1 Introduction

Let consider the following Schrödinger equation:

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

where Ω is a bounded smooth domain in 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

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

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

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

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

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

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 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

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

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 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

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

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)R such that


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


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

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

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

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

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:

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

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:


f: B 1 ×RR 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 \(s\leq 0\).


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


There exists a constant \(M>0\) such that

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

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 \((-\operatorname{div}(w(x)\nabla u), H_{0,{\mathrm{rad}}}^{1}(B_{1},w) )\).


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

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

There exist constants \(p> 2\) and \(C_{p}>0\) such that

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


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


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

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

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

Under the above assumptions on f, we consider the Euler–Lagrange functional J:ER 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 C 1 (E,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 ×RR 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 \(1\leq p<\infty \).


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

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


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

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


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


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

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

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

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


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

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

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

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

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


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


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

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

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

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

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

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

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

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 kN. Moreover, for \(\alpha >\alpha ^{*}_{\beta}\), we have

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


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

and the proof is complete. □

Corollary 2.3

Let \(\eta >0\). Then,

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

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

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

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

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

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

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

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

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


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)|ϵ|s | 2 +c|s | q exp ( 2 α 0 | u | 2 1 β + h ( | x | ) ) ,for all (x,s) B 1 ×R.

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

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

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

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

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

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

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.


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

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

where \(d>0\) can be characterized as

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


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

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.


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

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


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


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

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


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

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).


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

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

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,

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

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

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

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

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

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

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

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

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


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

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

|f(x,s)|Cexp ( ( α 0 + ϵ ) | s | 2 1 β + h ( | x | ) ) ,for all (x,s) B 1 ×R.

By Hölder and the above inequalities, we have

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

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

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

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

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

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

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

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

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

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

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


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

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.