## 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 [14, 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 [35, 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 $h:\left[0,1\right)\to \mathbb{R}$ 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:

$$(H_{1})$$:

$f:{B}_{1}×\mathbb{R}\to \mathbb{R}$ 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 ${\mathcal{C}}^{1}\left(E,\mathbb{R}\right)$ 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 $f:{B}_{1}×\mathbb{R}\to \mathbb{R}$ 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,

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

### 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

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

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.