A class of Schrödinger elliptic equations involving supercritical exponential growth

Abstract

This paper studies the existence of nontrivial solutions to the following class of Schrödinger equations:

$$ \textstyle\begin{cases} -\operatorname{div}(w(x)\nabla u) = f(x,u),&\ x \in B_{1}(0), \\ u = 0,&\ x \in \partial B_{1}(0), \end{cases} $$

where $w(x)= (\ln (1/|x|) )^{\beta}$ for some $\beta \in [0,1)$, the nonlinearity $f(x,s)$ behaves like ${\exp} (|s|^{\frac{2}{1-\beta}+h(|x|)} )$, and h is a continuous radial function such that $h(r)$ can be unbounded as r tends to 1. Our approach is based on a new Trudinger–Moser-type inequality for weighted Sobolev spaces and variational methods.

1 Introduction

Let consider the following Schrödinger equation:

$$ \textstyle\begin{cases} -\Delta u = f(x,u),& x \in \Omega , \\ u = 0,& x \in \partial \Omega , \end{cases} $$

(1.1)

where Ω is a bounded smooth domain in ${\mathbb{R}}^N$. In the case $N\geq 3$, some pioneering works developed by Brézis [7], Brézis & Nirenberg [8], Bartsh & Willem [6], and Capozzi, Fortunato & Palmieri [14] considered the assumption $|f(x,u)|\leq c(1+|u|^{q-1} )$, with $1< q\leq 2^{*}=2N/(N-2)$. The above growth of the nonlinearity f is related to the Sobolev embedding $H_{0}^{1}(\Omega )\subset L^{q}(\Omega )$ for $1\leq q\leq 2^{*}$. In the limiting case $N =2$, one has $2^{*}=+\infty$, that is, $H_{0}^{1}(\Omega )\subset L^{q}(\Omega )$ for $q\geq 1$, in particular, the nonlinear function f in (1.1) may have arbitrary polynomial growth. Also, some examples show that $H_{0}^{1}(\Omega )\not \subset L^{\infty}(\Omega )$. An important result found independently by Yudovich [37], Pohozaev [28], and Trudinger [35] showed that the maximal growth of the nonlinearity in the bivariate case is of exponential type. More precisely, it was stated that

$$ e^{\alpha u^{2}}\in L^{1}(\Omega ), \quad \text{for all } u\in H^{1}_{0}( \Omega ) \text{ and } \alpha >0. $$

(1.2)

Furthermore, Moser [26] stated the existence of a positive constant $C=C(\alpha ,\Omega )$ such that

$$ \mathop{\sup_{u\in H^{1}_{0}(\Omega ), }}_{\| \nabla u\|_{2}\leq 1} \int _{\Omega}e^{\alpha u^{2}}\,dx \textstyle\begin{cases} \leq C, & \alpha \leq 4\pi , \\ =+\infty , & \alpha >4\pi . \end{cases} $$

(1.3)

Estimates (1.2) and (1.3) from now on be referred to as Trudinger–Moser inequalities. The above results motivate us to say that the function f has subcritical exponential growth if

$$ \lim_{s\to +\infty}\frac {f(x,s)}{e^{\alpha s^{2}}}=0, \quad \text{for all } \alpha > 0, $$

and critical exponential growth if there exists $\alpha _{0}>0$ such that

$$ \lim_{s\to +\infty}\frac {f(x,s)}{e^{\alpha s^{2}}}= \textstyle\begin{cases} 0, & \alpha < \alpha _{0}, \\ +\infty , & \alpha >\alpha _{0}. \end{cases} $$

(1.4)

Equations of the type (1.1) considering nonlinearities involving subcritical and critical exponential growth were treated by Adimurthi [1], Adimurthi–Yadava [2], de Figueiredo, Miyagaki, and Ruf [18] (see also [1–4, 11, 13, 23, 27, 31]), and some results on Hamiltonian systems involving the above-mentioned growth can be found in [16, 17, 20, 24, 29, 33]. We shall write $g_{1}(s)\prec g_{2}(s)$ if there exist positive constants k and $s_{0}$ such that $g_{1}(s)\leq g_{2}(ks)$ for $s\geq s_{0}$. Additionally, we shall say that $g_{1}$ and $g_{2}$ are equivalent and write $g_{1}(s)\sim g_{2}(s)$ if $g_{1}(s)\prec g_{2}(s)$ and $g_{2}(s)\prec g_{1}(s)$. Therefore, f possesses critical exponential growth if only if $f(x,s)=g(s)$ with $g(s)\sim e^{|s|^{2}}$.

Several extensions of the Trudinger–Moser inequalities were obtained considering weighted Sobolev spaces, weighted Lebesgue measures, or Lorentz–Sobolev spaces (see [3–5, 13, 15, 19, 24, 25, 34] among others). In the above-mentioned papers, the growth of the nonlinearity is of the type $f(x,s)= Q(x)g(s)$ where $g(s)\sim e^{\lvert s\lvert ^{p}}$ with $p=2$ on Sobolev spaces and $p>1$ on Lorentz–Sobolev spaces and for some weight $Q(x)$. More precisely, on Lorentz–Sobolev spaces, Brezis and Wainger [9] have shown the following: Let Ω be a bounded domain in ${\mathbb{R}}^2$ and $s>1$. Then, $e^{\alpha |u|^{\frac{s}{s-1}}}$ belongs to $L^{1}(\Omega )$ for all $u\in W_{0}^{1}L^{2,s}(\Omega )$ and $\alpha >0$. Furthermore, Alvino [5] obtained the following refinement of (1.3): there exists a positive constant $C=C(\Omega ,s,\alpha )$ such that

$$ \mathop{\sup_{u\in W^{1}_{0}L^{2,s}(\Omega ), }}_{\|\nabla u\|_{2,s} \leq 1} \int _{\Omega}e^{\alpha |u|^{\frac{s}{s-1}}}\,dx \textstyle\begin{cases} \leq C, & \alpha \leq (4\pi )^{s/(s-1)}, \\ =+\infty , & \alpha >(4\pi )^{s/(s-1)}. \end{cases} $$

(1.5)

In order to extend equations (1.1), we will study Schrödinger equations involving a diffusion operator (see [10, 12, 32, 38, 39] among others). Let $B_{1}$ be the unit ball centered at the origin in ${\mathbb{R}}^2$ and $H^{1}_{0,{\mathrm{rad}}}(B_{1},w)$ be the subspace of the radially symmetric functions in the closure of $\mathcal{C}^{\infty}_{0}(B_{1})$ with respect to the norm

$$ \Vert u \Vert := \Vert \nabla u \Vert _{L^{2}(B_{1},w)}= \biggl( \int _{B_{1}}w(x) \vert \nabla u \vert ^{2}\,dx \biggr)^{\frac{1}{2}}. $$

(1.6)

In particular, if $w\equiv 1$, we denote the above space by $H^{1}_{0,{\mathrm{rad}}}(B_{1})$. Trudinger–Moser-type inequalities for radial Sobolev spaces with logarithmic weights were considered by Calanchi and Ruf in [11]. More precisely, the above-mentioned authors used the weight $w(x)= (\log{1}/{|x|} )^{\beta}$ for some fixed $0\leq \beta <1$, this logarithmic weight will be used in the rest of this article.

Proposition 1.1

(Calanchi–Ruf, [11])

Suppose that $w(x)= (\log{1}/{|x|} )^{\beta}$ and $0\leq \beta <1$. Then,

$$ \int _{B_{1}}e^{\alpha |u|^{\frac{2}{1-\beta}}}\,dx< +\infty ,\quad \textit{for all } u \in H^{1}_{0,{\mathrm{rad}}}(B_{1},w) \textit{ and } \alpha >0. $$

Furthermore, setting $\alpha _{\beta}^{*}=2 [2\pi (1-\beta ) ]^{\frac{1}{1-\beta}}$, there exists a positive constant $C=C(\alpha ,\beta )$ such that

$$ \mathop{\sup_{u\in H^{1}_{0, {\mathrm{rad}}}(B_{1}, w), }}_{\| u\|\leq 1} \int _{B_{1}}e^{\alpha |u|^{\frac{2}{1-\beta}}}\,dx \textstyle\begin{cases} \leq C, & \alpha \leq \alpha _{\beta}^{*}, \\ =+\infty , & \alpha >\alpha _{\beta}^{*}. \end{cases} $$

In order to establish a Trudinger–Moser inequality proved by Ngô and Nguyen [27], we consider a continuous radial function $h:[0,1)\to \mathbb{R}$ such that

($h_{1}$):

$h(0)=0$ and $h(r)>0$ for $r\in (0,1)$;

($h_{2}$):

there exists $c>0$ and $\gamma >2$ such that

$$ h(r)\leq \frac {c}{(-\ln r)^{\gamma}} \quad \text{near } 0. $$

Proposition 1.2

(Ngô–Nguyen, [27])

Suppose that h satisfies $(h_{1})$ and $(h_{2})$. Then, there exists a positive constant $C=C(\alpha , h)$ such that

$$ \mathop{\sup_{u\in H^{1}_{0, {\mathrm{rad}}}(B_{1}), }}_{ \Vert \nabla u \Vert _{2} \leq 1} \int _{B_{1}}\exp \bigl(\alpha \vert u \vert ^{2+h( \vert x \vert )} \bigr)\,dx \textstyle\begin{cases} \leq C, & \alpha \leq 4\pi , \\ =+\infty , & \alpha >4\pi . \end{cases} $$

Next we establish a new version of the Trudinger–Moser inequality which will be used throughout this paper.

Theorem 1.3

Suppose h satisfies $(h_{1})$ and $(h_{2})$ and $w(x)= (\log{1}/{|x|} )^{\beta}$ for some $\beta \in [0,1)$. Then, there exists a positive constant $C=C(\alpha ,\beta ,h)$ such that

$$ \sup_{ \Vert u \Vert \leq 1} \int _{B_{1}}\exp \bigl(\alpha \vert u \vert ^{ \frac{2}{1-\beta}+h( \vert x \vert )} \bigr)\,dx\leq C. $$

(1.7)

If $\alpha >\alpha _{\beta}^{*}$, then

$$ \sup_{ \Vert u \Vert \leq 1} \int _{B_{1}}\exp \bigl(\alpha \vert u \vert ^{ \frac{2}{1-\beta}+h( \vert x \vert )} \bigr)\,dx=+\infty . $$

(1.8)

The proof of Theorem 1.3 will be presented in the next section. In this work, we are interested in finding nontrivial weak solutions for the following class of Schrödinger equations:

$$ \textstyle\begin{cases} -\operatorname{div}(w(x)\nabla u) = f(x,u),& x \in B_{1}, \\ u = 0,& x \in \partial B_{1}, \end{cases} $$

(1.9)

where the growth of the nonlinearity of f is motivated by the Trudinger–Moser inequality given by Theorem 1.3. More precisely, we assume the following conditions on the nonlinearity f:

$(H_{1})$:

$f:B_1\times \mathbb{R}\to \mathbb{R}$ is a continuous and radially symmetric in the first variable function, that is, $f(x,s)=f(y,s)$ for $|x|=|y|$. Moreover, $f(x,s)=0$ for all $x\in B_{1}$ and $s\leq 0$.

$(H_{2})$:

There exists a constant $\mu >2$ such that

$$ 0< \mu F(x,s)\leq sf(x,s), \quad \text{for all } x\in B_{1} \text{ and } s> 0, $$

where $F(x,s)=\int _{0}^{s} f(x,t)\,dt$.

$(H_{3})$:

There exists a constant $M>0$ such that

$$ 0< F(x,s)\leq M f(x,s), \quad \text{for all } s>0. $$

$(H_{4})$:

There holds

$$ \limsup_{s \to 0}\frac {2F(x,s)}{s^{2}}< \lambda _{1},\quad \text{uniformly in } x\in B_{1}, $$

where $\lambda _{1}$ is the first eigenvalue associated to $(-\operatorname{div}(w(x)\nabla u), H_{0,{\mathrm{rad}}}^{1}(B_{1},w) )$.

$(H_{5})$:

There exists a constant $\alpha _{0}>0$ such that

$$ \lim_{s \to \infty} \frac {f(x,s)}{\exp (\alpha \vert u \vert ^{\frac{2}{1-\beta}+h( \vert x \vert )} )} = \textstyle\begin{cases} 0, & \alpha > \alpha _{0}, \\ +\infty , & \alpha < \alpha _{0}, \end{cases} $$

$(H_{6})$:

There exist constants $p> 2$ and $C_{p}>0$ such that

$$ f(x,s)\geq C_{p} s^{p-1}, \quad \text{for all } s \geq 0, $$

where

$$ C_{p}> \frac {(p-2)^{(p-2)/2}S_{p}^{p}}{p^{(p-2)/2}} \biggl( \frac {\alpha _{0}}{\alpha ^{*}_{\beta}} \biggr)^{(1-\beta )(p-2)/2} $$

and

$$ S_{p}:= \sup_{0\neq u\in H^{1}_{0, {\mathrm{rad}}}(B_{1}, w) } \frac { ( \int _{B_{1}}w(x) \vert \nabla u \vert ^{2}\,dx )^{1/2}}{ ( \int _{B_{1}} \vert u \vert ^{p}\,dx )^{1/p}}. $$

Throughout, we denote the space $E:=H^{1}_{0,\mathrm{rad}}(B_{1},w)$ endowed with the inner product

$$ \langle u, v\rangle _{E}= \int _{B_{1}}w(x)\nabla u \nabla v \,dx, \quad \text{for all } u, v\in E, $$

to which corresponds the norm

$$ \Vert u \Vert = \biggl( \int _{B_{1}}w(x) \vert \nabla u \vert ^{2}\,dx \biggr)^{1/2}. $$

Also, we denote by $E^{*}$ the dual space of E with its usual norm. We say that $u\in E$ is a weak solution of (1.9) if

$$ \int _{B_{1}}w(x)\nabla u \nabla \phi \,dx= \int _{B_{1}} f(x,u)\phi \,dx,\quad \text{for all } \phi \in E. $$

(1.10)

Under the above assumptions on f, we consider the Euler–Lagrange functional $J:E\to \mathbb{R}$ defined by

$$ J(u)=\frac {1}{2} \int _{B_{1}}w(x) \vert \nabla u \vert ^{2}\,dx- \int _{B_{1}} {F}(x,u)\,dx, \quad \text{for all } u \in E. $$

Furthermore, using standard arguments (see [21]), J belongs to ${\mathcal{C}}^1(E,\mathbb{R})$ and

$$ {J}'(u)\phi = \int _{B_{1}}w(x)\nabla u \nabla \phi \,dx- \int _{B_{1}} {f}(x,u)\phi \,dx, \quad \text{for all } u, \phi \in E. $$

Next, we present our existence result for the problem (1.9).

Theorem 1.4

Suppose that f satisfies $(H_{1})$–$(H_{6})$. Then, the problem (1.9) possesses a nontrivial weak solution.

Notice that the class of Schrödinger equations (1.9) represents a natural extension of the equation (1.1). Under assumption $(H_{5})$, the nonlinearity f behaves like ${\exp}((\alpha +h(|x|))|s|^{\frac{2}{1-\beta}})$ as s tends to infinity. Moreover, if $\beta =0$, we have that $w\equiv 1$ and the equation (1.9) is reduced to problem (1.1); the case with $\beta =0$ and $h(x)=|x|^{a}$ for some $a>0$ was studied in [27], and treated in many works considering $h=0$ (see [1, 2, 18] among others). Additionally, we observe that $(h_{1})$ and $(h_{2})$ are conditions near the origin, in particular, h can tend to infinity for values of $|x|$ close to 1. Also, if β is close to 1, the power of $|s|^{p}$ where $p=2/(1-\beta )$ can be sufficiently large. The above properties motivate us to say that f possesses supercritical exponential growth and represents an extension of other previously studied works. Finally, note that the class of functions which satisfies the conditions $(H_{1})$–$(H_{6})$ is not empty, for instance, consider the following function $f:B_1\times \mathbb{R}\to \mathbb{R}$ defined by

$$ f(x,s)= \textstyle\begin{cases} As^{p-1}+(p+ \vert x \vert ^{\eta})s^{p-1+ \vert x \vert ^{\eta}}e^{s^{p+ \vert x \vert ^{\eta}}},& s \geq 0, \\ 0,& s< 0. \end{cases} $$

for some positive constants η, $p=2/(1-\beta )$, and A sufficiently large.

2 Preliminaries

The space $H^{1}_{0,{\mathrm{rad}}}(B_{1},w)$ where $w(x)= (\log {1}/{|x|} )^{\beta}$ for some $0\leq \beta <1$, endowed with the norm given by (1.6), is a separable Banach space (see [22, Theorem 3.9]). Next, we present a compactness result.

Lemma 2.1

The embedding $H^{1}_{0,{\mathrm{rad}}}(B_{1},w)\hookrightarrow L^{p}(B_{1})$ is continuous and compact for $1\leq p<\infty$.

Proof

From the Cauchy–Schwarz inequality, we have

$$ \int _{B_{1}} \vert \nabla u \vert \,dx \leq \biggl( \int _{B_{1}}w(x) \vert \nabla u \vert ^{2}\,dx \biggr)^{1/2}\cdot \biggl( \int _{B_{1}} w(x)^{-1}\,dx \biggr)^{1/2}. $$

Using the change of variable $|x|=e^{-s}$, we get

$$ \frac {1}{2\pi} \int _{B_{1}} w(x)^{-1}\,dx= \int ^{+\infty}_{0} e^{-2s}s^{- \gamma}\,ds= \int ^{1}_{0} e^{-2s}s^{-\gamma}\,ds+ \int ^{+\infty}_{1} e^{-2s}s^{- \gamma}\,ds. $$

Note that

$$ \int _{0}^{1}e^{-2s}s^{-\gamma}\,ds \leq \int _{0}^{1} s^{-\gamma}\,ds = \frac {1}{1-\gamma} $$

and

$$ \int _{1}^{+\infty}e^{-2s}s^{-\gamma}\,ds \leq \int _{1}^{+\infty} e^{-2s}\,ds = \frac {e^{-2}}{2}. $$

Therefore, we can find a positive constant C such that

$$ \Vert \nabla u \Vert _{1}\leq C \biggl( \int _{B_{1}} \vert \nabla u \vert ^{2} w(x)\,dx \biggr)^{1/2}. $$

Thus, $H^{1}_{0}(B_{1},w)\hookrightarrow W_{0}^{1,1}(B_{1})$ continuously, which implies the continuous and compact embedding

$$ H^{1}_{0}(B_{1},w)\hookrightarrow L^{p}(B_{1}),\quad \text{for all } p \geq 1. $$

 □

Lemma 2.2

([11])

Let u be a function in $H^{1}_{0}(B_{1},w)$. Then,

$$ \bigl\vert u(x) \bigr\vert \leq \frac {(-\ln \vert x \vert )^{\frac{1-\beta}{2}}}{\sqrt{2\pi (1-\beta )}}\cdot \Vert u \Vert ,\quad \textit{for all } x\in B_{1}. $$

2.1 Proof of Theorem 1.3

Proof

To prove the first statement of the theorem, it is sufficient to consider $\alpha =\alpha _{\beta}^{*}$. From Lemma 2.2, for each $u\in E$ with $\| u\|\leq 1$, we have

$$ \alpha ^{*}_{\beta } \bigl\vert u(r) \bigr\vert ^{2/(1-\beta )}\leq -2\ln r,\quad \text{for all } 0< r< 1, $$

(2.1)

where $r=|x|$. Setting $r_{1}:=e^{-\alpha ^{*}_{\beta}/2}$, we have

$$ \bigl\vert u(r) \bigr\vert \leq 1, \quad \text{for all } r\geq r_{1}. $$

(2.2)

Thus,

$$ \int _{B_{1}\backslash{B_{r_{1}}}}\exp \bigl(\alpha _{\beta}^{*} \vert u \vert ^{ \frac{2}{1-\beta}+h( \vert x \vert )} \bigr)\,dx\leq \int _{B_{1}\backslash{B_{r_{1}}}}{ \rm exp} \bigl(\alpha _{\beta}^{*} \bigr)\,dx\leq \exp \bigl(\alpha _{ \beta}^{*}\bigr) \vert B_{1} \vert . $$

(2.3)

On the other hand, by (2.1), we can write

$$\begin{aligned} &\int _{B_{r_{1}}}\exp \bigl(\alpha _{\beta}^{*} \vert u \vert ^{ \frac{2}{1-\beta}+h( \vert x \vert )} \bigr)\,dx \\ &\quad \leq \int _{B_{r_{1}}}\exp \bigl(\alpha _{\beta}^{*} \vert u \vert ^{\frac{2}{1-\beta}} \vert u \vert ^{h( \vert x \vert )} \bigr)\,dx \\ &\quad \leq \int _{B_{r_{1}}}\exp \biggl(\alpha _{\beta}^{*} \vert u \vert ^{ \frac{2}{1-\beta}} \biggl(\frac {-2\ln r}{\alpha _{\beta}^{*}} \biggr)^{ \frac{(1-\beta )}{2}h( \vert x \vert )} \biggr)\,dx \\ &\quad \leq \int _{B_{r_{1}}}\exp \bigl(\alpha _{\beta}^{*} \vert u \vert ^{ \frac{2}{1-\beta}} \bigr) (\exp \biggl(\alpha _{\beta}^{*} \vert u \vert ^{ \frac{2}{1-\beta}} \biggl( \biggl( \frac {-2\ln r}{\alpha _{\beta}^{*}} \biggr)^{\frac{(1-\beta )}{2}h( \vert x \vert )}-1 \biggr)-1 \biggr)\,dx \\ & \qquad {}+ \int _{B_{r_{1}}}\exp \bigl(\alpha _{\beta}^{*} \vert u \vert ^{ \frac{2}{1-\beta}} \bigr)\,dx. \end{aligned}$$

Note that $-2\ln r/\alpha _{\beta}^{*}\geq 1$ for $0< r\leq r_{1}$. By $(h_{2})$, there exist $c>0$ and $0< r_{2}< r_{1}$ such that

$$ h\bigl( \vert x \vert \bigr)\leq \frac {c}{(-\ln r)^{\gamma}}, \quad \text{for all } 0< r< r_{2}. $$

(2.4)

Using (2.1) and (2.4), we have

$$ \begin{aligned} &\exp (\alpha _{\beta}^{*} \vert u \vert ^{\frac{2}{1-\beta}} \biggl( \biggl(\frac {-2\ln r}{\alpha _{\beta}^{*}} \biggr)^{ \frac{(1-\beta )}{2}h( \vert x \vert )}-1 \biggr)-1 \\ &\quad \leq \exp (-2\ln r \biggl( \biggl( \frac {-2\ln r}{\alpha _{\beta}^{*}} \biggr)^{ \frac{c(1-\beta )}{2(-\ln r)^{\gamma}}}-1 \biggr)-1:=k(r). \end{aligned} $$

Also, as $r\to 0^{+}$, one has

$$\begin{aligned} \biggl(\frac {-2\ln r}{\alpha _{\beta}^{*}} \biggr)^{ \frac{c(1-\beta )}{2(-\ln r)^{\gamma}}}&= \exp \biggl[{ \frac {c(1-\beta )}{2(-\ln r)^{\gamma}}}\ln \biggl( \frac {-2\ln r}{\alpha _{\beta}^{*}} \biggr) \biggr] \\ &= 1+ {\frac {c(1-\beta )}{2(-\ln r)^{\gamma}}}\ln \biggl( \frac {-2\ln r}{\alpha _{\beta}^{*}} \biggr)+o \biggl( \frac {1}{(-\ln r)^{\gamma}} \ln \biggl( \frac {-2\ln r}{\alpha _{\beta}^{*}} \biggr) \biggr). \end{aligned}$$

Therefore, as r is close to zero, we have

$$ \begin{aligned} -2\ln r \biggl( \biggl(\frac {-2\ln r}{\alpha _{\beta}^{*}} \biggr)^{\frac{c(1-\beta )}{2(-\ln r)^{\gamma}}}-1 \biggr) ={}& \frac {c(1-\beta )}{(-\ln r)^{\gamma -1}}\ln \biggl( \frac {-2\ln r}{\alpha _{\beta}^{*}} \biggr) \\ &{} +o \biggl(\frac {1}{(-\ln r)^{\gamma -1}} \ln \biggl( \frac {-2\ln r}{\alpha _{\beta}^{*}} \biggr) \biggr). \end{aligned} $$

Since $\gamma >2$, we obtain

$$ \frac {c(1-\beta )}{(-\ln r)^{\gamma -1}}\ln \biggl( \frac {-2\ln r}{\alpha _{\beta}^{*}} \biggr)\to 0, \quad \text{as } r\to 0^{+}. $$

(2.5)

Consequently,

$$\begin{aligned} k(r)&= \exp \biggl[\frac {c(1-\beta )}{(-\ln r)^{\gamma -1}} \ln \biggl(\frac {-2\ln r}{\alpha _{\beta}^{*}} \biggr)+o \biggl( \frac {1}{(-\ln r)^{\gamma -1}}\ln \biggl( \frac {-2\ln r}{\alpha _{\beta}^{*}} \biggr) \biggr) \biggr]-1 \\ &= \frac {c(1-\beta )}{(-\ln r)^{\gamma -1}}\ln \biggl( \frac {-2\ln r}{\alpha _{\beta}^{*}} \biggr)+o \biggl( \frac {1}{(-\ln r)^{\gamma -1}}\ln \biggl( \frac {-2\ln r}{\alpha _{\beta}^{*}} \biggr) \biggr). \end{aligned}$$

Set

$$ l(r)=\frac {c(1-\beta )}{(-\ln r)^{\gamma -1}}\ln \biggl( \frac {-2\ln r}{\alpha _{\beta}^{*}} \biggr). $$

In particular, k and l are continuous and positive in $(0,r_{2})$. Moreover, there exist $C>0$ and $0< r_{3}< r_{2}$ such that

$$ k(r)\leq Cl(r), \quad \text{for all } 0< r\leq r_{3}. $$

(2.6)

Therefore, by (2.1), (2.6), and the definition of $k(r)$, we have

$$\begin{aligned} &\int _{B_{r_{3}}} \exp \bigl(\alpha _{\beta}^{*} \vert u \vert ^{ \frac{2}{1-\beta}+h( \vert x \vert )} \bigr)\,dx \\ & \quad \leq \int _{B_{r_{3}}}\exp \bigl(\alpha _{\beta}^{*} \vert u \vert ^{ \frac{2}{1-\beta}} \bigr) k\bigl( \vert x \vert \bigr)\,dx+ \int _{B_{r_{3}}} \exp \bigl( \alpha _{\beta}^{*} \vert u \vert ^{\frac{2}{1-\beta}} \bigr)\,dx \\ &\quad \leq C_{1} \int _{B_{r_{3}}} \frac {1}{ \vert x \vert ^{2}} \ln \biggl( \frac {-2\ln \vert x \vert }{\alpha _{\beta}^{*}} \biggr){ \frac {c(1-\beta )}{(-\ln \vert x \vert )^{\gamma -1}}}\,dx+C_{2} \\ &\quad =2\pi C_{1}c(1-\beta ) \int _{0}^{\rho _{3}} \frac {1}{r} \ln \biggl(- \frac {2\ln r}{\alpha _{\beta}^{*}} \biggr) \frac {1}{(-\ln r)^{\gamma -1}}\,dr+C_{2} \\ &\quad = 2\pi C_{1}c(1-\beta ) \int _{-\ln \rho _{3}}^{+\infty} \ln \biggl( \frac {2s}{\alpha _{\beta}^{*}} \biggr) \frac {1}{ s^{\gamma -1}}\,ds+C_{2}, \end{aligned}$$

for some positive constants $C_{1}$ and $C_{2}$. Using the fact that $\gamma >2$, we have

$$ \int _{B_{r_{3}}} \exp \bigl(\alpha _{\beta}^{*} \vert u \vert ^{ \frac{2}{1-\beta}+h( \vert x \vert )} \bigr)\,dx\leq C_{2}. $$

(2.7)

On the other hand, using (2.1), we have

$$ 1\leq \bigl\vert u(r) \bigr\vert \leq \biggl(-\frac {2\ln r_{3}}{\alpha _{\beta}^{*}} \biggr)^{ \frac{1-\beta}{2}}, \quad \text{for all } r_{3}\leq r\leq r_{1} $$

Combining the above inequality with the boundedness of h in $B_{r_{1}}\backslash{B_{r_{3}}}$, we get

$$ \int _{B_{r_{1}}\backslash{B_{r_{3}}}}\exp \bigl(\alpha _{\beta}^{*} \vert u \vert ^{ \frac{2}{1-\beta}+h( \vert x \vert )} \bigr)\,dx\leq \vert B_{r_{1}} \vert M. $$

(2.8)

Consequently, from (2.3), (2.7), and (2.8), we obtain

$$ \int _{B_{1}}\exp \bigl(\alpha _{\beta}^{*} \vert u \vert ^{\frac{2}{1-\beta}+h( \vert x \vert )} \bigr)\,dx\leq C, $$

which implies the first assertion of the theorem. In order to prove the sharpness, we consider the following sequence given in [15]:

$$ \psi _{k}(x)= \biggl(\frac {1}{\alpha _{\beta}^{*}} \biggr)^{(1-\beta )/2} \textstyle\begin{cases} { k^{\frac{2}{1-\beta}}} \ln (\frac {1}{ \vert x \vert ^{2}} )^{1-\beta},&0 \leq \vert x \vert \leq e^{-k/2}, \\ k^{\frac{1-\beta}{2}},& e^{-k/2}\leq \vert x \vert \leq 1. \end{cases} $$

Then, $\|\psi _{k}\|=1$ for all $k\in \mathbb{N}$. Moreover, for $\alpha >\alpha ^{*}_{\beta}$, we have

$$ \int _{B_{1}}\exp \bigl(\alpha \vert \psi _{k} \vert ^{\frac{2}{1-\beta}+h( \vert x \vert } \bigr)\,dx\geq \int _{B_{1}}\exp \bigl(\alpha \vert \psi _{k} \vert ^{ \frac{2}{1-\beta}} \bigr)\,dx\geq \int _{e^{-k}/2}^{1} {\exp} \biggl( \frac {\alpha}{\alpha _{\beta}^{*}} k \biggr)r \,dr. $$

Then,

$$ \int _{B_{1}}\exp \bigl(\bigl(\alpha +h\bigl( \vert x \vert \bigr) \bigr) \vert \psi _{k} \vert ^{2/(1- \beta )} \bigr) \,dx\geq e^{k (\frac{ \alpha}{\alpha _{\beta}^{*}}-1 )} \bigl(e^{k}-1 \bigr)\to + \infty ,\quad \text{as } k\to \infty , $$

and the proof is complete. □

Corollary 2.3

Let $\eta >0$. Then,

$$ \int _{B_{1}}\exp \bigl(\alpha \vert \psi _{k} \vert ^{\frac{2}{1-\beta}+ \vert x \vert ^{ \eta}} \bigr)\,dx< +\infty ,\quad \textit{for all } u\in H^{1}_{0,{\mathrm{rad}}}(B_{1},w) \textit{ and } \alpha >0. $$

(2.9)

Furthermore, if $\alpha \leq \alpha ^{*}_{\beta}$, there exists a positive constant C such that

$$ \int _{B_{1}}\exp \bigl(\alpha \vert \psi _{k} \vert ^{\frac{2}{1-\beta}+ \vert x \vert ^{ \eta}} \bigr)\,dx\leq C. $$

(2.10)

If $\alpha >\alpha _{\beta}^{*}$, then

$$ \sup_{ \Vert u \Vert \leq 1} \int _{B_{1}}\exp \bigl(\alpha \vert \psi _{k} \vert ^{ \frac{2}{1-\beta}+ \vert x \vert ^{\eta}} \bigr)\,dx=+\infty . $$

(2.11)

As it was observed in [27], the statements of Theorem 1.3 and its corollary are no longer true if one considers the space of nonradial functions $H_{0}^{1}(B_{1},w)$. Additionally, using similar arguments as in Theorem 1.3, we can prove the natural extension of (1.2), that is, if $\alpha >0$ and $u\in H^{1}_{0,{\mathrm{rad}}}(B_{1},w)$, then

$$ \int _{B_{1}}\exp \bigl(\alpha \vert u \vert ^{\frac{2}{1-\beta}+h( \vert x \vert )} \bigr)\,dx< +\infty . $$

(2.12)

3 The geometry of the mountain pass theorem

This section is devoted to showing that the functional J satisfies the geometry of the mountain pass theorem.

Lemma 3.1

Suppose that $(H_{1})$, $(H_{4})$, and $(H_{5})$ hold. Then, there exist $\sigma ,\rho >0$ such that

$$ {J}(u)\geq \sigma , \quad \textit{for all } u \in E \textit{ with } \Vert u \Vert = \rho . $$

Proof

Consider $q>2$ and $0<\epsilon <{\lambda _{1}}/{2}$. From $(H_{1})$ and $(H_{4})$, we can find $c>0$ such that

$$|F(x,s)|\le ϵ|s{|}^2+c|s{|}^{q}exp(2{\alpha }_0|u{|}^{\frac{2}{1-\beta }+h(|x|)}),\text{for all }(x,s)\in B_1\times \mathbb{R}.$$

Integrating on $B_{1}$ and applying the Cauchy–Schwarz inequality, we obtain

$$ \int _{B_{1}}{F}(x,u)\,dx \leq \epsilon \Vert u \Vert _{2}^{2}+ c \Vert u \Vert _{2q}^{q} \biggl( \int _{B_{1}}\exp \bigl(4\alpha _{0} \vert u \vert ^{\frac{2}{1-\beta}+h( \vert x \vert )} \bigr)\,dx \biggr)^{1/2}. $$

(3.1)

Let $h_{0}=\max_{0\leq r\leq r_{1}}h(r)$ where $r_{1}$ is given by (2.2). By Theorem 1.3, we have

$$ \begin{aligned} \int _{B_{r_{1}}} \exp \bigl(4\alpha _{0} \vert u \vert ^{ \frac{2}{1-\beta}+h( \vert x \vert )} \bigr)\,dr &\leq \int _{B_{r_{1}}} \exp \biggl[4\alpha _{0} \Vert u \Vert ^{\frac{2}{1-\beta}+h( \vert x \vert )} \biggl( \frac{ \vert u \vert }{ \Vert u \Vert } \biggr)^{\frac{2}{1-\beta}+h( \vert x \vert )} \biggr] \,dx \\ &\leq \int _{B_{r_{1}}} \exp \biggl[4\alpha _{0} \Vert u \Vert ^{ \frac{2}{1-\beta}+h_{0}} \biggl(\frac{ \vert u \vert }{ \Vert u \Vert } \biggr)^{ \frac{2}{1-\beta}+h( \vert x \vert )} \biggr] \,dx \\ &\leq C_{1}, \end{aligned} $$

(3.2)

provided that $\|u\|\leq \rho _{0}$ for some $0<\rho _{0}<1$ such that $4\alpha _{0}\rho _{0}^{\frac{2}{1-\beta}+h_{0}}<\alpha _{\beta}^{*}$. Using (2.2), we have

$$ \int _{B_{1}\backslash{B_{r_{1}}}}\exp \bigl(4\alpha _{0} \vert u \vert ^{ \frac{2}{1-\beta}+h( \vert x \vert )} \bigr)\,dx\leq \int _{B_{1}\backslash{B_{r_{1}}}}{ \rm exp} (4\alpha _{0} )\,dx= C_{2}. $$

(3.3)

Replacing (3.2) and (3.3) in (3.1), we get some $c>0$ such that

$$ \int _{B_{1}}{F}(x,u)\,dx \leq \frac {\epsilon}{\lambda _{1}} \Vert u \Vert ^{2}+ c \Vert u \Vert ^{q}, $$

provided that $\| u\|\leq \rho _{0}$ for some $\rho _{0}>0$. Then,

$$ J(u)\geq \frac {1}{2} \Vert u \Vert ^{2}- \int _{B_{1}}{F}(x,u)\,dx \geq \biggl( \frac {1}{2}- \frac {\epsilon}{\lambda _{1}} \biggr) \Vert u \Vert ^{2}-c \Vert u \Vert ^{q}. $$

Therefore, we can find $\rho >0$ and $\sigma >0$ with $0<\rho <\rho _{0}$ sufficiently small such that ${J}(u)\geq \sigma >0$, for all $u\in E$ satisfying $\|u\|=\rho$. □

Lemma 3.2

Suppose that $(H_{1})$–$(H_{2})$ hold. Then, there exists $e\in E$ such that

$$ {J}(e)< \rho \quad \textit{and}\quad \Vert e \Vert >\rho , $$

where $\rho >0$ is given by Lemma 3.1.

Proof

It follows from $(H_{2})$, that there exist $C>0$ and $s_{0}>0$ such that

$$ F(x,s)\geq Ce^{s/M},\quad \text{for all } s\geq s_{0}. $$

Let $e_{0}\geq 0$ and $e_{0}\neq 0$ fixed. Then, there exists $\delta >0$ such that $|\{x\in B_{1}: e_{0}(x)\geq \delta \}|\geq \delta$. Thus, for $t\geq s_{0}/\delta$, we have

$$ {J}(te_{0})\geq \frac {t^{2}}{2} \Vert e_{0} \Vert ^{2}- \int _{\{x\in B_{1}:e_{0} \geq \delta \}}{F}(x,te_{0})\,dx\geq \frac {t^{2}}{2} \Vert e_{0} \Vert ^{2}- C \delta e^{t\delta /M}, $$

which implies that ${J}(te_{0})\to -\infty$, as $t\to +\infty$. Therefore, we can take $e=t_{0}e_{0}$ with $t_{0}>0$ sufficiently large such that ${J}(e)<0$ and $\|e\|>\rho$. □

4 Palais–Smale sequence

By Lemmas 3.1 and 3.2, in the mountain pass theorem (see [30, 36]), we can find a Palais–Smale sequence at level $d\geq \sigma$, where σ is given by Lemma 3.1, that is, there exists a sequence $(u_{n})\subset E$ such that

$$ {J}(u_{n})\to d \quad \text{and}\quad \bigl\Vert {J}'(u_{n}) \bigr\Vert _{E^{*}}\to 0, $$

(4.1)

where $d>0$ can be characterized as

$$ d=\inf_{\gamma \in \Gamma}\max_{t \in [0,1]} {J} \bigl(\gamma (t)\bigr), $$

(4.2)

and

$$ \Gamma =\bigl\{ \gamma \in \mathcal{C}\bigl([0,1],E\bigr): \gamma (0)=0, \gamma (1)=e \bigr\} . $$

Lemma 4.1

Let $(u_{n})\subset E$ be a Palais–Smale sequence for the functional J satisfying (4.1). Then, the sequence $(u_{n})$ is bounden in E.

Proof

From ($H_{2}$), we have

$$\begin{aligned} {J}(u_{n})-\frac {1}{\mu}{J}'(u_{n})u_{n}&= \biggl(\frac {1}{2}- \frac {1}{\mu} \biggr) \Vert u_{n} \Vert ^{2}-\frac {1}{\mu} \int _{B_{1}} \bigl(\mu {F}(x,u_{n})-{f}(x,u_{n})u_{n} \bigr)\,dx \\ &\geq \biggl(\frac {1}{2}-\frac {1}{\mu} \biggr) \Vert u_{n} \Vert ^{2}. \end{aligned}$$

Using (4.1), for n sufficiently large, we have

$$ {J}(u_{n})\leq d+1 \quad \text{and}\quad \bigl\Vert {J}'(u_{n}) \bigr\Vert _{E^{*}}\leq \mu . $$

Therefore, for n sufficiently large, we obtain

$$ \biggl(\frac {1}{2}-\frac {1}{\mu} \biggr) \Vert u_{n} \Vert ^{2}\leq d+1+ \Vert u_{n} \Vert , $$

which implies that the sequence $(u_{n})$ is bounded in E. □

Lemma 4.2

Let $(u_{n})$ be a Palais–Smale sequence for the functional J satisfying (4.1) and suppose that $u_{n}\rightharpoonup u$ weakly in E. Then, there exists a subsequence of $(u_{n})$, still denoted by $(u_{n})$, such that

$$ {f}(x,u_{n})\to {f}(x,u)\quad \textit{in } L^{1}(B_{1}) $$

(4.3)

and

$$ {F}(x,u_{n})\to {F}(x,u) \quad \textit{in } L^{1}(B_{1}). $$

(4.4)

Proof

From Lemma 2.1, we can suppose that $(u_{n})$ converges to u in $L^{1}(B_{1})$. By Theorem 1.3, $(H_{1})$, and $(H_{4})$, we have that ${f}(x,u_{n})\in L^{1}(B_{1})$. Using Lemma 4.1, the sequence $(\|u_{n}\|)$ is bounded and the fact that $\|{J}'(u_{n})\|_{E^{*}}\to 0$ allows us to obtain

$$ \bigl\vert {J}'(u_{n})u_{n} \bigr\vert \leq \bigl\Vert {J}'(u_{n}) \bigr\Vert _{E^{*}} \Vert u_{n} \Vert \to 0,\quad \text{as } n\to +\infty . $$

Thus,

$$ {J}'(u_{n})u_{n}=\frac { \Vert u_{n} \Vert ^{2}}{2}- \int _{B_{1}}{f}(x,u_{n})u_{n}\,dx\to 0, \quad \text{as } n\to +\infty . $$

Therefore, the sequence ${f}(x,u_{n})u_{n}$ is bounded in $L^{1}(B_{1})$. Due to [18, Lemma 2.10], we conclude that ${f}(x,u_{n})\to {f}(x,u)$ in $L^{1}(B_{1})$. On the other hand, by the convergence (4.3), there exists $p\in L^{1}(B_{1})$ such that

$$ f(x,u_{n})\leq p(x), \quad \text{almost everywhere in } B_{1} \text{ and for } n \text{ sufficiently large}. $$

From $(H_{3})$, we can write

$$ {F}(x,u_{n})\leq Mp(x), \quad \text{almost everywhere in } B_{1} \text{ and for } n \text{ sufficiently large}. $$

By Lebesgue’s dominated convergence theorem, the convergence (4.4) follows. □

Lemma 4.3

Let $(u_{n})\subset E$ be a Palais–Smale sequence for the functional J satisfying (4.1). Then,

$$ d< \frac {1}{2} \biggl(\frac {\alpha ^{*}_{\beta}}{\alpha _{0}} \biggr)^{1- \beta}, $$

where d is the minimax level given by (4.2).

Proof

Let $u_{p}\in E$ be a nonnegative function with $\|u_{p}\|_{p}=1$ such that

$$ S_{p}=\inf_{0\neq u\in H_{0,{\mathrm{rad}}}^{1}(B_{1},w)} \frac { ( \int _{B_{1}}w(x) \vert \nabla u \vert ^{2}\,dx )^{1/2}}{ ( \int _{B_{1}} \vert u \vert ^{p}\,dx )^{1/p}}= \Vert u_{p} \Vert . $$

From $(H_{6})$, we get

$$ J(t u_{p})=\frac {t^{2}}{2} \Vert u_{p} \Vert ^{2}- \int _{B_{1}}F(x,tu_{p})\,dx \geq \frac {t^{2}}{2} \Vert u_{p} \Vert ^{2}-\frac {C_{p}t^{p}}{p} \int _{B_{1}} \vert u_{p} \vert ^{p} \,dx. $$

Therefore, by the estimate on $C_{p}$, we have

$$ \sup_{t\geq 0} J(tu_{p})\leq \max _{t\geq 0} \biggl\{ \frac {t^{2}S_{p}^{2}}{2}-\frac {C_{p}t^{p}}{p} \biggr\} = \frac {(p-2)S_{p}^{2p/(p-2)}}{2pC_{p}^{2/(p-2)}}< \frac {1}{2} \biggl( \frac {\alpha ^{*}_{\beta}}{\alpha _{0}} \biggr)^{1-\beta}. $$

(4.5)

Take $e_{0}=u_{p}$ in Lemma 3.2, that is, we consider $e=t_{0}u_{p}$ with $t_{0}>0$ given by Lemma 3.2. Setting $\gamma _{0}(t)=tt_{0}u_{p}$, in particular, we have $\gamma _{0}\in \Gamma =\{\gamma \in \mathcal{C}([0,1],E) : \gamma (0)=0, \gamma (1)=e\}$. Using (4.2) and (4.5), we obtain

$$ d=\inf_{\gamma \in \Gamma}\max_{t \in [0,1]} {J}\bigl(\gamma (t) \bigr)\leq \max_{t \in [0,1]} {J}\bigl(\gamma _{0}(t)\bigr)= \max_{t \in [0,1]} {J}(tt_{0}u_{p}) \leq \max _{t \geq 0} {J}(tu_{p}) < \frac {1}{2} \biggl( \frac {\alpha ^{*}_{\beta}}{\alpha _{0}} \biggr)^{1-\beta}. $$

 □

5 Proof of Theorem 1.4

Let $(u_{n}) \subset E$ be a Palais–Smale sequence of the functional J satisfying (4.1). Then,

$$ {J}'(u_{n})\phi = \int _{B_{1}} w(x)\nabla u_{n} \nabla \phi \,dx - \int _{B_{1}}{f} (x,u_{n})\phi \,dx=o_{n}(1), $$

(5.1)

for all $\phi \in \mathcal{C}^{\infty}_{0,{\mathrm{rad}}}(B_{1})$. By Lemma 4.1, the sequence $(u_{n})$ is bounded in E. Thus, up to a subsequence, we can assume that there exists $u\in E$ such that $u_{n}\rightharpoonup u$ weakly in E, and replacing the above convergence in (5.1) yields

$$ \int _{B_{1}} w(x)\nabla u \nabla \phi \,dx - \int _{B_{1}}{f} (x,u) \phi \,dx=0,\quad \text{for all } \phi \in \mathcal{C}^{\infty}_{0,{ \mathrm{rad}}}(B_{1}). $$

Since $\mathcal{C}_{0,{\mathrm{rad}}}^{\infty}(B_{1})$ is dense in E, we obtain

$$ \int _{B_{1}} w(x)\nabla u \nabla \phi \,dx = \int _{B_{1}}{f}(x,u) \phi \,dx,\quad \text{for all } \phi \in E. $$

Therefore, $u\in E$ is a critical point of J. Now, we prove that u is nontrivial. Suppose, by contradiction, that $u\equiv 0$. From Lemma 2.1, we can assume that

$$ u_{n}\to 0 \quad \text{in } L^{p}(B_{1}), \text{for all } p \geq 1. $$

(5.2)

Using the fact that ${J}(u_{n})\to d$, we have

$$ {J}(u_{n})=\frac { \Vert u_{n} \Vert ^{2}}{2}- \int _{B_{1}}{F}(x,u_{n})\,dx=d+o_{n}(1). $$

(5.3)

Since, we suppose that $u_{n}\rightharpoonup 0$, by Lemma 4.2, we obtain

$$ \int _{B_{1}}{F}(x,u_{n})\,dx\to \int _{B_{1}}{F}(x,0)\,dx=0. $$

Replacing the above limit in (5.3), one has

$$ \frac { \Vert u_{n} \Vert ^{2}}{2}=d+o_{n}(1). $$

(5.4)

By Lemma 4.3, we get

$$ \Vert u_{n} \Vert ^{2}=2d+o_{n}(1)< \biggl( \frac {\alpha _{\beta}^{*}}{\alpha _{0}} \biggr)^{1-\beta}+o_{n}(1). $$

Thus, we can assume that there exists $\delta >0$ sufficiently small such that

$$ \Vert u_{n} \Vert ^{\frac{2}{1-\beta}} \leq \frac {\alpha _{\beta}^{*}}{\alpha _{0}}-2 \delta , \quad \text{for all } n\geq 1. $$

Now, we can find $\epsilon >0$ sufficiently small and $m>1$ sufficiently close to 1 such that

$$ \Vert u_{n} \Vert ^{\frac{2}{1-\beta}+\epsilon} \leq \frac {\alpha _{\beta}^{*}}{\alpha _{0}}-\delta ,\quad \text{for all } n\geq 1, $$

(5.5)

and

$$ m(\alpha _{0}+\epsilon ) \biggl(\frac {\alpha _{\beta}^{*}}{\alpha _{0}}- \delta \biggr)< \alpha _{\beta}^{*}. $$

(5.6)

From assumption $(H_{5})$ there exists a positive constant C such that

$$|f(x,s)|\le Cexp(({\alpha }_0+ϵ)|s{|}^{\frac{2}{1-\beta }+h(|x|)}),\text{for all }(x,s)\in B_1\times \mathbb{R}.$$

By Hölder and the above inequalities, we have

$$ \int _{B_{1}}{f}(x,u_{n})u_{n}\,dx \leq C \Vert u_{n} \Vert _{m'} \biggl( \int _{B_{1}}{ \rm exp} \bigl(m(\alpha _{0}+\epsilon ) \vert u_{n} \vert ^{\frac{2}{1-\beta}+h( \vert x \vert )} \bigr)\,dx \biggr)^{1/m}. $$

(5.7)

Since h is continuous and $h(0)=0$, there exists $r_{0}>0$ such that

$$ h\bigl( \vert x \vert \bigr)< \epsilon ,\quad \text{for all } \vert x \vert \leq r_{0}. $$

Using (5.5), (5.6), and Theorem 1.3, we obtain $C_{1}>0$ such that

$$ \begin{aligned} &\int _{B_{r_{0}}}\exp \bigl(m(\alpha _{0}+\epsilon ) \vert u_{n} \vert ^{\frac{2}{1-\beta}+h( \vert x \vert )} \bigr)\,dx \\ &\quad \leq \int _{B_{r_{0}}}\exp \biggl[m(\alpha _{0}+ \epsilon ) \Vert u_{n} \Vert ^{\frac{2}{1-\beta}+h( \vert x \vert )} \biggl( \frac { \vert u_{n} \vert }{ \Vert u_{n} \Vert } \biggr)^{\frac{2}{1-\beta}+h( \vert x \vert )} \biggr]\,dx \\ & \quad \leq \int _{B_{r_{0}}}\exp (m(\alpha _{0}+ \epsilon ) \Vert u_{n} \Vert ^{\frac{2}{1-\beta}+\epsilon} \biggl( \frac { \vert u_{n} \vert }{ \Vert u_{n} \Vert } \biggr)^{\frac{2}{1-\beta}+h( \vert x \vert )} ]\,dx \\ &\quad \leq \int _{B_{r_{0}}}\exp \biggl[\alpha _{\beta}^{*} \biggl( \frac { \vert u_{n} \vert }{ \Vert u_{n} \Vert } \biggr)^{\frac{2}{1-\beta}+h( \vert x \vert )} \biggr]\,dx\leq C_{1}. \end{aligned} $$

(5.8)

According to (2.2), we have $|u(x)|\leq 1$ for $r_{1}\leq |x|<1$. Thus, we can find $C_{2}>0$ such that

$$ \int _{B_{1}\backslash{B_{r_{1}}}}\exp \bigl(m(\alpha _{0}+ \epsilon ) \vert u \vert ^{\frac{2}{1-\beta}+h( \vert x \vert )} \bigr)\,dx\leq \int _{B_{1} \backslash{B_{r_{1}}}}\exp (m(\alpha _{0}+\epsilon )\,dx= C_{2}. $$

(5.9)

On the other hand, using the boundedness of $(\|u_{n}\|)$ and Lemma 2.2, we have

$$ \bigl\vert u_{n}(x) \bigr\vert \leq M_{0}, \quad \text{for all } r_{0}\leq \vert x \vert \leq r_{1} \text{ and } n\geq 1. $$

By the continuity of h, we can find $C_{3}>0$ such that

$$ \int _{B_{r_{1}}\backslash B_{r_{0}}}\exp \bigl(m(\alpha _{0}+ \epsilon ) \vert u_{n} \vert ^{\frac{2}{1-\beta}+h( \vert x \vert )} \bigr)\,dx\leq C_{3}. $$

(5.10)

Replacing (5.8), (5.9), and (5.10) in (5.7), we obtain

$$ \int _{B_{1}}{f}(x,u_{n})u_{n}\,dx \leq C \Vert u_{n} \Vert _{m'}. $$

By (5.2), we get

$$ \int _{B_{R}}{f}(x,u_{n})u_{n}\,dx \to 0, \quad \text{as } n\to + \infty . $$

(5.11)

Using the fact that $(\|u_{n}\|)$ is bounded and $\|{J}'(u_{n})\|_{E^{*}}\to 0$, we obtain $C>0$ such that

$$ \bigl\vert {J}'(u_{n})u_{n} \bigr\vert \leq \bigl\Vert {J}'(u_{n}) \bigr\Vert _{E^{*}} \Vert u_{n} \Vert \to 0, \quad \text{as } n\to +\infty . $$

(5.12)

Since,

$$ {J}'(u_{n})u_{n}= \Vert u_{n} \Vert ^{2}- \int _{B_{1}}{f}(x,u_{n})u_{n}\,dx. $$

By (5.11) and (5.12), we have

$$ \Vert u_{n} \Vert ^{2}={J}'(u_{n})u_{n}+ \int _{B_{1}}{f}(x,u_{n})u_{n}\,dx\to 0, \quad \text{as } n\to +\infty . $$

From (5.4), we have $\|u_{n}\|^{2}\to 2d$. Hence, $d=0$, which represents a contradiction with (4.2). Thus, u is a nontrivial critical point of J. Therefore, u is a nontrivial weak solution of the problem (1.9). This completes the proof.