Proof. See proof in [Reference Alves, Garain and Rădulescu8, Lemma 3.8]

Proof. The proof of $(i)$ can be done as in [Reference Alves, Garain and Rădulescu8, Lemma 3.9].

$(ii)$ Suppose for each $n\in \mathbb {N},$ there exists $u_n\in \hat {X}\setminus B_n(0)$ such that

(4.3)\begin{equation} I(u_n)>0. \end{equation}

From $(g_2)$, it follows that

(4.4)\begin{equation} 0<\chi F(x,t)\leq f(x,t)t,\,\forall\,(x,t)\in\Omega\times(\mathbb{R}\setminus\{0\}), \end{equation}

where $\chi =\min \{\theta,\beta,\zeta \}>q.$ Therefore, $f$ satisfies the Ambrosetti–Rabinowitz condition. This gives the existence of positive constants $C,D$ such that

(4.5)\begin{equation} F(x,t)\geq C|t|^{\chi}-D, \quad \forall t \in \mathbb{R} \quad \mbox{and} \quad \forall x \in \Omega. \end{equation}

Using Proposition 2.1-(ii) along with (4.5), we obtain

(4.6)\begin{equation} I(u_n)\leq ||u_n||^{q}-C\displaystyle\int_{\Omega}|u_n|^{\chi}\,{\rm d}x+D|\Omega|. \end{equation}

Since $\dim \,\hat {X}<\infty$ and $\chi >q$ letting $n\to \infty$ in (4.6), we arrive at a contradiction to our assumption (4.3). Hence $(ii)$ follows.

Proof. Let $d>0$ and $(u_n)$ be a $(PS)_d$ sequence for $I$. Then, there are constants $C_1,C_2>0$ such that

(4.7)\begin{equation} I(u_n)-\displaystyle\frac{1}{\chi}I'(u_n)u_n \leq C_1+C_2\|u_n\|, \quad \forall\,n \in \mathbb{N}. \end{equation}

If $f$ is of the type $(f_1)$ or $(f_2)$ it is easy to check that (4.4) holds. Hence, by $(\varphi _3)'$ and the definition of $I$,

\begin{align*} I(u_n)-\displaystyle\frac{1}{\chi}I'(u_n)u_n& \geq \displaystyle\int_{\Omega}\Phi(x,|\nabla u_n|)\,{\rm d}x-\frac{1}{\chi}\int_{\Omega}\varphi(x,|\nabla u_n|)|\nabla u_n|^{2}\,{\rm d}x\\ & \geq \left(1-\displaystyle\frac{q}{\chi}\right)\displaystyle\int_{\Omega}\Phi(x,|\nabla u_n|)\,{\rm d}x. \end{align*}

Therefore,

\[ \left(1-\frac{q}{\chi}\right)\int_{\Omega}\Phi(x,|\nabla u_n|)\,{\rm d}x \leq C_1+C_2\|u_n\|, \quad \forall \,n \in \mathbb{N}. \]

If there is $(u_{n_j}) \subset (u_n)$ such that $\|u_{n_j}\| \geq 1$, then Proposition 2.1-(i) leads to

\[ \left(1-\frac{q}{\chi}\right)\|u_{n_j}\|^{p}\leq C_1+C_2\|u_{n_j}\|, \quad \forall\,j \in \mathbb{N}, \]

from where it follows the boundedness of $(u_{n_j})$. This implies the boundedness of $(u_n)$.

Proof. First of all, we must recall that

\[ I(u_n)-\frac{1}{\chi}I'(u_n)u_n=d+o_n(1)\|u_n\|+o_n(1). \]

Therefore, by $(\varphi _5)$,

\begin{align*} d+o_n(1)\|u_n\|+o_n(1)& \geq \displaystyle\int_{\Omega}((\Phi(x,|\nabla u_n|)-\displaystyle\frac{1}{\chi}\varphi(x,|\nabla u_n|)|\nabla u_n|^{2})\,{\rm d}x\\ & \geq \displaystyle\frac{1}{N}\left(1-\frac{q}{\chi}\right)\displaystyle\int_{\Omega_N}|\nabla u_n|^{N}\,{\rm d}x. \end{align*}

Hence,

\[ \limsup_{n \to +\infty}\frac{1}{N}\left(1-\frac{q}{\chi}\right)\int_{\Omega_N}|\nabla u_n|^{N}\,{\rm d}x \leq d< \min\left(1-\frac{q}{\chi}\right)\left\{\frac{1}{N}\left(\frac{\alpha_N}{2^{N'}\alpha}\right)^{N-1},\frac{1}{p}S_p^{\frac{N}{p}}\right\} \]

leading to

\[ \limsup_{n \to +\infty}\int_{\Omega_N}|\nabla u_n|^{N}\,{\rm d}x < \left(\frac{\alpha_N}{2^{N'}\alpha}\right)^{N-1}, \]

which proves the lemma.

Proof. The proof of this lemma follows as in [Reference Alves, Garain and Rădulescu8, Lemma 3.13]; however, for the reader's convenience, we will write the proof for $(f_1)$, since for $(f_2)$, it follows with similar arguments. Let $(u_n)$ be a $(PS)_d$ sequence for $I$. Then, by Lemma 4.4, $(u_n)$ is bounded in $W_0^{1,\Phi }(\Omega )$. Since $W_0^{1,\Phi }(\Omega )$ is reflexive, we assume that for some subsequence, still denoted by itself, there is $u \in W_0^{1,\Phi }(\Omega )$ such that

\[ u_n \rightharpoonup u \quad \mbox{in} \quad W_0^{1,\Phi}(\Omega), \]

and

\[ u_n(x) \to u(x) \quad \mbox{a.e. in} \quad \Omega. \]

Let us set

\[ P_n=\int_{\Omega}\langle \varphi(x,|\nabla u_n|)\nabla u_n, \nabla u_n- \nabla u\rangle\,{\rm d}x, \]

that is,

\[ P_n=I'(u_n)u_n+\int_{\Omega}f(x,u_n)u_n\,{\rm d}x-I'(u_n)u-\int_{\Omega}f(x,u_n)u\,{\rm d}x. \]

Consequently

\[ P_n=\int_{\Omega}f(x,u_n)u_n\,{\rm d}x-\int_{\Omega}f(x,u_n)u\,{\rm d}x+o_n(1). \]

From the definition of $f$ together with embedding (2.9),

\begin{align*} & \lim_{n \to +\infty}\int_{\Omega}\tilde{\eta}_q(x)g(x,u_n)u_n\,{\rm d}x=\lim_{n \to +\infty}\int_{\Omega}\tilde{\eta}_q(x)g(x,u_n)u\,{\rm d}x=\int_{\Omega}\tilde{\eta}_q(x)g(x,u)u\,{\rm d}x,\\ & \lim_{n \to +\infty}\int_{\Omega}{\eta}_p(x)|u_n|^{\zeta}\,{\rm d}x=\lim_{n \to +\infty}\int_{\Omega}{\eta}_p(x)|u_n|^{\zeta-2}u_nu\,{\rm d}x=\int_{\Omega}{\eta}_p(x)|u|^{\zeta}\,{\rm d}x,\\ & \lim_{n \to +\infty}\int_{\Omega \setminus \Omega_N}{\eta}_N(x)|u_n|^{\beta}e^{\alpha|u_n|^{N'}}\,{\rm d}x=\int_{\Omega \setminus \Omega_N}{\eta}_N(x)|u|^{\beta}e^{\alpha|u|^{N'}}\,{\rm d}x,\\ & \lim_{n \to +\infty}\int_{\Omega \setminus \Omega_N}{\eta}_N(x)|u_n|^{\beta-2}u_{n}ue^{\alpha|u_n|^{N'}}\,{\rm d}x=\int_{\Omega \setminus \Omega_N}{\eta}_N(x)|u|^{\beta}e^{\alpha|u|^{N'}}\,{\rm d}x,\\ & \lim_{n \to +\infty}\int_{\Omega \setminus \Omega_p}{\eta}_p(x)|u_n|^{p^{*}}\,{\rm d}x=\int_{\Omega \setminus \Omega_p}{\eta}_p(x)|u|^{p^{*}}\,{\rm d}x, \end{align*}

and

\[ \lim_{n \to +\infty}\int_{\Omega \setminus \Omega_p}{\eta}_p(x)|u_n|^{p^{*}-2}u_nu\,{\rm d}x=\int_{\Omega \setminus \Omega_p}{\eta}_p(x)|u|^{p^{*}}\,{\rm d}x. \]

Consequently

\begin{align*} P_n& =\displaystyle \lambda \int_{\Omega_N}|u_n|^{\beta}e^{\alpha|u_n|^{N'}}\,{\rm d}x-\lambda \int_{\Omega_N }|u_n|^{\beta-2}u_nue^{\alpha|u_n|^{N'}}\,{\rm d}x +\displaystyle \int_{\Omega_p}|u_n|^{p^{*}}\,{\rm d}x \\ & \quad-\displaystyle \int_{\Omega_p}|u_n|^{p^{*}-2}u_nu\,{\rm d}x\,{\rm d}x+o_n(1). \end{align*}

By Corollary 4.5, the sequence $(u_n)$ satisfies

\[ \|\nabla u_n\|_{L^{N}(\Omega_N)}^{N'} \leq \frac{\tau \alpha_N}{2^{N'} \alpha}, \quad \forall n \in \mathbb{N}, \]

for some $\tau \in (0,1)$. Employing Corollary 3.6, there is $t>1$ and $t \approx 1$ such that the sequence $h_n(x)=e^{\alpha |u_n(x)|^{N'}}$ is weakly convergent to $h(x)=e^{\alpha |u(x)|^{N'}}$ in $L^{t}(\Omega _N)$ , that is,

(4.8)\begin{equation} \displaystyle\int_{\Omega_N}h_n \varphi \,{\rm d}x \to \int_{\Omega_N}h \varphi \,{\rm d}x, \quad \forall \varphi \in L^{t'}(\Omega_N). \end{equation}

As

\[ |u_n|^{\beta} \to |u|^{\beta} \quad \mbox{in} \quad L^{t'}(\Omega_N), \]

it follows that

\[ \int_{\Omega_N}h_n |u_n|^{\beta} \,{\rm d}x \to \int_{\Omega_N}h |u|^{\beta} \,{\rm d}x, \]

that is,

\[ \int_{\Omega_N}|u_n|^{\beta}e^{\alpha|u_n|^{N'}} \,{\rm d}x \to \int_{\Omega_N}|u|^{\beta}e^{\alpha|u|^{N'}} \,{\rm d}x. \]

Now, using the fact that

\[ |u_n|^{\beta-2}u_nu \to |u|^{\beta} \quad \mbox{in} \quad L^{t'}(\Omega_N), \]

we also derive that

\[ \int_{\Omega_N}|u_n|^{\beta-2}u_nue^{\alpha|u_n(x)|^{N'}} \,{\rm d}x \to \int_{\Omega_N}|u|^{\beta-2}uue^{\alpha|u(x)|^{N'}}\,{\rm d}x. \]

The above analysis ensures that

\begin{align*} & \lim_{n \to +\infty}\int_{\Omega_N}|u_n|^{\beta}e^{\alpha|u_n(x)|^{N'}}\,{\rm d}x\\& \quad =\lim_{n \to +\infty}\int_{\Omega_N}|u_n|^{\beta-2}u_nue^{\alpha|u_n(x)|^{N'}}\,{\rm d}x=\int_{\Omega_N}|u|^{\beta}e^{\alpha|u|^{N'}}\,{\rm d}x, \end{align*}

and then,

\[ P_n=\int_{\Omega_p}|u_n|^{p^{*}}\,{\rm d}x-\int_{\Omega_p}|u_n|^{p^{*}-2}u_nu\,{\rm d}x+o_n(1). \]

By [Reference Kavian35, Lemma 4.8],

\[ \lim_{n \to +\infty}\int_{\Omega_p}|u_n|^{p^{*}-2}u_nu\,{\rm d}x=\int_{\Omega_p}|u|^{p^{*}}\,{\rm d}x, \]

then

\[ P_n=\int_{\Omega_p}|u_n|^{p^{*}}\,{\rm d}x-\int_{\Omega_p}|u|^{p^{*}}\,{\rm d}x+o_n(1). \]

Now, we are going to use the Concentration Compactness Lemma 3.1 to the sequence $(u_n) \subset W^{1,p}(\Omega _p)$. From $(\varphi _6)$, for each open ball $B \subset (\Omega _q)_{\delta }$ we have that the embedding $W^{1,\Phi }(\Omega ) \hookrightarrow C(\overline {B})$ is compact, then as $(u_n)$ is a bounded $(PS)$ for $I$, it is possible to prove that for some subsequence, there holds

\[ \int_{B}\langle \varphi(x,|\nabla u_n|)\nabla u_n, \nabla u_n- \nabla u\rangle\,{\rm d}x \to 0. \]

Since from $(\varphi _5)-(\varphi _7)$, the embedding $W^{1,\Phi }(B) \hookrightarrow L^{\Phi }(B)$ is compact, the last limit together with the $\Delta _2$-condition (2.1) implies that

\[ u_n \to u \quad \mbox{in} \quad W^{1,\Phi}(B). \]

Now, recalling that the embedding $W^{1,\Phi }(B) \hookrightarrow W^{1,p}(B)$ is continuous, we derive that

\[ u_n \to u \quad \mbox{in} \quad W^{1,p}(B), \]

from where it follows that $x_i \in \overline {\Omega _p} \setminus (\Omega _q)_{\delta }$ for all $i \in J$. Now, our goal is proving that $J$ must be a finite set. Have this in mind, we will consider $J=J_1 \cup J_2$ where

\[ J_1=\{i \in J\,:\,x_i \in \overline{\Omega_p} \setminus \overline{(\Omega_q)_{\delta}} \} \]

and

\[ J_2=\{i \in J\,:\,x_i \in \partial (\Omega_q)_{\delta} \cap \Omega_p \}. \]

If $i \in J_1$, the condition $(\varphi _7)$ says that $c_3t^{p-2} \geq \varphi (x,t) \geq t^{p-2}$ for $x \in \overline {\Omega _p} \setminus \overline {(\Omega _q)_{\delta }}$. This fact permits us to repeat the same arguments explored in [Reference Garcia Azorero and Peral Alonso30, Lemma 2.3] to conclude that $J_1$ is finite. Now, if $i \in J_2$, the situation is more subtle and we must be careful. In what follows let us consider $\tilde {\psi } \in C_{0}^{\infty }(\mathbb {R}^{N})$ such that

\[ \tilde{\psi} \equiv 1 \text{ on } B(0,1) \text{ and } \tilde{\psi} \equiv 0 \text{ on } B(0,2)^{c}. \]

For each $\epsilon >0$, we set

\[ \psi(x)=\tilde{\psi}({(x-x_i)}/{\epsilon}), \quad \forall x \in \mathbb{R}^{N}. \]

Since $(u_n)$ is a bounded sequence in $W^{1,\Phi }(\Omega )$, the sequence $(\psi u_n)$ is also bounded in $W^{1,\Phi }(\Omega )$ and so, $I'(u_n)\psi u_n=o_n(1)$. Hence,

\[ \int_{\Omega}\varphi(x,|\nabla u_n|)\nabla u_n \nabla(\psi u_n)\,{\rm d}x=\int_{\Omega}\tilde{\eta}_q(x)g(x,u_n)\psi u_n\,{\rm d}x+\int_{\Omega}\eta_p(x)|u_n|^{p^{*}}\psi\,{\rm d}x+o_n(1). \]

Now, given $\xi >0$, the Young's inequality (2.2) combined with (2.3) and $\Delta _2$-condition (2.1) gives

\[ \int_{\Omega} |\varphi(x,|\nabla u_n|)|\nabla u_n||u_n||\nabla \psi|\,{\rm d}x \leq \xi \int_{\Omega}\Phi(x,|\nabla u_n|)\,{\rm d}x +C_{\xi}\int_{\Omega}\Phi(x,|\nabla \psi||u_n|)\,{\rm d}x. \]

for some $C_\xi >0$. Note that by $(\varphi _7)$,

\[ \int_{\Omega}\Phi(x,|\nabla \psi||u_n|)\,{\rm d}x \leq C_1\left(\int_{B(x_i,2\epsilon)}|\nabla \psi|^{p}||u_n|^{p}\,{\rm d}x+\int_{B(x_i,2\epsilon)}\tau_2(x)|\nabla \psi|^{q}||u_n|^{q}\,{\rm d}x \right). \]

By Hölder's inequality

\[ \limsup_{n \to +\infty}\int_{B(x_i,2\epsilon)} |u_n|^{p}|\nabla \psi|^{p}\,{\rm d}x \leq C_2\left(\int_{B(x_i,2\epsilon)}|u|^{p^{*}}\,{\rm d}x\right)^{\frac{N-p}{N}} \]

from where it follows that

(4.9)\begin{equation} \lim_{\epsilon \to 0}\left[\limsup_{n \to +\infty}\displaystyle\int_{B(x_i,2\epsilon)} |u_n|^{p}|\nabla \psi|^{p}\,{\rm d}x\right] \leq\lim_{\epsilon\to 0} C_2(\int_{B(x_i,2\epsilon)}|u|^{p^{*}}\,{\rm d}x)^{\frac{N-p}{N}}=0. \end{equation}

Arguing as above, we also have

\begin{align*} & \limsup_{n \to +\infty}\int_{\Omega}\tau_2(x)|u_n|^{q}|\nabla \psi|^{q}\,{\rm d}x\\ & \quad \leq \left(\int_{B(x_i,2\epsilon)}\left|\tau_2^{\frac{1}{q}}(x)\nabla \psi\right|^{\frac{qp^{*}}{p^{*}-q}}\,{\rm d}x \right)^{\frac{p^{*}-q}{p^{*}}}\left(\int_{B(x_i,2\epsilon)}|u|^{p^{*}} \,{\rm d}x\right)^{\frac{q}{p^{*}}}. \end{align*}

By change of variable,

\begin{align*} & \displaystyle\int_{B(x_i,2\epsilon)}\left|\tau_2^{\frac{1}{q}}(x)\nabla \psi\right|^{\frac{qp^{*}}{p^{*}-q}}\,{\rm d}x=\left(\frac{1}{\epsilon}\right)^{\frac{qp^{*}}{p^{*}-q}}\int_{B(0,2)}\left|\tau_2^{\frac{1}{q}}(\epsilon x+x_i)\nabla \tilde{\psi}\right|^{\frac{qp^{*}}{p^{*}-q}}\,{\rm d}x \\ & \quad\leq C_5\left(\frac{1}{\epsilon}\right)^{\frac{qp^{*}}{p^{*}-q}}\displaystyle \int_{B(0,2)}\left|\tau_2^{\frac{1}{q}}(\epsilon x+x_i)\right|^{\frac{qp^{*}}{p^{*}-q}}\,{\rm d}x . \end{align*}

Since $x_i \in \partial (\Omega _q)_{\delta } \cap \Omega _p$, it follows that

\[ \tau_2(\epsilon x+x_i) \leq c_4\epsilon^{s}|x|^{s} \]

and

\[ \int_{B(x_i,2\epsilon)}\left|\tau_2^{\frac{1}{q}}(x)\nabla \psi\right|^{\frac{qp^{*}}{p^{*}-q}}\,{\rm d}x \leq C_6\epsilon^{\frac{(s-q)p^{*}}{p^{*}-q}}. \]

As $s>q$, it follows that

(4.10)\begin{equation} \lim_{\epsilon \to 0}\left[\limsup_{n \to +\infty}\displaystyle\int_{\Omega}\tau_2(x)|u_n|^{q}|\nabla \psi|^{q}\,{\rm d}x\right]=0. \end{equation}

Now, the boundedness of $(u_n)$ in $W^{1,\Phi }(\Omega )$ together with Proposition (2.1), (4.9) and (4.10) ensures that

\[ \lim_{\epsilon \to 0}\left[\limsup_{n \to +\infty}\int_{\Omega}|\varphi(x,|\nabla u_n|)|\nabla u_n||u_n||\nabla \psi|\,{\rm d}x \right] \leq \xi C, \]

for some $C>0$. Since $\xi >0$ is arbitrary, we can deduce that

\[ \lim_{\epsilon \to 0}\left[\limsup_{n \to +\infty}\int_{\Omega}|\varphi(x,|\nabla u_n|)|\nabla u_n||u_n||\nabla \psi|\,{\rm d}x \right]=0. \]

The last limit together with the fact that $\varphi (x,t) \geq t^{p-2}$ for $x \in \Omega _p$ permit us to conclude as in [Reference Garcia Azorero and Peral Alonso30, Lemma 2.3], that $J_2$ is also finite. Consequently, $J$ is a finite set. However, in order to conclude the proof of the lemma, we need to show that $J$ is in fact an empty set. Seeking by a contradiction, assume that there is $i \in J$. In this case, the argument explored in [Reference Garcia Azorero and Peral Alonso30] also says for us that

\[ \nu_i \geq {S_p^{\frac{N}{p}}}. \]

Hence, by Lemma 3.1-(d),

\[ \mu_i \geq {S_p^{\frac{N}{p}}}. \]

As $|\nabla u_n|^{p} \to \mu$ weakly-$^{*}$ in the sense of measure, we have

\[ \liminf_{n \to +\infty}\int_{\Omega_p}|\nabla u_n|^{p}\,{\rm d}x \geq \mu_i \]

and so,

\[ \liminf_{n \to +\infty}\int_{\Omega_p}|\nabla u_n|^{p}\,{\rm d}x \geq {S_p^{\frac{N}{p}}}. \]

Now, using once more the equality

\[ I(u_n)-\frac{1}{\chi}I'(u_n)u_n=d+o_n(1)\|u_n\|+o_n(1), \]

we get

\[ d+o_n(1)\|u_n\|+o_n(1) \geq \frac{1}{p}\left(1-\frac{q}{\chi}\right)\int_{\Omega_p}|\nabla u_n|^{p}\,{\rm d}x. \]

Taking the limit of $n \to +\infty$, we find the inequality below

\[ d \geq \frac{1}{p}\left(1-\frac{q}{\chi}\right){S_p^{\frac{N}{p}}}, \]

which is a contradiction, showing that $J=\emptyset$. Thereby, by Lemma 3.1-$(a)$, $\nu =|u|^{p^{*}}$ and

\[ \int_{\Omega_p}|u_n|^{p^{*}}\,{\rm d}x \to \int_{\Omega_p}|u|^{p^{*}}\,{\rm d}x, \]

implying that $P_n=o_n(1)$, that is,

\[ \lim_{n \to +\infty}\int_{\Omega}\langle \varphi(x,|\nabla u_n|)\nabla u_n, \nabla u_n- \nabla u\rangle\,{\rm d}x=0. \]

Now, it is enough to apply Lemma 2.2 to finish the proof.

Proof. First, we claim that for some positive constant $C>0,$ we have

(4.11)\begin{equation} \min_{u\in K,\,||u||=1}\left\{\displaystyle\int_{\Omega_N}|u|^{\beta}\,{\rm d}x+\int_{\Omega_q}|u|^{q_2}\,{\rm d}x+\int_{\Omega_p}|u|^{\zeta}\,{\rm d}x\right\}\geq C, \end{equation}

where $K\subset X_m$ is compact such that $\dim \,X_m<\infty.$

Indeed, if (4.11) does not hold, then there exists a sequence $\{u_n\}\subset K$ with $||u_n||=1$ such that

(4.12)\begin{equation} \left\{\displaystyle\int_{\Omega_N}|u_n|^{\beta}\,{\rm d}x+\int_{\Omega_q}|u_n|^{q_2}\,{\rm d}x+\int_{\Omega_p}|u_n|^{\zeta}\,{\rm d}x\right\}\leq\displaystyle\frac{1}{n},\,\,\forall\,n\in\mathbb{N}. \end{equation}

Since $\dim \,X_m<\infty,$ there exists a subsequence of $\{u_n\}$ still denoted by $\{u_n\}$ and $u\in K$ with $||u||=1$ such that $u_{n_j}\to u$ in $X_m.$ Then, letting $n\to \infty$ in (4.12), we obtain

\[ \int_{\Omega_N}|u|^{\beta}\,{\rm d}x+\int_{\Omega_q}|u|^{q_2}\,{\rm d}x+\int_{\Omega_p}|u|^{\zeta}\,{\rm d}x=0. \]

Hence, we have $u=0$ a.e. in each of the sets $\Omega _N,\Omega _q$ and $\Omega _p.$ Now, since $\Omega =\Omega _N\cup \Omega _q\cup \Omega _p,$ we have $u=0$ a.e. in $\Omega.$ This contradicts the fact that $||u||=1.$ Hence the Claim (4.11) follows.

Now we choose $K=\overline {D}_m$ where $D_m$ is given by (4.1). Since $h=I_d \in G_m$, the definition of $\Phi$ combined with (2.4) and $(f_1)$ gives

\[ c_m^{\lambda,\mu,\tau}\leq\sup_{u\in K}\left\{\|u\|^{p}+\|u\|^{q}-\lambda\int_{\Omega_N }|u|^{\beta}\,{\rm d}x-\mu\int_{\Omega_q}|u|^{q_2}\,{\rm d}x -\tau\int_{\Omega_p}|u|^{\zeta}\,{\rm d}x\right\}, \]

or equivalently

\begin{align*} c_m^{\lambda,\mu,\tau}& \leq\sup_{u\in K}\left\{\|u\|^{p}+\|u\|^{q}-\lambda\|u\|^{\beta}\int_{\Omega_N}\left|\frac{u}{\|u\|}\right|^{\beta}\,{\rm d}x\right.\\& \left.\quad -\mu\|u\|^{q_2}\int_{\Omega_q}\left|\frac{u}{\|u\|}\right|^{q_2}\,{\rm d}x -\tau\|u\|^{\zeta}\int_{\Omega_p}\left|\frac{u}{\|u\|}\right|^{\zeta}\,{\rm d}x\right\}. \end{align*}

Now, when $||u||>1,$ we observe that

\begin{align*} & \|u\|^{p}+\|u\|^{q}-\lambda\|u\|^{\beta}\displaystyle\int_{\Omega_N }\left|\displaystyle\frac{u}{\|u\|}\right|^{\beta}\,{\rm d}x-\mu\|u\|^{q_2}\int_{\Omega_q}\left|\frac{u}{\|u\|}\right|^{q_2}\,{\rm d}x -\tau\|u\|^{r}\int_{\Omega_p}\left|\frac{u}{\|u\|}\right|^{\zeta}\,{\rm d}x\\ & \leq \|u\|^{p}+\|u\|^{q}-\|u\|^{l_1}\chi_1\left(\int_{\Omega_N }\left|\frac{u}{\|u\|}\right|^{\beta}\,{\rm d}x+\int_{\Omega_q}\left|\frac{u}{\|u\|}\right|^{q_2}\,{\rm d}x +\int_{\Omega_p}\left|\frac{u}{\|u\|}\right|^{\zeta}\,{\rm d}x\right)\\ & \leq \|u\|^{p}+\|u\|^{q}-C\chi_1\|u\|^{l_1}, \end{align*}

where $l_1=\min \{\beta,q_2,\zeta \}$, $\chi _1=\min \{\lambda,\mu,\tau \}$ and the constant $C$ is given by (4.11).

Moreover, when $\|u\|\leq 1,$ we get

\begin{align*} & \|u\|^{p}+\|u\|^{q}-\lambda\|u\|^{\beta}\displaystyle\int_{\Omega_N }\left|\displaystyle\frac{u}{\|u\|}\right|^{\beta}\,{\rm d}x-\mu\|u\|^{q_2}\int_{\Omega_q}\left|\frac{u}{\|u\|}\right|^{q_2}\,{\rm d}x -\tau\|u\|^{r}\int_{\Omega_p}\left|\frac{u}{\|u\|}\right|^{\zeta}\,{\rm d}x\\ & \leq \|u\|^{p}+\|u\|^{q}-\|u\|^{l_2}\chi_1\left(\int_{\Omega_N }\left|\frac{u}{\|u\|}\right|^{\beta}\,{\rm d}x+\int_{\Omega_q}\left|\frac{u}{\|u\|}\right|^{q_2}\,{\rm d}x +\int_{\Omega_p}\left|\frac{u}{\|u\|}\right|^{\zeta}\,{\rm d}x\right)\\ & \leq \|u\|^{p}+\|u\|^{q}-C\chi_1\|u\|^{l_2}, \end{align*}

where $l_2=\max \{\beta,q_2,\zeta \}$, $\chi _1=\min \{\lambda,\mu,\tau \}$ and the constant $C$ is given by (4.11). Hence,

\[ c_m^{\lambda,\mu,\tau}\leq \|u\|^{p}+\|u\|^{q}-C\chi\|u\|^{l}, \]

where $l=l_1$ or $l_2.$

Let, $w(t)=t^{p}+t^{q}-C\chi t^{l}$. Then using the fact that $l>q>p,$ it can be easily seen that $w$ achieves its maximum at $\hat {t}=\hat {t}(\lambda,\beta,\tau )>0$ which goes to $0$ as the parameters $\lambda,\mu,\tau$ goes to infinity. Hence there exists $\lambda _m,\mu _m,\tau _m$ such that for all $\lambda \geq \lambda _m,\mu \geq \mu _m$ and $\tau \geq \tau _m,$ we have

\[ c_m^{\lambda,\mu,\tau}< M, \]

where $M$ is given by Corollary 4.5. Hence, the Lemma follows.

Proof. We prove the result in two steps.

Step 1. We claim that for every fixed $1\leq k\leq \infty$ and $r>0$, there exists a constant $\eta >0$ such that

(5.1)\begin{equation} ||u||>\eta,\text{ for all }u\in\mathcal{M}_{k,r}. \end{equation}

Indeed, if (5.1) does not hold, there exists a sequence $(u_n)\in \mathcal {M}_{k,r}$ such that $||u_n||\to 0$ as $n\to \infty$. From $u_n\in \mathcal {M}_{k,r}$ we have $I'(u_n)u_n=0$. Hence

(5.2)\begin{equation} \displaystyle\int_{\Omega_r}\varphi(x,\nabla u_n)|\nabla u_n|^{2}\,{\rm d}x =\int_{\Omega_r} f(|x|,u_n)u_n \,{\rm d}x. \end{equation}

Due to the fact $||u_n||\to 0$ as $n\to \infty$, without loss of generality we may assume that $||u_n||<1$ for all $n \in \mathbb {N}$. Hence from $(\varphi _3)'$ and Proposition 2.1, we can estimate the left-hand side of (5.2) as follows:

(5.3)\begin{equation} p||u_n||^{q}\leq p\displaystyle\int_{\Omega_r}\Phi(x,u_n)\,{\rm d}x\leq \int_{\Omega_r}\varphi(x,\nabla u_n)|\nabla u_n|^{2}\,{\rm d}x. \end{equation}

Now, we estimate the right-hand side of (5.2). Indeed,

(5.4)\begin{equation} \begin{aligned} \displaystyle\int_{\Omega_r}f(|x|,u_n)u_n\,{\rm d}x & =\lambda\int_{\Omega_N}|u_n|^{\beta} e^{\alpha |u_n|^{N^{\prime}}}\,{\rm d}x+\int_{\Omega_r\setminus\Omega_N}\lambda \eta_{N}(|x|)|u_n|^{\beta} e^{\alpha |u_n|^{N^{\prime}}}\,{\rm d}x\\ & \quad+\displaystyle\int_{\Omega_q} g(|x|,u_n)u_n\,{\rm d}x+\int_{\Omega_r\setminus\Omega_q}\tilde{\eta}(|x|)g(|x|,u_n)u_n\,{\rm d}x\\ & \quad+\int_{\Omega_p}|u_n|^{p^{*}}\,{\rm d}x+\int_{\Omega_r\setminus\Omega_p}\eta_p(|x|)|u_n|^{p^{*}}\,{\rm d}x\\ & \leq\lambda\int_{\Omega_N}|u_n|^{\beta} e^{\alpha |u_n|^{N^{\prime}}}\,{\rm d}x+\int_{\Omega_q} g(|x|,u_n)u_n\,{\rm d}x+\int_{\Omega_p}|u_n|^{p^{*}}\,{\rm d}x\\ & \quad+\int_{(\overline{\Omega_q})_{\delta/2}} f(|x|,u_n)u_n\,{\rm d}x\\ & =I_1+I_2+I_3+I_4. \end{aligned} \end{equation}

Estimate of $I_1$: As $||u_n|| \to 0$, by Hölder's inequality and Lemma 3.5 for some constant $C_1>0$ (independent of $n$), we have

(5.5)\begin{equation} \begin{aligned} I_1 & =\lambda \displaystyle\int_{\Omega_N} |u_n|^{\beta} e^{\alpha |u_n|^{N^{\prime}}}\,{\rm d}x\\ & \leq \lambda\left(\displaystyle\int_{\Omega_N}|u_n|^{2\beta}\,{\rm d}x\right)^{\frac 12}\left(\int_{\Omega_N}e^{2\alpha|u_n|^{N^{\prime}}}\,{\rm d}x\right)^{\frac 12}\\ & \leq C_1||u_n||^{\beta}. \end{aligned} \end{equation}

Estimate of $I_2$: From the condition $(g_1)$ and the embedding (2.9) for some constant $C_2>0$ (independent of $n$) we deduce that

(5.6)\begin{equation} I_2=\displaystyle\int_{\Omega_q}g(|x|,u_n)u_n \,{\rm d}x\leq C_2 ||u_n||^{q_1}. \end{equation}

Estimate of $I_3$: It is clear that

(5.7)\begin{equation} I_3=\displaystyle\int_{\Omega_p}|u_n|^{p^{*}} \,{\rm d}x\leq C_3||u_n||^{p^{*}}, \end{equation}

for some constant $C_3>0$ (independent of $n$).

Estimate of $I_4$: By the embedding (2.9), the definition of $f_2$ and the condition $(g_1)$ we have that

(5.8)\begin{equation} I_4=\displaystyle\int_{(\overline{\Omega_q})_{\delta/2}}f(|x|,u_n)u_n \,{\rm d}x\leq C_4(||u_n||^{q_1}+||u_n||^{\beta}+||u_n||^{p^{*}}), \end{equation}

for some constant $C_4>0$ (independent of $n$).

Therefore, using the estimates (5.5), (5.6), (5.7) and (5.8) in (5.4), we obtain

(5.9)\begin{equation} \displaystyle\int_{\Omega_r}f(|x|,u_n)u_n \,{\rm d}x\leq C(||u_n||^{q_1}+||u_n||^{\beta}+||u_n||^{p^{*}}), \end{equation}

for some constant $C>0$ (independent of $n$). Using (5.9) in (5.3), we have

(5.10)\begin{equation} p||u_n||^{q}\leq p\displaystyle\int_{\Omega_r}\Phi(x,|\nabla u_n|)\,{\rm d}x\leq C(||u_n||^{q_1}+||u_n||^{\beta}+||u_n||^{p^{*}}). \end{equation}

Since all the parameters $\beta,q_1$ and $p^{*}$ are larger than $q$, from (5.10) for some constant $\widehat {C}>0$ (independent of $n$), we have

\[ ||u_n||\geq \widehat{C},\text{ for all }u_n\in \mathcal{M}_{k,r}, \]

which is a contradiction to the fact $||u_n||\to 0$. Hence (5.1) holds.

Step 2. From the definition of $I$, $(\varphi _3)'$ and Proposition 2.1 for any $u\in \mathcal {M}_{k,r}$, we have

\begin{align*} I(u)& =I(u)-\displaystyle\frac{1}{\chi}I'(u)u\\ & =\displaystyle\int_{\Omega_r}\Phi(x,|\nabla u|)\,{\rm d}x-\frac{1}{\chi}\int_{\Omega_r}\phi(x,|\nabla u_n|)|\nabla u_n|^{2}\,{\rm d}x\\ & \geq \left(1-\frac{q}{\chi}\right)\int_{\Omega_r}\Phi(x,|\nabla u|) \,{\rm d}x\\ & \geq\left(1-\frac{q}{\chi}\right)\max\{||u||^{q},||u||^{p}\}\\ & \geq\left(1-\frac{q}{\chi}\right)\max\{\eta^{q},\eta^{p}\}, \end{align*}

where in the last line, we have used the estimate (5.1) from Step 1 and $\chi =\min \{\theta,\beta,p^{*}\}$. This means that,

\[ J_{k,r}\geq \left(1-\frac{q}{r}\right)\max\{\eta^{q},\eta^{p}\}>0, \]

for every $1\leq k\leq \infty$ and all $r>0$. Hence the result follows.

Proof. Fix $1\leq k<\infty$. Due to $(\eta )$, there exists $\gamma =\gamma (k) < \min \left \{\frac {1}{2},\delta _1\right \}$ such that the ball $B_{\gamma,r}:=B_{\gamma }((\frac {2r+1}{2},0,\ldots,0))\subset \Omega _N\setminus {\overline {(\Omega _q)_\delta }}$ satisfies

\[ h^{i} B_{\gamma,r}\cap h^{j} B_{\gamma,r}=\emptyset,\text{ for all }h^{i}\in H_k, \, i\neq j\in\{0,1,\ldots,k-1\}. \]

Consider $v_r\in W^{1,\Phi }_0(B_{\delta,r})\setminus \{0\}$ and define

\[ v:=\sum_{h\in H_k} h v_r\in W^{1,\Phi}_{0,H_k}\left(\Omega_N\setminus{\overline{(\Omega_q)_\delta}}\right)\setminus\{0\}. \]

By definition of $I$, we observe that

\[ I'(tv)tv=\int_{{\Omega_N\setminus{\overline{(\Omega_q)_\delta}}}}\varphi(x,|t\nabla v|) t^{2}|\nabla v|^{2}\,{\rm d}x-\int_{\Omega_N\setminus{\overline{(\Omega_q)_\delta}}} f(x,tv)tv\,{\rm d}x. \]

Using

\[ f(x,tv)tv\geq \lambda t^{\beta}|v|^{\beta}, \forall x\in \Omega_N\setminus{\overline{(\Omega_q)_\delta}}\text{ and }t\geq 0. \]

Therefore, using $(\varphi _3)'$ and (2.4), for every $t\geq 0$,

\[ I'(tv)tv\leq q\xi_1(t)\int_{\Omega_N\setminus{\overline{(\Omega_q)_\delta}}}\Phi(x,|\nabla v|)\,{\rm d}x -\lambda t^{\beta} \int_{\Omega_N\setminus{\overline{(\Omega_q)_\delta}}} |v|^{\beta}\,{\rm d}x. \]

As $\beta >q> p$, we get that $I'(tv)tv\to -\infty$ as $t\to +\infty$ and $I'(tv)tv>0$ for $t \approx 0$.

So, there exists $t_v>0$ such that $t_v v\in W^{1,\Phi }_{0,H_k}({\Omega _N\setminus {\overline {(\Omega _q)_\delta }}})\setminus \{0\}$ with $I^{\prime }(t_v v)t_v v=0$. If we denote by $w=t_v v$, then

(5.11)\begin{equation} J_{k,r}\leq I(w)=kI(t_v v_r)=k\max_{t\geq 0} I(t v_r). \end{equation}

Following similar arguments as in the proof of [Reference Alves, Garain and Rădulescu8, Lemma 3.11], we have

\[ \max_{t\geq 0} I(t_v v_r)\leq \frac{1}{\lambda^{\frac{N}{\beta-N}}}\left(\frac 1N-\frac 1\beta\right)\frac{\left(c_1||\nabla v_r||^{N}_{L^{N}(\Omega_N)}\right)^{\frac{\beta}{\beta-N}}}{\left(||v_r||^{\beta}_{L^{\beta}(\Omega_N)}\right)^{\frac{N}{\beta-N}}}. \]

Now, we fix $\lambda _0=\lambda _0(k)>0$ such that for all $\lambda \geq \lambda _0$, we have

(5.12)\begin{equation} \displaystyle\frac{k}{\lambda^{\frac{N}{\beta-N}}}\left(\frac 1N-\frac 1\beta\right)\frac{(c_1||\nabla v_r||^{N}_{L^{N}(\Omega_N)})^{\frac{\beta}{\beta-N}}}{(||v_r||^{\beta}_{L^{\beta}(\Omega_N)})^{\frac{N}{\beta-N}}}<(1-\frac{q}{\chi})\min\left\{\frac 1N(\frac{\alpha_N}{2^{N'}\alpha})^{N-1},\frac{1}{p} {S_p^{N/p}}\right\}. \end{equation}

Therefore, from (5.11) and (5.12), the result follows.

Proof. Let $(v_n)\subset \mathcal {M}_{k,r}$ a minimizing sequence for $J_{k,r}$, i.e., $(v_n)\subset W^{1,\phi }_{0,H_k}(\Omega _r)\setminus \{0\}$ such that

\[ I'(v_n)v_n=0\text{ and }I(v_n)\to J_{k,r}. \]

We claim that $(v_n)$ is bounded. Assume there is some $n$ such that $||v_n||\geq 1$, since otherwise $(v_n)$ is bounded. Due to the fact $I(v_n)\to J_{k,r}$ and $J_{k,r} \leq M$, where $M$ was given in Lemma 5.2, it follows that

\begin{align*} M& \geq I(v_n)=I(v_n)-\displaystyle\frac{1}{\chi}I^{\prime}(v_n)v_n\\ & \geq\left(1-\displaystyle\frac{q}{\chi}\right)||v_n||^{p}. \end{align*}

Therefore, $||v_n||\leq c$ if $\|v_n\| >1$, for some constant $c>0$ independent of $n$. This shows that $(v_n)$ is bounded.

Claim:

\[ I'(v_n)\to 0\text{ in }(W^{1,\Phi}_{0,H_k}(\Omega_r))'. \]

Indeed, using the Ekeland variational Principle (see Willem [Reference Willem54]), there exists a sequence $(w_n)\subset \mathcal {M}_{k,r}$ such that

\[ w_n=v_n+o_n(1),\quad I(w_n)\to J_{k,r} \]

and

(5.13)\begin{equation} I'(w_n)-\ell_n E'(w_n)=o_n(1), \end{equation}

where $(\ell _n)\subset \mathbb {R}$ and $E(w)=I'(w)w$ for $w\in W^{1,\Phi }_{0,H_k}(\Omega _r)$. Since $(v_n)$ is bounded, we also have that $(w_n)$ is bounded. Now, we prove that there exists $C >0$ such that

(5.14)\begin{equation} |E'(w_n) w_n|>C\text{ for all } n\in\mathbb{N}. \end{equation}

Indeed, we observe that

(5.15)\begin{equation} \begin{aligned} -E^{\prime}(w_n)w_n & ={-}\displaystyle\int_{\Omega_r}\left[\displaystyle\frac{\partial}{\partial t}\varphi(x,|\nabla w_n|)|\nabla w_n|+2\varphi(x,|\nabla w_n|)\right]|\nabla w_n|^{2}\,{\rm d}x\\ & \quad+\int_{\Omega_r}\left[\frac{\partial}{\partial t}f(|x|,w_n)w_n^{2}+f(|x|,w_n)w_n\right]\,{\rm d}x\\ & \geq{-}q\int_{\Omega_r}\varphi(x,|\nabla w_n|)|\nabla w_n|^{2}\,{\rm d}x\\ & \quad+\int_{\Omega_r}\left[\frac{\partial}{\partial t}f(|x|,w_n)w_n^{2}+f(|x|,w_n)w_n\right]\,{\rm d}x\\ & =\int_{\Omega_r}\left[\frac{\partial}{\partial t}f(|x|,w_n)w_n^{2}-(q-1)f(|x|,w_n)w_n\right]\,{\rm d}x, \end{aligned} \end{equation}

where in the second step, we have used that the function $\dfrac {\varphi (x,t)}{|t|^{q-3}t}$ is non-increasing for $t\neq 0$, which is a consequence of $(\varphi _{3})$, while that in the third step we have use the fact that $I^{\prime }(w_n)w_n=0$, i.e.

\[ \int_{\Omega_r}\varphi(x,|\nabla w_n|)|\nabla w_n|^{2}\,{\rm d}x=\int_{\Omega_r}f(|x|,w_n)w_n\,{\rm d}x, \]

respectively. Since $(w_n)$ is bounded in $W_{0}^{1,\Phi }(\Omega _r)$ and $J_{k,r}>0$, there exists $w\in W^{1,\Phi }(\Omega _r)\setminus \{0\}$ such that $w_n\to w$ strongly in $L^{\Phi }(\Omega _r)$ and $w_n(x) \to w(x)$ a.e. in $\Omega _r$ for some subsequence. Then, by Fatou's lemma,

(5.16)\begin{equation} \begin{aligned} & \liminf_{n\to\infty}\displaystyle\int_{\Omega_r}\left[\displaystyle\frac{\partial}{\partial t}f(|x|,w_n)w_n^{2}-(q-1)f(|x|,w_n)w_n\right]\,{\rm d}x\\ & \geq\displaystyle\int_{\Omega_r}\left[\displaystyle\frac{\partial}{\partial t}f(|x|,w)w^{2}-(q-1)f(|x|,w)w\right]\,{\rm d}x. \end{aligned} \end{equation}

From the hypothesis $(g_4)$ and the definition of $f_2$, we obtain

(5.17)\begin{equation} \displaystyle\int_{\Omega_r}\left[\displaystyle\frac{\partial}{\partial t}f(|x|,w)w^{2}-(q-1)f(|x|,w)w\right]\,{\rm d}x>0. \end{equation}

By contradiction, suppose

\[ \lim_{n\to\infty}E^{\prime}(w_n)w_n=0. \]

Then letting $n\to \infty$ in (5.15) and using (5.16) along with (5.17), we obtain

\[ 0=\lim_{n\to\infty}E^{\prime}(w_n)w_n\geq\displaystyle\int_{\Omega_r}\left[\displaystyle\frac{\partial}{\partial t}f(|x|,w)w^{2}-(q-1)f(|x|,w)w\right]\,{\rm d}x>0, \]

which is absurd. Therefore, (5.14) holds.

From, (5.13)

\[ \ell_n E'(w_n)w_n=o_n(1), \]

and so, $\ell _n=o_n(1)$. Since $(w_n)$ is bounded we get $(E'(w_n))$ is bounded. Hence from (5.13)

\[ I'(w_n)\to 0\text{ in }\Big(W^{1,\Phi}_{0,H_k}(\Omega_r)\Big)'. \]

Thus, without loss generality, we may assume

\[ I(v_n)\to J_{k,r}\;\; \text{and} \;\; I'(v_n)\to 0. \]

Since $(v_n)$ is bounded, there exists $v\in W^{1,\Phi }_{0.H_k}(\Omega _r)$ such that, for a subsequence, we have

\[ \begin{cases} v_n\rightharpoonup v & \text{in } W^{1,\Phi}_{0,H_k}(\Omega_r),\\ v_n(x)\to v(x), & \text{a.e. in } \Omega_r. \end{cases} \]

Now following exactly the proof of Lemma 4.6, we get $I(v_n)\to I(v)=J_{k,r}$. Hence, $J_{k,r}$ is achieved.

Proof. We will establish the result for radial functions in $C_{0}^{\infty }(\mathbb {R}^{N})$.

Let $v\in C_0^{\infty }(\mathbb {R}^{N})$ be radial and let $|x|=r$, $w(r)=v(x)$. Then, from $(\varphi _{8})$

\[ \Phi\big(b,w(b)\big)-\Phi\big(r,w(r)\big)=\int_r^{b} \left(\frac{d}{ds} \Phi\big(s,w(s)\big)\right)\,ds,\text{ for all }b>r>0. \]

Since $w\in C_0^{\infty }([0,\infty ))$, for $b$ large enough,

(5.18)\begin{equation} \begin{aligned} \Phi(r,w(r)) & ={-}\displaystyle\int_r^{\infty}\displaystyle\frac{\partial}{\partial s}\Phi(s, w(s))\,ds-\int_r^{\infty}\varphi(s,w(s))w(s)w^{\prime}(s)\,ds\\ & \leq\displaystyle\int_r^{\infty}\Big|\displaystyle\frac{\partial}{\partial s}\Phi(s, w(s))\Big|\, ds+\int_r^{\infty}\varphi(s,|w(s)|) |w(s)| |w'(s)|\,ds. \end{aligned} \end{equation}

From (2.1), (2.2) and the $\Delta _2$ condition (2.1), for all $s\geq 0$,

(5.19)\begin{equation} \begin{aligned} \varphi(s,|w(s)|) |w(s)| |w^{\prime}(s)| & \leq \tilde{\Phi}(s,\varphi(s,|w(s)|)|w(s)|)+\Phi(s,|w'(s)|)\\ & \leq \Phi(s,2|w(s)|)+\Phi(x,|w^{\prime}(s)|)\\ & \leq K\Phi(s,|w(s)|)+\Phi(s,|w^{\prime}(s)|). \end{aligned} \end{equation}

From $(\varphi _{9})$, for all $s\geq 0$,

(5.20)\begin{equation} \Big| \displaystyle\frac{\partial}{\partial s} \Phi(s,w(s)) \Big| \leq \kappa \Phi (s,w(s)). \end{equation}

Now using (5.19) and (5.20) in (5.18), we obtain

\[ \Phi\big(r,w(r)\big)\leq (\kappa+K+1)\int_r^{\infty}\big[\Phi\big(s,|w(s)|\big)+\Phi\big(s,|w'(s)|\big)\big] \,ds. \]

Hence, we can conclude that

\[ \Phi\big(r,w(r)\big)\leq \frac{(\kappa+K+1)}{r^{N-1}}\int_r^{+\infty}\big[\Phi\big(s,|w(s)|\big)+\Phi\big(s,|w'(s)|\big)\big] s^{N-1}\,ds. \]

From this, there is $C>0$ such that

\[ \Phi\big(x,v(x)\big)\leq \frac{C}{|x|^{N-1}}\int_{\mathbb{R}^{N}}\big[\Phi\big(x,|v|\big)+\Phi\big(x,|\nabla v|\big)\big]\,{\rm d}x. \]

Since $\Phi$ is an even function, $\Phi (x,v(x))=\Phi (x,|v(x)|)$ for all $x\in \mathbb {R}^{N}$ and so,

\[ \Phi\big(x,|v(x)|\big)\leq \frac{C}{|x|^{N-1}}\int_{\mathbb{R}^{N}}\big[\Phi\big(x,|v|\big)+\Phi\big(x,|\nabla v|\big)\big]\,{\rm d}x. \]

From this,

\[ |v(x)|\leq \Phi^{{-}1}\left(x,\left(\frac{C}{|x|^{N-1}}\int_{\mathbb{R}^{N}}\big[\Phi\big(x,|v|\big)+\Phi\big(x,|\nabla v|\big)\big]\,{\rm d}x\right)\right),\text{ for all }x\in \mathbb{R}^{N}\setminus\{0\}, \]

where $\Phi ^{-1}(x, \cdot )$ denotes the inverse function of $\Phi (x, \cdot )$ restricted to $[0,+\infty )$. Now the result follows from the density of $C_0^{\infty }(\mathbb {R}^{N})$ in $W^{1,\Phi }(\mathbb {R}^{N})$, because $\Phi$ satisfies the $\Delta _2$ condition.

Proof. By contradiction, suppose there exists a sequence $(r_n)$ such that $r_n\to \infty$, satisfying

(5.21)\begin{equation} J_{\infty,r_n}<(1-\displaystyle\frac{q}{\chi})\min\left\{\frac 1N(\frac{\alpha_N}{2^{N'}\alpha})^{N-1},\frac{1}{p} {S_p^{N/p}}\right\},\text{ for all }n\in\mathbb{N}. \end{equation}

First, we claim that $J_{\infty,r_n}$ is attained, for all $n\in \mathbb {N}$. In fact, for a fixed $n$, let $(v_k)\subset \mathcal {M}_{\infty,r_n}$ be a minimizing sequence for $J_{\infty,r_n}$, i.e., $(v_k)\subset W^{1,\Phi }_{0,H_{\infty }}(\Omega _{r_n})\setminus \{0\}$ and satisfies

\[ I'(v_k)v_k=0,\text{ and }I(v_k)\to J_{\infty,r_n},\text{ as }k\to\infty. \]

Note that

(5.22)\begin{equation} \begin{aligned} o_k(1)+J_{\infty,r_n} & =I(v_k)-\displaystyle\frac{1}{\chi}I'(v_k)v_k\\ & \geq \left(1-\displaystyle\frac{q}{\chi}\right)\displaystyle\int_{\Omega_{r_n}}\Phi(x,|\nabla v_k|)\,{\rm d}x\\ & \geq \frac{1}{N}\left(1-\frac{q}{\chi}\right)\int_{\Omega_N}|\nabla v_k|^{N}\,{\rm d}x. \end{aligned} \end{equation}

Using (5.21) in (5.22),

\[ \limsup_{k\to+\infty }||\nabla v_k||_{W^{1,N}(\Omega_N)}^{N} <\left(\frac{\alpha_N}{2^{N'}\alpha}\right)^{N-1}. \]

Now, we can repeat the same arguments employed in the proof of Lemma 5.3 to conclude that

\[ I'(v_k)\to 0\,\text{in}\, (W^{1,\Phi}_{0,H_{\infty}}(\Omega_{r_n}))'\text{ and }v_k\to v\text{ in } W^{1,\Phi}_{0,H_{\infty}}(\Omega_{r_n}) \]

where $v\in W^{1,\Phi }_{0,H_{\infty }}(\Omega _{r_n})$ is the limit of $(v_k)$ in $W^{1,\Phi }_{0,H_{\infty }}(\Omega _{r_n})$. Then,

\[ I(v_k)\to I(v)=J_{\infty,r_n}\,\text{and}\, I'(v_k)\to I'(v)=0. \]

Hence $I(v)=J_{\infty,r_n}$. Note that $v\neq 0$, since by Lemma 5.1 $J_{\infty,r_n}>0$. Therefore, $v\in \mathcal {M}_{\infty,r_n}$ and $J_{\infty,r_n}$ is attained at $v$.

Therefore, for each $n\in \mathbb {N}$, we can choose a sequence $\{u_n\}\subset W^{1,\Phi }_{0,H_\infty }(\Omega _{r_n})\setminus \{0\}$ satisfying

\[ I'(u_n)u_n=0\text{ and }I(u_n)=J_{\infty,r_n}. \]

Proceeding as in (5.22)

\[ \frac{1}{N}\left(1-\frac{q}{\chi}\right)\left(\frac{\alpha_N}{2^{N'}\alpha}\right)^{N-1}>J_{\infty,r_n}=I(u_n)-\frac{1}{\chi} I'(u_n)u_n\geq \frac{1}{N}\left(1-\frac{q}{\chi}\right)\int_{\Omega_N}|\nabla u_n|^{N}\,{\rm d}x \]

which implies

(5.23)\begin{equation} \limsup_{k\to+\infty }||\nabla u_n||_{W^{1,N}(\Omega_N)}^{N} <(\displaystyle\frac{\alpha_N}{2^{N'}\alpha})^{N-1}. \end{equation}

Let $\{\tilde {u}_n\}$ be a sequence given by

\[ \tilde{u}_n(x)=\begin{cases} u_n(x),\text{ if }x\in\Omega_{r_n},\\ 0,\text{ if } x\notin \Omega_{r_n}. \end{cases} \]

Observe that the following properties hold:

1. (1) $\{\tilde {u}_n\}\subset W^{1,\Phi }_{H_{\infty }}(\mathbb {R}^{N})$;

2. (2) $||\tilde {u}_n||_{W^{1,\Phi }_{H_{\infty }}(\mathbb {R}^{N})}=||u_n||_{W^{1,N}_{0,H_{\infty }}(\Omega _{r_n})}$;

3. (3) $\tilde {u}_n \rightharpoonup 0\,\text {in}\, W^{1,\Phi }_{H_{\infty }}(\mathbb {R}^{N})\text { because }\tilde {u}_n(x)\to 0 \text { a.e. in }\mathbb {R}^{N}$.

Therefore, we have

(5.24)\begin{equation} \displaystyle\int_{\mathbb{R}^{N}}\varphi(x,|\nabla\tilde{u}_n|)|\nabla\tilde{u}_n|^{2}\,{\rm d}x=\int_{\mathbb{R}^{N}}f(x,\tilde{u}_n)\tilde{u}_n\,{\rm d}x. \end{equation}

From Lemma 5.4, we deduce that the sequence $\{\tilde {u}_n\}$ satisfies

\[ |\tilde{u}_n|_\infty \to 0. \]

Using the fact that

\[ \lim_{t \to 0}\frac{f(x,t)t}{\varphi(x,t)t^{2}}=0, \quad \forall\, x \in \mathbb{R}^{N}, \]

and $(\varphi _3)'$, it follows that given $\epsilon < \frac {p}{q \Upsilon }$, where $\Upsilon$ was given in Lemma 2.3, there exists $n_0 \in \mathbb {N}$ such that

\begin{align*} \int_{\mathbb{R}^{N}}f(x,\tilde{u}_n)\tilde{u}_n\,{\rm d}x & \leq \epsilon \int_{\mathbb{R}^{N}}\varphi(x,|\tilde{u}_n|)|\tilde{u}_n|^{2}\,{\rm d}x\\ & \leq \varepsilon q \int_{\mathbb{R}^{N}} \Phi(x,|\tilde{u}_n|)\,\, \,{\rm d}x=\varepsilon q \int_{\Omega_{r_n}} \Phi(x,|\tilde{u}_n|)\,\, \,{\rm d}x, \, n \geq n_0. \end{align*}

Since $r_n \to +\infty$, without loss of generality we can assume that $r_n \geq 1$ for all $n \in \mathbb {N}$. Therefore, by the Poincaré inequality from Lemma 2.3,

(5.25)\begin{equation} \displaystyle\int_{\mathbb{R}^{N}}f(x,\tilde{u}_n)\tilde{u}_n\,{\rm d}x \leq \epsilon q \Upsilon \int_{\Omega_{r_n}} \Phi(x,|\nabla \tilde{u}_n|)\,\, \,{\rm d}x, \quad \forall n \geq n_0. \end{equation}

From (5.24), (5.25) and $(\varphi _3)$

\begin{align*} p\displaystyle\int_{\Omega_{r_n}}\Phi(|\nabla \tilde{u}_n|)\,{\rm d}x& =p\int_{\mathbb{R}^{N}}\Phi(x,|\nabla \tilde{u}_n|)\,{\rm d}x\\ & \leq \displaystyle\int_{\mathbb{R}^{N}}\varphi(x,|\nabla\tilde{u}_n|)|\nabla\tilde{u}_n|^{2}\,{\rm d}x\\ & \leq \epsilon q \Upsilon \int_{\Omega_{r_n}} \Phi(x,|\nabla \tilde{u}_n|)\,\, \,{\rm d}x,\quad \forall n \geq n_0. \end{align*}

As $\tilde {u}_n \not =0$ for all $n \in \mathbb {N}$, we get $p \leq \epsilon q \Upsilon$, which is absurd. Hence, the result follows.

Proof. Consider $u\in \mathcal {M}_{km,r}$ such that $I(u)=J_{km,r}$. Let $x=(\theta,\rho )$ be the polar coordinates of $x\in \mathbb {R}^{2}$. Then, $u=u(\theta,\rho,|y|),\,y\in \mathbb {R}^{N-2}$. Note that

\[ \Phi\big(\sqrt{\rho^{2}+|y|^{2}},|\nabla u|\big)=\Phi\left(\sqrt{\rho^{2}+|y|^{2}},\left(\frac{1}{\rho^{2}}u_\theta^{2}+u_\rho^{2}+|\nabla_y u|^{2}\right)^{1/2}\right). \]

Define

\[ v(\theta,\rho,|y|):=u\left(\frac{\theta}{m},\rho,|y|\right), \]

We observe that,

1. (i) $v\in W^{1,\Phi }_{0,H_k}(\Omega _r)$;

2. (ii) $\Phi (\sqrt {\rho ^{2}+|y|^{2}},|\nabla v|)=\Phi (\sqrt {\rho ^{2}+|y|^{2}},(\frac {1}{\rho ^{2}m^{2}}u_\theta ^{2}+u_\rho ^{2}+|\nabla _y u|^{2})^{1/2})$;

3. (iii) $\int _{\Omega _r}F(|z|,v)\,{\rm d}x\,{\rm d}y=\int _{\Omega _r}F(|z|,u)\,{\rm d}x\,{\rm d}y$, where $z=(x,y) \in \Omega _r$.

Proceeding similarly as in the proof of Lemma 5.2, there exists $t_0>0$ such that $t_0v\in \mathcal {M}_{k,r}$. For simplicity, let $v:=t_0 v$. Now, since $v\in \mathcal {M}_{k,r}$,

\[ J_{k,r}\leq I(v)=\int_{\Omega_r}\Phi(|z|,|\nabla v|)\,{\rm d}x\,{\rm d}y-\int_{\Omega_r} F(|z|,v)\,{\rm d}x\,{\rm d}y. \]

Using $(ii)-(iii)$,

(5.26)\begin{equation} J_{k,r}\leq \displaystyle\int_{\Omega_r}\Phi(\sqrt{\rho^{2}+|y|^{2}},\left(\displaystyle\frac{1}{\rho^{2}m^{2}}u_\theta^{2}+u_\rho^{2}+|\nabla_y u|^{2}\right)^{1/2})\,{\rm d}x\,{\rm d}y-\int_{\Omega_r}F(|x|,u)\,{\rm d}x\,{\rm d}y. \end{equation}

If $I(u)=J_{km,r}< J_{\infty,r}$, then $u\notin W^{1,\Phi }_{0,H_{\infty }}(\Omega _r)$, then $u^{2}_{\theta }$ is not identically zero. Therefore, using that $m>1$,

\begin{align*} & \displaystyle\int_{\Omega_r}\Phi\left(\sqrt{\rho^{2}+|y|^{2}},\left(\displaystyle\frac{1}{\rho^{2}m^{2}}u_\theta^{2}+u_\rho^{2}+|\nabla_y u|^{2}\right)^{1/2}\right)\,{\rm d}x\,{\rm d}y\\ & \quad<\int_{\Omega_r}\Phi\left(\sqrt{\rho^{2}+|y|^{2}},\left(\frac{1}{\rho^{2}}u_\theta^{2}+u_\rho^{2}+|\nabla_y u|^{2}\right)^{1/2}\right)\,{\rm d}x\,{\rm d}y, \end{align*}

which together with (5.26) implies $J_{k,r}< I(u)=J_{km,r}$ and the proof is complete.