1. Introduction and main results

In this paper, we study nonlinear Schrödinger equations of the form

(P _{\unicode{x03BA}})\[\left\{ \begin{aligned} - & \Delta u+V(x)u+\displaystyle\frac{\kappa}{2}[\Delta(u^{2})]u=h(u)\quad \textrm{in} \quad \mathbb{R}^{2},\\ & u(x) \rightarrow 0 \quad \textrm{as} \quad |x|\rightarrow \infty, \end{aligned} \right. \]

where $V:\mathbb {R}^{2}\rightarrow \mathbb {R}$ is a given continuous potential bounded away from zero, which is asymptotically constant at infinity and $h:\mathbb {R}\rightarrow \mathbb {R}$ is a critical nonlinearity in the sense of the Trudinger–Moser inequality.

Problem ($P_{\kappa }$) allows for a variational formulation, which formally means that weak solutions arise as critical points of the energy functional

\[ J_{\kappa}(u) = \displaystyle\frac{1}{2}\displaystyle\displaystyle\int_{\mathbb{R}^{2}}(1-\kappa u^{2})|\nabla u|^{2} \; \mathrm{d} x + \frac{1}{2}\displaystyle\int_{\mathbb{R}^{2}}V(x)u^{2}\; \mathrm{d} x - \displaystyle\int_{\mathbb{R}^{2}}H(u)\; \mathrm{d} x. \]

The first difficulty one has to face is finding an appropriate function space where $J_{\kappa }$ is well defined. To avoid such a problem, we use a dual approach in combination with a mountain pass procedure to obtain the existence of positive solutions for ($P_{\kappa }$), depending on the behaviour of a small parameter $\kappa >0$. Precisely, applying a change of variable proposed in [Reference Alves, Wang and Shen5], the quasilinear equation ($P_{\kappa }$) is transformed to a nonlinear equation, whose respective associated functional is well defined in the usual Sobolev space $H^{1}(\mathbb {R}^{2})$ and satisfies the geometric conditions of the Mountain Pass Theorem. The Concentration–Compactness Principle due to P.L. Lions and a comparison argument allow us to verify that the problem possesses a non-trivial solution.

This class of problems arises naturally in many mathematical physics contexts, for example, in plasma physics, fluid mechanics in the classical and quantum theory of Heisenberg ferromagnets and magnons, and dissipative quantum mechanics. In particular, solutions of ($P_{\kappa }$) are motivated by the search of certain kinds of standing wave solutions for the following class of nonlinear Schrödinger equation:

(1.1)\begin{equation} i\displaystyle\frac{\partial\psi}{\partial t} ={-}\Delta \psi + W(x)\psi-\eta(|\psi|^{2})\psi+ \frac{\kappa}{2}[\Delta \rho(|\psi|^{2})]\rho'(|\psi|^{2})\psi, \end{equation}

where $\psi :\mathbb {R}^{2}\times \mathbb {R} \rightarrow \mathbb {C}$, $\kappa$ is a real constant, $W:\mathbb {R}^{2} \rightarrow \mathbb {R}$ is a given potential and $\eta :\mathbb {R}_+ \rightarrow \mathbb {R}$, $\rho :\mathbb {R}_+ \rightarrow \mathbb {R}$ are appropriate functions. Equations like (1.1) have been a researcher topic in some areas of physics depending on several types of nonlinear terms $\rho (s)$, see for instance [Reference Poppenberg, Schmitt and Wang28]. In our analysis, we are interested in the superfluid film equation in plasma physics, which corresponds to the case $\rho (s)=s$, see [Reference Kurihara22]. We also mention that if $\eta (s)$ is pure power, (1.1) appears in nonlinear optics, e.g., oscillating soliton instabilities during microwave and laser heating of plasma, see [Reference Goldman18, Reference Porkolab and Goldman29].

Problem ($P_{\kappa }$) has received significant attention from many researchers in the last decades motivated by afore mention physical applications. To prove existence and multiplicity results for equation ($P_{\kappa }$) variationally, we have to face some challenges. First, we cannot find a suitable function space on which the associated energy functional is well defined and belongs to $C^{1}$-class, except in dimension one (see [Reference Poppenberg, Schmitt and Wang28]). Moreover, we have to overcome the lack of compactness because the equation is defined in whole two-dimensional Euclidean space and involves nonlinearities with exponential critical growth. In [Reference Poppenberg, Schmitt and Wang28], by using a constrained minimization argument, the authors studied equation ($P_{\kappa }$) in the case when the parameter $\kappa$ is negative, for dimension one or when the potential $V$ is radially symmetric for high dimensions. See also [Reference Liu and Wang23] for the more general case. We mention that, more recently, to develop a suitable setting to use variational argument to study ($P_{\kappa }$), in [Reference Liu, Wang and Wang25, Reference Ruiz and Siciliano30], the authors used Nehari manifold and in [Reference Liu, Liu and Wang26], a perturbation method was considered. After that, in [Reference Liu, Wang and Wang24], the authors reduced the quasilinear Equation ($P_{\kappa }$) to a semilinear one, by using a change of variables (dual approach). For that, an Orlicz space framework was developed to obtain a positive solution via minimax methods.

To study the case when the parameter $\kappa$ is negative, in [Reference Colin and Jeanjean8], the authors adopted the dual approach by working in the usual Sobolev space environment. For that, they performed the change of variables: $v=f^{-1}(u)$, where $f(t)$ is defined by

(1.2)\begin{equation} \begin{aligned} f'(t) & =\dfrac{1}{[1-\kappa f^{2}(t)]^{1/2}} \text{ on }[0,+\infty),\\ f(t) & ={-}f({-}t) \text{ on } (-\infty,0]. \end{aligned} \end{equation}

Applying this change of variables, the authors reduced ($P_{\kappa }$) to the semilinear equation $-\Delta v = f'(v)[h(f(v)) - V(x)f(v)].$ Along this line, still in the case when the parameter $\kappa$ is negative, we refer to [Reference Adachi, Shibata and Watanabe2, Reference de Souza, Severo and Vieira10, Reference do Ó and Moameni13–Reference do Ó, Miyagaki and Soares15, Reference Moameni27, Reference Wang, Zhang and Shen33], for works on the existence and multiplicity of solutions for ($P_{\kappa }$) in the critical growth range.

In the mathematical literature, there are few known results about the existence of solutions for ($P_{\kappa }$) when the parameter $\kappa$ is positive. In [Reference Ambrosetti and Wang4], A. Ambrosetti and Z. -Q. Wang have considered perturbations of the quasilinear equation $-u''+V(x)u+\kappa (u^{2})''u=\lambda u^{q}$ in $\mathbb {R}$, where $q>1$, $\lambda >0$ and $\kappa$ is a real parameter. The authors have used variational methods to prove that there exists $\kappa _0>0$ such that for $\kappa <\kappa _0$, the equation has a positive solution $u\in H^{1}(\mathbb {R})$. Recently, C. Alves et al. in [Reference Alves, Wang and Shen5] have used another change of variables, different from (1.2), to lead with problem ($P_{\kappa }$) when $\kappa >0$ and $N\geq 3$. Considering nonlinearities of type $h(s)=|s|^{p-2}s,\,\ 2< p<2^{*}$ and $h(s)=[1-1/(1+s^{2})^{3}]s$, by using Mountain Pass Theorem and the well-known Nash–Moser iteration technique, they proved the existence of non-zero solution for ($P_{\kappa }$) provided that $\kappa >0$ is sufficiently small. After that, there are some results in this direction. In [Reference Aires and Souto3], the authors, by using the change of variables introduced in [Reference Alves, Wang and Shen5] and a penalization technique, consider potentials that may vanish at infinity, and they treated nonlinearities with subcritical growth. In [Reference Wang32], Y. Wang dealt with nonlinearities of the form $h(s) = \lambda |s|^{p-2}s+|s|^{q-2}s$ with $p \geq 2^{*}$, $2 < q < 2^{*}$ and he proved the existence of positive solution for ($P_{\kappa }$) with the restriction that $\kappa,\,\lambda >0$ are small enough. In [Reference Severo, Gloss and da Silva31], for $N\geq 3$, the authors studied the existence of positive solutions for ($P_{\kappa }$) with a general nonlinearity $h(t)$ asymptotically linear or superlinear at infinity. Moreover, they proved a Pohozaev type identity to justify that $2^{*}=2N(N-2)$ is the critical exponent for the case that $\kappa$ is positive, differently from the case when $\kappa$ is negative, where the critical exponent is $2.2^{*}$, for more details, see [Reference do Ó, Miyagaki and Soares17].

One of our central contributions in this work is to treat ($P_{\kappa }$) variationally when the nonlinearity $h(s)$ has maximal growth, and we complete the analysis of the works [Reference Alves, Wang and Shen5, Reference Severo, Gloss and da Silva31]. Precisely, we consider ($P_{\kappa }$) in the two-dimensional Euclidean space when the nonlinearity $h(s)$ has critical growth of Trudinger–Moser type, and the parameter $\kappa$ is positive.

1.1. Assumptions

To state our main results precisely, let us describe more accurately the assumptions on the nonlinearity $h(s)$. First, let us assume $h(s)=0$ for all $s<0$ because we are looking for non-negative solutions. Following some recent papers on this subject [1, Reference de Figueiredo, Miyagaki and Ruf9, Reference do Ó12], we say that the function $h(s)$ has critical growth of Trudinger–Moser type at $+ \infty$, if there exists $\alpha _0 >0$ such that

(c _{\unicode{x003B1}0})\[ \lim_{s \rightarrow + \infty} \frac{h(s)}{e^{\alpha s^{2}}} = \left\{ \begin{array}{@{}cl} 0, & \text{if }\quad\alpha > \alpha_0,\\ + \infty, & \text{if }\quad \alpha < \alpha_0 . \end{array} \right. \]

In this case, we say that $h(s)$ has $\alpha _0-$critical growth. We note that such notion is motivated by the following version of the Trudinger–Moser inequality in the whole space $\mathbb {R}^{2}$ proved in [Reference do Ó12]:

\[ e^{\alpha|u|^{2}} -1 \in L^{1}(\mathbb{R}^{2})\text{ for all } u\in H^{1}(\mathbb{R}^{2})\text{ and }\alpha >0. \]

Moreover, if $\alpha < 4\pi$ and $|u|_{L^{2}(\mathbb {R}^{2})}\leq M$, there exists a constant $C = C(M,\,\alpha )>0$ such that

(1.3)\begin{equation} \sup_{\|\nabla u\|_{L^{2}(\mathbb{R}^{2})}\leq 1}\displaystyle\int_{\mathbb{R}^{2}} (e^{\alpha|u|^{2}} - 1)\, \mathrm{d} x \leq C, \end{equation}

see also [Reference Cao7] for an equivalent version. We refer to [Reference de Figueiredo, do Ó and Ruf11] for a general discussion on problems involving critical growth of Trudinger–Moser type.

Besides the critical exponential growth ($c_{\alpha _0}$), we require that $h:\mathbb {R_+}\rightarrow \mathbb {R}$ is locally Hölder continuous and the primitive $H(s):=\int _0^{s}h(t) \mathrm {d} t$ has the following behaviour:

(\unicode{x210B})\[ \exists \, \mu>2 \text{ such that } \frac{H(s)}{s^{\mu}} \text{ is non-decreasing for } s>0 \textrm{ and } \lim_{s\rightarrow 0^{+}}\frac{H(s)}{s^{\mu}}=:\xi>0. \]

Hereafter, we assume that potential $V : \mathbb {R}^{2} \rightarrow \mathbb {R}$ is continuous and satisfies:

(V ₁)\[ V_0:=\inf_{x\in \mathbb{R}^{2}}V(x)>0. \]

(V ₂)\[ \exists \lim_{|x|\rightarrow \infty}V(x) =: V_\infty \geq V(x), \quad \forall \, x \in \mathbb{R}^{2}. \]

The following theorem contains the main result of this work:

Theorem 1.1 Assume ($c_{\alpha _0}$), ($\mathcal {H}$), ($V_{1}$)–($V_{2}$) and $h$ are differentiable (from the right side) at the origin. Then, there exists $\kappa _{0}>0$ such that for all $\kappa \in (0,\,\kappa _{0}),$ Problem ($P_{\kappa }$) has a positive solution $u_\kappa \in L^{\infty }(\mathbb {R}^{2})$ provided

\[ \xi>(3\alpha_0\mu)^{\frac{\mu-2}{2}}\left[\frac{4(1+V_\infty)}{\mu}\right]^{\frac{\mu}{2}}. \]

This is the first paper to investigate the existence of positive solutions of ($P_{\kappa }$) for the case that the parameter $\kappa$ is positive and the nonlinearity has critical exponential growth.

Let us give some examples to illustrate the above result.

Example 1.2 Given $\mu >2$ and $\xi,\, \alpha _0>0$, we can check that $h(s)=H'(s),$ with $H(s)=\xi s^{\mu } e^{\alpha _0s^{2}}$ for $s \geq 0$, verifies ($c_{\alpha _0}$), ($\mathcal {H}$) and $h_+'(0)=0$.

Example 1.3 A class of potentials that satisfies conditions ($V_{1}$)–($V_{2}$) is given by $V_{\alpha }(x,\,y)=3-[\cos ^{2}(x)+1]\mathrm {e}^{-\alpha (x^{2}+y^{2})}$ for $(x,\,y)\in \mathbb {R}^{2}$, where $\alpha >0$. We have $V_{\alpha }(x,\,y)\geq 1$ for all $(x,\,y)\in \mathbb {R}^{2}$ and $V_{\alpha }(x,\,y) \rightarrow 3$ as $|(x,\,y)|\rightarrow \infty$.

Here is a brief description of our plan to prove Theorem 1.1. We first establish the existence of a positive solution for the following modified quasilinear Schrödinger equation:

(AP _{\unicode{x03BA}})\[ -\textrm{div}(g^{2}(u)\nabla u) + g(u)g'(u)|\nabla u|^{2} + V(x)u = h(u) \quad \text{in}\quad \mathbb{R}^{2}, \]

with $g(t)=\sqrt {1-\kappa t^{2}}$ for $|t|<1/\sqrt {3\kappa }$ (see details in § 2). Observe that if we obtain a solution $u$ of ($AP_{\kappa }$) satisfying $\|u\|_{\infty }< 1/\sqrt {3\kappa }$, then $u$ is indeed a solution of our original problem ($P_{\kappa }$). To ensure that ($AP_{\kappa }$) has a non-zero solution with the desired $L^{\infty }-$estimate, we apply minimax techniques with the Nash–Moser iteration method. We emphasize that in our context, different from [Reference Aires and Souto3, Reference Alves, Wang and Shen5, Reference Severo, Gloss and da Silva31, Reference Yang, Wang and Abdelgadir35], the use of the well-known Nash–Moser iteration is more involved due to the critical exponential growth of $h(t)$. Moreover, the behaviour of the quotient $H(s)/s^{\mu }$ imposed by condition ($\mathcal {H}$) is sufficient to obtain our existence result.

Using the same approach as in Theorem 1.1, with some minor modifications, we can prove the existence of standing wave solutions for the following quasilinear Schrödinger equation when the parameter $\kappa =1$:

(Q _{\unicode{x03BB}})\[ \left\{ \begin{aligned} & -\Delta u+V(x)u+\displaystyle\frac{1}{2}[\Delta(u^{2})]u= \lambda h(u)\quad \textrm{in} \quad \mathbb{R}^{2},\\ & u(x) \rightarrow 0 \quad \textrm{as} \quad |x|\rightarrow \infty. \end{aligned} \right. \]

More precisely, we can obtain the following result:

Theorem 1.4 Assume ($c_{\alpha _0}$), ($\mathcal {H}$), ($V_{1}$)–($V_{2}$) and that $h$ is differentiable (from the right side) at the origin. Then, there exists $\lambda _{0}>0$ such that, for each $\lambda >\lambda _0$, Problem ($Q_{\lambda }$) has a positive solution $u_\lambda \in L^{\infty }(\mathbb {R}^{2})$.

The proof of this result is done by using analogous arguments as in Theorem 1.1 and therefore, we omit it. We estimate the minimax level of the energy functional and we control the $L^{\infty }$-norm of the solution by exploiting the fact that $\lambda$ is sufficiently large (see similar results in [Reference do Ó, Medeiros and Severo16, Reference Huang and Jia19]).

Remark 1.5 In [Reference do Ó, Miyagaki and Soares15], the authors have considered quasilinear elliptic equations of the second order with critical exponential growth of the form

(1.4)\begin{equation} -\Delta u+V(x)u - [\Delta(u^{2})]u= h(u)\quad \textrm{in} \quad \mathbb{R}^{2}. \end{equation}

Using the dual approach, it was proved the existence of positive solutions for (1.4), under the conditions ($V_{1}$)–($V_{2}$) and for nonlinearities with critical exponential growth, which contain the following one as an example: $h(s)=H'(s)$ with $H(s)= s^{\mu } e^{4 \pi s^{4}}$ for $s \geq 0$ and $\mu >4$, (see similar results in [Reference de Souza, Severo and Vieira10, Reference do Ó and Severo14, Reference do Ó, Miyagaki and Soares15, Reference Moameni27]). From these works, the maximal growth to treat variationally problems like (1.4) is of the form $e^{\alpha s^{4}},$ when the parameter $\kappa$ is negative. Thus, a natural question arises: what is the maximal growth to treat variationally problems like ($P_{\kappa }$) and ($Q_{\lambda }$), when the parameter $\kappa$ is positive? We emphasize that, in the present work, we have proved the existence of solutions for ($P_{\kappa }$) when $h(t)$ behaves like $e^{\alpha _0 t^{2}}$ at infinity.

2. Variational setting and an auxiliary problem

Here, we introduce an auxiliary problem, which will be a decisive point in the approach used to show Theorem 1.1. First, let us introduce the definition of weak solution for ($P_{\kappa }$). We say that a function $u:\mathbb {R}^{2}\rightarrow \mathbb {R}$ is a weak solution of problem ($P_{\kappa }$) if $u\in H^{1}(\mathbb {R}^{2})\cap L^{\infty }_{loc}(\mathbb {R}^{2})$ and for all $\phi \in C_{0}^{\infty }(\mathbb {R}^{2})$ it holds

(2.1)\begin{equation} \displaystyle\displaystyle\int_{\mathbb{R}^{2}}(1-\kappa u^{2})\nabla u\nabla\phi \; \mathrm{d} x - \kappa\displaystyle\int_{\mathbb{R}^{2}}u|\nabla u|^{2}\phi \mathrm{d} x + \displaystyle\int_{\mathbb{R}^{2}}V(x)u\phi \; \mathrm{d} x = \displaystyle\int_{\mathbb{R}^{2}}h(u)\phi \; \mathrm{d} x. \end{equation}

In § 3 (p. 13), we show that $h(0)=0$. Thus, $u=0$ is the trivial solution of ($P_{\kappa }$) and our goal is to prove by variational approach the existence of a non-trivial weak solution for ($P_{\kappa }$).

Formally ($P_{\kappa }$) is the Euler–Lagrange equation associated with the energy functional

(2.2)\begin{equation} J_{\kappa}(u) = \displaystyle\frac{1}{2}\displaystyle\displaystyle\int_{\mathbb{R}^{2}}(1-\kappa u^{2})|\nabla u|^{2} \; \mathrm{d} x + \frac{1}{2}\displaystyle\int_{\mathbb{R}^{2}}V(x)u^{2}\; \mathrm{d} x - \displaystyle\int_{\mathbb{R}^{2}}H(u)\; \mathrm{d} x. \end{equation}

As we already mentioned, because of the term $\int _{\mathbb {R}^{2}}u^{2}|\nabla u|^{2}\; \mathrm {d} x$, the functional $J_{\kappa }$ is not well defined in the usual Sobolev space. Moreover, the change of variables used in [Reference Colin and Jeanjean8, Reference Liu, Wang and Wang24] can not be considered to study this class of problems because the nonlinear term $T(u)=1-\kappa u^{2}$ in (2.2) changes sign. Thus, to study ($P_{\kappa }$) by using the variational method based on minimax argument, we have to use a different framework from the ones mentioned above. We propose here to argue as in [Reference Alves, Wang and Shen5]. We consider the auxiliary problem ($AP_{\kappa }$) with $g_{\kappa }:\mathbb {R}\rightarrow \mathbb {R}$ defined by

\[ g_{\kappa}(t) = \left\{ \begin{array}{rl} \begin{aligned} \sqrt{1-\kappa t^{2}} & ,\quad \text{if}\quad 0\leq t< 1/\sqrt{3\kappa}\\ \dfrac{1}{3\sqrt{2\kappa}}\dfrac{1}{t} + \dfrac{1}{\sqrt{6}} & , \quad\text{if}\quad t\geq 1/\sqrt{3\kappa}\\ g({-}t) & , \quad \text{if}\quad t<0. \end{aligned} \end{array} \right. \]

Although $g_{\kappa }(t)$ depends on the parameter $\kappa$ for simplicity of notation, we drop the subscript $\kappa$. A simple calculation shows that $g\in C^{1}(\mathbb {R},\,\mathbb {R})$. Furthermore, $g(t)$ is even, increasing in $(-\infty,\,0)$, decreasing in $(0,\,+\infty )$ and

(2.3)\begin{equation} 1/\sqrt{6} \leq g(t) \leq 1,\ \ |g'(t)|\leq \sqrt{\kappa/2},\quad \forall\ t\in\mathbb{R}. \end{equation}

The primitive $G(t)=\int _{0}^{t}g(s)\, \mathrm {d} s$ of $g(t)$ is an increasing function and therefore invertible. Furthermore, $G(t)$ e $G^{-1}(t)$ belong to $C^{2}(\mathbb {R},\,\mathbb {R})$ and $G^{-1}(t)$ is odd. Moreover, $g$ and $G^{-1}$ satisfy the following properties:

(g ₁)\[ \displaystyle\lim_{t\rightarrow 0}\dfrac{G^{{-}1}(t)}{t}=1; \]

(g ₂)\[ \displaystyle\lim_{t\rightarrow +\infty}\frac{G^{{-}1}(t)}{t}=\sqrt{6}; \]

(g ₃)\[ t\leq G^{{-}1}(t)\leq \sqrt{6}t, \quad \forall\ t\geq 0; \]

(g ₄)\[ -\dfrac{1}{2}\leq t\dfrac{g'(t)}{g(t)}\leq 0, \quad \forall\ t\in \mathbb{R}. \]

In [Reference Alves, Wang and Shen5], it was proved properties ($g_{1}$)–($g_{3}$) and property ($g_{4}$) for $t\geq 0$. To prove ($g_{4}$) for the case $t<0$, we can use the fact that $t g'(t)/g(t)$ is an even function.

We say that $u:\mathbb {R}^{2}\rightarrow \mathbb {R}$ is a weak solution of ($AP_{\kappa }$) if $u\in H^{1}(\mathbb {R}^{2})$ and for all $\varphi \in C_{0}^{\infty }(\mathbb {R}^{2})$ it holds

\[\displaystyle\int_{\mathbb{R}^{2}}g^{2}(u)\nabla u\nabla\varphi\ \mathrm{d} x + \int_{\mathbb{R}^{2}}g(u)g'(u)|\nabla u|^{2}\varphi\ \mathrm{d} x + \int_{\mathbb{R}^{2}}V(x)u\varphi\ \mathrm{d} x = \int_{\mathbb{R}^{2}}h(u)\varphi\ \mathrm{d} x . \]

The energy functional associated with Equation ($AP_{\kappa}$) is given by

(2.4)\begin{equation} I_{\kappa}(u) = \displaystyle\frac{1}{2}\displaystyle\displaystyle\int_{\mathbb{R}^{2}}g^{2}(u)|\nabla u|^{2} \; \mathrm{d} x + \frac{1}{2}\displaystyle\int_{\mathbb{R}^{2}}V(x)u^{2}\; \mathrm{d} x - \displaystyle\int_{\mathbb{R}^{2}}H(u)\; \mathrm{d} x. \end{equation}

We assume from now on that ($V_{1}$)–($V_{2}$), ($c_{\alpha _0}$) and ($\mathcal {H}$) hold. Taking into account condition ($\mathcal {H}$), we have $H(s)/s^{2} \rightarrow 0$ as $s\rightarrow 0$. From this fact and as a consequence of the continuity of $H(s)$ and properties ($g_{1}$)–($g_{4}$), it is not difficult to prove that given $q\geq 0$, $\alpha >\alpha _0$ and $\varepsilon >0$ there exists $C=C({\varepsilon },\,q,\,\alpha )>0$ verifying

(2.5)\begin{equation} |H(G^{{-}1}(s))| \leq \varepsilon s^{2}+C|s|^{q}(e^{6\alpha s^{2}}-1),\quad\forall\ s\in\mathbb{R}. \end{equation}

Next, we consider the Sobolev space $H^{1}(\mathbb {R}^{2})$ endowed with the norm

\[ \|u\|^{2}=\int_{\mathbb{R}^{2}}\left(|\nabla u|^{2}+V(x)u^{2}\right)\mathrm{d} x. \]

Note that this norm is equivalent to the standard norm of $H^{1}(\mathbb {R}^{2})$ because of assumptions ($V_{1}$)–($V_{2}$). Recalling that

\[ (G^{{-}1})'(t)=\frac{1}{g(G^{{-}1}(t))}\quad \text{for all} \quad t \in \mathbb{R}, \]

we introduce the functional $\mathcal {F}_{\kappa }:H^{1}(\mathbb {R}^{2})\rightarrow \mathbb {R}$ given by

(2.6)\begin{equation} \mathcal{F}_{\kappa}(v)=I_{\kappa}(G^{{-}1}(v))=\displaystyle\frac{1}{2}\displaystyle\int_{\mathbb{R}^{2}}[|\nabla v|^{2}\mathrm{d} x + V(x)|G^{{-}1}(v)|^{2}]\mathrm{d} x-\int_{\mathbb{R}^{2}}H(G^{{-}1}(v))\mathrm{d} x. \end{equation}

In view of properties ($g_{1}$)–($g_{4}$) and (2.5), we can see that $\mathcal {F}_{\kappa }\in C^{1}(H^{1}(\mathbb {R}^{2}),\,\mathbb {R})$ with derivative given by

(2.7)\begin{equation} \mathcal{F}_{\kappa}'(v)\varphi = \displaystyle\int_{\mathbb{R}^{2}}\nabla v\nabla\varphi\ \mathrm{d} x + \int_{\mathbb{R}^{2}}V(x)\displaystyle\frac{G^{{-}1}(v)}{g(G^{{-}1}(v))}\varphi\ \mathrm{d} x - \int_{\mathbb{R}^{2}}\frac{h(G^{{-}1}(v))}{g(G^{{-}1}(v))}\varphi\ \mathrm{d} x \end{equation}

for $v,\,\varphi \in H^{1}(\mathbb {R}^{2})$. We observe that critical points of $\mathcal {F}_{\kappa }$ are weak solutions of the semilinear elliptic equation

(2.8)\begin{equation} -\Delta v + V(x)\displaystyle\frac{G^{{-}1}(v)}{g(G^{{-}1}(v))} = \frac{h(G^{{-}1}(v))}{g(G^{{-}1}(v))}\quad \textrm{in}\quad \mathbb{R}^{2}. \end{equation}

We can prove that $u = G^{-1}(v)$ is a weak solution of ($AP_{\kappa}$) provided that $v\in H^{1}(\mathbb {R}^{2})$ is a critical point of $\mathcal {F}_{\kappa }$. Indeed, $u\in H^{1}(\mathbb {R}^{2})$ because by ($g_{3}$), we have

\[ u^{2}=|G^{{-}1}(v)|^{2}\leq 6|v|^{2}\quad \text{and} \quad |\nabla u|\leq\sqrt{6}|\nabla v|. \]

Moreover, since $v$ is a critical point of $\mathcal {F}_{\kappa }$, we can write

(2.9)\begin{equation} \displaystyle\int_{\mathbb{R}^{2}}\nabla v\nabla\varphi\ \mathrm{d} x + \int_{\mathbb{R}^{2}}V(x)\displaystyle\frac{G^{{-}1}(v)}{g(G^{{-}1}(v))}\varphi\ \mathrm{d} x=\int_{\mathbb{R}^{2}}\frac{h(G^{{-}1}(v))}{g(G^{{-}1}(v))}\varphi\ \mathrm{d} x ,\quad \forall \, \varphi \in H^{1}(\mathbb{R}^{2} ). \end{equation}

Let $\phi \in C^{\infty }_{0}(\mathbb {R}^{2})$ and $\varphi := g(G^{-1}(v))\phi$. Note that $|\varphi | \leq |\phi |$ and for $i=1,\,2$, we have

\begin{align*} \left|\displaystyle\frac{\partial\varphi}{\partial x_{i}}\right| & = \left|\frac{g'(G^{{-}1}(v))}{g(G^{{-}1}(v))}\frac{\partial v}{\partial x_{i}}\phi + g(G^{{-}1}(v))\frac{\partial\phi}{\partial x_{i}}\right| \\ & \leq \displaystyle\frac{\sqrt{6\kappa}}{2}\|\phi\|_{\infty}\left|\frac{\partial v}{\partial x_{i}}\right| + \left|\frac{\partial\phi}{\partial x_{i}}\right|\ \ \mathrm{a.e.\ \ in}\ \ \mathbb{R}^{2}. \end{align*}

Thus, $\varphi \in H^{1}(\mathbb {R}^{2})$ and taking $\varphi$ as a test function in (2.9), we obtain

\[ \int_{\mathbb{R}^{2}}\nabla v[g'(u)\phi\nabla u + g(u)\nabla\phi] \mathrm{d} x + \int_{\mathbb{R}^{2}}V(x)u\phi\ \mathrm{d} x=\int_{\mathbb{R}^{2}}h(u)\phi\ \mathrm{d} x . \]

Recalling that $v=G(u)$ and $\nabla v= g(u)\nabla u$, we obtain, for all $\phi \in C_{0}^{\infty }(\mathbb {R}^{2})$,

(2.10)\begin{equation} \displaystyle\int_{\mathbb{R}^{2}}g^{2}(u)\nabla u\nabla\phi\ \mathrm{d} x + \int_{\mathbb{R}^{2}}g(u)g'(u)|\nabla u|^{2}\phi\ \mathrm{d} x + \int_{\mathbb{R}^{2}}V(x)u\phi\ \mathrm{d} x = \int_{\mathbb{R}^{2}}h(u)\phi\ \mathrm{d} x, \end{equation}

which implies that $u = G^{-1}(v)$ is a weak solution of ($AP_{\kappa}$).

Now, let $v\in H^{1}(\mathbb {R}^{2})\cap L^{\infty }(\mathbb {R}^{2})$ be a critical point of $\mathcal {F}_{\kappa }$ satisfying

(2.11)\begin{equation} \|v\|_{\infty}< \displaystyle\frac{1}{\sqrt{3\kappa}\sqrt{6}}. \end{equation}

Using ($g_{3}$) and (2.11), we can see that $u=G^{-1}(v)$ satisfies $\|u\|_{\infty }< 1/\sqrt {3\kappa }$, which implies that

(2.12)\begin{equation} g(u)=\sqrt{1-\kappa u^{2}}\quad \text{and}\quad g'(u)=\displaystyle\frac{-\kappa u}{\sqrt{1-\kappa u^{2}}}. \end{equation}

Since (2.10) holds for all $\phi \in C_{0}^{\infty }(\mathbb {R}^{2})$, we get

\[ \displaystyle\int_{\mathbb{R}^{2}}(1-\kappa u^{2})\nabla u\nabla\phi \; \mathrm{d} x - \kappa\displaystyle\int_{\mathbb{R}^{2}}u|\nabla u|^{2}\phi \; \mathrm{d} x + \displaystyle\int_{\mathbb{R}^{2}}V(x)u\phi \; \mathrm{d} x = \displaystyle\int_{\mathbb{R}^{2}}h(u)\phi \; \mathrm{d} x, \]

which implies that $u$ is a weak solution of ($P_{\kappa }$).

We have proved until now that to show the existence of a weak solution of ($P_{\kappa }$), we can look for a critical point $v\in H^{1}(\mathbb {R}^{2})\cap L^{\infty }(\mathbb {R}^{2})$ of $\mathcal {F}_{\kappa }$ satisfying (2.11). Furthermore, one can show that critical points of $\mathcal {F}_{\kappa }$ belong to the class $C^{2,\alpha }_{loc}(\mathbb {R}^{2})$, see [Reference do Ó and Severo14, p. 304] for more details.

3. Palais–Smale sequences for $\mathcal {F}_{\kappa }$

In this section, we have the primary step in our argument to obtain a non-trivial critical point of $\mathcal {F}_{\kappa }$. We establish its main properties, which are essential in our approach. For that, let us begin proving that $\mathcal {F}_{\kappa }$ possesses the mountain pass structure.

Proof. For any $\varepsilon >0$, $q>2$ and $\alpha >\alpha _0$, by (2.5), we have

\[ H(G^{{-}1}(s))\leq\varepsilon s^{2}+C_1|s|^{q}(e^{6\alpha s^{2}}-1),\quad \forall\ s\in \mathbb{R}, \]

for some $C_1>0$. Thus,

\begin{align*} \mathcal{F}_{\kappa}(v)& \geq\displaystyle\frac{1}{2}\|v\|^{2}-\varepsilon\displaystyle\int_{\mathbb{R}^{2}}v^{2}\mathrm{d} x- C_1\int_{\mathbb{R}^{2}}|v|^{q}(e^{6\alpha v^{2}}-1)\mathrm{d} x\\ & \geq\frac{1}{2}\|v\|^{2}-\varepsilon C_2\|v\|^{2}- C_1\bigg(\int_{\mathbb{R}^{2}}|v|^{2q}\mathrm{d} x\bigg)^{1/2}\left[\int_{\mathbb{R}^{2}}(e^{12\alpha v^{2}}-1)\mathrm{d} x\right]^{1/2}\\ & \geq\bigg(\frac{1}{2}-\varepsilon C_2\bigg)\|v\|^{2}-C_3\|v\|^{q}\left[\int_{\mathbb{R}^{2}}(e^{12\alpha\|\nabla v\|_2^{2} (v/\|\nabla v\|_2)^{2}}-1)\mathrm{d} x\right]^{1/2}\\ \end{align*}

for all $v\in H^{1}(\mathbb {R}^{2})$. For $\rho >0$ satisfying $12\alpha \rho ^{2}<4\pi$, Trudinger–Moser inequality implies

\[ \left[\int_{\mathbb{R}^{2}}\left(e^{12\alpha\|\nabla v\|_2^{2} (v/\|\nabla v\|_2)^{2}}-1\right)\mathrm{d} x\right]^{1/2}\leq \tilde{C},\quad\forall\ v\in H^{1}(\mathbb{R}^{2})\quad\mbox{with}\quad \|v\|\leq\rho, \]

for some positive constant $\tilde {C}$. Hence, for $\|v\|\leq \rho$ we get

\[ \mathcal{F}_{\kappa}(v)\geq\bigg(\frac{1}{2}-\varepsilon C_2\bigg)\|v\|^{2}-C_4\|v\|^{q}\,\geq\,\|v\|^{2}\bigg(\frac{1}{2}-\varepsilon C_2-C_4\rho^{q-2}\bigg). \]

Choosing $\varepsilon >0$ such that $2a_\kappa :=(1/2)-\varepsilon C_2>0$ and $\rho _\kappa \in (0,\,\rho )$ such that $C_4\rho _\kappa ^{q-2}\leq a_\kappa$, we obtain Item (1). To prove Item (2), we consider a non-negative and non-trivial function $\varphi \in H^{1}(\mathbb {R}^{2})$. Due to condition ($\mathcal {H}$), one has $H(s)\geq \xi s^{\mu }$ for all $s\geq 0$. Using again that $s^{2}\leq (G^{-1}(s))^{2}\leq 6s^{2}$ for all $s\in \mathbb {R}$, we reach

\[ \mathcal{F}_{\kappa}(t\varphi) \leq 3t^{2}\|\varphi\|^{2}-\xi t^{\mu}\displaystyle\int_{\mathbb{R}^{2}}\varphi^{\mu}\mathrm{d} x \quad \textrm{for all}\quad t>0. \]

Since $\mu >2$, there is $t_0>0$ such that $\mathcal {F}_{\kappa }(t\varphi )<0$ for all $t\geq t_0$ and $\kappa >0$. Taking $e=t_0\varphi$, Item (2) is proved.

Lemma 3.2 Assume ($V_{1}$)–($V_{2}$) ($c_{\alpha _0}$) and ($\mathcal {H}$) hold. Let $(u_{n})$ be a Palais–Smale sequence for $\mathcal {F}_{\kappa }$ at level $c \in \mathbb {R}$ $($in short, $(PS)_c),$ that is,

\[ \mathcal{F}_{\kappa}(u_{n}) \rightarrow c\quad \mathrm{and}\quad \mathcal{F}'_{\kappa}(u_{n}) \rightarrow 0. \]

Then, $(u_{n})$ is bounded in $H^{1}(\mathbb {R}^{2})$.

Proof. Let $(u_{n})\subset H^{1}(\mathbb {R}^{2})$ be a Palais–Smale sequence for $\mathcal {F}_{\kappa }$ at the level $c\in \mathbb {R}$, that is, $\mathcal {F}_{\kappa }(u_n) = c + o_{n}(1)$ and $\|\mathcal {F}'_{\kappa }(u_n)\|=o_{n}(1)$. Thus, for each $\psi \in H^{1}(\mathbb {R}^{2}),$ we have

(3.1)\begin{equation} \displaystyle\int_{\mathbb{R}^{2}}\nabla u_{n}\nabla\psi \ \mathrm{d} x + \int_{\mathbb{R}^{2}}V(x)\displaystyle\frac{G^{{-}1}(u_{n})}{g(G^{{-}1}(u_{n}))}\psi\ \mathrm{d} x - \int_{\mathbb{R}^{2}}\frac{h(G^{{-}1}(u_{n}))}{g(G^{{-}1}(u_{n}))}\psi\ \mathrm{d} x = o_{n}(1)\|\psi\|. \end{equation}

At this moment, we consider $\psi _n :=G^{-1}(u_{n})g(G^{-1}(u_{n}))$. Using properties ($g_{1}$)–($g_{4}$), we see that

\[ |\psi_n|\leq\sqrt{6}|u_n|\quad\mbox{and}\quad|\nabla\psi_n|=\left|1+G^{{-}1}(u_{n})\displaystyle\frac{g'(G^{{-}1}(u_{n}))}{g(G^{{-}1}(u_{n}))}\right||\nabla u_{n}|\leq|\nabla u_{n}|, \]

which implies $\psi _n\in H^{1}(\mathbb {R}^{2})$ and $\|\psi _n\|\leq \sqrt {6}\|u_n\|$. Moreover, for all $n\in \mathbb {N}$, it holds

\[ [G^{{-}1}(u_{n})/g(G^{{-}1}(u_n))]\psi_n=[G^{{-}1}(u_{n})]^{2}\geq(u_n)^{2}. \]

Next, since

\[ \frac{d}{ds}\left(\frac{H(s)}{s^{\mu}}\right)=\frac{s^{\mu-1}[h(s)s-\mu H(s)]}{s^{2\mu}}\quad\textrm{for}\quad s>0, \]

as a consequence of assumption ($\mathcal {H}$), we get

(3.2)\begin{equation} h(s)s-\mu H(s)\geq 0\quad\textrm{for all}\quad s\geq 0. \end{equation}

In particular, $h(s)>0$ for all $s>0$. Taking $\psi =\psi _n$ as the test function in (3.1) and using (3.2), we reach

(3.3)\begin{equation} \begin{aligned} c+o_{n}(1)+o_{n}(1)\|u_{n}\| & \geq\mathcal{F}_{\kappa}(u_{n})-\displaystyle\frac{1}{\mu}\mathcal{F}_{\kappa}'(u_{n})\psi_n\\ & \geq\left(\displaystyle\frac{1}{2}-\frac{1}{\mu}\right)\displaystyle\int_{\mathbb{R}^{2}}|\nabla u_{n}|^{2} \ \mathrm{d} x+\frac{1}{2}\int_{\mathbb{R}^{2}}V(x)|G^{{-}1}(u_n)|^{2}\ \mathrm{d} x\\ & \quad-\frac{1}{\mu}\int_{\mathbb{R}^{2}}V(x)\frac{G^{{-}1}(u_{n})}{g(G^{{-}1}(u_n))}\psi_n\ \mathrm{d} x\\ & \geq\bigg(\frac{\mu-2}{2\mu}\bigg)\|u_n\|^{2}. \end{aligned} \end{equation}

Therefore, $(u_n)$ is bounded in $H^{1}(\mathbb {R}^{2})$ and the proof is finished.

Corollary 3.3 Under the conditions and notation of Lemma 3.2, if $(u_n)$ is a $(PS)_c$ sequence for $\mathcal {F}_{\kappa }$ then

\[ \int_{\mathbb{R}^{2}}|\nabla u_{n}|^{2}\ \mathrm{d} x\leq\bigg(\frac{2\mu}{\mu-2}\bigg)c+o_n(1). \]

In view of Lemma 3.1, $\mathcal {F}_{\kappa }$ has the mountain pass geometry and, therefore, we can apply the Mountain Pass Theorem without the Palais–Smale condition (see [Reference Willem34]) to obtain a $(PS)_{c_\kappa }$ sequence, where $c_\kappa$ is the minimax level for $\mathcal {F}_{\kappa }$. In other words, there exists a sequence $(u_n)$ in $H^{1}(\mathbb {R}^{2})$ satisfying $\mathcal {F}_{\kappa }(u_n)\to c_\kappa$ and $\mathcal {F}'_{\kappa }(u_n)\to 0$ as $n\to \infty$, where

(3.4)\begin{equation} c_\kappa=\inf_{\gamma\in\Gamma_\kappa}\max_{t\in[0,1]}\mathcal{F}_{\kappa}(\gamma(t)) \end{equation}

and

\[ \Gamma_\kappa=\{\gamma\in C([0,1],H^{1}(\mathbb{R}^{2})):\gamma(0)=0\text{ and }\mathcal{F}_{\kappa}(\gamma(1))<0\}. \]

Due to Lemma 3.2, $(u_n)$ is a bounded sequence in $H^{1}(\mathbb {R}^{2})$. So, up to a subsequence, there exists $v_\kappa \in H^{1}(\mathbb {R}^{2})$ such that

(3.5)\begin{equation} u_n\rightharpoonup v_\kappa\text{ in } H^{1}(\mathbb{R}^{2}),\quad u_n \rightarrow v_\kappa\text{ in } L^{q}_{loc}(\mathbb{R}^{2}),\quad \forall\ q\geq 2 \text{ and }u_n(x) \rightarrow v_\kappa(x)\text{ a.e. } x\in \mathbb{R}^{2}. \end{equation}

In order to prove that $v_k$ is a non-trivial critical point of $\mathcal {F}_{\kappa }$, we will show that $(c_\kappa )$ is uniformly bounded.

Lemma 3.4 If condition ($\mathcal {H}$) holds with

\[ \xi>(3\alpha_0\mu)^{\frac{\mu-2}{2}}\left[\frac{4(1+V_\infty)}{\mu}\right]^{\frac{\mu}{2}}, \]

then, for some $c_0=c_0(\xi )>0,$ we have

\[ c_\kappa\leq c_0<\frac{\pi(\mu-2)}{6\alpha_0\mu},\quad\forall\ \kappa>0. \]

Proof. Consider $\varphi \in C^{\infty }_{0}(\mathbb {R}^{2},\,[0,\,1])$ satisfying

\[ \varphi(x)=1,\quad\forall\ x\in B_1(0),\ \varphi(x)=0,\ \forall\ x\in \mathbb{R}^{2}\backslash B_2(0)\quad\text{and}\quad |\nabla\varphi(x)|\leq 1,\ \forall x\in\mathbb{R}^{2}. \]

Let $t_0>0$ be such that $\mathcal {F}_{\kappa }(t_0\varphi )<0$ and $t_0\|\varphi \|>\rho _0$. Thus, $\gamma (t):=tt_0\varphi \in \Gamma _\kappa$ and, since $H(s)\geq \xi s^{\mu }$ for all $s\geq 0$, we reach

\begin{align*} c_\kappa\leq\max_{t\in[0,1]}\mathcal{F}_{\kappa}(tt_0\varphi)& \leq\max_{t\geq 0}\mathcal{F}_{\kappa}(t\varphi)\\ & \leq\max_{t\geq 0}\left\{\frac{t^{2}}{2}\left[\int_{B_2(0)}\left(|\nabla\varphi|^{2}+V_\infty\varphi^{2}\right)\mathrm{d} x\right]-\xi\int_{B_2(0)}\left[G^{{-}1}(t\varphi)\right]^{\mu}\mathrm{d} x\right\}\\ & \leq\max_{t\geq 0}\left\{\frac{t^{2}}{2}\left(1+V_\infty\right)|B_2(0)|-\xi t^{\mu}\int_{B_1(0)}\varphi^{\mu}\ \mathrm{d} x\right\}\\ & =\max_{t\geq 0}\left\{2\pi\left(1+V_\infty\right)t^{2}-\xi\pi t^{\mu}\right\}\\ & = \pi\max_{t\geq 0}\left\{2\left(1+V_\infty\right)t^{2}-\xi t^{\mu}\right\}, \end{align*}

where we have used $(V_2)$. Thus, calculating the maximum, we get

(3.6)\begin{equation} c_\kappa\leq\pi\left(\displaystyle\frac{\mu-2}{2}\right)\left[\frac{4(1+V_\infty)}{\mu}\right]^{\mu/(\mu-2)}\frac{1}{\xi^{2/(\mu-2)}}=:c_0(\xi). \end{equation}

Now, using the assumption on $\xi$, we obtain

\[ \xi^{{-}2/(\mu-2)}<\frac{1}{\left[\frac{4(1+V_\infty)}{\mu}\right]^{\mu/(\mu-2)}3\alpha_0\mu}, \]

and hence the result follows from (3.6).

Now we are able to show that the weak limit $v_\kappa$ is a non-trivial critical point of $\mathcal {F}_{\kappa }$.

Proof. At first, we will prove that $v_\kappa$ is a critical point for $\mathcal {F}_{\kappa }$. By density, it is enough to show that $\mathcal {F}'_{\kappa }(v_\kappa )\phi =0$ for all $\phi \in C^{\infty }_0(\mathbb {R}^{2})$. For $\phi \in C^{\infty }_0(\mathbb {R}^{2})$, we denote $\varOmega =\mathrm {supp}(\phi )$ and $M=\|\phi \|_\infty$. Since

\begin{align*} & \int_{\mathbb{R}^{2}}\nabla u_{n}\nabla(\phi u_n) \; \mathrm{d} x + \int_{\mathbb{R}^{2}}V(x)\frac{G^{{-}1}(u_n)}{g(G^{{-}1}(u_n))} \phi u_n \; \mathrm{d} x -\int_{\mathbb{R}^{2}}\frac{h(G^{{-}1}(u_n))}{g(G^{{-}1}(u_n))} \phi u_n \; \mathrm{d} x \\ & \quad= \mathcal{F}'_{\kappa}(u_n)(\phi u_n)=o_n(1), \end{align*}

it is easy to see that

\[ \left| \int_{\Omega}\frac{h(G^{{-}1}(u_n))}{g(G^{{-}1}(u_n))} \phi u_n\mathrm{d} x \right|\leq C, \]

for some positive constant $C$. Thus, from [Reference de Figueiredo, Miyagaki and Ruf9, Lemma 2.1], we get

\[ \int_{\Omega}\frac{h(G^{{-}1}(u_n))}{g(G^{{-}1}(u_n))} \phi \mathrm{d} x \longrightarrow \int_{\Omega}\frac{h(G^{{-}1}(v_\kappa))}{g(G^{{-}1}(v_\kappa))} \phi \mathrm{d} x. \]

Moreover, since $u_n\rightharpoonup v_\kappa \text { in } H^{1}(\mathbb {R}^{2})$ and $u_n \rightarrow v_\kappa \text { in } L^{2}_{loc}(\mathbb {R}^{2})$, it follows that

\[ \int_{\mathbb{R}^{2}}\nabla u_{n}\nabla\phi \mathrm{d} x\longrightarrow\int_{\mathbb{R}^{2}}\nabla v_{\kappa}\nabla\phi \mathrm{d} x \]

and

\[ \int_{\mathbb{R}^{2}}V(x)\frac{G^{{-}1}(u_n)}{g(G^{{-}1}(u_n))} \phi \mathrm{d} x\longrightarrow \int_{\mathbb{R}^{2}}V(x)\frac{G^{{-}1}(v_\kappa)}{g(G^{{-}1}(v_\kappa))} \phi \mathrm{d} x. \]

Hence, we conclude that $\mathcal {F}'_\kappa (v_\kappa )\phi =0$ for all $\phi \in C^{\infty }_0(\mathbb {R}^{2})$ and therefore $v_\kappa$ is a critical point of $\mathcal {F}_\kappa$. Furthermore, it holds

\[ -\Delta u_n={-}V(x) \frac{G^{{-}1}( u_n)}{g(G^{{-}1}(u_n))}+\frac{h(G^{{-}1}( u_n))}{g(G^{{-}1}( u_n))}+\mathcal{F}_{\kappa}'( u_n), \]

in the sense of continuous linear functional on $H^{1}(\mathbb {R}^{2})$. Now, invoking a result due to L. Boccardo and F. Murat (see [Reference Boccardo and Murat6, Theorem 2.1]), we can see that $\nabla u_n(x)\to \nabla v_\kappa (x)$ almost everywhere in $x\in \mathbb {R}^{2}$. Since

\begin{align*} c_\kappa & = \liminf_{n\to\infty}\left[\mathcal{F}_{\kappa}(u_n)-\frac{1}{2}\mathcal{F}_{\kappa}'( u_n)\left(G^{{-}1}( u_n)g(G^{{-}1}(u_n))\right)\right]\\ & =\liminf_{n\to\infty}\int_{\mathbb{R}^{2}}\left[\left(-\frac{G^{{-}1}(u_n)g'(G^{{-}1}( u_n))}{2g(G^{{-}1}( u_n))}\right)|\nabla u_n|^{2}\right.\\ & \quad\left.+\frac{1}{2}\left(h(G^{{-}1}( u_n))G^{{-}1}( u_n)-2H(G^{{-}1}( u_n))\right)\right]\!\mathrm{d} x, \end{align*}

from Fatou's Lemma, we obtain

\begin{align*} c_\kappa& \!\geq\! \int_{\mathbb{R}^{2}}\left[\left(-\frac{G^{{-}1}(v_\kappa)g'(G^{{-}1}( v_\kappa))}{2g(G^{{-}1}( v_\kappa))}\right)|\nabla v_\kappa|^{2}\!+\!\frac{1}{2}\left(h(G^{{-}1}( v_\kappa))G^{{-}1}( v_\kappa)\!-\!2H(G^{{-}1}( v_\kappa))\right)\right]\,\mathrm{d} x\\ & = \mathcal{F}_{\kappa}( v_\kappa)-\frac{1}{2}\mathcal{F}_{\kappa}'( v_\kappa)\left(G^{{-}1}( v_\kappa)g(G^{{-}1}( v_\kappa))\right) = \mathcal{F}_{\kappa}( v_\kappa). \end{align*}

To conclude this proof, we must show that $v_\kappa \neq 0$. Arguing by contradiction, let us assume that $v_\kappa =0$. As in [Reference Alves, Wang and Shen5], we see that $(u_n)$ is also a $(PS)_{c_\kappa }$ sequence for the functional $\mathcal {F}_{\kappa,\infty }: H^{1}(\mathbb {R}^{2})\to \mathbb {R}$ defined by

\[ \mathcal{F}_{\kappa,\infty}(v)=\frac{1}{2}\int_{\mathbb{R}^{2}}|\nabla v|^{2} \mathrm{d} x+\frac{V_\infty}{2}\int_{\mathbb{R}^{2}}(G^{{-}1}(v))^{2} \mathrm{d} x-\int_{\mathbb{R}^{2}}H(G^{{-}1}(v)) \mathrm{d} x. \]

We claim that, for any $R>0$, it holds

(3.7)\begin{equation} \liminf_{n\to\infty}\sup_{y\in\mathbb{R}^{2}}\displaystyle\int_{B_R(y)}u_n^{2} \mathrm{d} x > 0. \end{equation}

If this statement is false, from a result due to P.L. Lions (see[Reference Willem34, Lemma 1.21]), it follows, up to subsequences, that $u_n \rightarrow 0$ in $L^{q}(\mathbb {R}^{2})$ for all $q>2$. Since $(u_n)$ is a $(PS)_{c_\kappa }$ sequence for $\mathcal {F}_{\kappa }$, by Corollary 3.3 and Lemma 3.4, we have

\[ \|\nabla u_n\|_2^{2}\leq\bigg(\frac{2\mu}{\mu-2}\bigg)c_\kappa+ o_n(1) \leq\frac{\pi}{3\alpha_0}+ o_n(1) . \]

Hence, there exist $\beta _0>0$ and $n_0\in \mathbb {N}$ such that

\[ \|\nabla u_n\|_2^{2}\leq\beta_0<\frac{2\pi}{3\alpha_0},\quad\forall\ n\geq n_0., \]

which allows us to choose $\beta >\beta _0$, $\alpha >\alpha _0$ and $q>1$ satisfying

(3.8)\begin{equation} 6\alpha q\|\nabla u_n\|_2^{2}\leq\beta<4\pi, \quad\forall\ n\geq n_0.\end{equation}

Once $(u_n)$ is bounded in $L^{2}(\mathbb {R}^{2})$, the Trudinger–Moser inequality implies

(3.9)\begin{equation} \displaystyle\int_{\mathbb{R}^{2}}(e^{6\alpha q u_n^{2}}-1) \mathrm{d} x\leq C_1,\quad\forall\ n\in\mathbb{N}, \end{equation}

for some $C_1>0$.

Next, we claim that $h(s)/s\rightarrow 0$ as $s\rightarrow 0^{+}$. First, we shall show that $\liminf _{s\to 0{{}^{+}}}h(s)/s=0$. Since $h(s)>0$ for all $s>0$, suppose otherwise that $\liminf _{s\to 0{{}^{+}}}h(s)/s>0$. Then, there exist $c,\, \delta >0$ such that $h(s)\geq cs$ for all $s\in (0,\,\delta )$. Consequently, $H(s)/s^{\mu }\geq cs^{2-\mu }/2$ for all $s\in (0,\,\delta )$. Taking the limit as $s\rightarrow 0^{+}$, we obtain a contradiction and therefore we must have $\liminf _{s\to 0{{}^{+}}}h(s)/s=0$. By continuity, this fact implies, in particular, that $h(0)=0$. From this and since $h$ is differentiable (from the right side) at the origin, the proof of the claim is done. By this claim and ($c_{\alpha _0}$), for any $\varepsilon >0$ and $\alpha > \alpha _0$, there exists a positive constant $C=C(\alpha,\,\varepsilon )$ such that

\[ h(s)s\leq \frac{\varepsilon}{12M}s^{2}+Cs^{2}\left( e^{\alpha s^{2}}-1 \right), \quad \textrm{for all}\ s\in \mathbb{R}, \]

where $M>0$ is such that $\|u_n\|_2^{2}\leq M$. From this estimate and by invoking the Hölder inequality, ($g_{3}$) and (3.9), we reach

\begin{align*} & \int_{\mathbb{R}^{2}}|h(G^{{-}1}(u_n))G^{{-}1}(u_n)| \mathrm{d} x \\ & \quad\leq \frac{\varepsilon}{12M}\int_{\mathbb{R}^{2}}(G^{{-}1}(u_n))^{2} \mathrm{d} x+C\int_{\mathbb{R}^{2}}\left(e^{\alpha(G^{{-}1}(u_n))^{2}}-1\right)(G^{{-}1}(u_n))^{2} \mathrm{d} x\\ & \quad \leq \frac{\varepsilon}{2M} \int_{\mathbb{R}^{2}}u_n^{2} \mathrm{d} x+C\left[\int_{\mathbb{R}^{2}}\left(e^{6\alpha q u_n^{2}}-1\right) \mathrm{d} x\right]^{1/q}\left[\int_{\mathbb{R}^{2}}|u_n|^{2q'}\mathrm{d} x\right]^{1/q'}\\ & \quad\leq \frac{\varepsilon}{2}+C_2\|u_n\|_{2q'}^{2}\leq \varepsilon \end{align*}

for all $n$ sufficiently large, where $q'=q/(q-1)>1$ and the last inequality holds because $u_n\to 0$ in $L^{2q'}(\mathbb {R}^{2})$. Thus,

(3.10)\begin{equation} \displaystyle\int_{\mathbb{R}^{2}}|h(G^{{-}1}(u_n))G^{{-}1}(u_n)| \mathrm{d} x\to 0\quad\mbox{as}\quad n\to\infty. \end{equation}

Taking $w_n:=G^{-1}(u_n)g(G^{-1}(u_n))$ as a test function and recalling properties ($g_{1}$)–($g_{4}$), we obtain

\begin{align*} o_n(1)& = \mathcal{F}_{\kappa,\infty}'(u_n)w_n \!=\!\int_{\mathbb{R}^{2}}\left[1+\frac{G^{{-}1}(u_n)g'(G^{{-}1}(u_n))}{g(G^{{-}1}(u_n))}\right]|\nabla u_n|^{2} \mathrm{d} x+V_\infty\int_{\mathbb{R}^{2}}(G^{{-}1}(u_n))^{2}\mathrm{d} x \\ & \quad-\int_{\mathbb{R}^{2}}h(G^{{-}1}(u_n))G^{{-}1}(u_n) \mathrm{d} x \\ & \geq\frac{1}{2}\int_{\mathbb{R}^{2}}|\nabla u_n|^{2} \mathrm{d} x+\frac{V_\infty}{2}\int_{\mathbb{R}^{2}}u_n^{2}\mathrm{d} x+o_n(1), \end{align*}

which shows that

\[ \frac{1}{2}\int_{\mathbb{R}^{2}}|\nabla u_n|^{2} \mathrm{d} x+\frac{V_\infty}{2}\int_{\mathbb{R}^{2}}u_n^{2}\mathrm{d} x\to 0 \quad\mbox{as}\quad n\to\infty. \]

On the other hand, (3.2) together with (3.10) provides

\[ \int_{\mathbb{R}^{2}}H(G^{{-}1}(u_n)) \mathrm{d} x\to 0 \quad\mbox{as}\quad n\to\infty \]

and consequently $\mathcal {F}_{\kappa,\infty }(u_n)\to 0$, which is a contradiction with $c_\kappa >0$. Therefore, (3.7) must occur for any $R>0$. Taking $R=1$, due to (3.7), there exist $\rho >0$ and $(y_n)\subset \mathbb {R}^{2}$ verifying

(3.11)\begin{equation} \liminf_{n\to\infty}\displaystyle\int_{B_1(y_n)}u_n^{2} \mathrm{d} x\geq\rho>0. \end{equation}

Defining $\tilde u_n(x)=u_n(x+y_n)$ and since $(u_n)$ is a $(PS)_{c_\kappa }$ sequence for $\mathcal {F}_{\kappa,\infty }$, it is easy to see that $(\tilde u_n)$ is also a $(PS)_{c_\kappa }$ sequence for $\mathcal {F}_{\kappa,\infty }$. Let $\tilde v_\kappa$ be the weak limit of $(\tilde u_n)$ in $H^{1}(\mathbb {R}^{2})$. Arguing similarly as in the beginning of this proof, we can show that $\tilde v_\kappa$ is a critical point of $\mathcal {F}_{\kappa,\infty }$. Moreover, (3.11) implies that

\[ \int_{B_1(0)}\tilde v_\kappa^{2} \mathrm{d} x=\lim_{n\to\infty}\int_{B_1(0)}\tilde u_n^{2} \mathrm{d} x=\lim_{n\to\infty}\int_{B_1(y_n)}u_n^{2} \mathrm{d} x\geq\rho>0 \]

and hence $\tilde v_\kappa \neq 0$.

The same arguments used in this proof to show that $\mathcal {F}_{\kappa }(v_\kappa )\leq c_\kappa$ can be applied again to conclude that $\mathcal {F}_{\kappa,\infty }(\tilde v_\kappa )\leq c_\kappa$.

Next, we are going to show that $\mathcal {F}_{\kappa,\infty }(\tilde v_\kappa )>c_{\kappa }$ and this contradiction ensures that $v_\kappa \neq 0$. With this aim, we consider the equation associated with the functional $\mathcal {F}_{\kappa,\infty }$, namely,

\[ -\Delta v+V_\infty\frac{G^{{-}1}(v)}{g(G^{{-}1}(v))}=\frac{h(G^{{-}1}(v))}{g(G^{{-}1}(v))}\quad\mbox{in}\quad\mathbb{R}^{2}. \]

Since $\tilde v_\kappa$ is a weak solution of this equation, by the same arguments as in [Reference do Ó, Miyagaki and Soares15, Reference Severo, Gloss and da Silva31], we can infer that

\[ \tilde v_\kappa\in C^{2,\alpha}_{loc}(\mathbb{R}^{2}),\quad \tilde v_\kappa>0 \text{ in }\mathbb{R}^{2}\quad\mbox{and}\quad\lim_{|x|\to\infty}\tilde v_\kappa(x)=0. \]

From this, we consider a continuous path $\gamma :[0,\,1]\to H^{1}(\mathbb {R}^{2})$ satisfying

(3.12)\begin{equation} \left\{ \begin{aligned} & \gamma(0)=0,\quad\mathcal{F}_{\kappa,\infty}(\gamma(1))<0, \tilde v_\kappa\in\gamma([0,1])\\ & \gamma(t)(x)>0\ \forall x\in\mathbb{R}^{2},\quad t\in(0,1]\\ & \max_{t\in[0,1]}\mathcal{F}_{\kappa,\infty}(\gamma(t))=\mathcal{F}_{\kappa,\infty}(\tilde v_\kappa). \end{aligned} \right. \end{equation}

The existence of this path $\gamma$ satisfying (3.12) can be proved by using the ideas developed in [Reference do Ó, Miyagaki and Soares15, Reference Jeanjean and Tanaka20]. We present here just a sketch of the construction of $\gamma$. Defining

\[ \tilde v_t(x)=\left\{ \begin{array}{@{}ccl} \tilde v_\kappa(x/t) & \mbox{if} & t>0,\\ 0 & \mbox{if} & t=0, \end{array} \right. \]

we see that $t\mapsto \tilde v_t$ is continuous from $[0,\,\infty )$ to $H^{1}(\mathbb {R}^{2})$ and $\tilde v_t(x)>0$ for all $x\in \mathbb {R}^{2}$ and $t>0$. The path $\gamma$ satisfying (3.12) is the juxtapotion of the following three paths

\[ \begin{array}{cccl} \gamma_1: & [0,1] & \to & H^{1}(\mathbb{R}^{2}),\\ & \theta & \mapsto & \theta \tilde v_{t_0} \end{array} \begin{array}{cccl} \gamma_2: & [t_0,t_1] & \to & H^{1}(\mathbb{R}^{2})\\ & t & \mapsto & \tilde v_{t} \end{array} \quad\mbox{and}\quad \begin{array}{cccl} \gamma_3: & [1,\theta_1] & \to & H^{1}(\mathbb{R}^{2})\\ & \theta & \mapsto & \theta\tilde v_{t_1} \end{array} \]

where we have chosen the points $t_0\in (0,\,1)$, $t_1\in (1,\,\infty )$ and $\theta _1>t_1$ properly.

Here we can assume $V\not \equiv V_\infty$ (otherwise there is nothing to prove). Thus, since $V(x)\leq V_\infty$ for all $x\in \mathbb {R}^{2}$ and $\gamma (t)(x)>0$ for all $x\in \mathbb {R}^{2}$ and $t\in (0,\,1]$, we have $\mathcal {F}_{\kappa }(\gamma (t))<\mathcal {F}_{\kappa,\infty }(\gamma (t))$ for all $t\in (0,\,1]$. In particular, $\gamma \in \Gamma _\kappa$ and

\[ c_\kappa\leq\max_{t\in[0,1]}\mathcal{F}_{\kappa}(\gamma(t))= \mathcal{F}_{\kappa}(\gamma(\bar t))<\mathcal{F}_{\kappa,\infty}(\gamma(\bar t))\leq \max_{t\in[0,1]}\mathcal{F}_{\kappa,\infty}(\gamma(t))=\mathcal{F}_{\kappa,\infty}(\tilde v_\kappa)\leq c_\kappa. \]

This absurd occurs because we are assuming $v_\kappa \equiv 0$. Therefore, we should have $v_\kappa \not \equiv 0$ and this ends the proof.

Lemma 3.6 The critical point $v_\kappa$ of $\mathcal {F}_{\kappa }$ satisfies the inequality

\[ \|v_\kappa\|^{2}\leq\frac{2\mu}{\mu-2}c_\kappa. \]

Proof. As in Lemma 3.2, by using (3.2), we get

\begin{align*} c_\kappa& \geq\mathcal{F}_{\kappa}(v_\kappa)=\mathcal{F}_{\kappa}(v_\kappa)-\frac{1}{\mu}\mathcal{F}'_{\kappa}(v_\kappa)[G^{{-}1}(v_\kappa)g(G^{{-}1}(v_\kappa))]\\ & \geq\bigg(\frac{\mu-2}{2\mu}\bigg)\int_{\mathbb{R}^{2}}\left(|\nabla v_\kappa|^{2}+V(x)v_\kappa^{2}\right)\mathrm{d} x \end{align*}

and the result follows.

4. The Nash–Moser iteration technique

In this section, we apply Nash–Moser iteration technique to prove some regularity results on quasilinear elliptic equations. The main point here is to ensure that any critical point $v$ of $\mathcal {F}_{\kappa }$ belongs to $L^{\infty }(\mathbb {R}^{2})$, providing a precise behaviour for the norm $\|v\|_{\infty }$ depending on $\kappa > 0$. Soon after, we will show that $u = G^{-1}(v)$ satisfies $\|u\|_{\infty } \leq 1/\sqrt {3 \kappa }$ for $\kappa >0$ sufficiently small, proving that $u$ is a weak solution for problem ($P_{\kappa }$).

At first, we observe that any non-zero critical point $v$ of $\mathcal {F}_{\kappa }$ is a positive function. Indeed, since $\mathcal {F}_{\kappa }'(v)\varphi =0$ for all $\varphi \in H^{1}(\mathbb {R}^{2})$, taking $\varphi =v^{-}$, we obtain

\[ \int_{\mathbb{R}^{2}}|\nabla v^{-}|^{2}\mathrm{d} x + \int_{\mathbb{R}^{2}}V(x)\frac{G^{{-}1}(v^{-})}{g(G^{{-}1}(v^{-}))}(v^{-})\; \mathrm{d} x=0. \]

Observing that

\[ \int_{\mathbb{R}^{2}}V(x)\frac{G^{{-}1}(v^{-})}{g(G^{{-}1}(v^{-}))}(v^{-})\; \mathrm{d} x\geq 0, \]

we have $v^{-}\equiv 0$ and therefore $v=v^{+}\geq 0$.

Now, arguing by contradiction, assume that there exists $x_0\in \mathbb {R}^{2}$ such that $v(x_0)=0$. Equation (2.8) can be rewritten in the form

\[ -\Delta v +c(x)v =V(x)\left[6v-\frac{G^{{-}1}(v)}{g(G^{{-}1}(v))}\right] +\frac{h(G^{{-}1}(v))}{g(G^{{-}1}(v))} \]

where $c(x)=6V(x)\geq 6V_0>0$ for all $x\in \mathbb {R}^{2}$. Using that $g(t)\geq 1/\sqrt {6}$, ($g_{3}$) and $h(t)\geq 0$, one has

\[ -\Delta v +c(x)v\geq 0. \]

By applying the Strong Maximum Principle in a arbitrary ball centred in $x_0$, we can conclude that $v=0$, which is impossible and hence $v$ is positive in all $\mathbb {R}^{2}$.

Proposition 4.1 Let $v \in H^{1}(\mathbb {R}^{2})$ be a critical point of $\mathcal {F}_{\kappa }$. Under the assumptions ($V_{1}$)–($V_{2}$) ($c_{\alpha _0}$) and ($\mathcal {H}$), we have $u \in L^{\infty }(\mathbb {R}^{2})$ and

\[ \|v\|_{\infty} \leq (C_0I_{\alpha,q})^{q/(4-2q)} (\displaystyle\frac{2}{q})^{2q/(2-q)^{2}}\|v\|_4, \]

where $q\in (1,\,2),$ $C_0>0$ does not depend on $v$ or $\kappa,$ and

\[ I_{\alpha,q}:=\left[\int_{\mathbb{R}^{2}}\left(e^{[6\alpha q/(q-1)]v^{2}}-1\right)\mathrm{d} x\right]^{(q-1)/q}. \]

Proof. Let $\beta > 1$ and $m \in \mathbb {N}$. Define the sets $A_{m} = \{ x \in \mathbb {R}^{2},\, (v)^{\beta -1} \leq m \}$ and $B_{m} = \mathbb {R}^{N}\backslash A_m$. Consider the sequence

\[ v_{m} = \left\{ \begin{array}{@{}l} v^{2\beta -1}\,\, \text{in}\,\, A_{m},\\ m^{2}v\,\, \text{in}\,\, B_{m}. \\ \end{array} \right. \]

It is easy that $v_m\in H^{1}(\mathbb {R}^{2})$ and $|v_{m}| \leq |v|^{2\beta -1}$ in $\mathbb {R}^{2}$. Moreover,

\[ \nabla v_{m} = \left\{\begin{array}{@{}l} (2 \beta - 1)v^{2(\beta -1)} \nabla v \,\, \text{in} \,\, A_{m},\\ m^{2} \nabla v \,\, \text{in} \,\, B_{m}. \\ \end{array} \right. \]

Since $v$ is a critical point of $\mathcal {F}_{\kappa }$, we have

(4.1)\begin{equation} \displaystyle\int_{\mathbb{R}^{2}} \nabla v \nabla v_{m} \ \mathrm{d} x + \int_{\mathbb{R}^{2}} V(x) \dfrac{G^{{-}1}(v)}{g(G^{{-}1}(v))}v_{m} \ \mathrm{d} x = \int_{\mathbb{R}^{2}} \dfrac{h(G^{{-}1}(v))}{g(G^{{-}1}(v))} v_{m} \ \mathrm{d} x. \end{equation}

Note that

(4.2)\begin{equation} \displaystyle\int_{\mathbb{R}^{2}} \nabla v \nabla v_{m}\ \mathrm{d} x = (2 \beta -1) \int_{A_{m}}v^{2(\beta -1)} |\nabla v|^{2}\ \mathrm{d} x + m^{2} \int_{B_{m}} |\nabla v|^{2}\ \mathrm{d} x. \end{equation}

Next, define the sequence

\[ w_{m} = \left\{\begin{array}{@{}l} v^{\beta}\,\, \text{in }\,\, A_{m},\\ m v \,\, \text{in }\,\, B_{m}. \\ \end{array} \right. \]

It is not difficult to verify that $w_{m}$ belongs to $H^{1}(\mathbb {R}^{2})$ and $w_{m}^{2} = v v_{m} \leq v^{2 \beta }$ in $\mathbb {R}^{2}$. Moreover,

\[ \nabla w_{m} = \left\{ \begin{array}{@{}l} \beta v^{\beta -1} \nabla v\,\, \text{in }\,\, A_{m},\\ m \nabla v \,\, \text{in }\,\, B_{m}. \\ \end{array} \right. \]

Hence,

(4.3)\begin{equation} \displaystyle\int_{\mathbb{R}^{2}} |\nabla w_{m}|^{2}\ \mathrm{d} x = \beta^{2} \int_{A_{m}}v^{2(\beta -1)} |\nabla v|^{2}\ \mathrm{d} x + m^{2} \int_{B_{m}} |\nabla v|^{2}\ \mathrm{d} x. \end{equation}

It follows from (4.2) and (4.3) that

(4.4)\begin{equation} \displaystyle\int_{\mathbb{R}^{2}} (|\nabla w_{m}|^{2} - \nabla v \nabla v_{m})\ \mathrm{d} x = (\beta - 1)^{2} \int_{A_{m}} v^{2(\beta -1)} |\nabla v|^{2}\ \mathrm{d} x. \end{equation}

From (4.2),

(4.5)\begin{equation} \displaystyle\int_{A_{m}} v^{2(\beta -1)} |\nabla v|^{2}\ \mathrm{d} x\leq \displaystyle\frac{1}{(2 \beta - 1)}\int_{\mathbb{R}^{2}} \nabla v \nabla v_{m}\ \mathrm{d} x, \end{equation}

which together with (4.4)–(4.5) implies

\[ \displaystyle\int_{\mathbb{R}^{2}} |\nabla w_{m}|^{2}\ \mathrm{d} x \leq \left[1 + \dfrac{(\beta - 1)^{2}}{2 \beta - 1}\right]\int_{\mathbb{R}^{2}} \nabla v \nabla v_{m}\ \mathrm{d} x. \]

In view of (4.1) and $\beta >1$, we get

(4.6)\begin{equation} \displaystyle\int_{\mathbb{R}^{2}} |\nabla w_{m}|^{2}\ \mathrm{d} x \leq\beta^{2}\left[ \int_{\mathbb{R}^{2}} \dfrac{h(G^{{-}1}(v))}{g(G^{{-}1}(v))} v_{m}\ \mathrm{d} x- \int_{\mathbb{R}^{2}} V(x) \dfrac{G^{{-}1}(v)}{g(G^{{-}1}(v))} v_{m}\ \mathrm{d} x\right]. \end{equation}

By $(c_{\alpha _0})$ and since $h(s)/s\rightarrow 0$ as $s\rightarrow 0$, given $\alpha >\alpha _0$ and $\varepsilon >0,$ there exists $C=C(\alpha,\,\varepsilon )>0$ such that

\[ h(t)\leq\varepsilon t+Ct(e^{\alpha t^{2}}-1),\quad\text{for all } t\geq 0. \]

Taking $\varepsilon =V_0/2$ and using $(V_1)$, from (4.6), we obtain

\begin{align*} \int_{\mathbb{R}^{2}} |\nabla w_{m}|^{2}\ \mathrm{d} x \!+\!\frac{V_0}{2}\int_{\mathbb{R}^{2}} \dfrac{G^{{-}1}(v)}{g(G^{{-}1}(v)} v_{m}\ \mathrm{d} x & \!\leq\!\int_{\mathbb{R}^{2}} |\nabla w_{m}|^{2}\ \mathrm{d} x \!+\!\frac{\beta^{2}}{2}\int_{\mathbb{R}^{2}} V(x)\dfrac{G^{{-}1}(v)}{g(G^{{-}1}(v))} v_{m}\ \mathrm{d} x\\ & \leq\beta^{2}C\int_{\mathbb{R}^{2}} \dfrac{G^{{-}1}(v)}{g(G^{{-}1}(v))}\left[e^{\alpha(G^{{-}1}(v))^{2}}-1\right] v_{m}\ \mathrm{d} x \end{align*}

and once $0< g(t)\leq 1$ and $t\leq G^{-1}(t)\leq \sqrt {6}t$ for all $t\geq 0$, we get

\[ \displaystyle\int_{\mathbb{R}^{2}} |\nabla w_{m}|^{2}\ \mathrm{d} x +\displaystyle\frac{V_0}{2}\int_{\mathbb{R}^{2}} v v_{m}\ \mathrm{d} x \leq\beta^{2}C\sqrt{6}\int_{\mathbb{R}^{2}} (e^{6\alpha v^{2}}-1)v v_{m}\ \mathrm{d} x. \]

Now, fixing $q\in (1,\,2)$ and recalling that $vv_m=w_m^{2}$, the Hölder inequality implies

\[ \min\left\{1,\frac{V_0}{2}\right\}\int_{\mathbb{R}^{2}} (|\nabla w_{m}|^{2}\!+\!w_m^{2})\ \mathrm{d} x \!\leq\! \beta^{2}C\sqrt{6}\left[\int_{\mathbb{R}^{2}} \left(e^{6\alpha q' v^{2}}-1\right)\ \mathrm{d} x\right]^{1/q'} \bigg(\int_{\mathbb{R}^{2}} w_m^{2q}\ \mathrm{d} x\bigg)^{1/q} \]

where $q'=q/(q-1)$. Denoting

\[ I_{\alpha,q}:=\left[\int_{\mathbb{R}^{2}}\left(e^{6\alpha q'v^{2}}-1\right)\mathrm{d} x\right]^{(q-1)/q} \]

and $\tilde C=\sqrt {6}C/\min \left \{1,\,V_0/{2}\right \}$, we can write

\[ 2\|\nabla w_m\|_2\|w_m\|_2\leq\int_{\mathbb{R}^{2}} (|\nabla w_{m}|^{2}+w_m^{2})\ \mathrm{d} x \leq \tilde C\beta^{2}I_{\alpha,q}\bigg(\int_{\mathbb{R}^{2}} w_m^{2q}\ \mathrm{d} x\bigg)^{1/q}. \]

By applying the Gagliardo–Nirenberg inequality (see for instance [Reference Kavian21, Proposition 8.12]), we get

\[ \|w_m\|_4 \leq C_1\|\nabla w_m\|_2^{1/2}\|w_m\|_2^{1/2}\quad\forall\ m\in\mathbb{N}. \]

Thus,

\[ \|w_m\|_4^{2}\leq C_2\beta^{2}I_{\alpha,q}(\displaystyle\int_{\mathbb{R}^{2}} w_m^{2q}\ \mathrm{d} x)^{1/q}\quad\forall\ m\in\mathbb{N}, \]

where $C_2>0$ does not depend on $\beta >1$ or $v$ or $\kappa >0$. Once $w_m=v^{\beta }$ in $A_m$ and $w_m\leq v^{\beta }$ in $\mathbb {R}^{2}$, it follows that

(4.7)\begin{equation} \left(\displaystyle\int_{A_{m}} v^{4\beta}\mathrm{d} x\right)^{1/2} \leq C_2\beta^{2} I_{\alpha,q}\bigg(\int_{\mathbb{R}^{2}} w_m^{2q}\ \mathrm{d} x\bigg)^{1/q}\quad\forall\ m\in\mathbb{N}. \end{equation}

Taking the limit as $m\rightarrow \infty$ and using Fatou's lemma, we reach

(4.8)\begin{equation} \|v\|_{4\beta}^{2\beta} \leq C_2\beta^{2} I_{\alpha,q}\|v\|_{2q\beta}^{2\beta}. \end{equation}

Considering $\sigma =2/q$, we have $\sigma >1$ because $q\in (1,\,2)$. Defining $\beta _j=\sigma ^{j}$ for $j\in \mathbb {N}$, one has

\[ 2q\beta_{j+1}=2q\sigma\beta_j=4\beta_j,\quad\forall\ j\in\mathbb{N}. \]

Hence, for $\beta =\beta _1=\sigma$, as a consequence of (4.8), we obtain

\[ \|v\|_{4\sigma}\leq \left(C_2 I_{\alpha,q}\right)^{1/(2\sigma)}\sigma^{1/\sigma}\|v\|_{4}. \]

Now, taking $\beta =\beta _2=\sigma ^{2}$, from (4.8), we get

\[ \|v\|_{4\sigma^{2}}\leq \left(C_2 I_{\alpha,q}\right)^{1/(2\sigma^{2})}(\sigma^{2})^{1/\sigma^{2}}\|v\|_{4\sigma} \leq\left(C_2 I_{\alpha,q}\right)^{(1/2)(1/\sigma+1/\sigma^{2})}\sigma^{(1/\sigma+2/\sigma^{2})}\|v\|_{4\sigma}. \]

Applying this iteration procedure $j$ times, with $\beta _j = \sigma ^{j}$, we can infer

(4.9)\begin{equation} \|v\|_{4\sigma^{j}} \leq (C_2 I_{\alpha,q})^{\frac{1}{2}\sum_{i =1}^{j} 1/\sigma^{i}} \sigma^{\sum_{i=1}^{j} i/\sigma^{i}} \|v\|_{ 4}, \end{equation}

which ensures that $v\in L^{4\sigma ^{j}}(\mathbb {R}^{2})$ for all $j\in \mathbb {N}$. Since

(4.10)\begin{equation} \displaystyle\sum_{j =1}^{\infty} \displaystyle\frac{1}{\sigma^{j}} = \dfrac{1}{\sigma - 1}\quad \textrm{and}\quad \sum_{j=1}^{\infty} \frac{j}{\sigma^{j}} = \dfrac{\sigma}{(\sigma -1)^{2}}, \end{equation}

taking the limit in (4.9) as $j \rightarrow \infty$ and using (4.10), we reach

\[ \|v\|_{\infty}\leq \left(C_2 I_{\alpha,q}\right)^{1/2(\sigma -1)} \sigma^{\sigma/(\sigma -1)^{2}} \|v\|_{4}, \]

which finalizes the proof, taking into account that $\sigma =2/q$.

Now, we are in a position to obtain a uniform estimate in $\kappa$ for the norm $\|v_{\kappa }\|_\infty$.

Lemma 4.2 Let $v_\kappa \in H^{1}(\mathbb {R}^{2})$ be the non-trivial critical point of $\mathcal {F}_\kappa$ given in Proposition 3.5. Then, there exists a positive constant $d_0$ satisfying

\[ \|v_k\|_\infty\leq d_0\quad\forall\ \kappa>0. \]

Proof. By the continuous embedding $H^{1}(\mathbb {R}^{2})\hookrightarrow L^{4}(\mathbb {R}^{2})$ and using Proposition 4.1, we reach

(4.11)\begin{equation} \|v_\kappa\|_{\infty} \leq [C_0I_{\alpha,q}(v_\kappa)]^{q/(4-2q)} (\displaystyle\frac{2}{q})^{2q/(2-q)^{2}}\|v_\kappa\|, \end{equation}

where $q\in (1,\,2)$, $\alpha >\alpha _0$,

\[ I_{\alpha,q}(v_\kappa):=\left[\int_{\mathbb{R}^{2}}\left(e^{[6\alpha q/(q-1)]v_\kappa^{2}}-1\right)\mathrm{d} x\right]^{(q-1)/q}, \]

and $C_0>0$ does not depend on $v_\kappa$ or $\kappa$. Due to Lemma 3.6, $\|v_\kappa \|^{2}\leq [2\mu /(\mu -2)]c_\kappa$ and according to Lemma 3.4, it follows that

\[ \|\nabla v_\kappa\|_2^{2}\leq\|v_\kappa\|^{2}\leq[2\mu/(\mu-2)]c_0<\frac{\pi}{3\alpha_0},\quad\forall\ \kappa>0. \]

In particular,

(4.12)\begin{equation} \|v_\kappa\|\leq \sqrt{\pi/(3\alpha_0)}\quad\forall\ \kappa>0. \end{equation}

Since $q\in (1,\,2)$ and $\alpha >\alpha _0$ are arbitraries, we can choose $q$ close to $2$ and $\alpha$ close to $\alpha _0$ in order to get still

\[ \alpha\frac{q}{q-1}\|\nabla v_\kappa\|_2^{2}\leq\hat c_0<\frac{2\pi}{3}\ \Longleftrightarrow\ 6\alpha\frac{q}{q-1}\|\nabla v_\kappa\|_2^{2}\leq 6\hat c_0<4\pi. \]

Thus, by applying (1.3), we have $(I_{\alpha,q}(v_\kappa ))$ uniformly bounded on $\kappa >0$ and, by (4.11) and (4.12), the conclusion of the lemma follows.

As a consequence of Proposition 4.1 and Lemma 4.2, we obtain the following result:

Corollary 4.3 Let $v_{\kappa } \in H^{1}(\mathbb {R}^{2})$ be a non-trivial critical point of $\mathcal {F}_{\kappa }$ obtained in Proposition 3.5. Then, there exists $\kappa _{0} > 0$ such that

\[ \|v_{\kappa}\|_{\infty} < \dfrac{1}{\sqrt{3 \kappa} \sqrt{6}} \ \forall\ \kappa \in (0, \kappa_{0}). \]

Proof. By Lemma 4.2, there exists $d_0>0$, which does not depend on $\kappa$, such that $\|v_{\kappa }\|_{\infty } \leq d_0$ for all $\kappa >0$. Defining $\kappa _0=1/(18d_0^{2})$, we have that if $\kappa \in (0,\,\kappa _0)$ then

\[ \|v_{\kappa}\|_{\infty} \leq d_0=\frac{1}{\sqrt{3\kappa_0}\sqrt{6}}<\frac{1}{\sqrt{3\kappa}\sqrt{6}} \]

and the proof is concluded.

Now, we can guarantee that $u_{\kappa }=G^{-1}(v_{\kappa })$ is indeed a weak solution for ($P_{\kappa }$), where $v_{\kappa }$ is the critical point of $\mathcal {F}_{\kappa }$ obtained in Proposition 3.5.

5. Proof of Theorem 1.1 completed

From ($g_{3}$) and Corollary 4.3, it follows that

\[ \|u_{\kappa}\|_{\infty} = \|G^{{-}1}(v_{\kappa})\|_{\infty} \leq \sqrt{6} \|v_{\kappa}\|_{\infty} < 1/\sqrt{3 \kappa},\text{ for all } \kappa\in (0,\kappa_0). \]

Therefore, as we have proved in the end of § 2, $u_\kappa$ is a weak solution of ($P_{\kappa }$), which is positive since $u_{\kappa }=G^{-1}(v_{\kappa })> 0$. This ends the proof of Theorem 1.1.