\begin{equation*}\Phi(st) \leq C \Phi(s)\Phi(t), \ \forall s,t \gt 0.\end{equation*}Theorem 1.7. Suppose that 
$(\phi_{1})$–
$(\phi_{4})$, 
$(H_1)$–
$(H_2)$, and 
$\Phi \in \Delta^{'}$ hold. Assume that 
$p \lt l^{*}_{s} - (m - l)$. Then any weak solution for Problem (Pλ) belongs to 
$L^{\infty}(\Omega)$.
1.2. Notation
In the present work, we shall use the following notation:
•
$K, K_1, C, C_1, C_2 \cdots,$ denotes the positive constants (whose values are irrelevant).• The norm in
$L^t(\Omega)$ and
$L^{\infty}(\Omega)$ will be denoted by
$\|\cdot\|_t, t \in [1, \infty]$.•
$\|\cdot\|$ denotes the Luxemburg norm of
$W^{s,\Phi}_0(\Omega)$.•
$[\ \cdot\ ]_{s,\Phi}$ denotes the Gagliardo seminorm.• The open ball in
$\mathbb{R}^N$ centred at
$x \in \mathbb{R}^N$ with radius
$r \gt 0$ is denoted by
$B_r(x)$.• Given
$A \subset \mathbb{R}^N$, we define
$A^c = \{x \in \mathbb{R}^N: x \notin A\}$.• Given any measurable set
$A \subset \mathbb{R}^N$, the Lebesgue measure of
$A$ is denoted by
$\mu(A)$.
1.3. Outline
The remainder of this paper is organized as follows: In the forthcoming section, we consider some preliminaries results involving the fractional 
$\Phi$-Laplacian operator. In § 3, we consider some results related to the nonlinear Rayleigh quotient together with the Nehari method. Section 4 is devoted to the proof of our main theorems for each 
$\lambda \in (0, \lambda^*)$. In § 5, we consider some existence and multiplicity of solutions for our main problem assuming that 
$\lambda = \lambda^*$. In § 6, we prove that any weak solution of our main problem is bounded assuming extra conditions on 
$\ell, m , p$, and 
$q$.
2. Basic results
In the present section, we provide a brief overview for fractional Orlicz–Sobolev spaces which is essential for our work. For a more comprehensive discussion, we refer the interested reader to [Reference Adams1, Reference Alberico, Cianchi, Pick and Slavíková2, Reference Bonder and Salort11, Reference Fukagai and Narukawa25, Reference Gossez29, Reference Gossez30, Reference Krasnosel’ski and Ruticki34, Reference Rao and Ren39]. We begin with by defining an 
$N$-function 
$\Phi:\mathbb{R}\to (0,\infty)$ as follows:
\begin{equation*}
\Phi(t) = \int_0^{|t|} \phi(\tau) \tau d\tau, \quad t \geq 0,
\end{equation*}where 
$\phi:(0,\infty)\to\mathbb{R})$ satisfies the following assertions:
(1)
$\displaystyle\lim_{t\to 0}\phi(t)t=0$;(2)
$\displaystyle\lim_{t \rightarrow \infty}\phi(t)t=\infty$;(3)
$\phi(t)t \gt 0$ for
$t \gt 0$;(4)
$t\mapsto \phi(t)t$ is right-continuous at any point
$t \geq 0$;(5)
$t\mapsto \phi(t)t$ is non-decreasing in
$(0,\infty)$.
Notice that, assuming 
$(\phi_1)$ and 
$(\phi_2)$, the function 
$\Phi$ is an 
$N$-function. Furthermore, 
$\Phi$ satisfies the 
$\Delta_2$-condition if there exists 
$K \gt 1$ such that
\begin{equation*}
\Phi(2t) \leq K\Phi(t), \quad \forall t \geq 0.
\end{equation*} Recall also that 
$\Phi$ satisfies the 
$\Delta_2$-condition if and only if there exist 
$l \geq 1$ and 
$m \gt 1$ such that
\begin{equation}
1 \leq l \leq \frac{\phi(t)t^2}{\Phi(t)} \leq m.
\end{equation} It is straightforward to show that 
$(\phi_4)$ implies that (4) is satisfied. It is important to emphasize that 
$\Phi$ satisfies the global 
$\Delta^{'}$ condition if there exists a constant 
$C \gt 0$ such that
\begin{equation*}\Phi(st) \leq C\Phi(s)\Phi(t), \ s,t \gt 0.\end{equation*} The conjugate N-function 
$\tilde{\Phi}:\mathbb{R}\to [0,\infty)$ of the 
$N$-function 
$\Phi$ is defined by
\begin{equation*}
\tilde{\Phi} (s) = \sup \{st - \Phi(t) ; \ \ t \gt 0 \}.
\end{equation*} Hence, we show that 
$\widetilde{\Phi}(\phi(t)t) \leq \Phi(2t)$ holds for all 
$t \in \mathbb{R}$. It is important to point out that if 
$\Phi$ satisfies (4) the function 
$\widetilde{\Phi}$ may not satisfy 
$\Delta_2$ condition.
Next, we introduce some partial order relations between 
$N$-functions. We say that 
$\Psi$ is stronger (essentially stronger) than 
$\Phi$, denoted by 
$\Phi \lt \Psi$ (respectively, 
$\Phi \ll \Psi$), if
\begin{equation*}
\Phi(t) \leq \Psi(kt), \quad \text{for some (for all) } k \gt 0, \ t \geq t_0,
\end{equation*}where 
$t_0 \gt 0$ depends on 
$k \gt 0$. Notice also that 
$\Phi \ll \Psi$ is equivalent to
\begin{equation*}
\lim_{t \to +\infty} \frac{\Phi(kt)}{\Psi(t)} = 0, \quad \text{for all } k \gt 0.
\end{equation*}2.1. Orlicz and Orlicz–Sobolev spaces
Let 
$\Omega$ be open domain of 
$\mathbb{R}^N$ and 
$\Phi$ an 
$N$-function. The space
\begin{equation*}L^{\Phi}(\Omega) = \left\{u: \Omega \rightarrow \mathbb{R} \ \textrm{mensurable}; \ \int_{\Omega} \Phi\left(\frac{|u|}{\lambda}\right)dx \lt + \infty, \mbox{for some }\lambda \gt 0\right\}\end{equation*}is called Orlicz-space. This linear space is a Banach space equipped with the Luxemburg Norm given by
\begin{equation*}||u||_{\Phi}= \inf \left\{\lambda \gt 0; \int_{\Omega} \Phi\left(\frac{|u(x)|}{\lambda}\right)dx \leq 1\right\}.\end{equation*} It is worth noting that if 
$\Phi$ satisfies the 
$\Delta_2$ condition we obtain that
\begin{equation*}\int_\Omega\Phi\left(\frac{u}{\|u\|_\Phi}\right)dx=1.\end{equation*}In [Reference Fukagai, Ito and Narukawa24], the authors proved the following result:
Proposition 2.1. Suppose that 
$(\phi_{1})$, 
$(\phi_{2})$, and 
$(\phi_{4})$ are verified. Then, the 
$N$-function 
$\Phi$ satisfies
\begin{equation*}\displaystyle\zeta_0 (t)\Phi(\rho) \leq \Phi(\rho t) \leq \zeta_1 (t)\Phi(\rho) \ \ \ \rho, t \gt 0,\end{equation*}
\begin{equation*}\displaystyle\zeta_0 (||u||_{\Phi}) \leq \int_{\Omega}\Phi(x) dx \leq \zeta_1 (||u||_{\Phi}), \ \ \ \ u\in L_{\Phi}(\Omega)\end{equation*} where 
$\zeta_0 (t) = \min \{t^l,t^m\}$ and 
$\zeta_1 (t) = \max \{t^l,t^m\}$, 
$t \geq 0$.
Now, we consider the set 
$E^\Phi(\Omega)=\overline{C_0^\infty(\Omega)}^{L^\Phi(\Omega)}$. Recall that 
$E^\Phi(\Omega)=L^\Phi(\Omega)$ if, and only if, 
$\Phi \in \Delta_2$. Furthermore, we emphasize that
\begin{equation*}(E^{\widetilde\Phi}(\Omega))'=L^\Phi(\Omega)\qquad\mbox{and}\qquad(E^\Phi(\Omega))'=L^{\widetilde\Phi}(\Omega).\end{equation*}2.2. Fractional Orlicz–Sobolev spaces
This subsection is spirit in part due to the work of J. F. Bonder and A. M. Salort introduced, see [Reference Bonder and Salort11], where the Fractional Orlicz–Sobolev space 
$W^{s,\Phi}(\Omega)$ with 
$0 \lt s \lt 1$ was considered. Namely, we consider the following space:
\begin{equation*}W^{s,\Phi}(\Omega)=\left\{u \in L^{\Phi}(\Omega); D_su \in L^{\Phi}(\mathbb{R}^N \times \mathbb{R}^N, d\mu) \right\}\end{equation*}which is a Banach space endowed with the norm
\begin{equation*}||u||_{s,\Phi}=||u||_{\Phi} + [u]_{s,\Phi}.\end{equation*} Recall also that 
$[.]_{s,\Phi}$ denotes the 
$(s,\Phi)$-seminorm of Gagliardo given by
\begin{equation*}[u]_{s,\Phi}:= \inf\left\{\lambda \gt 0; \ \displaystyle\iint\limits_{\mathbb{R}^N \times \mathbb{R}^N} \Phi \left( \frac{|D_su|}{\lambda} \right)d\mu \leq 1\right\}\end{equation*}and
\begin{equation*}D_{s}u =\frac{u(x) - u (y)}{|x - y|^{s}}\qquad\mbox{and}\qquad \frac{dxdy}{|x - y|^{N}} = d\mu, \ \ \ \forall (x,y) \in (\mathbb{R}^N \times \mathbb{R}^N).\end{equation*}Now, using the ideas discussed in the proof of Proposition 2.1, we prove the following result:
Proposition 2.2. Suppose that 
$(\phi_{1})$, 
$(\phi_{2})$, and 
$(\phi_{4})$ are satisfied. Then, the 
$N$-function 
$\Phi(t)$ satisfies
\begin{equation*}\displaystyle\zeta_0 ([u]_{s,\Phi}) \leq \displaystyle\iint\limits_{\mathbb{R}^N \times \mathbb{R}^N}\Phi(|D_s u|) d\mu \leq \zeta_1 ([u]_{s,\Phi}), \ \ \ \ u\in W^{s,\Phi}(\Omega).\end{equation*}Proposition 2.3. (See [Reference Salort and Vivas42])
Let 
$\Phi$ be an 
$N$-function satisfying the 
$\Delta_2$-condition. Then the functional
\begin{equation*}\rho_{s,\Phi}(u) = \displaystyle\iint\limits_{\mathbb{R}^N \times \mathbb{R}^N}\Phi(|D_s u|) d\mu\end{equation*}is Fréchet differentiable. Furthermore, 
$u \mapsto \rho_{s,\Phi}(u)$ is lower semicontinuous in the weak
$^*$ topology.
Now, we shall consider a result proved by [Reference de Albuquerque, de Assis, Carvalho and Salort19, Proposition 3.7] which is called as the Brezis–Lieb lemma. This kind of result is also verified for the case 
$l=1$. Namely, we obtain the following result
Proposition 2.4. (Brezis–Lieb type lemma)
Assume that 
$(\phi_1)$–
$(\phi_4)$ hold. Let 
$\{u_n\}_{n\in\mathbb{N}} $ be a bounded sequence in 
$ W_0^{s,\Phi}(\Omega)$ such that 
$u_n(x) \rightarrow u(x)$ a.e. in 
$\mathbb{R}^N$. Then, 
$u \in W_0^{s,\Phi}(\Omega)$ and
\begin{equation*}\lim_{n\rightarrow +\infty} \left( \rho_{s,\Phi}(u_n) - \rho_{s,\Phi}(u_n - u) \right) = \rho_{s,\Phi}(u).\end{equation*}The next result will be crucial in order to prove the strong convergence of minimizer sequences in the Nehari set. In fact, we shall prove the following result:
Proposition 2.5. Assume that 
$(\phi_{1})$–
$(\phi_{4})$, 
$(H_1)$, and 
$(H_2)$ hold. Let 
$(u_k)_{k \in \mathbb{N}}$ be a sequence in 
$W^{s,\Phi}_0(\Omega)$ such that 
$(u_k) \stackrel{*}{\rightharpoonup} u$ and 
$(u_k)\not \to u$ in 
$W^{s,\Phi}_0(\Omega)$. Then
(i)
$\displaystyle\iint\limits_{\mathbb{R}^N \times \mathbb{R}^N} \Phi(|D_s u|)d\mu \lt \displaystyle\liminf_{k \rightarrow \infty} \displaystyle\iint\limits_{\mathbb{R}^N \times \mathbb{R}^N} \Phi(|D_s u_k |)d\mu $;(ii)
$\displaystyle\iint\limits_{\mathbb{R}^N \times \mathbb{R}^N} \phi(|D_s u|)|D_s u|^2 d\mu \lt \displaystyle\liminf_{k \rightarrow \infty} \displaystyle\iint\limits_{\mathbb{R}^N \times \mathbb{R}^N} \phi(|D_s u_k |)|D_s u_k |^2 d\mu$.
Proof. The proof of item 
$(i)$ follows from Proposition 2.3 by using the fact that 
$ u \mapsto \rho_{s,\Phi}$ is lower semicontinuous in the weak
$^*$ topology. Let us assume by contradiction that 
$(u_k) \nrightarrow u$ in 
$W^{s,\Phi}_0(\Omega)$ and
\begin{equation}
\displaystyle\iint\limits_{\mathbb{R}^N \times \mathbb{R}^N} \Phi(|D_s u|)d\mu = \displaystyle\liminf_{k \rightarrow \infty} \displaystyle\iint\limits_{\mathbb{R}^N \times \mathbb{R}^N} \Phi(|D_s u_k |)d\mu.
\end{equation} Recall that 
$\Phi$ is convex and 
$ u_k\stackrel{*}{\rightharpoonup} u$ in 
$W^{s,\Phi}_0(\Omega)$. Define 
$(v_k)_{k \in \mathbb{N}}$ in the following form 
$u_k = u + v_k$. Hence, 
$v_k \rightarrow 0$ 
$\mu-$ a.e. in 
$\Omega$. Now, by using Proposition 2.4, we have
\begin{equation*}\displaystyle\liminf_{k \rightarrow \infty} \displaystyle\iint\limits_{\mathbb{R}^N \times \mathbb{R}^N} \big(\Phi(|D_s u_k |) - \Phi(|D_s (u_k - u)|) - \Phi(|D_s u|)\big)d\mu = 0.\end{equation*}Therefore, we obtain that
\begin{equation*}\displaystyle\lim_{k \rightarrow \infty} \displaystyle\iint\limits_{\mathbb{R}^N \times \mathbb{R}^N} \Phi(|D_s (u_k - u)|)d\mu = 0.\end{equation*} Since 
$\Phi \in \Delta_2$, we conclude that 
$[ u_k- u]_{s,\Phi} \rightarrow 0$. This is a contradiction showing the desired result.
$(ii)$ Let us we assume by contradiction that 
$ u_k \stackrel{*}{\rightharpoonup} u$ and 
$ u_k \nrightarrow u$ in 
$W_{0}^{s, \Phi}(\Omega)$ in such way that
\begin{align}
\displaystyle\left \lt (-\Delta_{\Phi})^s u, u \right \gt & =\iint\limits_{\mathbb{R}^N \times \mathbb{R}^N} \phi(|D_s u|)|D_s u|^2 d\mu = \displaystyle\liminf_{k \rightarrow \infty} \displaystyle\iint\limits_{\mathbb{R}^N \times \mathbb{R}^N} \phi(|D_s u_k |)|D_s u_k |^2 d\mu\nonumber\\ &=\liminf_{k \rightarrow \infty} \left \lt (-\Delta_{\Phi})^s u_k , u_k \right \gt .
\end{align}It is not hard to see that
\begin{equation*}\lim_{k \rightarrow \infty} \left \lt (-\Delta_{\Phi})^s u_k, u_k - u \right \gt = \lim_{k \rightarrow \infty} (\left \lt (-\Delta_{\Phi})^s u_k, u_k \right \gt - \left \lt (-\Delta_{\Phi})^s u_k, u \right \gt ) = 0.\end{equation*}Here, we used the fact that
\begin{equation*}\displaystyle\lim_{k \rightarrow \infty} \left \lt (-\Delta_{\Phi})^s u_k, u \right \gt = \left \lt (-\Delta_{\Phi})^s u, u \right \gt .\end{equation*}This can be done proving that
\begin{equation*}\phi(|D_s u_k|)D_s u_k \stackrel{*}\rightharpoonup \phi(|D_s u|)D_s u\qquad\mbox{in}~ L_{\widetilde\Phi}(\Omega;d \mu),\end{equation*}where 
$L_{\widetilde\Phi}(\Omega;d \mu)$ is the Orlicz space with respect to the measure 
$\mu$. It follows from [Reference Fukagai, Ito and Narukawa24, Lemma A.2] that
\begin{align*}
\tilde{\Phi}(\phi(|D_s u_k|)D_s u_k) \leq \Phi(2|D_s u_k|) \leq 2^m \Phi(|D_s u_k|) \leq 2^m \phi(|D_s u_k|)|D_s u_k|^2 = f_k.
\end{align*} Now, by using (6), we deduce that 
$f_k \to f := 2^m \phi(|D_s u|)|D_s u|^2$. Since 
$t \mapsto \phi(t)t^2$ is continuous and increasing, we infer that 
$D_s u_k \rightarrow D_s u \ \mu$-a.e. in 
$\Omega$. Consequently, we see that
\begin{equation}
\phi(|D_s u_k|)D_s u_k \rightarrow \phi(|D_s u|)D_s u \ \ \mu\text{-a.e.}
\end{equation}According to Lemma in [Reference Gossez30, Page 63], we infer also that
\begin{equation*}\displaystyle\lim_{k \rightarrow \infty} \left \lt (-\Delta_{\Phi})^s u_k, u_k \right \gt = \left \lt (-\Delta_{\Phi})^s u, u \right \gt .\end{equation*} It follows also from [Reference de Albuquerque, de Assis, Carvalho and Salort19, Theorem 3.14] that 
$u_k \rightarrow u$ in 
$W_{0}^{s, \Phi}(\Omega)$ which leads to a contradiction. This ends the proof.
Now, we shall introduce some important tools that are fundamental for our research on continuous and compact embedding from the fractional Orlicz–Sobolev space into the Orlicz spaces. Let us assume that 
$\Phi$ satisfies 
$(\phi_1)$–
$(\phi_4)$. Define the critical function 
$t \mapsto \Phi_*(H^{-1}(t)),~t\in\mathbb{R}$ where
\begin{equation*}
H(t)=\left(\int_{0}^{|t|}\left(\frac{\tau}{\Phi(\tau)}\right)^{\frac{s}{N-s}}\, d\tau\right)^{\frac{N-s}{N}}, \;\; t\in \mathbb{R}.
\end{equation*}Under these conditions, we shall prove the following result:
Proposition 2.6. Suppose 
$(\phi_1)$–
$(\phi_4)$. Let 
$\Omega \subset \mathbb{R}^N$ be a bounded Lipschitz domain and an 
$N$-function 
$\Psi$ satisfying 
$\Psi\ll \Phi_\ast$. Then the embedding 
$W^{s,\Phi}(\Omega)\hookrightarrow L^{\Psi}(\Omega)$ is compact and 
$W^{s,\Phi}(\Omega)\hookrightarrow L^{\Phi_\ast}(\Omega)$ is continuous.
At this stage, we introduce our workspace 
$W_0^{s,\Phi}(\Omega)$, which is defined by
\begin{equation*}W^{s,\Phi}_0(\Omega)=\left\{u \in W^{s,\Phi}(\mathbb{R}^N):~ u=0 \ \ \mbox{a.e. in } \mathbb{R}^N\setminus \Omega\right\}.\end{equation*}Furthermore, we consider the Poincaré-type inequality for fractional Orlicz–Sobolev spaces in the following form:
Proposition 2.7. ([Reference Salort and Vivas42])
Let 
$\Omega \subset \mathbb{R}^N$ be a bounded domain. Consider 
$\Phi$ be an 
$N$-function. Then, for 
$s\in(0, 1)$, it holds that
\begin{equation*}\|u\|_\Phi\leq C[u]_{s,\Phi},~\forall u\in W^{s,\Phi}_0(\Omega),\end{equation*}for some 
$C=C(s,n,\Omega) \gt 0$.
As a consequence, we obtain that 
$[.]_{s,\Phi}$ is an equivalent norm in 
$ W^{s,\Phi}_0(\Omega)$.
3. Nehari method and the nonlinear Rayleigh quotient
In this section, we shall consider the Nehari method taking into the fibering maps. The main idea is to look for the fibering map 
$t \mapsto I(t u)$ with 
$t \gt 0$ and 
$u \in W^{s,\Phi}(\Omega)$ proving that critical points for the fibering map provide us critical points for the energy functional 
$I$. More precisely, we shall consider the map 
$\beta : (0, \infty) \rightarrow (0, \infty)$ given by
\begin{equation*}\beta(t):=I_{\lambda}(tu)=\displaystyle\iint\limits_{\mathbb{R}^N \times \mathbb{R}^N} \Phi(t|D_{s}u|) d\mu - \frac{\lambda}{1-\gamma} t^{1-\gamma}\int_{\Omega}|u|^{1 - \gamma} dx - \frac{t^{p}}{p}\int_{\Omega}|u|^{p} dx,\end{equation*}where 
$u \in W_{0}^{s,\Phi} (\Omega)\setminus\{0\}$. Moreover, we observe that
\begin{eqnarray*}
\beta^{'}(t) & = &\displaystyle\iint\limits_{\mathbb{R}^N \times \mathbb{R}^N} \phi(t|D_{s}u|)(D_{s}u)^2 d\mu - \lambda t^{-\gamma} \int_{\Omega}|u|^{1 - \gamma} dx - t^{p-1}\int_{\Omega}|u|^{p} dx =0
\end{eqnarray*}if, and only if, 
$tu \in \mathcal{N}_{\lambda}$. In particular, 
$u \in \mathcal{N}_{\lambda}$ if, and only if, 
$\beta^{'}(1) = 0$. It is important to stress also that
\begin{eqnarray*}
\beta^{''}(t)& = & \displaystyle\iint\limits_{\mathbb{R}^N \times \mathbb{R}^N} \phi^{'}(t|D_su|)(D_su)^2 + \phi(t|D_su|)|D_su|^3 + \lambda \gamma t^{-\gamma - 1}\displaystyle\int_{\Omega} |u|^{1-\gamma}dx - (p-1)t^{p-2}\\ &&\quad\times \displaystyle \int_{\Omega} |u|^p dx.
\end{eqnarray*}3.1. Nonlinear Rayleigh quotients
The present subsection relies on nonlinear generalization of Rayleigh quotients whose their critical points provide us some extreme values in order to apply the Nehari method. This method has been extensively studied in recent years, see for instance [Reference Carvalho, Gasiński, Júnior and Silva14, Reference Il’yasov31–Reference Il’yasov and Silva33]. Our primary challenge comes from the lack of homogeneity of the 
$\Phi$-Laplacian operator. In order to overcome this difficulty, we introduced hypothesis 
$(H_2)$, which implies that the nonlinear Rayleigh quotient has a unique critical point.
Firstly, we consider the functionals 
$R_n, R_e: W_{0}^{s,\Phi}(\Omega) \setminus \{0\} \to \mathbb{R}$ associated with the parameter 
$\lambda$ defined by (3). Based on the definitions of 
$R_n$ and 
$R_e$, we consider the following statements:
Remark 3.1. Let 
$t \gt 0$ and 
$u \in W_{0}^{s,\Phi}(\Omega) \setminus \{0\}$ be fixed. Then 
$tu \in \mathcal{N}_\lambda$ if and only if 
$R_n(tu) = \lambda$. More specifically, we obtain the following assertions:
(i)
$R_n(tu)=\lambda$ if, and only if,
$I_{\lambda}^{'}(tu)tu=0$;(ii)
$R_n(tu) \gt \lambda$ if, and only if,
$I_{\lambda}^{'}(tu)tu \gt 0$;(iii)
$R_n(tu) \lt \lambda$ if, and only if,
$I_{\lambda}^{'}(tu)tu \lt 0$.
Remark 3.2. Similarly, we obtain that
(i)
$R_e(tu)=\lambda$ if, and only if,
$I_{\lambda}(tu)tu=0$;(ii)
$R_e(tu) \gt \lambda$ if, and only if,
$I_{\lambda}(tu)tu \gt 0$;(iii)
$R_e(tu) \lt \lambda$ if, and only if,
$I_{\lambda}(tu)tu \lt 0$;(iv)
$R_n(tu)-R_e(tu)=t\frac{t}{dt}R_e(tu),~t \gt 0,~u\in W_0^{s,\Phi}(\Omega)\setminus\{0\}$.
In order to understand the behaviour of 
$R_n$ and 
$R_e$, we consider the fibres 
$Q_{n}: (0,+\infty) \rightarrow \mathbb{R}$ and 
$Q_{e}: (0,+\infty) \rightarrow \mathbb{R}$ defined as follows:
\begin{equation*}Q_{n}(t):= R_n(tu)=\frac{t^{1 + \gamma} \displaystyle\iint\limits_{\mathbb{R}^N \times \mathbb{R}^N} \phi(|D_{s}tu|)|D_{s}u|^2 d\mu - t^{p + \gamma -1}||u||^{p}_{p}}{||u||^{1-\gamma}_{1 - \gamma}}\end{equation*}and
\begin{equation*}\displaystyle Q_{e}(t):= R_e(tu)=\displaystyle (1-\gamma)\frac{t^{\gamma - 1}\displaystyle\iint\limits_{\mathbb{R}^N \times \mathbb{R}^N} \Phi(|D_{s}tu|) d\mu - \frac{1}{p} t^{p + \gamma - 1}||u||_{p}^{p}}{||u||_{1-\gamma}^{1-\gamma}}\end{equation*}where 
$u\in W_0^{s,\Phi}(\Omega)\setminus\{0\}$ is fixed.
Proposition 3.3. Suppose that 
$(\phi_1)$–
$(\phi_4)$ and 
$(H_1)$–
$(H_2)$ are hold. Then the function 
$t \mapsto Q_n(t)=R_n(tu)$ satisfies
(i)
$\displaystyle\lim_{t \rightarrow 0^{+}} \frac{Q_n(t)}{t^{m+\gamma - 1}} \gt 0 \ \ \mbox{and } \displaystyle\lim_{t \rightarrow 0^{+}} \frac{Q_n'(t)}{t^{m+\gamma -2}} \gt 0$;(ii)
$\displaystyle\lim_{t \rightarrow +\infty} \frac{Q_n(t)}{t^{p+\gamma - 1}} \lt 0 \ \ \mbox{and } \displaystyle\lim_{t \rightarrow +\infty} \frac{Q_n'(t)}{t^{p+\gamma -2}} \lt 0$.
Proof. 
$(i)$ In view of condition (4) and Proposition 2.1, we deduce that
\begin{align*}
\lim_{t \rightarrow 0^{+}}\displaystyle\frac{Q_{n}(t)}{t^{m +\gamma -1}} &= \lim_{t \rightarrow 0^{+}}\frac{\displaystyle t^{2-m} \iint\limits_{\mathbb{R}^N \times \mathbb{R}^N} \phi(|D_{s}tu|)|D_{s}u|^2 d\mu - t^{p - m}||u||^{p}_{p}}{||u||^{1-\gamma}_{1 - \gamma}}\\
&\geq \lim_{t \rightarrow 0^{+}}\frac{t^{-m} \displaystyle\iint\limits_{\mathbb{R}^N \times \mathbb{R}^N} \Phi(t|D_{s}u|) d\mu }{||u||^{1-\gamma}_{1 - \gamma}}~~\geq ~~\frac{\displaystyle\iint\limits_{\mathbb{R}^N \times \mathbb{R}^N} \Phi(|D_{s}u|) d\mu }{||u||^{1-\gamma}_{1 - \gamma}} \gt 0.
\end{align*} Similarly, by using hypothesis 
$(\phi_4)$, we obtain that
\begin{equation*}\displaystyle\lim_{t \rightarrow 0^{+}} \frac{Q_n'(t)}{t^{m+\gamma -2}} \geq \displaystyle\frac{(2 + \gamma) \displaystyle\iint\limits_{\mathbb{R}^N \times \mathbb{R}^N} \Phi(|D_{s}u|) d\mu }{||u||^{1-\gamma}_{1 - \gamma}} \gt 0.\end{equation*} This ends the proof of item 
$(i)$. The proof of item 
$(ii)$ follows arguing as done in the previous item. The details are omitted.
In order to apply the Rayleigh quotient method, we need to determine the critical points of the fibres 
$t \mapsto Q_{n}(t)$ and 
$t \mapsto Q_{e}(t)$. For this reason, we mention that
\begin{eqnarray*}
{Q_{n}'(t)} &=&\frac{(1+\gamma)t^{\gamma}\displaystyle\iint\limits_{\mathbb{R}^N \times \mathbb{R}^N} \phi(|tD_{s}u|)(D_{s}u)^2 d\mu +t^{1+\gamma}\displaystyle\int\int_{{\mathbb{R}^N}\times {\mathbb{R}^N} } \phi'(|tD_{s}u |)|D_{s}u|^3 d\mu}{||u||_{1-\gamma}^{1-\gamma}}\\ &&\quad -(p + \gamma -1)\frac{||u||^{p}_{p}}{||u||_{1-\gamma}^{1-\gamma}}t^{p + \gamma - 2}.
\end{eqnarray*} Thus, the critical points of 
$t \mapsto Q_{n}(t)$ are ensured solving the following equation
\begin{equation}
{(p + \gamma - 1)}||u||^{p}_{p} = \displaystyle\iint\limits_{\mathbb{R}^N \times \mathbb{R}^N}\frac{[(1 + \gamma) \phi(|tD_{s}u|) + \phi'(|tD_{s}u|)(|tD_{s}u|)|D_{s}u|^p}{|tD_{s}u|^{p-2}}d\mu.
\end{equation} Similarly, we find the critical points of 
$t \mapsto Q_{e}(t)$ by solving the following identity
\begin{equation*}\frac{(p + \gamma - 1)}{p}||u||^{p}_{p} = t^{-p}\displaystyle\iint\limits_{\mathbb{R}^N \times \mathbb{R}^N} \left[\Phi(|tD_{s}u|) + \phi(|tD_{s}u|) |tD_{s}u|^2\right]d\mu.\end{equation*}Under these conditions, we are able to prove the following result:
Proposition 3.4. Assume that 
$(\phi_1)$–
$(\phi_4)$ and 
$(H_1)$–
$(H_2)$ hold. Then, for each 
$u \in W_{0}^{s,\Phi}(\Omega)\backslash \{0\}$, there exists a unique 
$t_{n}(u) \gt 0$ such that 
$Q_n'(t)=0$. Moreover, 
$t_{n}: W_{0}^{s,\Phi}(\Omega)\backslash {0} \rightarrow (0, \infty)$ is a continuous functional.
Proof. According to Proposition 3.3, there is at least one 
$t_n(u) \gt 0$ such that 
$Q_n'(t)=0$ is satisfied. Moreover, 
$Q_n'(t)=0$ if, and only if, 
$t \gt 0$ satisfies the identity given by (8). Now, by using hypothesis 
$(H_2)$, we obtain that the function 
$h: (0,\infty) \rightarrow \mathbb{R}$ given by
\begin{equation*}h(t)=\displaystyle\iint\limits_{\mathbb{R}^N \times \mathbb{R}^N} \frac{[(1+\gamma) \phi(t|D_s u|) + \phi^{'}(t|D_s u|)|tD_s u| ]|D_s u|^p}{|tD_s u|^{p-2}} d\mu\end{equation*}is strictly decreasing. Hence, 
$t_n(u)$ is unique.
Now, we shall prove that 
$t_{n}: W_{0}^{s,\Phi}(\Omega)\backslash \{0\} \rightarrow (0, \infty)$ is a 
$C^{0}$-functional. Let 
$u_0 \in W_{0}^{s,\Phi}(\Omega)\backslash \{0\}$ be fixed. Consider the auxiliary functional 
$F: W_{0}^{s,\Phi} (\Omega)\backslash \{0\} \rightarrow \mathbb{R}$ given by 
$F(v,t)=R_n^{'}(tv)tv$. It is easy to verify that
\begin{equation*}F_t(v,t)=R_n{''}(tv)tv^2 + R_n(tv)v.\end{equation*}This assertion implies that
\begin{equation*}F_t(u_0,t_n(u_0))=\frac{R_n{''}(t_n(u_0)u_0)(t_n(u_0)u_0)^2}{t_n(u_0)} \lt 0.\end{equation*} In light of the Implicit Function Theorem (see [Reference Drábek and Milot22]), there exists an open set 
$U$ containing 
$u_0$ and a 
$C^0$-functional 
$t: U \rightarrow (0,\infty)$ such that
\begin{equation*}F(v,t(v))=R_n^{'}(t(v)v)t(v)v=0,~~\forall v \in U.\end{equation*} On the other hand, for each 
$v\in U$, there exists a unique 
$t_n(v)$ such that 
$R_n^{'}(t_n(v)v)t_n(v)v=0$. As a consequence, we obtain that 
$t(v)=t_n(v), ~\forall v \in U$. Then, 
$t_n \in C^0(U;(0,\infty))$. Since 
$u_0$ is arbitrary the desired result follows. This ends the proof.
Fibres of 
$Q_n(t)$ and 
$Q_e(t)$.
In view of the Propositions 3.3 and 3.4, the fibering maps of the functional 
$R_n$ are shown in Figure 1. Furthermore, we define the functional 
$\Lambda_n: W_0^{s,\Phi}(\Omega)\setminus\{0\}\to \mathbb{R}$ given by
\begin{equation*}\Lambda_{n}(u):= \displaystyle\max_{t \gt 0} R_{n}(tu)=R_n(t_n(u)u),~u\in W_0^{s,\Phi}(\Omega)\setminus\{0\}.\end{equation*}Now, we introduce the extreme value, see [Reference Il’yasov32], in the following form
\begin{equation*}\lambda^{*}:= \displaystyle\inf_{u \in W_{0}^{s,\Phi}(\Omega)\backslash \{0\}} \Lambda_{n}(u).\end{equation*}As a consequence, we prove the following result:
Proposition 3.5. Assume that 
$(\phi_{1})$–
$(\phi_{4})$ and 
$(H_1)$–
$(H_2)$ hold. Then we obtain the following assertions:
(i)
$\Lambda_{n}(u)$ is
$0$-homogeneous, i.e.,
$\Lambda_{n}(su) = \Lambda_{n}(u), \forall s \gt 0$ and
$u \in W_{0}^{s,\Phi}(\Omega) \backslash \{0\};$(ii) The functional
$\Lambda_{n}: W_{0}^{s,\Phi}(\Omega) \backslash \{0\} \rightarrow (0, \infty) $ is weakly
$^*$ lower semicontinuous;(iii) There exists
$c \gt 0$ such that
$[t_n(u)u]_{s,\Phi} \geq c$ for all
$u\in W_{0}^{s,\Phi}(\Omega)\backslash \{0\}$;(iv) There exists
$C=C(l, m, p, \gamma, \Omega) \gt 0$ such that
$\Lambda_{n}(u) \geq C$ for all
$u \in W_{0}^{s,\Phi}(\Omega) \backslash \{0\}$;(v) There exists
$u^{*} \in W_{0}^{s,\Phi}(\Omega) \backslash \{0\} $ such that
$\lambda^{*}= \Lambda_{n}(u^{*})$.
Proof. 
$(i)$ The proof of the fact that 
$\Lambda_{n}(u)$ is 
$0$-homogeneous follows directly. 
$(ii)$ Now, we shall prove that 
$\Lambda_n$ is weakly
$^*$ lower semicontinuous. Let 
$(u_k)$ be a sequence such that 
$u_k \stackrel{*}{\rightharpoonup} u$ in 
$W^{s,\Phi}_0(\Omega)$ for some 
$u \in W^{s,\Phi}_0(\Omega)$. Since 
$t \mapsto R_{n}^{'} (tu)tu$ is sequentially weakly
$^*$ lower semicontinuous we infer that
\begin{equation*}0= R_{n}^{'} (t_n(u) u)t_n(u)u \leq \displaystyle\liminf_{k \rightarrow +\infty} R_{n}^{'} (t_n(u)u_k)t_n(u)u_k.\end{equation*} This implies that for 
$k$ large enough 
$R_{n}^{\prime}(t_n(u)u_k) \geq 0$. Hence, 
$0 \lt t_n(u) \leq t_n(u_k)$ for 
$k$ large enough. Since 
$R_n(tu)$ is non-decreasing and 
$t_n(u) \in (0, t_n(u_k))$, we infer that
\begin{equation*}
\Lambda_n (u) := R_{n} (t_n(u)u) \leq \displaystyle\liminf_{k \rightarrow 0} R_{n} (t_n(u)u_k) \leq \liminf_{k \rightarrow +\infty} R_{n} (t_n(u_k)u_k) = \liminf_{k \rightarrow 0} \Lambda_n (u_k).
\end{equation*} This shows that the functional 
$\Lambda_{n}$ is weakly
$^*$ lower semicontinuous.
$(iii)$ Now, we shall consider 
$u\in W_0^{s,\Phi}(\Omega)\setminus\{0\}$. It follows from (8), 
$(\phi_4)$ and Proposition 2.2 that
\begin{align}
||\tilde{u}||_{p}^{p} &\geq \frac{\gamma + l -1}{p + \gamma -1}\displaystyle\iint\limits_{\mathbb{R}^N \times \mathbb{R}^N} \phi(|D_s \tilde{u}|)|D_s \tilde{u}|^2 d\mu ~\geq ~ \frac{l(\gamma + l - 1)}{p+\gamma -1} \displaystyle\iint\limits_{\mathbb{R}^N \times \mathbb{R}^N} \Phi(|D_s \tilde{u}|) d \mu\nonumber\\ &\quad\times \geq ~ \frac{l(\gamma + l - 1)}{p+\gamma -1} \min\{
[\tilde{u}]_{s,\Phi}^l, [\tilde{u}]_{s,\Phi}^m \},
\end{align}where 
$\tilde{u}=t_n(u)u$. Now, taking into account the embedding 
$W_{0}^{s, \Phi}(\Omega) \hookrightarrow L^p(\Omega)$, we have
\begin{equation}
||\tilde{u}||_{p}^{p} \geq \frac{l(\gamma + l - 1)}{p+\gamma -1} \min\{
S_{p}^{l}, S_{p}^{m}\} \min\{
||\tilde{u}||_{p}^{l}, ||\tilde{u}||_{p}^{m} \}.
\end{equation} Thus, there exists 
$c \gt 0$ such that 
$\|\tilde{u}\|_p,[\tilde{u}]_{s,\Phi}\geq c$.
$(iv)$ Let us consider 
$u\in W_0^{s,\Phi}(\Omega)\setminus\{0\}$, 
$t_n(u)$ that satisfies
\begin{equation*}\Lambda_n(u):=R_n(t_n(u)u)=\frac{\displaystyle\iint\limits_{\mathbb{R}^N \times \mathbb{R}^N} \phi(t_n(u)|D_s u|)|t_n(u)D_s u|^2 d\mu - ||t_n(u)u||_{p}^{p}}{t_n(u)^{1-\gamma}||u||_{1-\gamma}^{1-\gamma}}.\end{equation*}Now, by using the same steps as in the previous item, we deduce the following inequality
\begin{equation*}||t_n(u)u||^{p}_{p} \leq \frac{m + \gamma - 1}{p + \gamma -1}\displaystyle\iint\limits_{\mathbb{R}^N \times \mathbb{R}^N} \phi(t_n(u)|D_s u|)|t_n(u)D_s u|^2 d\mu.\end{equation*}Hence, by using Equation 4, we have
\begin{align}
\Lambda_n(u) &\geq \frac{p-m}{p + \gamma -1}\displaystyle\iint\limits_{\mathbb{R}^N \times \mathbb{R}^N} \phi(t_n(u)|D_s u|)|t_n(u)D_s u|^2 d\mu
\geq l \frac{p-m}{p + \gamma -1}\nonumber\\ &\quad\times \frac{ \min\{
[t_n(u)u]_{s,\Phi}^l, [t_n(u)u]_{s,\Phi}^m \}}{||t_n(u)u||_{1-\gamma}^{1-\gamma}}.
\end{align}This estimate implies that
\begin{equation*}\Lambda_n(u) \geq l \frac{p-m}{p + \gamma -1}min\{S_{p}^{l}, S_{p}^{m}\}\min\{||t_n(u)u||^{l + \gamma - 1}_{p}, ||t_n(u)u||^{m + \gamma - 1}_{p}\}.\end{equation*} As a consequence, by using item 
$(iii)$, we infer that
\begin{equation}
\Lambda_n(u) \geq l \frac{p-m}{p + \gamma -1}min\{S_{p}^{l}, S_{p}^{m}\}\min\{c^{l + \gamma - 1}, c^{m + \gamma - 1}\} =: C.
\end{equation} 
$(v)$ Let us consider a minimizing sequence 
$(u_k) \in W^{s,\Phi}_0(\Omega)$ for the extreme 
$\lambda^{*}$. Since 
$\Lambda_n$ is 
$0$-homogeneous, we assume that 
$t_n(u_k)=1$. From (11), we deduce that the sequence 
$(u_k)$ is bounded in 
$W_{0}^{s,\Phi}(\Omega)$. Hence, there is a subsequence of 
$(u_k)$ and 
$u\in W_{0}^{s,\Phi}(\Omega)$ such that 
$u_{k} \stackrel{*}{\rightharpoonup} u$ in 
$W_{0}^{s,\Phi}(\Omega)$. Now, by using (10), we obtain that 
$u\not\equiv 0$. Since 
$\Lambda_n$ is weakly
$^*$ lower semicontinuous, we have
\begin{equation*}\lambda^{*} \leq \Lambda_n(u) \leq \displaystyle\liminf_{k \rightarrow \infty} \Lambda_n(u_k)=\lambda^{*}.\end{equation*}The proof is now complete.
Now, the next results are on concerned the behaviour of 
$Q_e$. Namely, we consider the following result:
Proposition 3.6. Suppose that 
$(\phi_1)-\phi_4)$ and 
$(H_1)-(H_2)$ are satisfied. Then the function 
$t \mapsto Q_e(t)=R_e(tu)$ has the following properties:
(i)
$\displaystyle\lim_{t \rightarrow 0^{+}} \frac{Q_e(t)}{t^{m+\gamma - 1}} \gt 0\qquad\mbox{and}\qquad\displaystyle\lim_{t \rightarrow 0^{+}} \frac{\frac{d}{dt}Q_e(t)}{t^{m+\gamma -2}} \gt 0$;(ii)
$\displaystyle\lim_{t \rightarrow +\infty} \frac{Q_e(t)}{t^{p+\gamma - 1}} \lt 0 \qquad\mbox{and}\qquad\displaystyle\lim_{t \rightarrow +\infty} \frac{\frac{d}{dt}Q_e(t)}{t^{p+\gamma -2}} \lt 0$.
Proof. The proof follow the same ideas discussed in the proof of Proposition 3.3. We omit the details.
Now, we shall consider the behaviour of 
$Q_n$ and 
$Q_e$. The proof of this kind of result can be obtained [Reference Carvalho, Gasiński, Júnior and Silva14].
Proposition 3.7. Assume that 
$(\phi_1)-(\phi_4)$ and 
$(H_1)-(H_2)$ are satisfied. Then the function
\begin{equation*}t \mapsto \frac{\phi(t)t^2 - (1-\gamma)\Phi(t)}{t^p}\end{equation*}is strictly decreasing.
Proposition 3.8. Suppose that 
$(\phi_1)$–
$(\phi_4)$ and 
$(H_1)$–
$(H_2)$ are satisfied. Then for each 
$u \in W_{0}^{s,\Phi}(\Omega)\backslash \{0\}$ there exists a unique 
$t_{e}(u) \gt 0$ such that 
$Q_e'(t)=0$. Furthermore, 
$t_{e}: W_{0}^{s,\Phi}(\Omega)\backslash \{0\} \rightarrow (0, \infty)$ is a functional 
$C^{0}$.
Proof. The proof is similar the proof of Proposition 3.4. The details are omitted.
Proposition 3.9. Suppose that 
$(\phi_1)$–
$(\phi_4)$ and 
$(H_1)$–
$(H_2)$ are satisfied. Assume also that 
$u\in W_{0}^{s,\Phi}(\Omega)\backslash \{0\}$ satisfies 
$Q_n(t)=\lambda$ for some 
$t \gt 0$. Then 
$Q_n^{'}(t) \gt 0$ if, and only if, 
$I_{\lambda}^{''}(tu)(tu,tu) \gt 0$. Furthermore, we obtain that 
$Q_n^{'}(t) \lt 0$ if, and only if, 
$I_{\lambda}^{''}(tu)(tu,tu) \lt 0$ and 
$Q_n^{'}(t) = 0$ if, and only if, 
$I_{\lambda}^{''}(tu)(tu,tu) = 0$.
Proof. Let us consider 
$u\in W_{0}^{s,\Phi}(\Omega)\backslash \{0\}$ such that 
$R_n(tu)=\lambda$. It is not hard to see that
\begin{equation}
Q_{n}^{'}(t)= \frac{t^{\gamma - 2} I_{\lambda}^{''}(tu)(tu,tu)}{||u||_{1-\gamma}^{1-\gamma}}=\frac{1}{t}\frac{I_{\lambda}^{''}(tu)(tu,tu)}{||tu||_{1-\gamma}^{1-\gamma}}.
\end{equation}Hence, we obtain the desired result by using the last assertion. This finishes the proof.
In view of Propositions 3.6 and 3.8, we define the functional 
$\Lambda_e: W_0^{s,\Phi}(\Omega)\setminus\{0\}\to \mathbb{R}$ given by
\begin{equation*}\Lambda_{e}(u):= \displaystyle\max_{t \gt 0} R_{e}(tu)=R_e(t_e(u)u),~u\in W_0^{s,\Phi}(\Omega)\setminus\{0\}.\end{equation*} This function is related with to the sign of 
$I_\lambda(u)$ for any 
$u\in W_{0}^{s,\Phi}(\Omega)\backslash \{0\}$.
Now, we introduce another Rayleigh extreme value as follows:
\begin{equation*}\lambda_{*}:= \displaystyle\inf_{u \in W_{0}^{s,\Phi}(\Omega)\backslash \{0\}} \Lambda_{e}(u).\end{equation*}As a consequence, we shall prove the following result:
Proposition 3.10. Suppose that 
$(\phi_1)$–
$(\phi_4)$ and 
$(H_1)$–
$(H_2)$ holds. Assume also that 
$u\in W_{0}^{s,\Phi}(\Omega)\backslash \{0\}$ satisfies 
$Q_e(t)=\lambda$ for some 
$t \gt 0$. Then 
$Q_e^{'}(t) \gt 0$ if, and only if, 
$I_{\lambda}^{'}(tu)(tu,tu) \gt 0$. Furthermore, we obtain that 
$Q_e^{'}(t) \lt 0$ if, and only if, 
$I_{\lambda}^{'}(tu)(tu,tu) \lt 0$ and 
$Q_e^{' }(t) = 0$ if, and only if, 
$I_{\lambda}^{'}(tu)(tu,tu) = 0$.
Proof. The proof follows in a similar way to the proof of Proposition 3.9. The details are omitted.
Proposition 3.11. Suppose that 
$(\phi_{1})$–
$(\phi_{4})$, 
$(H_1)$, and 
$(H_2)$ are verified. Then the functional 
$\Lambda_{e}$ satisfies the following statements:
(i)
$\Lambda_{e}$ is
$0$-homogeneous, i.e.,
$\Lambda_{e}(su) = \Lambda_{e}(u)$ for all
$s \gt 0$ and
$u \in W_{0}^{s,\Phi} (\Omega)\backslash \{0\};$(ii) The functional
$\Lambda_{e}: W_{0}^{s,\Phi}(\Omega) \backslash \{0\} \rightarrow (0, \infty)$ is weakly
$^*$ lower semicontinuous;(iii) There exists
$c \gt 0$ such that
$\|t_e(u)u\| \geq c$ for all
$u\in W_{0}^{s,\Phi}\backslash \{0\}$;(iv) There exists
$C=C(l, m, p, \gamma, \Omega) \gt 0$ such that
$\Lambda_{e}(u) \geq C$ for all
$u \in W_{0}^{s,\Phi} \backslash \{0\}$;(v) The number
$\lambda_{*}$ is attained, i.e., there is
$u_{*} \in W_{0}^{s,\Phi} (\Omega) \backslash \{0\} $ such that
$\lambda_{*}= \Lambda_{e}(u_{*})$;(vi) For
$\lambda=\lambda_*$ it holds that
$u_{*}$ is a solution for Problem (Pλ).
Proof. The proof of items 
$(i)--(v)$ follows the same steps as in the proof of Proposition 3.5. The proof of item 
$(vi)$ can be found in [Reference Il’yasov32]. The details are omitted.
Proposition 3.12. Suppose that 
$(\phi_1)$–
$(\phi_4)$ and 
$(H_1)$–
$(H_2)$ hold. Assume also that 
$\lambda \in (0, \lambda^*)$ and consider 
$u\in W_{0}^{s,\Phi}(\Omega)\backslash \{0 \}$. Then the fibering map 
$\beta(t)=I_{\lambda}(tu)$, 
$t \gt 0$ satisfies the following assertions:
(i) There exist
$t^{n,+}(u) \lt t_n(u) \lt t^{n,-}(u)$ such that
$t^{n,+}(u) u \in \mathcal{N}_{\lambda}^{+}$ and
$t^{n,-}(u)u \in \mathcal{N}_{\lambda}^{-}$. Furthermore,
$t^{n,+}(u)$ is a local minimum point of the fibre
$\beta$;(ii) The point
$t^{n,-}(u)$ is a global maximum of the function
$\beta$ for each
$\lambda \in (0,\lambda_{*})$. For
$\lambda \in [\lambda_{*}, \lambda^{*})$ we obtain that
$t^{n,-}(u)$ is a local maximum point of
$\beta$;(iii) The functions
$t \mapsto t^{n,+}(u)$ and
$t \mapsto t^{n,-}(u)$ are in
$C^{0}$ class.
Proof. 
$(i)$ Consider 
$\lambda \in (0,\lambda^{*})$ and 
$u\in W_{0}^{s,\Phi}(\Omega)\backslash \{0\}$. Firstly, we observe that 
$tu \in \mathcal{N}_{\lambda}$ if, and only if 
$Q_{n}(t)=\lambda$. Furthermore, we obtain that
\begin{equation*}0 \lt \lambda \lt \lambda^{*}= \displaystyle\inf_{w \in W_{0}^{s,\Phi}(\Omega)\backslash \{0\}} \Lambda_{n}(w)\leq \Lambda_{n}(u)=\max_{t \gt 0}Q_{n}(t).\end{equation*} Now, by using Propositions 3.3 and 3.4, the identity 
$Q_{n}(t)=\lambda$ has exactly two roots which are denoted by 
$t^{n,+}(u)$ and 
$t^{n,-}(u)$. Notice also that 
$t^{n,+}(u) \lt t_n(u) \lt t^{n,-}(u)$. Moreover, we infer that 
$t^{n,+}(u)$ and 
$t^{n,-}(u)$ are critical points of the fibre 
$\beta(t)=I_{\lambda}(tu)$. It follows from Proposition 3.9 that 
$Q_{n}^{'}(t)$ has the same sign as 
$I_{\lambda}^{''}(tu)(tu,tu)$. Thus, 
$Q_ {n}^{'}(t^{n,+}(u)) \gt 0$ and 
$Q_{n}^{'}(t^{n,-}(u)) \lt 0$ which ensure that 
$t^{n,+ }(u)u \in \mathcal{N}_{\lambda}^{+}$ and 
$t^{n,-}(u)u \in \mathcal{N}_{\lambda}^{- }$, see Figure 2. Therefore, 
$t^{n,+}(u)$ is a local minimum point of 
$\beta$.
Fibre critical points 
$\beta(t)$.
$(ii)$ Now, we shall use the same steps as in the previous item obtaining that 
$t^{n,-}(u)$ is a point of local maximum of 
$\beta$ for all 
$\lambda \in (0,\lambda^{*})$. Assuming that 
$\lambda \in (0,\lambda_{*})$ we deduce that
\begin{equation*}Q_{n}(t^{n,-}(u)) = \lambda \lt \lambda_{*} \leq Q_{e}(t^{n,-}(u)). \end{equation*} Furthermore, 
$\beta(t^{n,-}(u))= I_{\lambda}(t^{n,-}(u)u) \gt I_{Q_{e}(t^{n,-}(u))}(t^{n,-}(u)u)$ =0. Therefore, 
$t^{n,-}(u)$ is a global maximum point of the fibre for each 
$\lambda \in (0,\lambda_{*})$.
$(iii)$ Now, we consider the function 
$F^{\pm}: W_{0}^{s,\Phi}(\Omega)\backslash \{0\} \times (0,\infty) \rightarrow \mathbb{R}$ given by 
$F(u,t)=I_{\lambda}^{'}(tu)tu.$ Notice that 
$F(u,t)=0$ if, and only if, 
$tu \in \mathcal{N}_{\lambda}$. Furthermore, by using the fact that 
$tu \in \mathcal{N}_{\lambda}^{\pm}$, we have that 
$F^{\pm}_{t}(u,t)\neq 0$. Since 
$t^{n,\pm}(u)u\in \mathcal{N}_{\lambda}^{\pm}$, we obtain that 
$F_{t}(u,t^{n,\pm}(u)) \neq 0$. Once again by using the Implicit Function Theorem, see [Reference Drábek and Milot22], we can argue as was done in the proof of Proposition 3.4 which implies that 
$t \rightarrow t^{n,\pm}(u)$ are functions in 
$C^{0}$ class.
Now, we shall consider situations where 
$\mathcal{N}_\lambda^0$ is empty or non-empty. This kind of result can be proved using the same arguments employed in [Reference Alves, Santos and Silva5]. Hence, we can consider the following result:
Lemma 3.13. Suppose that 
$(\phi_{1})$–
$(\phi_{4})$ and 
$(H_1)$–
$(H_2)$ hold. Then the following statements:
(1) For every
$\lambda \in (0,\lambda^{*})$, we deduce that
$\mathcal{N}_{\lambda}^{0} = \emptyset$;(2) For
$\lambda\geq\lambda^*$, we obtain that
$\mathcal{N}_\lambda^0\neq\emptyset$.
Proposition 3.14. Suppose that 
$(\phi_{1})$–
$(\phi_{4})$ and 
$(H_1)$–
$(H_2)$ are satisfied. Then 
$\overline{\mathcal{N}_{\lambda}^{+}} = \mathcal{N}_{\lambda}^{+} \cup \{0\}$ for each 
$\lambda \in (0, \lambda^{*})$.
Proof. Firstly, we shall prove that 
$\overline{\mathcal{N}_{\lambda}^{+}} \subset \mathcal{N}_{\lambda}^{+} \cup \{0\}$. Consider a sequence 
$(u_k) \subset \mathcal{N}_{\lambda}^{+}$ in such way that 
$u_k \rightarrow u$ in 
$W^{s,\Phi}_0(\Omega)$ holds for some 
$u \in W_{0}^{s,\Phi} (\Omega) $. Since 
$u\mapsto I_\lambda'(u)u$ is continuous, we infer that 
$I_{\lambda}^{'}(u_k)u_k \rightarrow I_{\lambda}^{'}(u)u=0$. Similarly, we prove that 
$I_{\lambda}^{''}(u_k)(u_k,u_k) \rightarrow I_{\lambda}^{''}(u)(u,u) \geq 0$. In view of the fact that 
$\mathcal{N}_\lambda^0=\emptyset$ holds we obtain that 
$u \in \mathcal{N}_{\lambda}^{+} \cup \{0\}$. As a consequence, 
$\overline{\mathcal{N}_{\lambda}^{+}} \subset \mathcal{N}_{\lambda}^{+} \cup \{0\}$.
Now, we shall prove that 
$\mathcal{N}_{\lambda}^{+} \cup \{0\} \subset \overline{\mathcal{N}_{\lambda}^{+}} $. The proof follows by using the Josefson–Nissenzwiiez theorem, see [Reference Diestel21], which implies that there exists a sequence 
$(w_k) \in W_{0}^{s,\Phi}(\Omega)$ such that 
$w_k \stackrel{*}{\rightharpoonup} 0 $ where 
$[w_k]_{s,\Phi}=1$ and 
$w_k \nrightarrow 0$ in 
$W_{0}^{s,\Phi}(\Omega)$. According to Proposition 3.12, there exists 
$t^{n,+}(w_k)$ such that 
$u_k = t^{n,+}(w_k)w_k \in \mathcal{N}_{\lambda}^{+}.$ Now, we claim that 
$(u_k)$ is a bounded in 
$W_{0}^{s,\Phi}(\Omega)$. Indeed, taking into account that 
$R_n(u_k)=\lambda$ and 
$R_n'(u_k)u_k \gt 0$, we deduce that
\begin{align*}
& \displaystyle\iint\limits_{\mathbb{R}^N \times \mathbb{R}^N} \phi'(|D_{s}u_k |)|D_{s}u_k|^3 d\mu - (p-2)\displaystyle\iint\limits_{\mathbb{R}^N \times \mathbb{R}^N} \phi(|D_{s}u_k|)|D_{s}u_k|^2 d\mu +\lambda (p + \gamma -1)\\ & \quad\times \displaystyle\int_{\Omega}|u_k|^{1-\gamma} dx \gt 0.
\end{align*} As a consequence, by using 
$(\phi_4)$, Hölder’s inequality and the embedding of 
$W_{0}^{s,\Phi}(\Omega)$ into 
$L^{1}(\Omega)$, we obtain that
\begin{align}
l(p-m)\displaystyle\iint\limits_{\mathbb{R}^N \times \mathbb{R}^N} \Phi(|D_{s}u_k|) d\mu& \leq(p-m)\displaystyle\iint\limits_{\mathbb{R}^N \times \mathbb{R}^N} \phi(|D_{s}u_k|)|D_{s}u_k|^2 d\mu \lt \lambda (p + \gamma -1)\nonumber\\ &\quad \times S_1|\Omega|^{\gamma}[u_k]^{1-\gamma}_{s,\Phi}.
\end{align}Therefore, by using Proposition 2.2, we have that
\begin{equation*}l(p-m)\min \{[u_k]^l,[u_k]^m\} \lt \lambda (p+\gamma -1)S_1|\Omega|^{\gamma}[u_k]^{1-\gamma}. \end{equation*} This assertion proves that 
$(u_k)$ is a bounded in 
$W_{0}^{s,\Phi}(\Omega).$ Now, we claim that 
$I_{\lambda}(u_k) \lt 0$. According to Propositions 3.12 and 3.15, we obtain that 
${1=t^{n,+}(u_k)} \lt t_n(u_k) \lt t_{e}(u_k) $. Furthermore, we see that 
$\Lambda_{n}(u_k) \lt \Lambda_{e}(u_k)$. It follows from Remark 3.2-(iv) that 
$\lambda=R_{n}(u_k) \lt R_{e}(u_k)$. In particular, we that
\begin{equation*}I_{\lambda}(u_k) \lt I_{R_e(u_k)}(u_k)=0.\end{equation*} Now, we shall prove that 
$u_k \rightarrow 0$ in 
$W^{s,\Phi}_0(\Omega)$. Indeed, we observe that
\begin{align*}
0 \gt I_{\lambda}(u_k) - \frac{1}{p}I^{'}_{\lambda}(u_k)u_k & = \displaystyle\iint\limits_{\mathbb{R}^N \times \mathbb{R}^N} \Phi (|D_s u_k|)d\mu - \frac{1}{p}\displaystyle\iint\limits_{\mathbb{R}^N \times \mathbb{R}^N} \phi(|D_su_k)|D_s u_k|^2 d\mu \\ &\quad - \lambda\left( \frac{1}{1-\gamma} - \frac{1}{p}\right)\int_{\Omega}|u_k|^{1-\gamma} dx.
\end{align*} Hence, by using 
$(\phi_4)$, Proposition 2.2, Hölder’s inequality and the compact embedding 
$ W_{0}^{s,\Phi}(\Omega)\hookrightarrow L^{1}(\Omega)$, we infer that
\begin{equation*}
C||u_k||_{1}^{1-\gamma} \geq \left( 1- \frac{m}{p} \right)\displaystyle\iint\limits_{\mathbb{R}^N \times \mathbb{R}^N} \Phi (|D_s(t^{n,+}(w_k)(w_k))|)d\mu \geq \min\{[u_k]_{s,\phi}^l,[u_k]_{s,\Phi}^m\}.
\end{equation*} As a consequence, by using the fact 
$w_k \stackrel{*}{\rightharpoonup} 0 $ in 
$W^{s,\Phi}_0(\Omega)$, we deduce that
\begin{equation*}\min\{(t^{n,+}(w_k))^{l-1+\gamma},(t^{n,+}(w_k))^{m-1+\gamma}\}\leq C\|w_k\|_1 \to 0.\end{equation*} Therefore, 
$t^{n,+}(w_k) \rightarrow 0$ where 
$(u_k) \in \mathcal{N}_{\lambda}^{+}$ which satisfies 
$u_k \rightarrow 0$ in 
$W_{0}^{s,\Phi}(\Omega)$. This assertion proves that 
$\overline{\mathcal{N}_{\lambda}^{+}} = \mathcal{N}_{\lambda}^{+} \cup \{0\}$ for each 
$\lambda \in (0, \lambda^{*})$. This ends the proof.
Proposition 3.15. Suppose that 
$(\phi_{1})$–
$(\phi_{4})$, 
$(H_1)$, and 
$(H_2)$ hold. Then we obtain the following statements:
(i) It holds
$t_{n}(u) \lt t_{e}(u)$ for all
$u \in W_{0}^{s,\Phi}(\Omega)\backslash \{0\}$.(ii) There holds
$\Lambda_{e}(u) \lt \Lambda_{n}(u)$ for all
$u \in W_{0}^{s,\Phi}(\Omega)\backslash \{0\}$.(iii) It holds
$0 \lt \lambda_{*} \lt \lambda^{*} \lt \infty$.
Proof. 
$(i)$ It is important to emphasize that
\begin{equation*}Q_n(t) - Q_e(t) = \frac{1}{t(1-\gamma)} \frac{d}{dt}Q_e(t)\end{equation*}holds for all 
$t \gt 0$ where 
$u \in W_{0}^{s,\Phi}(\Omega) \setminus \{0\}$. In particular, for each 
$t \in (0, t_e(u))$, we have 
$\frac{d}{dt}Q_e(t) \gt 0$ which implies that 
$Q_n(t) \gt Q_e(t)$ with 
$t \in (0, t_e(u))$. Moreover, 
$Q_n(t) = Q_e(t)$ if and only if 
$t = t_e(u)$. Similarly, 
$Q_n(t) \lt Q_e(t)$ for each 
$t \gt t_e(u)$. Therefore, we obtain that 
$t_n(u) \lt t_e(u)$ holds for all 
$u \in W_{0}^{s,\Phi}(\Omega) \setminus \{0\}$, see Figure 1.
$(ii)$ In light of the previous assertion in infer that
\begin{equation*}
\Lambda_n(u) = \max_{t \gt 0} Q_n(t_n(u)) \gt Q_n(t_e(u)) = Q_e(t_e(u)) = \Lambda_e(u).
\end{equation*} This fact establishes the proof of item 
$(ii)$.
$(iii)$ Now, by using Proposition 3.5, there exists 
$u^{*} \in W_{0}^{s, \Phi}(\Omega) \setminus \{0\}$ such that 
$\lambda^{*} = \Lambda_n(u^{*})$. Moreover, using the item 
$(ii)$ given just above, we deduce that
\begin{equation*}
\lambda^{*} = \Lambda_n(u^{*}) \gt \Lambda_e(u^{*}) \geq \lambda_{*}.
\end{equation*} Therefore, 
$0 \lt \lambda_{*} \lt \lambda^{*} \lt \infty$ is now satisfied. This ends the proof.
Remark 3.16. Since 
$Q_n(t) = Q_e(t)$ if and only if 
$t = t_e$ we obtain that 
$\Lambda_n(u) \gt \Lambda_e(u)$ for each 
$u \in X \setminus \{0\}$.
Proposition 3.17. Suppose 
$(\phi_1)$–
$(\phi_4)$, 
$(H_1)$, and 
$(H_2)$ hold. Then for each 
$\lambda \gt 0$, the functional 
$I_\lambda$ is coercive in the Nehari set 
$\mathcal{N}_\lambda$.
Proof. Let 
$u \in \mathcal{N}_{\lambda}$ be fixed. Recall that 
$I_{\lambda}'(u)u = 0$. Now, by using hypothesis 
$(\phi_4)$, we deduce that
\begin{equation*}I_{\lambda}(u) \geq \left( 1- \frac{m}{p}\right)\displaystyle\iint\limits_{\mathbb{R}^N \times \mathbb{R}^N} \Phi(|D_{s}u|) d\mu + \lambda\left( -\frac{1}{1-\gamma} + \frac{1}{p}\right) \int_{\Omega}|u|^{1 - \gamma} dx.\end{equation*} Hence, by using Hölder’s inequality and the embedding 
$W_{0}^{s,\Phi}(\Omega) \hookrightarrow L^{l_s^*}(\Omega)$, there exists 
$\tilde{c} \gt 0$ such that
\begin{equation*}I_{\lambda}(u) \geq \left( 1- \frac{m}{p}\right)\displaystyle\iint\limits_{\mathbb{R}^N \times \mathbb{R}^N} \Phi(|D_{s}u|) d\mu + \lambda\left( -\frac{1}{1-\gamma} + \frac{1}{p}\right) \tilde{c}[u]_{s,\Phi}^{1-\gamma}.\end{equation*} Therefore, there exist constants 
$c_{1} \gt 0$ and 
$c_{3} \gt 0$ such that
\begin{equation*}I_{\lambda}(u) \geq c_{1}\min\{[u]_{s,\Phi}^l, [u]_{s,\Phi}^m\} - \lambda c_{3}[u]_{s,\Phi}^{1-\gamma}.\end{equation*} The last assertion implies that the functional 
$I_\lambda$ is coercive in the Nehari set. This finishes the proof.
Now, following some ideas provided by [Reference Salort and Vivas42, Proposition 3.2], we can prove the following result:
Proposition 3.18. Suppose that hypotheses 
$(\phi_1)$–
$(\phi_4)$, 
$(H_1)$, and 
$(H_2)$ hold. Then, for each 
$\lambda \gt 0$, the functional 
$I_\lambda$ is weakly
$^*$ lower semicontinuous in the Nehari set 
$\mathcal{N}_\lambda$.
In what follows, we find how the sign of the energy functional behaves restricted to the Nehari set 
$\mathcal{N}_{\lambda}^+$. More specifically, we shall prove the following result:
Proposition 3.19. Suppose 
$(\phi_1)$–
$(\phi_4)$, 
$(H_1)$, and 
$(H_2)$ hold. Also assume that 
$\lambda \gt 0$. Then 
$\mathcal{C}_{\mathcal{N}_{\lambda}^{+}} \lt 0$.
Proof. Let 
$u \in \mathcal{N}_{\lambda}^{+}$ be a fixed function. Notice also that 
$R_n(u) = \lambda$ and 
$1 = t_n^+(u) \lt t_n(u) \lt t_e(u)$. Hence, by using Proposition 3.12, we infer that 
$\lambda = R_n(u) \gt R_e(u)$. Under these conditions, we obtain that
\begin{equation*}\mathcal{C}_{\mathcal{N}_{\lambda}^{+}} \leq I_{\lambda}(u) \lt I_{R_e(u)}(u)=0.\end{equation*}This ends the proof.
Proposition 3.20. Suppose 
$(\phi_1)$–
$(\phi_4)$, 
$(H_1)$, and 
$(H_2)$ hold. Assume also that 
$\lambda \gt 0$. Then there exists 
$c \gt 0$ such that 
$c\leq \|u\|_p,[u]_{s,\Phi}$ holds for all 
$u \in \mathcal{N}_{\lambda}^{-} \cup \mathcal{N}_{\lambda}^{0}$. Furthermore, we obtain the following statements:
(1)
$\mathcal{N}_{\lambda}^{-}$ is closed for all
$\lambda \in (0, \lambda^*)$;(2)
$\mathcal{N}_{\lambda}^{0}$ is closed for all
$\lambda \gt 0$.
Proof. Let 
$u \in \mathcal{N}_{\lambda}^{-} \cup \mathcal{N}_{\lambda}^{0}$ be fixed. In view of 
$I_{\lambda}^{'}(u)u = 0$ and 
$I_{\lambda}^{''}(u)(u,u) \leq 0$ we deduce that
\begin{equation*}
0 \geq \displaystyle\iint\limits_{\mathbb{R}^N \times \mathbb{R}^N} \left[\phi^{'}(|D_{s}u|)(D_{s}u)^3 + (1 + \gamma)\phi(|D_{s}u|)|D_{s}u|^2 \right]d\mu -(p + \gamma -1)\int_{\Omega}|u|^{p} dx.
\end{equation*} In light of (4) and the embedding 
$ W_{0}^{s,\Phi}(\Omega)\hookrightarrow L^{p}(\Omega)$ together with Proposition 2.1, there exists 
$C \gt 0$ such that
\begin{equation}
C\displaystyle \zeta_0(\|u\|_p)\leq \zeta_0([u]_{s,\Phi}) \leq \frac{p + \gamma -1}{l + \gamma - 1}\|u\|_p^p.
\end{equation} Recall also that 
$l\leq m \lt p$. Hence, we deduce that 
$\|u\|_p\geq c$ holds for some positive constant 
$c \gt 0$. From embedding 
$ W_{0}^{s,\Phi}(\Omega)\hookrightarrow L^{p}(\Omega)$, we obtain that 
$[u]_{s,\Phi}\geq c$.
Now, we shall prove that 
$\mathcal{N}_{\lambda}^{-}$ is closed in 
$W_{0}^{s,\Phi}(\Omega)$ for 
$\lambda \in (0, \lambda^*)$. Let 
$(u_k)$ be a sequence in 
$\mathcal{N}_{\lambda}^{-}$ such that 
$u_k \to u$ in 
$W_{0}^{s,\Phi}(\Omega)$ as 
$k \to \infty$. It is important to emphasize that 
$[u_k] \geq c_3 \gt 0$. Thus
\begin{equation*}\displaystyle\lim_{k \rightarrow \infty} [u_k]_{s,\Phi}=[u]_{s,\Phi} \geq c \gt 0.\end{equation*} Consequently, 
$u \neq 0$ is verified. Since 
$(u_k) \subset \mathcal{N}_{\lambda}^{-}$ and 
$I_{\lambda}$ is continuous, we infer that
\begin{equation*}\displaystyle\lim_{k \rightarrow \infty}I_{\lambda}^{'}(u_k)u_k=I_{\lambda}^{'}(u)u=0\qquad\mbox{and}\qquad\displaystyle\lim_{k \rightarrow \infty}I_{\lambda}^{''}(u_k)(u_k, u_k)=I_{\lambda}^{''}(u)(u,u)\leq 0.\end{equation*} This implies that 
$u \in \mathcal{N}_{\lambda}^{-} \cup \mathcal{N}_{\lambda}^{0}$ for 
$\lambda \in (0, \lambda^*)$. However, by Proposition 3.13, 
$\mathcal{N}_{\lambda}^{0} = \emptyset$ for each 
$\lambda \in (0, \lambda^*)$. Therefore, 
$u \in \mathcal{N}_{\lambda}^{-}$. This ends the proof.
At this stage, we consider some results showing that Problems (1) and (2) admit weak solutions in 
$\mathcal{N}_{\lambda}^{+} $ and 
$\mathcal{N}_{\lambda}^{-}$, respectively.
Proposition 3.21. Suppose 
$(\phi_1)$–
$(\phi_4)$, 
$(H_1)$, and 
$(H_2)$ hold. Assume that 
$0 \lt \lambda \lt \lambda^*$. Let 
$(u_k)$ be a minimizing sequence for 
$\mathcal{C}_{\mathcal{N}_{\lambda}^{+}}$. Then, there exists 
$u \in W_{0}^{s,\Phi}(\Omega) \setminus \{0\}$ such that 
$u_k \to u$ in 
$W_{0}^{s,\Phi}(\Omega)$. Moreover, 
$\mathcal{C}_{\mathcal{N}_{\lambda}^{+}} = I_{\lambda}(u) \lt 0$ and 
$u \in \mathcal{N}_{\lambda}^{+}$.
Proof. Let 
$(u_k) \subset \mathcal{N}_{\lambda}^{+}$ be a minimizer sequence in the Nehari set 
$\mathcal{C}_{\mathcal{N}_{\lambda}^{+}}$. In light of Proposition 3.17, the functional 
$I_{\lambda}$ is coercive in the set 
$\mathcal{N}_{\lambda}$. This fact implies that 
$u_k$ is bounded. Moreover, given that 
$W_0^{s,\Phi}(\Omega)$ is continuously embedded in 
$\prod L_\Phi(\Omega) = \prod (E_\Phi(\Omega))^*$, and that 
$\prod E_\Phi(\Omega)$ is a separable space, it follows that there exists a subsequence, also denoted 
$u_k$, such that 
$u_k \stackrel{*}\rightharpoonup u$ in 
$W_{0}^{s,\Phi}(\Omega)$ for some 
$u \in W_{0}^{s,\Phi}(\Omega)$. From now on, we prove that 
$u \neq 0$. Indeed, taking into account the fact that 
$(u_k)$ is a minimizing sequence, we have
\begin{equation*}
I_{\lambda}(u) \leq \liminf_{k \to \infty} I_{\lambda}(u_k) = \mathcal{C}_{\mathcal{N}_{\lambda}^{+}}.
\end{equation*} Now, by using Proposition 3.19, we see that 
$\mathcal{C}_{\mathcal{N}_{\lambda}^{+}} \lt 0$. Therefore, 
$u \neq 0$ as was claimed.
Now, we shall prove that 
$u_k \rightarrow u$ in 
$W_{0}^{s,\Phi}(\Omega)$. The proof follows arguing by contradiction. Assume that 
$u_k \nrightarrow u$ in 
$W^{s,\Phi}_0(\Omega)$. It follows from Proposition 2.5 that 
$R_n(tu) \lt \liminf_{k \rightarrow \infty} R_n(u_k)$. Hence, by using Proposition 3.5, we deduce that 
$\Lambda_n(u) \lt \liminf_{k \rightarrow \infty} \Lambda_n(u_k)$. It follows that 
$\Lambda_n(u) \lt \Lambda_n(u_k)$ holds for 
$k \in \mathbb{N}$ large enough. Hence, we infer that
\begin{equation*}t_{\lambda}^{n,+}(u) \gt t_{\lambda}^{n,+}(u_k)=1.\end{equation*} Since 
$t \mapsto I_{\lambda}(tu)$ is strictly decreasing on 
$(0, t_{\lambda}^{n,+}(u))$ and 
$1 \in (0, t_{\lambda}^{n,+}(u))$, we conclude that
\begin{equation*}
I_{\lambda}(t_{\lambda}^{n,+}(u)u) \lt I_{\lambda}(u) \leq \liminf_{k \rightarrow \infty} I_{\lambda}(u_k) = \inf_{v \in \mathcal{N}_{\lambda}^{+}} I_{\lambda}(v).
\end{equation*} This leads to a contradiction. Therefore, 
$u_k \to u$ in 
$W^{s,\Phi}_0(\Omega)$ where 
$u \in \mathcal{N}_{\lambda}^{-} \cup \mathcal{N}_{\lambda}^{0}$. Recall also that 
$\mathcal{N}_{\lambda}^{0} = \emptyset$. It follows from the last assertion that 
$u \in \mathcal{N}_{\lambda}^{-}$ for all 
$\lambda \in (0, \lambda^*)$. This finishes the proof.
Proposition 3.22. Suppose 
$(\phi_1)$–
$(\phi_4)$, 
$(H_1)$, and 
$(H_2)$ hold. Assume 
$0 \lt \lambda \lt \lambda^*$. Let 
$(v_k)_{k \in \mathbb{N}} \subset \mathcal{N}_\lambda^-$ be a minimizing sequence for Problem (2). Then there exists 
$v \in W_0^{s,\Phi}(\Omega) \setminus \{0\}$ such that 
$v_k \rightarrow v$ in 
$W_0^{s,\Phi}(\Omega)$. Furthermore, 
$\mathcal{C}_{\mathcal{N}_\lambda^-} = I_\lambda(v)$ and 
$v \in \mathcal{N}_\lambda^- $.
Proof. Consider a sequence 
$(v_k)_{k \in \mathbb{N}} \subset \mathcal{N}_\lambda^-$. Recall that the functional 
$I_\lambda$ is coercive in the set 
$\mathcal{N}_\lambda$. Hence, we see that 
$v_k \stackrel{*}\rightharpoonup v$ in 
$W_0^{s,\Phi}(\Omega)$. Furthermore, by using Proposition 3.20, 
$v\neq 0$ is verified. Now, using the same ideas discussed in the proof of Proposition 3.21, we show that 
$v_k \to v$ in 
$W_0^{s,\Phi}(\Omega)$ and 
$v \in \mathcal{N}_\lambda^-$ for all 
$\lambda \in (0, \lambda^*)$. This ends the proof.
Now, we shall consider some useful results in order to prove that minimizer in the Nehari set provide us critical point for the energy functional. As a first step, in the spirit of (Q1) and [Reference Alves, Santos and Silva5, Lemma 3.3], we are able to prove the following result:
Lemma 3.23. Suppose 
$(\phi_1)$–
$(\phi_4)$, 
$(H_1)$, and 
$(H_2)$ hold. Let 
$0 \lt \lambda \lt \lambda^*$. Consider 
$u\in \mathcal{N}^+_\lambda$ and 
$v\in \mathcal{N}^-_\lambda$ the functions obtained from Propositions 3.21 and 3.22, respectively. Then there exists 
$\varepsilon_0 \gt 0$ such that
(i)
$I_\lambda(u + \varepsilon \varphi) \geq I_\lambda(u)$;(ii)
$t_{\lambda}^{n,-}(v + \varepsilon \varphi) \to 1$ as
$\varepsilon \downarrow 0$, where
$t_{\lambda}^{n,-}(v + \varepsilon \varphi)$ is the unique positive real number given by Proposition 3.12 satisfying
$t_{\lambda}^{n,-}(v + \varepsilon \varphi)(v + \varepsilon \varphi) \in \mathcal{N}_\lambda^-$ for all
$\varphi \geq 0$,
$\varphi \in W_0^{s,\Phi}(\Omega)$, and
$0 \leq \varepsilon \leq \varepsilon_0$.
Now, we shall consider another result proving that the minimizers in 
$\mathcal{N}_{\lambda}^+$ and 
$\mathcal{N}_\lambda^-$ are weak solutions for Problem (Pλ).
Proposition 3.24. Suppose 
$(\phi_1)$–
$(\phi_4)$, 
$(H_1)$, and 
$(H_2)$ hold. Assume also that 
$0 \lt \lambda \lt \lambda^*$. Let 
$u \in \mathcal{N}_{\lambda}^+ $ and 
$v \in \mathcal{N}_\lambda^- $ be minimizers sequences for Problems (1) and (2), respectively. Then 
$u$ and 
$v$ are solutions for Problem (Pλ).
Proof. Notice that the energy functional is even. Recall also that 
$ u \mapsto I_\lambda(u)$, 
$u \mapsto I_\lambda'(u)u$ and 
$ u \mapsto I_\lambda''(u)(u,u)$ are even functions. Hence, we assume without loss of generality that 
$u \geq 0$ in 
$\Omega$. Firstly, we shall prove that 
$I_{\lambda}'(u)\psi \geq 0$ holds for all non-negative 
$\psi \in W_0^{s,\Phi}(\Omega)$. In fact, consider the test function 
$\omega = u + \varepsilon \psi$. Now, by using Lemma 3.23, we deduce that
\begin{eqnarray}
0 &\leq & I_{\lambda}(u+\varepsilon \psi) - I_{\lambda}(u)\\ \nonumber
&=& \displaystyle\iint\limits_{\mathbb{R}^N \times \mathbb{R}^N}(\Phi(|D_s (u+\varepsilon\psi)|) - \Phi(|D_s u|))d\mu -\frac{\lambda}{1-\gamma} \int_{\Omega}(|u+\varepsilon\psi|^{1-\gamma} - |u|^{1-\gamma}) dx \\
& & \quad - \frac{1}{p} \int_{\Omega}(|u+\varepsilon\psi|^{p} - |u|^{p})dx. \nonumber \end{eqnarray}This assertion implies that
\begin{align*}
\frac{\lambda}{1-\gamma}\displaystyle \liminf_{\varepsilon \rightarrow 0^+}\int_{\Omega}\frac{(u+\varepsilon\psi)^{1-\gamma} - u^{1-\gamma}}{\varepsilon} dx &\leq -\frac{1}{p}\lim_{\varepsilon \rightarrow 0^+} \int_{\Omega} \frac{(u+\varepsilon\psi)^{p} - u^{p}}{\varepsilon}dx \\ \nonumber
&\quad + \displaystyle\lim_{\varepsilon \rightarrow 0^+}\iint\limits_{\mathbb{R}^N \times \mathbb{R}^N}\frac{\Phi(|D_s (u+\varepsilon\psi)|) - \Phi(|D_s u|)}{\varepsilon}d\mu.
\end{align*}Using Fatou’s Lemma and Proposition 2.3, we obtain also that
\begin{eqnarray}
0\leq\lambda \int_{\Omega}u^{-\gamma}\psi dx & \leq &\frac{\lambda}{1-\gamma}\displaystyle \liminf_{\varepsilon \rightarrow 0}\int_{\Omega}\frac{(u+\varepsilon\psi)^{1-\gamma} - u^{1-\gamma}}{\varepsilon} dx \\ &\leq& \displaystyle\iint\limits_{\mathbb{R}^N \times \mathbb{R}^N} \phi(|D_s u|)D_s uD_s \psi d\mu - \int_{\Omega} u^{p-1}\psi dx \lt \infty.\nonumber\end{eqnarray} Now, we shall prove that 
$u \gt 0$ almost everywhere in 
$\Omega$. The proof follows arguing by contradiction. Let us assume that the measure of the set 
$[u=0]$ is positive. Note that for any 
$\varepsilon \gt 0$, 
$[u=0] \subseteq [u \lt \varepsilon]$. From (17), for 
$\psi \gt 0$, we have that
\begin{align*}
\infty &= \lim_{\varepsilon \to 0} \frac{1}{\varepsilon^{\gamma}} \int_{[u=0]} \!\psi \, dx\leq \lim_{\varepsilon \to 0} \int_{[u \lt \varepsilon]} \!\epsilon^{-\gamma} \psi \, dx \leq \lim_{\varepsilon \to 0} \int_{[u \lt \varepsilon]} \!u^{-\gamma} \psi \, dx \nonumber\\
&\leq \int_{\Omega}\! u^{-\gamma} \psi \, dx \leq \iint\limits_{\mathbb{R}^N \times \mathbb{R}^N} \!\phi(|D_s u|)D_s uD_s \psi d\mu - \int_{\Omega} u^{p-1}\psi dx \lt \infty.
\end{align*} This is a contradiction proving that 
$u \gt 0$ a.e. in 
$\Omega$.
It follows from (17) that 
$I_\lambda'(u)\varphi \geq 0$ hold for all non-negative 
$\varphi \in W_0^{s,\Phi}(\Omega)$. Moreover, we conclude that 
$u^{-\gamma}w = u^{-\gamma}w^{+} - u^{-\gamma}w^{-} \in L^1(\Omega)$ for all 
$w \in W_0^{s,\Phi}(\Omega)$.
At this stage, we consider 
$\omega=(u+\varepsilon \psi)^{+}$ as a test function in (17). Notice also that 
$\Omega_\epsilon:=[u+\epsilon\psi\geq 0]$. Hence, we deduce that
\begin{eqnarray*}
0 & \leq & \displaystyle\iint\limits_{\mathbb{R}^N \times \mathbb{R}^N} \phi(|D_s u|)D_s udD_s (u+\epsilon\psi)d\mu-\lambda\int_\Omega u^{-\gamma}(u+\epsilon\psi)dx-\int_\Omega u^{p-1}(u+\epsilon\psi) dx\\
&&\quad + \displaystyle\iint\limits_{\mathbb{R}^N \times \mathbb{R}^N} \phi(|D_s u|)D_s uD_s (u+\epsilon\psi)^-d\mu+\int_{[u+\epsilon \psi \lt 0]} [\lambda u^{-\gamma}+u^{p-1}](u+\epsilon\psi)dx\\
& = & \left[~\displaystyle\iint\limits_{\mathbb{R}^N \times \mathbb{R}^N} \phi(|D_s u|)|D_s u|^2d\mu -\lambda\int_\Omega u^{1-\gamma}dx-\int_\Omega u^p dx\right]\\
&&\quad +\int_{[u+\epsilon \psi \lt 0]} [\lambda u^{-\gamma}+u^{p-1}](u+\epsilon\psi)dx\\
&&\quad - \displaystyle\iint\limits_{\mathbb{R}^N \times \mathbb{R}^N} \phi(|D_s u|)D_s uD_s [(u+\epsilon\psi)\chi_{\Omega_\epsilon^c}]d\mu+\varepsilon\\ &&\quad \times \left[~\displaystyle\iint\limits_{\mathbb{R}^N \times \mathbb{R}^N} \phi(|D_s u|)D_s uD_s \psi d\mu -\lambda\int_\Omega u^{-\gamma} \psi dx -\int_\Omega u^{p-1}\psi dx\right].
\end{eqnarray*} Recall that 
$u\in \mathcal{N}_\lambda$ and 
$[\lambda u^{-\gamma}+u^{p-1}](u+\epsilon\psi)\leq 0$ a.e. in 
$[u+\epsilon\psi \lt 0]$. As a consequence, we obtain that
\begin{align}
&0 \leq \left[~\displaystyle\iint\limits_{\mathbb{R}^N \times \mathbb{R}^N} \phi(|D_s u|)D_s uD_s \psi d\mu -\lambda\int_\Omega u^{-\gamma} \psi dx -\int_\Omega u^{p-1}\psi dx\right]\nonumber\\ &\quad -\iint\limits_{\mathbb{R}^N \times \mathbb{R}^N} \phi(|D_s u|)D_s uD_s (\psi\chi_{\Omega_\epsilon^c})d\mu\notag\\
&\quad - \frac 1\varepsilon\iint\limits_{\mathbb{R}^N \times \mathbb{R}^N} \phi(|D_s u|)|D_s (u\chi_{\Omega_\epsilon^c})|^2d\mu\notag\\
&\leq \left[~\displaystyle\iint\limits_{\mathbb{R}^N \times \mathbb{R}^N} \phi(|D_s u|)D_s uD_s \psi d\mu -\lambda\int_\Omega u^{-\gamma} \psi dx -\int_\Omega u^{p-1}\psi dx\right]\nonumber\\ &\quad -\iint\limits_{\mathbb{R}^N \times \mathbb{R}^N} \phi(|D_s u|)D_s uD_s (\psi\chi_{\Omega_\epsilon^c})d\mu.
\end{align} Therefore, by using the Dominated Convergence Theorem, we obtain 
$I_\lambda'(u)\psi \geq 0$ holds for each 
$\psi \in W^{s, \Phi}_0(\Omega)$. The last assertion says that 
$u$ is a weak positive solution to Problem (Pλ).
In order to prove that 
$v$ is a weak positive solution we mention that 
$t_{\lambda,\varepsilon}^{n,-} := t_{\lambda}^{n,-}(v + \varepsilon\psi)$ is a local maximum of the fibering maps 
$\beta_{\lambda,v}$. Hence, we have
\begin{equation*}I_{\lambda}(t_{\lambda,\varepsilon}^{n,-}(v+\varepsilon \psi)) \geq I_{\lambda}(v) = \beta_{\lambda,v}(1) \geq \beta_{\lambda,v}(t_{\lambda,\varepsilon}^{n,-}) = I_{\lambda}(t_{\lambda,\varepsilon}^{n,-}v).\end{equation*} Now, using the same ideas discussed above using the function 
$v$ instead of 
$u$, we obtain 
$I_{\lambda}'(v)\psi \geq 0$ holds for any 
$\psi \geq 0$. This ends the proof.
Proposition 3.25. Suppose that 
$(\phi_{1})$–
$(\phi_{4})$, 
$(H_1)$, and 
$(H_2)$ hold. Assume that 
$\lambda \gt 0$. Let 
$v \in W_{0}^{s,\Phi}(\Omega)$ be the function obtained in Proposition 3.22. Then, we obtain the following statements
(i) Assume that
$\lambda \in (0,\lambda_{*})$. Then
$\mathcal{C}_{\mathcal{N}_{\lambda}^{-}} = I_{\lambda}(v) \gt 0$ (see Figure 3);(ii) Suppose
$\lambda = \lambda_{*}$. Then
$\mathcal{C}_{\mathcal{N}_{\lambda}^{-}} = I_{\lambda}(v) = 0$;(iii) Suppose
$\lambda \in (\lambda_{*}, \lambda^{*})$. Then
$\mathcal{C}_{\mathcal{N}_{\lambda}^{-}} = I_{\lambda}(v) \lt 0$.
Geometry of the fibering map 
$t \mapsto I_{\lambda}(tu)$.
Figure 3 Long description
A line plot titled Geometry of the fibering map displays three curves, each representing c subscript N lambda plotted against an unlabeled horizontal parameter with no units shown. The vertical axis is labeled with reference levels: c subscript N lambda superscript minus greater than 0, c subscript N lambda superscript minus equals 0 and c subscript N lambda superscript minus less than 0, along with two lower levels labeled c subscript N lambda superscript plus. No numeric tick values appear on either axis. Three lambda regimes are shown. The first curve corresponds to lambda between 0 and lambda-star. Its local maximum rises above the zero reference line, marked with a filled dot at a positive vertical level labeled c subscript N lambda superscript minus greater than 0. Its local minimum occurs at a lower horizontal position, marked with a filled dot at a level labeled c subscript N lambda superscript plus. The second curve corresponds to lambda equals lambda-star. Its local maximum sits exactly at the zero reference line, marked with a filled dot at c subscript N lambda superscript minus equals 0. Its local minimum is marked at a level labeled c subscript N lambda superscript plus, positioned below the first curve′s minimum. The third curve corresponds to lambda between lambda-star and lambda superscript asterisk. Its local maximum falls below the zero reference line, marked with a filled dot at c subscript N lambda superscript minus less than 0. Its local minimum is the lowest of all three curves, marked at a level labeled c subscript N lambda superscript plus at the lowest reference line. All three maxima occur at the same horizontal position. All three minima occur at a second, distinct horizontal position to the left of the maxima. Dotted vertical and horizontal lines connect each marked point to its corresponding axis reference level.
Proof. The proof follows similar the same ideas employed in [Reference Il’yasov31, Theorem 1.2]. We omit the details.
4. The proof of the main theorems
4.1. The proof of Theorem 1.2
According to Propositions 3.5 and 3.11, there exist 
$\lambda_{*}$ and 
$\lambda^{*}$ such that 
$0 \lt \lambda_{*} \lt \lambda^{*}$, see for instance Proposition 3.15. Let us consider 
$\lambda \in (0, \lambda^*)$. Since the energy functional 
$I_{\lambda}$ is coercive, see Proposition 3.17, there exists 
$u \in \mathcal{N}_{\lambda}^{+}$ such that 
$I_{\lambda}(u) = \mathcal{C}_{\mathcal{N}_{\lambda}^{+}} \lt 0$, see Proposition 3.21. Furthermore, by using Propositions 3.22 and 3.24, we conclude that 
$u$ is a critical point of the energy functional 
$I_{\lambda}$. Therefore, 
$u \in W_{0}^{s, \Phi}(\Omega)$ is a positive weak solution.
From now on, we prove item (ii). Note that the inequality 
$I_\lambda''(u)(u,u) \gt 0$ and condition 
$(\phi_4)$ imply that
\begin{equation*}
(m-1)\iint\limits_{\mathbb{R}^N \times \mathbb{R}^N} \phi(|D_s u|)|D_s u|d\mu+\lambda\gamma\int_\Omega u^{1-\gamma}dx-(p-1)\int_\Omega u^pdx\geq 0.
\end{equation*} Hence, by using the fact that 
$I_\lambda'(u)u=0$ and the Hölder inequality, we deduce that
\begin{equation*}
(m+\gamma-1)|\Omega|^{\frac{p}{p+\gamma-1}}\lambda\|u\|_p^{1-\gamma}\geq(m+\gamma-1)\lambda\int_\Omega u^{1-\gamma}dx\geq (p-m)\|u\|_{p}^p.
\end{equation*}Consequently,
\begin{equation*}
\|u\|_p^{p+\gamma+1}\leq \frac{(m+\gamma-1)|\Omega|^{\frac{p}{p+\gamma-1}}}{m-p}\lambda.
\end{equation*} This proves that 
$\|u\|_p\to 0$ as 
$\lambda\to 0^+$. Since
\begin{equation*}
\int_\Omega u^{1-\gamma}dx\leq |\Omega|^{\frac{p}{p+\gamma-1}}\|u\|_p^{1-\gamma},
\end{equation*}we obtain that 
$\int_\Omega u^{1-\gamma}dx\to 0$ as 
$\lambda\to 0^+$. Furthermore, using that 
$\Phi\in \Delta_2$ and taking the limit as 
$\lambda\to 0^+$ in 
$I_\lambda'(u)u=0$, we obtain that 
$[u]_{s,\Phi}\to 0$. This ends the proof.
4.2. The proof of Theorem 1.3
Let us consider 
$\lambda \in (0,\lambda^{*})$. Using Proposition 3.22, there exists 
$v \in \mathcal{N}_{\lambda}^{-}$ such that 
$I_{\lambda}(v)=\mathcal{C}_{\mathcal{N}_{\lambda}^{-}}$ and 
$I_{\lambda}'(v)(v,v) \lt 0$. This proves the item 
$(i)$. Moreover, by Proposition 3.24, we show that 
$v$ is a critical point of the functional 
$I_{\lambda}$. The proof of 
$(ii)$–
$(iv)$ is a direct consequence of Proposition 3.25.
5. Existence of solutions at the level
$\lambda=\lambda^{*}$
In this section, we shall consider the case where the parameter 
$\lambda$ assumes the extreme value. In other words, we consider the case 
$\lambda = \lambda^*$ where inflextion points for the fibering map are allowed. To this end, we consider a sequence 
$\lambda_k \lt \lambda^*$ such that 
$\lambda_k \to \lambda^*$ as 
$k \to \infty$. Under these conditions, we consider a sequence 
$(u_k) := (u_{\lambda_k})$ which are weak solutions in 
$\mathcal{N}_{\lambda_k}^+$ for Problem (1.1) with 
$\lambda = \lambda_k$. This leads us to a non-trivial solution in 
$\mathcal{N}_{\lambda^*}^+ \cup \mathcal{N}_{\lambda^*}^0$. In order to do that, we shall consider an auxiliary result.
Lemma 5.1. Assume that 
$(\phi_{1})$–
$(\phi_{4})$, 
$(H_1)$, and 
$(H_2)$ hold. Let 
$0 \lt \lambda_k \lt \lambda^{*}$ such that 
$\lambda_k\to \lambda^*$. Then 
$\limsup_{k \to \infty} I_{\lambda_k}(u_k) \lt 0$.
Proof. Assume that 
$(\lambda_k)$ is a sequence such that 
$0 \lt \lambda_1 \leq \lambda_k \lt \lambda^*$ for all 
$k \in \mathbb{N}$. Let 
$u_1 \in \mathcal{N}_{\lambda_1}^{+}$ be a solution to Problem (Pλ) at the level 
$\lambda_1$. Then 
$\mathcal{C}_{\mathcal{N}_{\lambda_1}^{+}} = I_{\lambda_1}(u_1) \lt 0$, see for instance Theorem 1.2. Let 
$u_k \in \mathcal{N}_{\lambda_k}^{+}$ be a solution to Problem (Pλ) at the level 
$\lambda_k$. Thus, 
$\mathcal{C}_{\mathcal{N}_{\lambda_k}^{+}}=I_{\lambda_k}(u_k)$. It follows that
\begin{equation*}\mathcal{C}_{\mathcal{N}_{\lambda_k}^{+}}=I_{\lambda_k}(u_k) \leq I_{\lambda_k}(t^{n,+}_{\lambda_k}(u_1)u_1).\end{equation*} Since 
$t^{n,+}_{\lambda_1}(u_k)=1 \leq t^{n,+}_{\lambda_k}(u_1)$ and 
$t\mapsto I_{\lambda_k}(t)u_1$ is strictly decreasing in 
$(0,t^{n,+}_{\lambda_k}(u_1))$, we obtain
\begin{equation}
I_{\lambda_k}(u_k) \leq I_{\lambda_k}(u_1) = I_{\lambda_1}(u_1) - \frac{(\lambda_k - \lambda_1)}{1-\gamma} \displaystyle\int_{\Omega}|u|^{1-\gamma}dx
\leq I_{\lambda_1}(u_1) \lt 0.
\end{equation} Therefore, 
$\limsup_{k \to \infty} I_{\lambda_k}(u_k) \lt 0$. This ends the proof.
5.1. The proof of Theorem 1.5
Consider a sequence 
$(\lambda_k)$ such that 
$\lambda_k \lt \lambda^*$ for each 
$k \in \mathbb{N}$ and 
$\lambda_k \to \lambda^*$ as 
$k \to \infty$. According to Theorem 1.2, there exists a sequence of solutions 
$u_k := u_{\lambda_k}$ in 
$\mathcal{N}_{\lambda_k}^{+}$ for Problem (Pλ). Here, we shall prove that 
$(u_k)$ is bounded in 
$W^{s,\Phi}_0(\Omega)$. In fact, by using the fact that 
$u_k \in \mathcal{N}_{\lambda_k}^{+}$, we obtain that
\begin{equation*}
R_n(u_k) = \lambda_k\qquad \mbox{and}\qquad R_n'(u_k)u_k \gt 0.
\end{equation*}This assertion implies that
\begin{align}
& \displaystyle\iint\limits_{\mathbb{R}^N \times \mathbb{R}^N} \phi'(|D_{s}u_k |)|D_{s}u_k|^3 d\mu - (p-2)\displaystyle\iint\limits_{\mathbb{R}^N \times \mathbb{R}^N} \phi(|D_{s}u_k|)(D_{s}u_k)^2 d\mu
+\lambda_k (p + \gamma -1)\nonumber\\ &\quad\times \displaystyle\int_{\Omega}|u_k|^{1-\gamma} dx \gt 0.
\end{align} According to 
$(\phi_4)$, Holder’s inequality and the embedding of 
$W_{0}^{s,\Phi}(\Omega)$ into 
$L^{1}(\Omega)$, we infer that
\begin{equation}
(p-m)\displaystyle\iint\limits_{\mathbb{R}^N \times \mathbb{R}^N} \phi(|D_{s}u_k|)(D_{s}u_k)^2 d\mu \lt \lambda_k (p + \gamma -1)\displaystyle\int_{\Omega}|u_k|^{1-\gamma} dx\leq (p+\gamma -1)S_1|\Omega|^{\gamma}[u_k]_{s,\Phi}^{1-\gamma}.
\end{equation}Now, by using Proposition 2.2, we see that
\begin{equation*}l(p-m)\min \{[u_k]_{s,\Phi}^{\ell+\gamma-1},[u_k]_{s,\Phi}^{m+\gamma-1}\} \lt \lambda^* (p+\gamma -1)S_1|\Omega|^{\gamma}.\end{equation*} Consequently, 
$(u_k)$ is a bounded sequence in 
$W^{s,\Phi}_0(\Omega)$ and there exists 
$u_{\lambda^*} \in W^{s,\Phi}_0(\Omega)$ such that 
$u_k \stackrel{*}{\rightharpoonup} u_{\lambda^*}$ in 
$W^{s,\Phi}_0(\Omega)$.
At this stage, we shall prove that 
$u_{\lambda^*} \neq 0$. Indeed, by using the fact that 
$u_k$ is a weak solution for (Pλ) at the level 
$\lambda = \lambda_k$, we can apply Lemma 5.1 showing that
\begin{equation*}0 \gt \displaystyle\liminf_{k \rightarrow \infty}I_{\lambda}(u_k) \geq I_{\lambda}(u_{\lambda^{*}}),\end{equation*}which implies that 
$I_{\lambda}(u_{\lambda^{*}}) \lt 0$, therefore 
$u_{\lambda^{*}} \neq 0$. Now, using the same ideas considered in the proof of Proposition 3.21, we deduce that 
$u_k \to u_{\lambda^*}$ in 
$W^{s,\Phi}_0(\Omega)$. Once again, using the last assertion and the continuity of 
$u \mapsto I_{\lambda}(u)$ and 
$u \mapsto I_{\lambda}''(u)(u,u)$, we have that 
$u_{\lambda^*} \in \mathcal{N}_{\lambda^*}^+ \cup \mathcal{N}_{\lambda^*}^0$. Recall also that 
$u_k \geq 0$ in 
$\Omega$ for each 
$k \in \mathbb{N}$. In particular, we obtain that 
$u_{\lambda^*} \geq 0$ in 
$\Omega$ and 
$\int_{\Omega} u_{\lambda^*}^{-\gamma} \psi dx \lt \infty$ holds for each 
$k \in \mathbb{N}$ with 
$\psi \geq 0 $ in 
$\Omega$ and 
$\psi \in W^{s,\Phi}_0(\Omega)$. Here, we used the same ideas as those employed in the proof of Proposition 3.24. As a consequence, we know that 
$u \gt 0$ a.e. in 
$\Omega$. Since 
$u_k \to u_{\lambda^*}$ in 
$W^{s,\Phi}_0(\Omega)$ it follows that 
$I'_{\lambda^*}(u_{\lambda^*}) \psi = 0$ for each 
$\psi \in W^{s,\Phi}_0(\Omega)$. This ends the proof.
6.
$L^{\infty}$ regularity results for weak solutions of Problem (Pλ)
In the present section, we shall discuss some important conditions in order to obtain 
$L^\infty$ estimates for weak solutions of Problem (Pλ). Here we refer the interested reader to [Reference Carvalho, Silva, de Albuquerque and Bahrouni15]. The main point here is to prove Theorem 1.7. To this end, we consider a function 
$\theta \in C^0(\mathbb{R},\mathbb{R})$ be such that 
$\theta'$ exists in 
$\mathbb{R} \setminus \Gamma_\theta$. Notice also that 
$\Gamma_\theta$ is a finite set. The set of all functions satisfying this assumption is denoted by 
$T_\theta.$
Lemma 6.1. (See [Reference Carvalho, Silva, de Albuquerque and Bahrouni15])
Let 
$\Phi$ be an 
$N$-function such that 
$\Phi \in \Delta^{'}$ and 
$s \in (0,1)$. Define the auxiliary function
\begin{equation*}\Gamma_{\theta}(t):= \int_{0}^{t}\Phi^{-1}(\theta^{'}(\tau))d\tau \ t \gt 0 \end{equation*}where 
$\theta \in T_{\theta}$ is a non-decreasing function. Then there exists a constant 
$C \gt 0$ such that for all 
$a,b \in \mathbb{R}$, 
$x \neq y \in \mathbb{R}^N$, we obtain that
\begin{equation*}C\Phi \left(\frac{\Gamma_{\theta}(b) - \Gamma_{\theta}(a)}{|x-y|^s}\right) \leq \left(\frac{\theta(b) - \theta(a)}{|x-y|^s}\right)\phi\left( \frac{b-a}{|x-y|^s}\right)\left( \frac{b-a}{|x-y|^s}\right).\end{equation*} Let 
$L,\beta_k \gt 1$ be fixed. Define
\begin{equation*}\theta_k(t) = t\Phi\left(\frac{t_L^{\beta_k - 1}}{A}\right) \quad \text{and} \quad \Gamma_{\theta_k}(t) = \int_0^t \Phi^{-1}(\theta_k'(\tau)) d\tau\end{equation*}where 
$t_L = \min\{t, L\}$ and 
$A$ will be defined later.
Now, we shall prove some fine estimates in order to get prove some 
$L^\infty$ estimates. Firstly, we shall prove the following result:
Lemma 6.2. (See [Reference Carvalho, Silva, de Albuquerque and Bahrouni15])
The function 
$\Gamma_{\theta}$ satisfies the following estimate
\begin{equation}
\Gamma_{\theta}(t) \geq \frac{tt_{L}^{\beta_{k}-1}}{\beta_{k}A}, \ t \geq 0.
\end{equation}Lemma 6.3. (See [Reference Carvalho, Silva, de Albuquerque and Bahrouni15])
Suppose that 
$\tilde{\Psi} \circ \Phi \prec \Phi^{*}_{s}$. Let 
$C^{*} \gt 0$ be fixed such that
\begin{equation*}||v||_{\tilde{\Psi} \circ \Phi } \leq C^{*}||v||_{\Phi^{*}_{s}} \ \forall v \ in W_{0}^{s,\Phi}(\Omega).\end{equation*}Then
\begin{equation*}\left|\left| \Phi\left(\frac{v}{C^{*}||v||_{\Phi^{*}_{s}}}\right) \right|\right|_{\tilde{\Psi}} \leq 1 \ \forall v \in W_{0}^{s,\Phi}(\Omega). \end{equation*}Lemma 6.4. (See [Reference Carvalho, Silva, de Albuquerque and Bahrouni15])
Let 
$u \in W_{0}^{s,\Phi}(\Omega)$ be fixed. Then the function 
$\theta_k (u) \in W_{0}^{s,\Phi}(\Omega)$.
6.1. Proof of Theorem 1.7
Let
\begin{equation*}\theta_k(t) = t \Phi\left(\frac{t_{L}^{\beta_k - 1}}{A}\right)\end{equation*}be a sequence with 
$A = \|u^{\beta_k}\|_{\Phi_{s}^{*}}$, 
$\beta_k \gt 1$ where 
$u$ is a weak solution of Problem (Pλ). In view of Lemma 6.4, we obtain that 
$\theta_k(u) \in W^{s,\Phi}_0(\Omega)$ for each 
$u \in W^{s,\Phi}_0(\Omega)$. Now, invoking the 
$\Delta_2$ condition and Proposition 2.1, we deduce
\begin{equation}
1 = \iint\limits_{\mathbb{R}^N \times \mathbb{R}^N} \Phi\left(\frac{|D_s w|}{[w]_{s,\Phi}}\right) d\mu \leq \zeta_1\left(\frac{1}{[ w]_{s,\Phi}}\right) \iint\limits_{\mathbb{R}^N \times \mathbb{R}^N} \Phi(|D_s w|) d\mu,
\end{equation}where 
$w=\Gamma_{\theta_k} (u)$. Hence, by using Lemma 6.1 and (23), we obtain
\begin{equation}
C\leq \zeta_1 \left( \frac{1}{[\Gamma_{\theta_k}(u)]_{s,\Phi}}\right)\displaystyle\iint\limits_{\mathbb{R}^N \times \mathbb{R}^N} \phi\left(|D_su|)D_suD_s(\theta_k(u)\right)d\mu.
\end{equation} Furthermore, by using the embedding 
$W_{0}^{s, \Phi} \hookrightarrow L^{\Phi^{*}_{s}}(\Omega)$, there exists a constant 
$S_{\Phi}^{*}$ defined by
\begin{equation*}S_{\Phi}^{*} = \inf_{v \in W_{0}^{s, \Phi}(\Omega)} \frac {\|v\|_{s, \Phi}} {\|v\|_{\Phi_{s}^{*}}} \gt 0.\end{equation*}Hence, by using (22) and (24), we obtain that
\begin{equation}
C \leq \zeta_1 \left( S_{\Phi}^{*}\beta_k\right)\left[\lambda \displaystyle\int_{\Omega}\theta_{k}(u)u^{-\gamma} dx + \displaystyle\int_{\Omega}\theta_{k}(u)u^{p-1}dx \right]=: \zeta_1\left( S_{\Phi}^{*}\beta_k\right)(I_1+I_2).
\end{equation}It is important to emphasize that
\begin{eqnarray*}
I_1 & = &\lambda \displaystyle\left[ \int_{|u| \lt 1}+\int_{|u|\geq 1}\right]u^{1 -\gamma}\Phi\left(\frac{u ^{\beta_k -1}}{A}\right)dx
\leq \lambda \int_{|u| \lt 1} u^{z_k}\Phi\left(\frac{1}{A}\right)dx\\ &&\quad +\lambda \int_{|u|\geq 1}u^{1-\gamma}\Phi\left(\frac{u^{\beta_k - 1}}{A}\right)dx,
\end{eqnarray*}where 
$z_k = 1 - \gamma + (\beta_k - 1)l \gt 1$ for 
$k$ large enough. Notice also that, for each 
$|u| \lt 1$, we obtain that 
$v_k:=u^{z_k} \lt 1$. As a consequence, we see that
\begin{equation*}
\lambda \int_{|u| \lt 1} u^{z_k} \Phi \left( \frac{1}{A} \right) dx \leq \lambda |\Omega| \Phi \left( \frac{1}{A} \right)\leq \lambda |\Omega| \Phi(1) \zeta_1 \left( \frac{1}{\|v_k\|_{\Phi_{s}^{*}}} \right).
\end{equation*} On the other hand, for the set 
$|u| \geq 1$, we infer that 
$1 \leq |u|^{1-\gamma} \lt |u|^l = \zeta_0(|u|)$. Hence, by using Equation 4 and the embedding 
$L^{\Phi_{s}^{*}}(\Omega) \hookrightarrow L^{\Phi}(\Omega)$, we obtain
\begin{equation}
\lambda \int_{|u| \geq 1} u^{1 - \gamma} \Phi \left( \frac{u^{\beta_k - 1}}{A} \right) dx \leq \lambda \zeta_1 \left( \frac{1}{C^{*}} \right) \zeta_1 \left( \frac{C^{*} \|v_k\|_{\Phi}}{\|v_k\|_{\Phi_{s}^{*}}} \right)=\lambda \zeta_1 \left( \frac{1}{C^{*}} \right) \left( \frac{C^{*} \|v_k\|_{\Phi}}{\|v_k\|_{\Phi_{s}^{*}}} \right)^l,
\end{equation}where 
$\frac{C^{*} \|v_k\|_{\Phi}}{\|v_k\|_{\Phi_{s}^{*}}} \leq 1$. As a consequence, we obtain that
\begin{equation*}
I_1 \leq \lambda |\Omega| \Phi(1) \zeta_1 \left( \frac{1}{\|v_k\|_{\Phi_{s}^{*}}} \right) + \lambda \zeta_1 \left( \frac{1}{C^{*}} \right) \left( \frac{C^{*} \|v_k\|_{\Phi}}{\|v_k\|_{\Phi_{s}^{*}}} \right)^l.
\end{equation*}Now, we consider the following integral
\begin{equation*}
I_2 = \left[ \int_{|u| \lt 1} + \int_{|u| \geq 1} \right] u^p \Phi \left( \frac{u^{\beta_k - 1}}{A} \right) dx.
\end{equation*}It is not hard to see that
\begin{eqnarray*}
\int_{|u| \lt 1} u^p \Phi \left( \frac{u^{\beta_k - 1}}{A} \right) dx \leq \int_{|u| \lt 1} u^{p + (\beta_k - 1)l} \Phi \left( \frac{1}{A} \right) dx
\leq |\Omega| \Phi(1) \zeta_1 \left( \frac{1}{\|v_k\|_{\Phi_{s}^{*}}} \right).
\end{eqnarray*} On the other hand, fixing 
$R \gt 1$ and using the same ideas as employed in (26), Proposition 2.1, and Hölder’s inequality with exponents 
$\frac{l_{s}^{*}}{p - l}$ and its conjugate 
$\frac{l_{s}^{*}}{l_{s}^{*} - p + l}$, we deduce
\begin{align}
&\int_{|u| \geq 1} u^p \Phi \left( \frac{u^{\beta_k - 1}}{A} \right) dx = \left[ \int_{[|u| \geq 1] \cap [|u| \gt R]} + \int_{[|u| \geq 1] \cap [|u| \leq R]} \right] u^p \Phi \left( \frac{u^{\beta_k - 1}}{A} \right) dx\notag\\
&\leq \zeta_1 (C^{*})\int_{[|u| \geq 1] \cap [|u| \gt R]} u^{p - l} \Phi \left( \frac{u^{\beta_k}}{C^{*} A} \right) dx+ R^{p - l} \zeta_{1} \left( \frac{1}{C^{*}} \right) \left( \frac{C^{*} \|v_k\|_{\Phi}}{\|v_k\|_{\Phi^{*}_{s}}} \right)^l\notag\\
&\leq \zeta_1 (C^{*}) \|u\|^{p - l}_{l^{*}_{s}} \left\| \Phi \left( \frac{u^{\beta_k}}{C^{*} A} \right) \right\|_{\frac{l^{*}_{s}}{l^{*}_{s} - p + l}} + R^{p - l} \zeta_{1} \left( \frac{1}{C^{*}} \right) \left( \frac{C^{*} \|v_k\|_{\Phi}}{\|v_k\|_{\Phi^{*}_{s}}} \right)^l.
\end{align} Recall also that 
$p \leq l_{s}^{*} - (m - l)$. Hence, we choose 
$\tilde{\Psi}(t) = t^{\frac{l_{s}^{*}}{l_{s}^{*} - p + l}}$. It follows from Lemma 6.3 and (27) that
\begin{equation*}
I_2 \leq |\Omega| \Phi(1) \zeta_1 \left( \frac{1}{\|v_k\|_{\Phi^{*}_{s}}} \right) + R^{p - l} \zeta_{1} \left( \frac{1}{C^{*}} \right) \left( \frac{C^{*} \|v_k\|_{\Phi}}{\|v_k\|_{\Phi^{*}_{s}}} \right)^l + \zeta_1 (C^{*}) \|u\|^{p - l}_{\Psi, [|u| \geq R]}.
\end{equation*}As a consequence, by using (25), we see that
\begin{eqnarray*}
C &\leq& \zeta_1 \left( S_{\Phi}^{*}\beta_k\right)\left[ (\lambda + 1) |\Omega|\Phi(1)\zeta_1\left(\frac{1}{||v_k||_{\Phi^{*}_{s}}}\right) + (\lambda + R^{p-l})\zeta_{1}\left(\frac{1}{C^{*}}\right) \left(\frac{C^{*}||v_k||_{\Phi}}{||v_k||_{\Phi^{*}_{s}}}\right)^l\right. \\ &&\quad + \left. \zeta_1 (C^{*})||u||^{p-l}_{\Psi,[|u|\geq R]} \right].
\end{eqnarray*} Moreover, taking 
$R \gt \gt 1$ such that
\begin{equation*} \zeta_1(S_{\Phi}^{*})\beta_1(C^{*})||u||^{p-l}_{\Psi,[|u|\geq R]} \leq \frac{C}{2},\end{equation*}we infer that
\begin{eqnarray*}
\frac{C}{2} &\leq& \zeta_1 \left( S_{\Phi}^{*}\right)\beta_k^m\left[ (\lambda + 1) |\Omega|\Phi(1)\zeta_1\left(\frac{1}{||v_k||_{\Phi^{*}_{s}}}\right) + (\lambda + R^{p-l})\zeta_{1}\left(\frac{1}{C^{*}}\right) \left(\frac{C^{*}||v_k||_{\Phi}}{||v_k||_{\Phi^{*}_{s}}}\right)^l\right].
\end{eqnarray*} Since 
$\zeta_0(t)=\min \{t^l, t^m\}$, we conclude that
\begin{equation*}\zeta_0(||v_k||_{\Phi^{*}_{s}}) \leq D\beta_k^m(1+(\lambda + R^{p-l})||v_k||_{\Phi}^l).\end{equation*} Moreover, by using the embeddings 
$L^{\Phi_{s}^{*}}(\Omega) \hookrightarrow L^{l_{s}^{*}}(\Omega)$ and 
$L^m(\Omega) \hookrightarrow L^{\Phi}(\Omega)$, we obtain that
\begin{equation}
\zeta_0(||u||_{\beta_kl_{s}^{*}}^{\beta_k}) \leq \bar{D}\beta_k^m(1+(\lambda + R^{p-l})||u||_{\beta_k m}^{\beta_k l})
\end{equation} From now on, we consider the recursive sequence 
$\beta_k$ as follows: for each 
$k \geq 1$, take 
$\beta_0 = 1$, 
$\beta_1=\frac{l^{*}_{s}}{m}$, and 
$\beta_{k+1}=\beta_k \frac{l^{*}_{s}}{m}$. Notice also that 
$\beta_k=\beta_1^k$.
Define the sequence 
$F_{k+1}$ by 
$F_{k+1}=\beta_{k+1}\ln (||u||_{\beta_{k+1}l_s^{*}})$ for 
$k \geq 1$ and
\begin{equation*}N_{+}(u)=\{k \in \mathbb{N}; ||u||_{\beta_k m} \geq 1\}.\end{equation*} It follows from (28) and 
$k \in N_{+}(u)$ that
\begin{eqnarray*}
\nonumber
F_{k+1} & \leq & \ln(\bar{D}\beta_{k+1}^{m})^{\frac{1}{\alpha}} + \frac{1}{\alpha}\ln \left[(1+(\lambda + R^{p-l}))||u||_{m\beta_{k+1}}^{l\beta_{k+1}}\right] \\
&\leq & \ln(\bar{D}\beta_{k+1}^{m})^{\frac{1}{\alpha}} + \frac{1}{\alpha}\ln \left[2(\lambda + R^{p-l})||u||_{m\beta_{k+1}}^{l\beta_{k+1}}\right] \\
&\leq & \ln(\bar{D}\beta_{k+1}^{m}2(\lambda + R^{p-l}))^{\frac{1}{\alpha}} + \beta_1 F_k, \ \ k \in N_{+}(u).
\end{eqnarray*} Similarly, for each 
$k \in \mathbb{N} \backslash N_{+}(u)$, we obtain
\begin{equation}
F_{k+1} \leq \ln(\bar{D}\beta_{k+1}^{m}2(\lambda + R^{p-l}))^{\frac{1}{\alpha}},
\end{equation}where 
$\alpha \in \{l, m \}$. Therefore, by using (29), we infer that
\begin{equation}
\begin{array}{rl}
\displaystyle F_{k+1} \leq \left\{\begin{array}{lcr}
J_k + \beta_{1}F_k, & \text{if} & k \in N_{+}(u), \\
J_k, & \text{if} & k \in \mathbb{N}\backslash N_{+}(u),
\end{array}\right.
\end{array}
\end{equation}where 
$J_k = \ln(\bar{D}\beta_{k+1}^{m}2(\lambda + R^{p-l}))^{\frac{1}{\alpha}}$. Consequently,
\begin{equation*}
\frac{J_k}{\beta_{1}^k} = \frac{\ln(\bar{D}\beta_{k+1}^{m}2(\lambda + R^{p-l}))^{\frac{1}{\alpha}}}{\beta_{1}^k} \leq \frac{\ln (\bar{D}2 (\lambda + R^{p-l}))^\frac{1}{\alpha}}{\beta_1^k} + \frac{m}{\alpha}\frac{\ln (\beta_{k+1})}{\beta_1^k}
= \frac{d_1}{\beta_1^k} + \tilde{d}\frac{k}{\beta_1^k}, \nonumber
\end{equation*}where 
$d_1= \ln(\bar{D}2(\lambda + R^{p-l}))^{\frac{1}{\alpha}} + \frac{m}{\alpha}\ln \left( \frac{l_{s}^{*}}{m}\right)$ and 
$\tilde{d}= \frac{m}{\alpha}\ln\left( \frac{l_{s}^{*}}{m} \right)$. This implies that
\begin{equation}
F_k \leq \beta_1^{k-1}|F_1|+J_{k-1}+\beta_1J_{k-2} + \beta_1^2J_{k-3}+ ... +\beta_1^{k-2}J_1.
\end{equation}Furthermore, by using (31), we see that
\begin{eqnarray*}
\ln(||u||_{\beta_kl_{s}^{*}}) &=&\frac{F_k}{\beta_k} \leq \frac{1}{\beta_1} \left[ |F_1|+ \sum_{j=0}^{k-1}\frac{J_j}{\beta_1^{j}}\right]
\leq \frac{1}{\beta_1}\left[ |F_1|+ \sum_{j=0}^{k-1}\left(\frac{d_1}{\beta_1^{j}} + \frac{\tilde{d}}{\beta_1^{j}}\right)\right]\\ &=& \frac{1}{\beta_1} \left[ |F_1|+d_1\sum_{j=0}^{k-1} \frac{1}{\beta_1^{j}} + \tilde{d}\sum_{j=0}^{k-1} \frac{j}{\beta_1^{j}}\right].
\end{eqnarray*}As a consequence, we obtain that
\begin{equation*}\lim_{k \rightarrow \infty}\ln(||u||_{\beta_kl_{s}^{*}}) \leq \lim_{k \rightarrow \infty} \frac{1}{\beta_1} \left[ |F_1|+d_1\sum_{j=0}^{k-1} \frac{1}{\beta_1^{j}} + \tilde{d}\sum_{j=0}^{k-1} \frac{j}{\beta_1^{j}}\right]=d_0.\end{equation*} Thus, 
$0 \lt u \lt e^{d_0}$ a.e. in 
$\Omega$. This ends the proof.
Acknowledgments
M.C. was partially supported by CNPq with grant 300411/2025-1. A.M. was partially supported by Capes with grant 88887.804697/2023-00. E.S. was partially supported by CNPq with grant 310440/2025-4.
