Fourth order weighted elliptic problem under exponential nonlinear growth

Brahim Dridi (1) and Rached Jaidane (2)

(1) Department of Mathematics, El Manar Preparatory Institute for Engineering
Studies Tunis, University of Tunis El Manar, Tunisia. Address e-mail:
<EMAIL_ADDRESS>
(2) Department of Mathematics, Faculty of Science of Tunis, University of
Tunis El Manar, Tunisia.
Address e-mail<EMAIL_ADDRESS>

###### Abstract

We deal with nonlinear weighted biharmonic problem in the unit ball of
$\mathbb{R}^{4}$. The weight is of logarithm type. The nonlinearity is
critical in view of Adam’s inequalities in the weighted Sobolev space
$W^{2,2}_{0}(B,w)$. We prove the existence of non trivial solutions via the
critical point theory. The main difficulty is the loss of compactness due to
the critical exponential growth of the nonlinear term $f$. We give a new
growth condition and we point out its importance for checking the Palais-Smale
compactness condition.

_Keywords:_ Adam’s inequality, Moser-Trudinger’s inequality, Nonlinearity of
exponential growth, Mountain pass method, Compactness level.
$2010$ Mathematics Subject classification: $35$J$20$, $35$J$30$, $35$K$57$,
$35$J$60$.

##  1 Introduction and Main results

In this paper, we consider the following elliptic nonlinear problem:

$\displaystyle\left\\{\begin{array}[]{rclll}L(u):=\Delta(w(x)\Delta
u)&=&f(x,u)&\mbox{ in }&B,\\\ u=\frac{\partial u}{\partial n}&=&0&\mbox{ on
}&\partial B,\end{array}\right.$ (1.1)

where $B=B(0,1)$ is the unit open ball in $\mathbb{R}^{4}$. The weight is
given by

$w(x)=\big{(}\log\frac{e}{|x|}\big{)}^{\beta},\beta\in(0,1),$ (1.2)

The nonlinearity $f(x,t)$ is continuous in $B\times\mathbb{R}$ and behaves
like $\exp\\{\alpha t^{\frac{2}{1-\beta}}\\}$ as $t\rightarrow+\infty$, for
some $\alpha>0$ and where $\frac{\partial u}{\partial n}$ denotes the outer
normal derivative of $u$ on $\partial B$.
Problems of critical exponential growth in second order elliptic equations in
dimension $N=2$

$-\Delta u=f(x,u)~{}~{}\mbox{in}~{}~{}\Omega\subset\mathbb{R}^{2}.$

have been studied considerably [2, 19, 22, 26]. In dimension $N\geq 2$, the
critical exponential growth is given by the well known Trudinger-Moser
inequality [28, 33]

$\displaystyle\sup_{\int_{\Omega}|\nabla u|^{N}\leq
1}\int_{\Omega}e^{\alpha|u|^{\frac{N}{N-1}}}dx<+\infty~{}~{}\mbox{if and only
if}~{}~{}\alpha\leq\alpha_{N},$

where $\alpha_{N}=\omega_{N-1}^{\frac{1}{N-1}}$ with $\omega_{N-1}$ is the
area of the unit sphere $S^{N-1}$ in $\mathbb{R}^{N}$.
Later, the Trudinger-Moser inequality was improved to weighted inequalities
[9, 10]. The influence of the weight in the Sobolev norm was studied as the
compact embedding [21].
When the weight is of logarithmic type, Calanchi and Ruf [11] extend the
Trudinger-Moser inequality and proved the following results in the weighted
Sobolev space for radial functions

$W_{0,rad}^{1,N}(B,\rho)=cl\\{u\in
C_{0,rad}^{\infty}(B)~{}~{}|~{}~{}\int_{B}|\nabla u|^{N}\rho(x)dx<\infty\\}:$

######  Theorem 1.1.

[10]

  * $(i)$

Let $\beta\in[0,1)$ and let $\rho$ given by
$\rho(x)=\big{(}\log\frac{1}{|x|}\big{)}^{\beta}$, then

$\int_{B}e^{|u|^{\gamma}}dx<+\infty,~{}~{}\forall~{}~{}u\in
W_{0,rad}^{1,N}(B,\rho),~{}~{}\mbox{if and only
if}~{}~{}\gamma\leq\gamma_{N,\beta}=\frac{N}{(N-1)(1-\beta)}=\frac{N^{\prime}}{1-\beta}$

and

$\sup_{\begin{subarray}{c}u\in W_{0,rad}^{1,N}(B,\rho)\\\ \int_{B}|\nabla
u|^{N}w(x)dx\leq
1\end{subarray}}\int_{B}~{}e^{\alpha|u|^{\gamma_{N,\beta}}}dx<+\infty~{}~{}~{}~{}\Leftrightarrow~{}~{}~{}~{}\alpha\leq\alpha_{N,\beta}=N[\omega^{\frac{1}{N-1}}_{N-1}(1-\beta)]^{\frac{1}{1-\beta}}$

where $\omega_{N-1}$ is the area of the unit sphere $S^{N-1}$ in
$\mathbb{R}^{N}$ and $N^{\prime}$ is the H$\ddot{o}$lder conjugate of $N$.

  * $(ii)$

Let $\rho$ given by $\rho(x)=\big{(}\log\frac{e}{|x|}\big{)}^{N-1}$, then

$\int_{B}exp\\{e^{|u|^{\frac{N}{N-1}}}\\}dx<+\infty,~{}~{}~{}~{}\forall~{}~{}u\in
W_{0,rad}^{1,N}(B,\rho)$

and

$\sup_{\begin{subarray}{c}u\in W_{0,rad}^{1,N}(B,\rho)\\\ \|u\|_{\rho}\leq
1\end{subarray}}\int_{B}exp\\{\beta
e^{\omega_{N-1}^{\frac{1}{N-1}}|u|^{\frac{N}{N-1}}}\\}dx<+\infty~{}~{}~{}~{}\Leftrightarrow~{}~{}~{}~{}\beta\leq
N,$

where $\omega_{N-1}$ is the area of the unit sphere $S^{N-1}$ in
$\mathbb{R}^{N}$ and $N^{\prime}$ is the H$\ddot{o}$lder conjugate of $N$.

These results opened the way to study second order weighted elliptic problems
in dimension $N\geq 2$ . We cite the work of Calanchi et all [12], $N=2$

$\displaystyle\left\\{\begin{array}[]{rclll}-\nabla.(\nu(x)\nabla u)&=&\
f(x,u)&\mbox{in}&B\\\ u&>&0&\mbox{in }&B\\\ u&=&0&\mbox{on }&\partial
B,\end{array}\right.$

with the weight $\nu(x)=\log(\frac{e}{|x|})$ and where the function $f(x,t)$
is continuous in $B\times\mathbb{R}$ and behaves like
$\exp\\{e^{\alpha{t^{2}}}\\}~{}~{}\mbox{as}~{}~{}t\rightarrow+\infty$, for
some $\alpha>0$.
Also, recently, Deng et all [15] and Zhang[35] studied the following problem

$\displaystyle\left\\{\begin{array}[]{rclll}-\textmd{div}(\rho(x)|\nabla
u|^{N-2}\nabla u)&=&\displaystyle f(x,u)&\mbox{in}&B\\\ u&=&0&\mbox{on
}&\partial B,\end{array}\right.$

where $N\geq 2$, the function $f(x,t)$ is continuous in $B\times\mathbb{R}$
and behaves like
$\exp\\{e^{\alpha{t^{\frac{N}{N-1}}}}\\}~{}~{}\mbox{as}~{}~{}t\rightarrow+\infty$,
for some $\alpha>0$. The authors proved that there is a non-trivial solution
to this problem using Mountain Pass theorem.
Also, we mention that Baraket et all [6] studied the following non-autonomous
weighted elliptic equations

$\displaystyle\left\\{\begin{array}[]{rclll}L:=-\textmd{div}(\rho(x)|\nabla
u|^{N-2}\nabla u)+\xi(x)|u|^{N-2}u&=&\displaystyle f(x,u)&\mbox{in}&B\\\
u&>&0&\mbox{in }&B\\\ u&=&0&\mbox{on }&\partial B,\end{array}\right.$

where B is the unit ball of $\mathbb{R}^{N}$, $N>2$ , $f(x,t)$ is continuous
in $B\times\mathbb{R}$ and behaves like $\exp\\{e^{\alpha
t^{\frac{N}{N-1}}}\\}$ as $t\rightarrow+\infty$, for some $\alpha>0$.
$\xi:B\rightarrow\mathbb{R}$ is a positive continuous function satisfying some
conditions. The weight $\rho(x)$ is given by
$\rho(x)=\big{(}\log\frac{e}{|x|}\big{)}^{N-1}\cdot$
The biharmonic equation in dimension $N>4$

$\Delta^{2}u=f(x,u)~{}~{}\mbox{in}~{}~{}\Omega\subset\mathbb{R}^{N},$

where the nonlinearity $f$ has subcritical and critical polynomial growth of
power less than $\frac{N+4}{N-4}$, have been extensively studied [7, 17, 20,
31] .

For bounded domains $\Omega\subset\mathbb{R}^{4}$, in [1, 29] the authors
proved the following Adams’ inequality

$\sup_{\begin{subarray}{c}u\in S\end{subarray}}\int_{\Omega}~{}(e^{\alpha
u^{2})}-1)dx<+\infty~{}~{}~{}~{}\Leftrightarrow~{}~{}~{}~{}\alpha\leq
32\pi^{2}$

where

$S=\\{u\in
W^{2,2}_{0}(\Omega)~{}~{}|~{}~{}\displaystyle\big{(}\int_{\Omega}|\triangle
u|^{2}dx\big{)}^{\frac{1}{2}}\leq 1\\}.$

This last result opened the way to study fourth-order problems with
subcritical or critical nonlinearity (see [30] , [13]).
We study the existence of the nontrivial solutions when the nonlinear terms
have the critical exponential growth in the sense of Adams inequalities [34].
Our approach is variational methods such as the Mountain Pass Theorem with
Palais-Smale condition combining with a concentration compactness result.
More precisely, let $\Omega\subset\mathbb{R}^{4}$ be a bounded domain and
$w\in L^{1}(\Omega)$ be a nonnegative function, the weighted sobolev space is
defined as $W_{0}^{2,2}(\Omega,w)=cl\\{u\in
C_{0}^{\infty}(\Omega)~{}~{}|~{}~{}\displaystyle\int_{\Omega}|\triangle
u|^{2}w(x)dx<\infty\\}.$ We will restrict our attention to radial functions
and then consider the subspace

$\mathbf{W}=W_{0,rad}^{2,2}(B,w)=cl\\{u\in
C_{0,rad}^{\infty}(B)~{}~{}|~{}~{}\int_{B}|\triangle
u|^{2}w(x)~{}~{}dx<\infty\\},$ (1.3)

equipped with norm

$\|u\|=\displaystyle\big{(}\int_{B}|\triangle
u|^{2}w(x)dx\big{)}^{\frac{1}{2}},~{}~{}w(x)=\big{(}\log\frac{e}{|x|}\big{)}^{\beta}$

which comes from the scalar product

$<u,v>=\int_{B}\Delta u.\Delta v~{}(\log\frac{e}{|x|})^{\beta}~{}dx.$

The norm $\|u\|=\displaystyle\big{(}\int_{B}|\triangle
u|^{2}w(x)dx\big{)}^{\frac{1}{2}},$ and

$\|u\|_{W_{0,rad}^{2,2}(B,w)}=\displaystyle\big{(}\int_{B}u^{2}dx+\int_{B}|\nabla
u|^{2}~{}dx+\int_{B}|\triangle u|^{2}w(x)dx\big{)}^{\frac{1}{2}}$

are equivalent (see Lemma 1).
The choice of the weight and the space $W_{0,rad}^{2,2}(B,w)$ are motivated by
the following inequality of Adam’s type.

######  Theorem 1.2.

[34] Let $\beta\in(0,1)$ and let $w$ given by (1.2), then

$\sup_{\begin{subarray}{c}u\in W_{0,rad}^{2,2}(B,w)\\\ \|u\|\leq
1\end{subarray}}\int_{B}~{}e^{\displaystyle\alpha|u|^{\frac{2}{1-\beta}}}dx<+\infty~{}~{}~{}~{}\Leftrightarrow~{}~{}~{}~{}\alpha\leq\alpha_{\beta}=4[8\pi^{2}(1-\beta)]^{\frac{1}{1-\beta}}$
(1.4)

Let $\gamma:=\displaystyle\frac{2}{1-\beta}$. In view of inequality (1.4), we
say that $f$ has critical growth at $+\infty$ if there exists some
$\alpha_{0}>0$,

$\lim_{s\rightarrow+\infty}\frac{|f(x,s)|}{e^{\alpha
s^{\gamma}}}=0,~{}~{}~{}\forall~{}\alpha~{}~{}\mbox{such
that}~{}~{}\alpha>\alpha_{0}~{}~{}~{}~{}\mbox{and}~{}~{}~{}~{}\lim_{s\rightarrow+\infty}\frac{|f(x,s)|}{e^{\alpha
s^{\gamma}}}=+\infty,~{}~{}\forall~{}\alpha<\alpha_{0}.$ (1.5)

Let us now state our results. We suppose that $f(x,t)$ satisfies the following
hypothesis:

  1. $(H_{1})$

$f:B\times\mathbb{R}\rightarrow\mathbb{R}$ is continuous, positive, radial in
$x$ and $f(x,t)=0$ for $t\leq 0;$

  2. $(H_{2})$

There exists $t_{0}>0$ and $M>0$ such that for all $t>t_{0}$ and for all $x\in
B$ we have

$0<F(x,t)\leq Mf(x,t),$

where

$F(x,t)=\displaystyle\int_{0}^{t}f(x,s)ds;$

  3. $(H_{3})$

$0<F(x,t)\leq\displaystyle\frac{1}{2}f(x,t)t$, $\forall t>0,\forall x\in B.$

  4. $(H_{4})$

$~{}~{}~{}\displaystyle\limsup_{t\rightarrow
0}\frac{2F(x,t)}{t^{2}}<\lambda_{1}~{}~{}~{}~{}\mbox{uniformly in}~{}~{}x,$

We denote by

$\lambda_{1}=\inf_{\begin{subarray}{c}u\in\mathbf{W}\\\ u\neq
0\end{subarray}}\displaystyle\frac{\displaystyle\int_{B}|\triangle
u|^{2}w(x)dx}{\displaystyle\int_{B}|u|^{2}dx},$

the first eigenvalue of $(L,\mathbf{W})$. It is well known that $\lambda_{1}$
is isolated simple positive eigenvalue and has a positive bounded associated
eigenfunction, [16].
We say that u is a solution to the problem (1.1), if u is a weak solution in
the following sense:

######  Definition 1.1.

We say that a function $u\in\mathbf{W}$ is a solution of the problem (1.1) if

$\int_{B}\triangle u.\triangle\varphi~{}w(x)dx=\int_{B}f(x,u)\varphi
dx,~{}~{}\forall~{}\varphi\in\mathbf{W}.$

Let $\mathcal{J}:\mathbf{W}\rightarrow\mathbb{R}$ be the functional given by

$\mathcal{J}(u)=\frac{1}{2}\int_{B}|\triangle u|^{2}w(x)dx-\int_{B}F(x,u)dx,$
(1.6)

where

$F(x,t)=\displaystyle\int_{0}^{t}f(x,s)ds.$

It is well-known that seeking a weak solution of (1.1) is equivalent to
finding a nonzero critical point of $\mathcal{J}$.
Our Euler–Lagrange functional does not satisfy the Palais–Smale condition at
all level anymore. To overcome the verification of compactness of
Euler–Lagrange functional at some suitable level, we construct Adams type
functions, which are extremal to the inequality (1.4). Our result is as
follows :

######  Theorem 1.3.

Assume that $f(x,t)$ has a critical growth at $+\infty$ for some $\alpha_{0}$
and satisfies the conditions $(H_{1}),(H_{2}),(H_{3})$ and $(H_{4})$. If in
addition $f(x,t)$ satisfies the asymptotic condition

$(H_{5})~{}~{}~{}~{}~{}~{}\displaystyle\lim_{t\rightarrow\infty}\frac{f(x,t)t}{e^{\alpha_{0}t^{\gamma}}}\geq\gamma_{0}~{}~{}~{}~{}\mbox{
uniformly
in}~{}~{}x,~{}~{}\mbox{with}~{}~{}~{}~{}\gamma_{0}>\frac{1024(1-\beta)}{\alpha^{1-\beta}_{0}},$

then the problem (1.1) has a nontrivial solution.

In general the study of fourth order partial differential equations is
considered an interesting topic. The interest in studying such equations was
stimulated by their applications in micro-electro-mechanical systems, phase
field models of multi-phase systems, thin film theory,surface diffusion on
solids, interface dynamics, flow in Hele-Shaw cells, see [14, 18, 27]. However
many applications are generated by the weighted elliptic problems, such as the
study of traveling waves in suspension bridges, radar imaging (see, for
example [4, 23]).
The main reason for this study is that, to our knowledge, there are few
research taking into account both this type of non-linearity for a non-linear
fourth order elliptic equation in the framework of Sobolev weighted spaces.

This paper is organized as follows:
In Section 2, we present some necessary preliminary knowledge about working
space, and we prove that the energy $\mathcal{J}$ satisfied the two geometric
properties. Section 3 is devoted for the compactness analysis. More precisely,
we prove a concentration compactness result of Lions type and identify the
first compactness level of the energy $\mathcal{J}$. Finally, we fulfil the
proof of the main results in section 4.
In this work, the constant C may change from line to another and sometimes we
index the constants in order to show how they change.

##  2 Functional setting and Variational formulation

###  2.1 Functional setting

Let $\Omega\subset\mathbb{R}^{N}$, $N\geq 2$, be a bounded domain in
$\mathbb{R}^{N}$ and let $w\in L^{1}(\Omega)$ be a nonnegative function. To
deal with weighted operator, we need to introduce some functional spaces
$L^{p}(\Omega,w)$, $W^{m,p}(\Omega,w)$, $W_{0}^{m,p}(\Omega,w)$ and some of
their properties that will be used later. Let $S(\Omega)$ be the set of all
measurable real-valued functions defined on $\Omega$ and two measurable
functions are considered as the same element if they are equal almost
everywhere.

Following Drabek et al. and Kufner in [16], the weighted Lebesgue space
$L^{p}(\Omega,w)$ is defined as follows:

$L^{p}(\Omega,w)=\\{u:\Omega\rightarrow\mathbb{R}~{}\mbox{measurable};~{}~{}\int_{\Omega}w(x)|u|^{p}~{}dx<\infty\\}$

for any real number $1\leq p<\infty$.
This is a normed vector space equipped with the norm

$\|u\|_{p,w}=\Big{(}\int_{\Omega}w(x)|u|^{p}~{}dx\Big{)}^{\frac{1}{p}}.$

For $m\geq 2$, let $w$ be a given family of weight functions
$w_{\tau},~{}~{}|\tau|\leq m,$
$w=\\{w_{\tau}(x)~{}~{}x\in\Omega,~{}~{}|\tau|\leq m\\}.$

In [16], the corresponding weighted Sobolev space was defined as

$W^{m,p}(\Omega,w)=\\{u\in L^{p}(\Omega),D^{\tau}u\in
L^{p}(\Omega)~{}~{}\forall~{}~{}1\leq|\tau|\leq m-1,D^{\tau}u\in
L^{p}(\Omega,w)~{}~{}\forall~{}~{}|\tau|=m\\}$

endowed with the following norm:

$\|u\|_{W^{m,p}(\Omega,w)}=\bigg{(}\sum_{|\tau|\leq
m-1}\int_{\Omega}|D^{\tau}u|^{p}dx+\displaystyle\sum_{|\tau|=m}\int_{\Omega}|D^{\tau}u|^{p}\omega(x)dx\bigg{)}^{\frac{1}{p}},$

where $w_{\tau}=1~{}~{}\mbox{for all}~{}~{}|\tau|<k,$
$w_{\tau}=\omega~{}~{}\mbox{for all}~{}~{}|\tau|=k$.

If we suppose also that $w(x)\in L^{1}_{loc}(\Omega)$, then
$C^{\infty}_{0}(\Omega)$ is a subset of $W^{m,p}(\Omega,w)$ and we can
introduce the space

$W^{m,p}_{0}(\Omega,w)$

as the closure of $C^{\infty}_{0}(\Omega)$ in $W^{m,p}(\Omega,w).$
$(L^{p}(\Omega,w),\|\cdot\|_{p,w})$ and
$(W^{m,p}(\Omega,w),\|\cdot\|_{W^{m,p}(\Omega,w)})$ are separable, reflexive
Banach spaces provided that $w(x)^{\frac{-1}{p-1}}\in L^{1}_{loc}(\Omega)$.
For $w(x)=1$, one finds the standard Sobolev spaces $W^{m,p}(\Omega)$,
$W_{0}^{m,p}(\Omega)$ and the Lebesgue spaces $L^{p}(\Omega)$.
Our space setting is

$\mathbf{W}=\\{u\in
W_{0,rad}^{2,2}(B,w)~{}~{}|~{}~{}\displaystyle\int_{\Omega}|\triangle
u|^{2}w(x)dx<\infty\\}.$

$\mathbf{W}$ is equipped with norm

$\|u\|=\displaystyle\big{(}\int_{B}|\triangle
u|^{2}w(x)dx\big{)}^{\frac{1}{2}},$

which comes from the scalar product

$<u,v>=\int_{B}\Delta u.\Delta v~{}(\log\frac{e}{|x|})^{\beta}~{}dx.$

We have the following result:

######  Lemma 1.

$(\mathbf{W},\|.\|_{W_{0,rad}^{2,2}(B,w)})$ is a Banach space and the norm
$\|.\|$ is equivalent in $\mathbf{W}$ to the norm
$\|.\|_{W_{0,rad}^{2,2}(B,w)}$.

###### Proof.

The Sobolev weighted space $(\mathbf{W},\|.\|_{W_{0,rad}^{2,2}(B,w)})$ is a
normed linear space. In order to prove that it is a Banach space, let
$\\{u_{n}\\}$ be a Cauchy sequence that

$\|u_{n}-u_{m}\|_{\mathbf{W}_{0,rad}^{2,2}(B,w)}\rightarrow
0~{}~{}\mbox{as}~{}~{}n,m\rightarrow+\infty.$

Therefore $\\{u_{n}\\}$ is also a Cauchy sequence in
$(W_{0,rad}^{2,2}(B,w),\|.\|_{W_{0,rad}^{2,2}(B,w)})$.
By the completeness of the last space, there exists $u\in
W_{0,rad}^{2,2}(B,w)$ such that

$\|u_{n}-u\|_{W_{0,rad}^{2,2}(B,w)}\rightarrow
0~{}~{}\mbox{as}~{}~{}n\rightarrow+\infty.$ (2.1)

Since $\|u\|^{2}_{W_{0,rad}^{1,2}(B)}=\displaystyle\int_{B}|\nabla u|^{2}dx$,
then

$\|u\|_{W_{0,rad}^{1,2}(B)}\leq\|u\|_{W_{0,rad}^{2,2}(B,w)},$

for all $u\in\mathbf{W}$, the sequence $\\{u_{n}\\}$ is also a Cauchy sequence
in $(W^{1,2}_{0,rad}(B),\|.\|_{W^{1}(B)})$. By the completeness of
$((W^{1,2}_{0,rad}(B),\|.\|_{W^{1,2}_{0,rad}(B)}$ there exists $v\in
W^{1,2}_{0,rad}(B)$ such that

$\|u_{n}-v\|_{W^{1,2}_{0,rad}(B)}\rightarrow
0~{}~{}\mbox{as}~{}~{}n\rightarrow+\infty$ (2.2)

Since $u\in W^{2,2}_{0,rad}(B,w)$, $u\in W^{1,2}_{0,rad}(B)$ and by (2.1) we
obtain

$\|u_{n}-u\|_{W^{1,2}_{0,rad}(B)}\rightarrow
0~{}~{}\mbox{as}~{}~{}n\rightarrow+\infty,$ (2.3)

and from (3.2), (2.3),we have

$\|u-v\|_{W^{1,2}_{0,rad}(B)}\leq\|u_{n}-u\|_{W^{1,2}_{0,rad}(B)}+\|u_{n}-v\|_{W^{1,2}_{0,rad}(B)}\rightarrow
0~{}~{}\mbox{as}~{}~{}n\rightarrow+\infty,$

so $u=v~{}~{}\mbox{a .e in }~{}~{}B$ , $u\in\mathbf{W}$ and satisfies

$\|u_{n}-u\|_{W_{0,rad}^{2,2}(B,w)}\rightarrow
0~{}~{}\mbox{as}~{}~{}n\rightarrow+\infty.$

Now we prove that $\|.\|$ is equivalent to $\|.\|_{W_{0,rad}^{2,2}(B,w)}$ in
$\mathbf{W}$.

$\|u\|^{2}_{W_{0}^{2,2}}=\|u\|^{2}_{2}+\|\nabla
u\|^{2}_{2}+\displaystyle\int_{B}|\triangle u|^{2}w(x)dx.$

For all $u\in W_{0,rad}^{2,2}(B)$, we have

$\|u\|^{2}=\displaystyle\int_{B}|\triangle u|^{2}w(x)dx\leq\|u\|_{2}+\|\nabla
u\|^{2}_{2}+\displaystyle\int_{B}|\triangle u|^{2}w(x)dx$

On the other hand, for all $u\in W_{0,rad}^{2,2}(B,w)$, by Poincaré
inequality,

$\|u\|^{2}_{2}\leq C\|\nabla u\|^{2}_{2},$

and using the Green formula we get

$\displaystyle\int_{B}\nabla u\nabla u=-\displaystyle\int_{B}u\Delta
u+\underbrace{\displaystyle\int_{\partial B}u\frac{\partial u}{\partial
n}}_{=0}~{}\leq\displaystyle\Big{|}\int_{B}u\Delta u\Big{|},$

By Young inequality, we get for all $\varepsilon>0$

$\displaystyle\Big{|}\int_{B}u\Delta
u\Big{|}\leq\displaystyle\frac{1}{2\varepsilon}\displaystyle\int_{B}|\Delta
u|^{2}+\displaystyle\frac{\varepsilon}{2}\displaystyle\int_{B}|u|^{2},$

Again, by the Poincarré inequality and using the fact that $w(x)\geq
1,~{}~{}\mbox{for all}~{}~{}x\in B,$ we get

$\displaystyle\int_{B}\nabla u\nabla
udx\leq\displaystyle\frac{1}{2\varepsilon}\displaystyle\int_{B}|\Delta
u|^{2}dx+\displaystyle\frac{\varepsilon}{2}C^{2}\displaystyle\int_{B}|\nabla
u|^{2}dx\leq\displaystyle\frac{1}{2\varepsilon}\displaystyle\int_{B}|\Delta
u|^{2}w(x)dx+\displaystyle\frac{\varepsilon}{2}C^{2}\displaystyle\int_{B}|\nabla
u|^{2}dx.$

Hence

$(1-\displaystyle\frac{\varepsilon}{2}C^{2})\displaystyle\int_{B}|\nabla
u|^{2}dx~{}\leq~{}\displaystyle\frac{1}{2\varepsilon}\displaystyle\int_{B}|\Delta
u|^{2}w(x)dx,$

wich implies that

$\|\nabla u\|^{2}_{2}\leq C\displaystyle\int_{B}|\Delta u|^{2}w(x)dx$ (2.4)

and it is easy to conclude. ∎

###  2.2 The pass mountain geometry

Since the nonlinearity $f(x,t)$ is critical at $+\infty$, there exist
$a,~{}C>0$ positive constants and there exists $t_{1}>1$ such for that

$|f(x,t)|\leq Ce^{a~{}t^{\gamma}},~{}~{}~{}~{}~{}~{}\forall|t|>t_{1}.$ (2.5)

So, the functional $\mathcal{J}$ given by (1.6) is well defined and of class
$C^{1}$ .
In order to prove the existence of nontrivial solution to the problem (1.1),
we will prove the existence of nonzero critical point of the functional
$\mathcal{J}$ by using the following theorem which is introduced by Ambrosetti
and Rabionowitz in [5] (Mountain Pass Theorem).

######  Definition 2.1.

Let $(u_{n})$ be a sequence in a Banach space $E$ and $J\in
C^{1}(E,\mathbb{R})$ and let $c\in\mathbb{R}$. We say that the sequence
$(u_{n})$ is a Palais-Smale sequence at level $c$ ( or $(PS)_{c}$ sequence )
for the functional $J$ if

$J(u_{n})\rightarrow
c~{}~{}\mbox{in}~{}~{}\mathbb{R},~{}~{}\mbox{as}~{}~{}n\rightarrow+\infty$

and

$J^{\prime}(u_{n})\rightarrow
0~{}~{}\mbox{in}~{}~{}E^{\prime},~{}~{}\mbox{as}~{}~{}n\rightarrow+\infty.$

We say that the functional $J$ satisfies the Palais-Smale condition $(PS)_{c}$
at the level $c$ if every $(PS)_{c}$ sequence $(u_{n})$ is relatively compact
in $E$.

######  Theorem 2.1.

[5] Let $E$ be a Banach space and $J:E\rightarrow\mathbb{R}$ a $C^{1}$
functional satisfying $J(0)=0$. Suppose that

  1. (i)

There exist $\rho,~{}~{}\beta>0$ such that $\forall u\in\partial
B(0,\rho),J(u)\geq\beta$;

  2. (ii)

There exists $x_{1}\in E$ such that $\|x_{1}\|>\rho~{}~{}\text{and
}~{}~{}J(x_{1})<0$;

  3. (iii)

$J$ satisfies the Palais-Smale condition $(PS)$, that is for all sequence
$(u_{n}$ in $E$ satisfying

$J(u_{n})\rightarrow d~{}~{}~{}~{}\mbox{as}~{}~{}n\rightarrow+\infty$ (2.6)

for some $d\in\mathbb{R}$ and

$\|J^{\prime}(u_{n})\|_{\ast}~{}\rightarrow
0~{}~{}~{}~{}\mbox{as}~{}~{}n\rightarrow+\infty,$ (2.7)

the sequence $(u_{n})$ is relatively compact.

Then, $J$ has a critical point $u$ and the critical value $c=J(u)$ verifies

$c:=\displaystyle{\inf_{\gamma\in\Gamma}}\displaystyle{\max_{t\in[0,1]}}J(\gamma(t))$

where $\Gamma:=\\{\gamma\in C([0,1],X)~{}~{}\mbox{such
that}~{}~{}\gamma(0)=0~{}~{}\mbox{ and }~{}~{}\gamma(1)=x_{1}\\}$ and
$c\geq\beta$.

Before starting the proof of the geometric properties for the function
$\mathcal{J}$, we recall the following radial Lemma introduced in [34].

######  Lemma 2.

[34] Let $u$ be a radially symmetric function in $C_{0}^{2}(B)$. Then, we have

$|u(x)|\leq\displaystyle\frac{1}{2\sqrt{2}\pi}\bigg{(}\frac{~{}~{}|\log(\frac{e}{|x|})|^{1-\beta}-1}{1-\beta}\bigg{)}^{\frac{1}{2}}\|u\|\cdot$

Since $w(x)\geq 1,~{}~{}\mbox{for all}~{}~{}x\in B$, then the following
embedding $W_{0}^{2,2}(B,w)\hookrightarrow W_{0}^{2,2}(B)\hookrightarrow
L^{q}(B)$ are continuous and also compact for all $q\geq 2$. So there exists a
constant $C>0$ such that $\|u\|_{2q}\leq c\|u\|$, for all $u\in\mathbf{W}$.
In the next Lemma, we prove that the $\mathcal{J}$ satisfies the first
geometric property.

######  Lemma 3.

Suppose that $(H_{1})~{}~{}\mbox{and}~{}~{}(H_{4})$ hold. Then, there exist
$\rho,~{}\beta_{0}>0$ such that $\mathcal{J}(u)\geq\beta_{0}$ for all
$u\in\mathbf{W}$ with $\|u\|=\rho$.

###### Proof.

It follows from the hypothesis $(H_{4})$ that there exists $t_{0}>0$ and there
exists $\varepsilon\in(0,1)$ such that

$F(x,t)\leq\frac{1}{2}\lambda_{1}(1-\varepsilon_{0})|t|^{2},~{}~{}~{}~{}~{}~{}\mbox{for}~{}~{}|t|<t_{2}.$
(2.8)

Indeed,

$\displaystyle\limsup_{t\rightarrow 0}\frac{2F(x,t)}{t^{2}}<\lambda_{1}$

or

$\displaystyle\inf_{\tau>0}\sup\\{\frac{2F(x,t)}{t^{2}},~{}~{}0<t<\tau\\}<\lambda_{1}$

Since this inequality is strict, then there exists $\varepsilon_{0}>0$ such
that

$\displaystyle\inf_{\tau>0}\sup\\{\frac{2F(x,t)}{t^{2}},~{}~{}0<t<\tau\\}<\lambda_{1}-\varepsilon_{0},$

hence, there exists $t_{2}>0$ such that

$\sup\\{\frac{2F(x,t)}{t^{N}},~{}~{}0<t<t_{2}\\}<\lambda_{1}-\varepsilon_{0}.$

Hence

$\forall|t|<t_{2}~{}~{}~{}F(x,t)\leq\frac{1}{2}\lambda_{1}(1-\varepsilon_{0})t^{2}.$

From $(H_{3})$ and (2.5) and for all $q>2$, there exist a constant $C>0$ such
that

$F(x,t)\leq C|t|^{q}e^{a~{}t^{\gamma}},~{}~{}\forall~{}|t|>t_{1}.$ (2.9)

So

$F(x,t)\leq\frac{1}{2}\lambda_{1}(1-\varepsilon_{0})|t|^{2}+C|t|^{q}e^{a~{}t^{\gamma}},~{}~{}~{}~{}~{}~{}\mbox{for
all}~{}~{}t\in\mathbb{R}.$ (2.10)

Since

$\mathcal{J}(u)=\frac{1}{2}\|u\|^{2}-\int_{B}F(x,u)dx,$

we get

$\mathcal{J}(u)\geq\frac{1}{2}\|u\|^{2}-\frac{1}{2}\lambda_{1}(1-\varepsilon_{0})\|u\|^{2}-C\int_{B}|u|^{q}e^{a~{}u^{\gamma}}~{}dx.$

But $\lambda_{1}\|u\|_{2}^{2}\leq\|u\|^{2}$ and from the Hölder inequality, we
obtain

$\mathcal{J}(u)\geq\frac{\varepsilon_{0}}{2}\|u\|^{2}-C(\int_{B}e^{2a~{}|u|^{\gamma}}dx\Big{)}^{\frac{1}{2}}\|u\|^{q}_{2q}\cdot$
(2.11)

From the Theorem 1.2, if we choose $u\in\mathbf{W}$ such that

$2a\|u\|^{\gamma}\leq\alpha_{\beta},$ (2.12)

we get

$\int_{B}e^{2a|u|^{\gamma}}dx=\int_{B}e^{2a\|u\|^{\gamma}(\frac{|u|}{\|u\|})^{\gamma}}dx<+\infty.$

On the other hand $\|u\|_{2q}\leq C\|u\|$, so

$\mathcal{J}(u)\geq\frac{\varepsilon_{0}}{2}\|u\|^{2}-C\|u\|^{q},$

for all $u\in\mathbf{W}$ satisfying (2.12). Since $2<q$, we can choose
$\rho=\|u\|>0$ as the maximum point of the function
$g(\sigma)=\frac{\varepsilon_{0}}{2}\sigma^{2}-C\sigma^{q}$ on the interval
$[0,(\frac{\alpha_{\beta}}{2a})^{\frac{1}{\gamma}}]$ and $\beta_{0}=g(\rho)$ ,
$\mathcal{J}(u)\geq\beta_{0}>0$. ∎

By the following Lemma, we prove the second geometric property for the
functional $\mathcal{J}$.

######  Lemma 4.

Suppose that $(H_{1})$ and $(H_{2})$ hold. Let $\varphi_{1}$ be a normalized
eigenfunction associated to $\lambda_{1}$ in $\mathbf{W}$. Then,
$\mathcal{J}(t\varphi_{1})\rightarrow-\infty,~{}~{}\mbox{as}~{}~{}t\rightarrow+\infty$.

###### Proof.

It follows from the condition $(H_{2})$ that

$f(x,t)=\frac{\partial}{\partial t}F(x,t)\geq\frac{1}{M}F(x,t),$

for all $t\geq t_{0}$. So

$F(x,t)\geq C~{}e^{\frac{t}{M}},~{}~{}\forall~{}~{}t\geq t_{0}.$

It follows that, there exist $b>\lambda_{1}$ and $C>0$ such that
$F(x,t)\geq\frac{b}{2}t^{2}+C$ for all $t>0$.

$\mathcal{J}(t\varphi_{1})\leq\frac{t^{2}}{2}\|\varphi_{1}\|^{2}-\frac{b}{2}t^{2}\|\varphi_{1}\|_{2}^{2}-C|B|,$

where $|B|=meas(B)=Vol(B)$. Then, from the definition of $\lambda_{1}$, we get

$\mathcal{J}(t\varphi_{1})\leq
t^{2}\frac{\lambda_{1}-b}{2}\|\varphi_{1}\|_{2}^{2}<0~{}~{}\forall t>0.$

So, the Lemma 4 follows. ∎

##  3  The compactness analysis

###  3.1 Concentration Compactness Theorem

In order to prove that the functional $\mathcal{J}$ satisfies the $(PS)$
condition, we need a lions type result [25] about an improved Adam’s
inequality.

######  Theorem 3.1.

Let $(u_{k})_{k}$ be a sequence in $\mathbf{W}$. Suppose that,
$\|u_{k}\|=1$, $u_{k}\rightharpoonup u$ weakly in $\mathbf{W}$,
$u_{k}(x)\rightarrow u(x)~{}~{}a.e~{}x\in B$, and $u\not\equiv 0$. Then

$\displaystyle\sup_{k}\int_{B}e^{p~{}\alpha_{\beta}|u_{k}|^{\gamma}}dx<+\infty,~{}~{}\mbox{where}~{}~{}\alpha_{\beta}=4[8\pi^{2}(1-\beta)]^{\frac{1}{1-\beta}},$

for all $1<p<U(u)$ where $U(u)$ is given by:

$U(u):=\displaystyle\left\\{\begin{array}[]{rcll}&\displaystyle\frac{1}{(1-\|u\|^{2})^{\frac{\gamma}{2}}}&\mbox{
if }\|u\|<1\\\ &+\infty&\mbox{ if }\|u\|=1\\\ \end{array}\right.$

###### Proof.

Since $\left\|u\right\|\leq\underset{k}{\liminf}\left\|u_{k}\right\|=1$, we
will split the evidence into two cases.

Case $1$ $:\left\|u\right\|<1$. We assume by contradiction for some
$p_{1}<U(u)$, we have

$\sup_{k}\int_{B}\exp\left(\alpha_{\beta}p_{1}u_{k}^{\gamma}\right)dx=+\infty.$

Set

$B_{\mathcal{L}}^{k}=\left\\{x\in B:u_{k}(x)\geq\mathcal{L}\right\\}$

where $\mathcal{L}$ is a constant that we will choose later. Let
$v_{k}=u_{k}-\mathcal{L}$. we have

$(1+a)^{q}\leq(1+\varepsilon)a^{q}+(1-\frac{1}{(1+\varepsilon)^{\frac{1}{q-1}}})^{1-q},~{}~{}\forall
a\geq 0,~{}~{}\forall\varepsilon>0~{}~{}\forall q>1.$ (3.1)

So, using (3.2), we get

$\begin{array}[]{rcll}|u_{k}|^{\gamma}&=&|u_{k}-\mathcal{L}+\mathcal{L}|^{\gamma}\\\
&\leq&(|u_{k}-\mathcal{L}|+|\mathcal{L}|)^{\gamma}\\\
&\leq&(1+\varepsilon)|u_{k}-\mathcal{L}|^{\gamma}+\big{(}1-\frac{1}{(1+\varepsilon)^{\frac{1}{\gamma-1}}}\big{)}^{1-\gamma}|\mathcal{L}|^{\gamma}\\\
&\leq&(1+\varepsilon)v_{k}^{\gamma}+C(\varepsilon,\gamma)\mathcal{L}^{\gamma}\cdot\end{array}$
(3.2)

We have

$\displaystyle\int_{B}\exp\left(\alpha_{\beta}p_{1}u_{k}^{\gamma}\right)dx$
$\displaystyle=\int_{B_{L}^{k}}\exp\left(\alpha_{\beta}p_{1}u_{k}^{\gamma}\right)dx+\int_{B\backslash
B_{\mathcal{L}}^{k}}\exp\left(\alpha_{\beta}p_{1}u_{k}^{\gamma}\right)dx$
$\displaystyle\leq\int_{B_{\mathcal{L}}^{k}}\exp\left(\alpha_{\beta}p_{1}u_{k}^{\gamma}\right)dx+c\exp\left(\alpha_{\beta}p_{1}\mathcal{L}^{\gamma}\right)$
$\displaystyle\leq\int_{B_{\mathcal{L}}^{k}}\exp\left(\alpha_{\beta}p_{1}u_{k}^{\gamma}\right)dx+c(\mathcal{L},\gamma,|B|),$

and then

$\sup_{k}\int_{B_{\mathcal{L}}^{k}}\exp\left(\alpha_{\beta}p_{1}u_{k}^{\gamma}\right)dx=\infty.$

By (3.2) we have

$\displaystyle\int_{B_{\mathcal{L}}^{k}}\exp\left(\alpha_{\beta}p_{1}u_{k}^{\gamma}\right)dx\leq$
$\displaystyle\exp\left(\alpha_{\beta}p_{1}C(\varepsilon,\gamma)\mathcal{L}^{\gamma}\right)\int_{B_{\mathcal{L}}^{k}}\exp\left((1+\varepsilon)\alpha_{\beta}p_{1}v_{k}^{\gamma}\right)dx.$

Since, $p_{1}<U(u)$, there exists $\varepsilon>0$ such that
$\tilde{p}_{1}=(1+\varepsilon)p_{1}<U(u)$. Thus

$\sup_{k}\int_{B_{\mathcal{L}}^{k}}\exp\left(\tilde{p}_{1}\alpha_{\beta}v_{k}^{\gamma}\right)dx=\infty$
(3.3)

Now, we define

$T^{\mathcal{L}}(u)=\min\\{\mathcal{L},u\\}\mbox{ and
}T_{\mathcal{L}}(u)=u-T^{\mathcal{L}}(u)$

and choose $\mathcal{L}$ such that

$\frac{1-\left\|u\right\|^{2}}{1-\left\|T^{\mathcal{L}}u\right\|^{2}}>\left(\frac{\tilde{p}_{1}}{U(u)}\right)^{\frac{2}{\gamma}}.$
(3.4)

We claim that

$\lim\sup_{k}\int_{B_{\mathcal{L}}^{k}}\omega(x)\left|\triangle
v_{k}\right|^{2}dx<\left(\frac{1}{\tilde{p}_{1}}\right)^{\frac{2}{\gamma}}.$

If this is not the case, then up to a subsequence, we get

$\int_{B_{\mathcal{L}}^{k}}\omega(x)\left|\triangle
v_{k}\right|^{2}dx=\int_{B}\omega(x)\left|\triangle
T_{\mathcal{L}}u_{k}\right|^{2}dx\geq\left(\frac{1}{\tilde{p}_{1}}\right)^{\frac{2}{\gamma}}+o_{k}(1).$

Thus,

$\displaystyle\left(\frac{1}{\tilde{p}_{1}}\right)^{\frac{2}{\gamma}}+\int_{B}\omega(x)\left|\triangle
T^{\mathcal{L}}u_{k}\right|^{2}dx+o_{k}(1)$
$\displaystyle\leq\int_{B}\omega(x)\left|\triangle
T_{\mathcal{L}}u_{k}\right|^{2}dx+\int_{B\backslash
B_{\mathcal{L}}^{k}}\omega(x)\left|\triangle u_{k}\right|^{2}dx$
$\displaystyle=\int_{B_{\mathcal{L}}^{k}}\omega(x)\left|\triangle
u_{k}\right|^{2}dx+\int_{B\backslash
B_{\mathcal{L}}^{k}}\omega(x)\left|\triangle u_{k}\right|^{2}dx=1.$

For $\mathcal{L}>0$ fixed, $T^{\mathcal{L}}u_{k}$ is also bounded in
$\mathbf{X}$. Therefore, up to a subsequence,
$T^{\mathcal{L}}u_{k}\rightharpoonup$ $T^{\mathcal{L}}u$ weakly in
$\mathbf{X}$ and $T^{\mathcal{L}}u_{k}\rightarrow T^{\mathcal{L}}u$ almost
everywhere in $B$. By the lower semicontinuity of the norm in $\mathbf{X}$ and
the last inequality, we have

$\tilde{p}_{1}\geq\frac{1}{\left(1-\underset{k\rightarrow+\infty}{\lim\inf}\left\|T^{\mathcal{L}}u_{k}\right\|^{2}\right)^{\frac{\gamma}{2}}}\geq\frac{1}{\left(1-\left\|T^{\mathcal{L}}u\right\|^{2}\right)^{\frac{\gamma}{2}}},$

combining with (3.4), we obtain

$\tilde{p}_{1}\geq\frac{1}{\left(1-\left\|T^{\mathcal{L}}u\right\|^{2}\right)^{\frac{\gamma}{2}}}>\frac{\tilde{p}_{1}}{U(u)}\frac{1}{\left(1-\left\|T^{\mathcal{L}}u\right\|^{2}\right)^{\frac{\gamma}{2}}}=\tilde{p}_{1},$

which is a contradiction. Therefore

$\limsup_{k}\int_{B_{\mathcal{L}}^{k}}\omega(x)\left|\triangle
v_{k}\right|^{2}dx<\left(\frac{1}{\tilde{p}_{1}}\right)^{\frac{2}{\gamma}}.$

By the Adam’s inequality (1.4), we deduce that

$\sup_{k}\int_{B_{\mathcal{L}}^{k}}\exp\left(\tilde{p}_{1}\alpha_{\beta}v_{k}^{\gamma}\right)dx<\infty$

which is also a contradiction. The proof is finished in this case.

Case $2$ $:\left\|u\right\|=1$. We can then proceed as in case 1 and obtain

$\sup_{k}\int_{B_{\mathcal{L}}^{k}}\exp\left(\tilde{p}_{1}\alpha_{\beta}v_{k}^{\gamma}\right)dx=\infty$

where $\tilde{p}_{1}=(1+\varepsilon)p_{1}$. Then we have

$\lim\sup_{k}\int_{B_{L}^{k}}\omega(x)\left|\triangle
v_{k}\right|^{2}dx=\limsup_{k}\int_{B}\omega(x)\left|\triangle
T_{\mathcal{L}}u_{k}\right|^{2}dx\geq\left(\frac{1}{\tilde{p}_{1}}\right)^{\frac{2}{\gamma}}$

thus,

$\displaystyle\left\|T^{\mathcal{L}}u\right\|^{2}\leq\liminf_{k}\int_{B}\omega(x)\left|\triangle
T^{\mathcal{L}}u_{k}\right|^{2}dx\leq 1-\limsup_{k}\int_{B}\left|\triangle
T_{\mathcal{L}}u_{k}\right|^{2}dx\leq
1-\left(\frac{1}{\tilde{p}_{1}}\right)^{\frac{2}{\gamma}}.$

On the other hand, since $\left\|u\right\|=1$, we can take $\mathcal{L}$ large
enough such that

$\left\|T^{\mathcal{L}}u\right\|^{2}>1-\frac{1}{2}\left(\frac{1}{\tilde{p}_{1}}\right)^{\frac{2}{\gamma}}$

which is a contradiction, and the proof is complete in this case.

∎

###  3.2 The Palais-Smale sequence

The main difficulty in the approach to the critical problem of growth is the
lack of compactness. Precisely, the overall condition of Palais-Smale does not
hold except for a certain level of energy. In the following proposition, we
identify the first level of compactness.

######  Proposition 3.1.

Suppose that $(H_{1}),(H_{2})$ and $(H_{3})$ hold. If the function $f(x,t)$
satisfies the condition (1.5) for some $\alpha_{0}>0$, then the functional
$\mathcal{J}$ satisfies the Palais-Smale condition $(PS)_{c}$ for any

$c<\displaystyle\frac{1}{2}(\frac{\alpha_{\beta}}{\alpha_{0}})^{\frac{2}{\gamma}},$

where $\alpha_{\beta}=4[8\pi^{2}(1-\beta)]^{\frac{1}{1-\beta}}.$

###### Proof.

Consider a $(PS)_{c}$ sequence in $\mathbf{W}$, for some $c\in\mathbb{R}$,
that is

$\mathcal{J}(u_{n})=\frac{1}{2}\|u_{n}\|^{2}-\int_{B}F(x,u_{n})dx\rightarrow
c,~{}~{}n\rightarrow+\infty$ (3.5)

and

$|\langle\mathcal{J}^{\prime}(u_{n}),\varphi\rangle|=\Big{|}\int_{B}w(x)\Delta
u_{n}.\Delta\varphi dx-\int_{B}f(x,u_{n})\varphi
dx\Big{|}\leq\varepsilon_{n}\|\varphi\|,$ (3.6)

for all $\varphi\in\mathbf{W}$, where $\varepsilon_{n}\rightarrow 0$, when
$n\rightarrow+\infty$.
Also, inspired by [12], it follows from $(H_{2})$ that for all $\varepsilon>0$
there exists $t_{\varepsilon}>0$ such that

$F(x,t)\leq\varepsilon tf(x,t),~{}~{}~{}~{}\mbox{for
all}~{}~{}|t|>t_{\varepsilon}~{}~{}\mbox{and uniformly in}~{}~{}x\in B,$ (3.7)

and so, by (3.5), for all $\varepsilon>0$ there exists a constant $C>0$

$\displaystyle\frac{1}{2}\|u_{n}\|^{2}\leq
C+\displaystyle\int_{B}F(x,u_{n})dx,$

hence

$\displaystyle\frac{1}{2}\|u_{n}\|^{2}\leq C+\displaystyle\int_{{|u_{n}|\leq
t_{\varepsilon}}}F(x,u_{n})dx+\varepsilon\int_{B}f(x,u_{n})u_{n}dx$

and so, from (3.6), we get

$\displaystyle\frac{1}{2}\|u_{n}\|^{2}\leq
C_{1}+\varepsilon\varepsilon_{n}\|u_{n}\|+\varepsilon\|u_{n}\|^{2},$

for some constant $C_{1}>0$. Since

$\displaystyle(\frac{1}{2}-\varepsilon)\|u_{n}\|^{2}\leq
C_{1}+\varepsilon\varepsilon_{n}\|u_{n}\|,$ (3.8)

we deduce that the sequence $(u_{n})$ is bounded in $\mathbf{W}$. As
consequence, there exists $u\in\mathbf{W}$ such that, up to subsequence,
$u_{n}\rightharpoonup u$ weakly in $\mathbf{W}$, $u_{n}\rightarrow u$ strongly
in $L^{q}(B)$, for all $1\leq q<4$ and $u_{n}(x)\rightarrow u(x)$ a.e. in $B$.
Furthermore, we have, from (3.5) and (3.6), that

$0<\int_{B}f(x,u_{n})u_{n}\leq C,$ (3.9)

and

$0<\int_{B}F(x,u_{n})\leq C.$ (3.10)

Since by Lemma 2.1 in [19], we have

$f(x,u_{n})\rightarrow
f(x,u)~{}~{}\mbox{in}~{}~{}L^{1}(B)~{}~{}as~{}~{}n\rightarrow+\infty,$ (3.11)

then, it follows from $(H_{2})$ and the generalized Lebesgue dominated
convergence theorem that

$F(x,u_{n})\rightarrow
F(x,u)~{}~{}\mbox{in}~{}~{}L^{1}(B)~{}~{}as~{}~{}n\rightarrow+\infty.$ (3.12)

So,

$\displaystyle\lim_{n\rightarrow+\infty}\|u_{n}\|^{2}=2(c+\int_{B}F(x,u)dx).$
(3.13)

Using (3.5), we have

$\displaystyle\lim_{n\rightarrow+\infty}\int_{B}f(x,u_{n})u_{n}dx=2(c+\int_{B}F(x,u)dx).$
(3.14)

Then by $(H_{3})$ and (3.6), we get

$\displaystyle\lim_{n\rightarrow+\infty}2\int_{B}F(x,u_{n})dx\leq\displaystyle\lim_{n\rightarrow+\infty}\int_{B}f(x,u_{n})u_{n}dx=2(c+\int_{B}F(x,u)dx).$
(3.15)

As a direct consequence from (3.11) and (3.12), we get $c\geq 0$.
Also, it follows from (3.5), (3.6), (3.11) and (3.12), by passing to the
limit, we obtain that $u$ is a weak solution of the problem (1.1) that is

$\int_{B}\triangle u.\triangle\varphi~{}w(x)dx=\int_{B}f(x,u)\varphi
dx,~{}~{}\forall~{}\varphi\in\mathbf{W}.$

Taking $\varphi=u$ as a test function, we get

$\int_{B}|\triangle u|^{2}~{}w(x)dx=\int_{B}f(x,u)udx\geq
2\int_{B}F(x,u)dx\cdot$

Hence $\mathcal{J}(u)\geq 0$ . We also have by the Fatou’s lemma and (3.12)

$0\leq\mathcal{J}(u)\leq\frac{1}{2}\liminf_{n\rightarrow\infty}\|u_{n}\|^{2}-\int_{B}F(x,u)dx=c.$

So, we will finish the proof by considering three cases for the level $c$.

Case 1. $c=0$. In this case

$0\leq\mathcal{J}(u)\leq\liminf_{n\rightarrow+\infty}\mathcal{J}(u_{n})=0.$

So,

$\mathcal{J}(u)=0$

and then by (3.12)

$\displaystyle\lim_{n\rightarrow+\infty}\frac{1}{2}\|u_{n}\|^{2}=\int_{B}F(x,u)dx=\frac{1}{2}\|u\|^{2}.$

It follows that $u_{n}\rightarrow u~{}~{}\mbox{in}~{}~{}\mathbf{W}$.
Case 2. $c>0$ and $u=0$. We prove that this case cannot happen.
From (3.5) and (3.6) with $v=u_{n}$, we have

$\displaystyle\lim_{n\rightarrow+\infty}\|u_{n}\|^{2}=2c~{}~{}\mbox{and}~{}~{}\displaystyle\lim_{n\rightarrow+\infty}\int_{B}f(x,u_{n})u_{n}dx=2c.$

Again by (3.6) we have

$\big{|}\|u_{n}\|^{2}-\int_{B}f(x,u_{n})u_{n}dx\big{|}\leq C\varepsilon_{n}.$

First we claim that there exists $q>1$ such that

$\int_{B}|f(x,u_{n})|^{q}dx\leq C,$ (3.16)

so

$\|u_{n}\|^{2}\leq
C\varepsilon_{n}+\big{(}\int_{B}|f(x,u_{n})|^{q}\big{)}^{\frac{1}{q}}dx(\int_{B}|u_{n}|^{q^{\prime}}\big{)}^{\frac{1}{q^{\prime}}}$

where $q^{\prime}$ the conjugate of $q$. Since $(u_{n})$ converge to $u=0$ in
$L^{q^{\prime}}(B)$,

$\displaystyle\lim_{n\rightarrow+\infty}\|u_{n}\|^{2}=0$

which in contradiction with $c>0$.

For the proof of the claim, since $f$ has subcritical or critical growth, for
every $\varepsilon>0$ and $q>1$ there exists $t_{\varepsilon}>0$ and $C>0$
such that for all $|t|\geq t_{\varepsilon}$, we have

$|f(x,t)|^{q}\leq Ce^{\alpha_{0}(1+\varepsilon)t^{\gamma}}.$ (3.17)

Consequently,

$\begin{array}[]{rcll}\displaystyle\int_{B}|f(x,u_{n})|^{q}dx&=&\displaystyle\int_{\\{|u_{n}|\leq
t_{\varepsilon}}|f(x,u_{n})|^{q}dx+\int_{\\{|u_{n}|>t_{\varepsilon}\\}}|f(x,u_{n})|^{q}dx\\\
&\leq&\displaystyle
2\pi^{2}\max_{B\times[-t_{\varepsilon},t_{\varepsilon}]}|f(x,t)|^{q}+C\int_{B}e^{\alpha_{0}(1+\varepsilon)|u_{n}|^{\gamma}}\big{)}dx.\end{array}$

Since
$2c<\displaystyle(\frac{\alpha_{\beta}}{\alpha_{0}})^{\frac{2}{\gamma}}$,
there exists $\eta\in(0,\frac{1}{2})$ such that
$2c=\displaystyle(1-2\eta)\displaystyle(\frac{\alpha_{\beta}}{\alpha_{0}})^{\frac{2}{\gamma}}$.
On the other hand, $\|u_{n}\|^{\gamma}\rightarrow(2c)^{\frac{\gamma}{2}}$, so
there exists $n_{\eta}>0$ such that for all $n\geq n_{\eta}$, we get
$\|u_{n}\|^{\gamma}\leq(1-\eta)\frac{\alpha_{\beta}}{\alpha_{0}}$. Therefore,

$\alpha_{0}(1+\varepsilon)(\frac{|u_{n}|}{\|u_{n}\|})^{\gamma}\|u_{n}\|^{\gamma}\leq(1+\varepsilon)(1-\eta)\alpha_{\beta}\cdot$

We choose $\varepsilon>0$ small enough to get

$\alpha_{0}(1+\varepsilon)\|u_{n}\|^{\gamma}\leq\alpha_{\beta}\cdot$

Therefore, the second integral is uniformly bounded in view of (1.4) and the
claim is proved.

Case 3. $c>0$ and $u\neq 0$. In this case, we claim that $\mathcal{J}(u)=c$
and therefore, we get

$\lim_{n\rightarrow+\infty}\|u_{n}\|^{2}=2\big{(}c+\int_{B}F(x,u)dx\big{)}=2\big{(}\mathcal{J}(u)+\int_{B}F(x,u)dx\big{)}=\|u\|^{2}.$

Do not forgot that

$\mathcal{J}(u)\leq\frac{1}{2}\liminf_{n\rightarrow+\infty}\|u_{n}\|^{2}-\int_{B}F(x,u)dx=c.$

Suppose that $\mathcal{J}(u)<c$. Then,

$\|u\|^{\gamma}<(2\big{(}c+\int_{B}F(x,u)dx\big{)}\big{)}^{\frac{\gamma}{2}}.$
(3.18)

Set

$v_{n}=\displaystyle\frac{u_{n}}{\|u_{n}\|}$

and

$v=\displaystyle\frac{u}{(2\big{(}c+\displaystyle\int_{B}F(x,u)dx\big{)})^{\frac{1}{2}}}\cdot$

We have $\|v_{n}\|=1$, $v_{n}\rightharpoonup v$ in $\mathbf{W}$, $v\not\equiv
0$ and $\|v\|<1$. So, by Theorem 3.1, we get

$\displaystyle\sup_{n}\int_{B}e^{p\alpha_{\beta}|v_{n}|^{\gamma}}dx<\infty,$

for $1<p<U(v)=(1-\|v\|^{2})^{\frac{-\gamma}{2}}$.
As in the case $(2)$, we are going to estimate
$\displaystyle\int_{B}|f(x,u_{n})|^{q}dx$.
For $\varepsilon>0$, one has

$\begin{array}[]{rclll}\displaystyle\int_{B}|f(x,u_{n})|^{q}dx&=&\displaystyle\int_{\\{|u_{n}|\leq
t_{\varepsilon}}|f(x,u_{n})|^{q}dx+\int_{\\{|u_{n}|>t_{\varepsilon}\\}}|f(x,u_{n})|^{q}dx\\\
&\leq&\displaystyle
2\pi^{2}\max_{B\times[-t_{\varepsilon},t_{\varepsilon}]}|f(x,t)|^{q}+C\int_{B}e^{\alpha_{0}(1+\varepsilon)|u_{n}|^{\gamma}}dx\\\
&\leq&C_{\varepsilon}+C\displaystyle\int_{B}e^{\alpha_{0}(1+\varepsilon)\|u_{n}\|^{\gamma}|v_{n}|^{\gamma}}\big{)}dx\leq
C,\end{array}$

provided that $\alpha_{0}(1+\varepsilon)\|u_{n}\|^{\gamma}\leq
p~{}~{}\alpha_{\beta}$ and $1<p<U(v)=(1-\|v\|^{2})^{\frac{-\gamma}{2}}$.
Since

$\displaystyle(1-\|v\|^{2})^{\frac{-\gamma}{2}}=\displaystyle\big{(}\frac{2(c+\int_{B}F(x,u)dx)}{2(c+\int_{B}F(x,u)dx)-\|u\|^{2})}\big{)}^{\frac{\gamma}{2}}=\big{(}\frac{c+\int_{B}F(x,u)dx}{c-\mathcal{J}(u)}\big{)}^{\frac{\gamma}{2}}$

and

$\displaystyle\lim_{n\rightarrow+\infty}\|u_{n}\|^{\gamma}=(2\big{(}c+\int_{B}F(x,u)dx)\big{)}^{\frac{\gamma}{2}},$

then, for large $n$,

$\alpha_{0}(1+\varepsilon)\|u_{n}\|^{\gamma}\leq\alpha_{0}(1+2\varepsilon)(2\big{(}c+\int_{B}F(x,u)dx\big{)}^{\frac{\gamma}{2}}.$

But $\mathcal{J}(u)\geq 0$ and
$c<\displaystyle\frac{1}{2}(\frac{\alpha_{\beta}}{\alpha_{0}})^{\frac{2}{\gamma}}$
, then if we choose $\varepsilon>0$ small enough such that

$\frac{\alpha_{0}}{\alpha_{\beta}}(1+2\varepsilon)<\displaystyle\big{(}\frac{1}{2(c-\mathcal{J}(u))}\big{)}^{\frac{\gamma}{2}},$

we get,

$(1+2\varepsilon)\displaystyle\big{(}(c-\mathcal{J}(u)\big{)}^{\frac{\gamma}{2}}<\frac{\alpha_{\beta}}{2^{\frac{\gamma}{2}}\alpha_{0}}\cdot$

So, the sequence $(f(x,u_{n}))$ is bounded in $L^{q}$, $q>1$.
Since $\langle\mathcal{J}^{\prime}(u_{n}),(u_{n}-u)\rangle=o(1)$, we have from
the boundedness of $\\{f(x,u_{n})\\}$ in $L^{q}(B)$ for $q>1$, we can prove
that $u_{n}\rightarrow u$ strongly in $\mathbf{W}$. Indeed, we have

$\|u_{n}-u\|^{2}=\langle u_{n},u_{n}-u\rangle-\langle u,u_{n}-u\rangle=\langle
u_{n},u_{n}-u\rangle+o_{n}(1)\rightarrow 0~{}~{}as~{}~{}n\rightarrow+\infty.$

From (3.6) and using the Hölder inequality, we get

$\begin{array}[]{rclllll}\displaystyle|<u_{n},u_{n}-u>|&\leq&\displaystyle\varepsilon_{n}\|u_{n}-u\|+\big{|}\int_{B}f(x,u_{n})(u_{n}-u)dx\big{|}\\\
&\leq&C\displaystyle\varepsilon_{n}+\big{(}\int_{B}|f(x,u_{n}|^{q}dx\big{)}^{\frac{1}{q}}(\int_{B}|u_{n}-u|^{q^{\prime}})^{\frac{1}{q^{\prime}}}dx\rightarrow
0~{}~{}as~{}~{}n\rightarrow+\infty.\end{array}$

Hence,

$\displaystyle\lim_{n\rightarrow+\infty}\|u_{n}\|^{2}=2(c+\int_{B}F(x,u)dx)=\|u\|^{2}$

and this contradicts (3.18). So, $\mathcal{J}(u)=c$ and consequently,
$u_{n}\rightarrow u$.

∎

##  4 Proof of Theorem 1.3

In the sequel, we will estimate the minimax level of the energy $\mathcal{J}$.
We will prove that the mountain pass level $c$ in Theorem 2.1 satisfies

$c<\displaystyle\frac{1}{2}(\frac{\alpha_{\beta}}{\alpha_{0}})^{\frac{2}{\gamma}}\cdot$

For this purpose , we will prove that there exists $v_{0}\in\mathbf{W}$ such

$\max_{t\geq
0}\mathcal{J}(tv_{0})<\displaystyle\frac{1}{2}(\frac{\alpha_{\beta}}{\alpha_{0}})^{\frac{2}{\gamma}}\cdot$
(4.1)

###  4.1 Adams functions

Now, we will construct particular functions, namely the Adams functions. We
consider the sequence defined for all $n\geq 3$ by

(4.2)
$w_{n}(x)=\displaystyle\left\\{\begin{array}[]{rcllll}&\displaystyle\bigg{(}\frac{\log(e\sqrt[4]{n})}{\alpha_{\beta}}\bigg{)}^{\frac{1}{\gamma}}-\frac{|x|^{2(1-\beta)}}{2\big{(}\frac{\alpha_{\beta}}{4n}\big{)}^{\frac{1}{\gamma}}\big{(}\log(e\sqrt[4]{n})\big{)}^{\frac{\gamma-1}{\gamma}}}+\frac{1}{2(\frac{\alpha_{\beta}}{4})^{\frac{1}{\gamma}}\big{(}\log(e\sqrt[4]{n})\big{)}^{\frac{\gamma-1}{\gamma}}}&\mbox{
if }0\leq|x|\leq\frac{1}{\sqrt[4]{n}}\\\ \\\ \\\
&\displaystyle\frac{\bigg{(}\log(\frac{e}{|x|})\bigg{)}^{1-\beta}}{\bigg{(}\frac{\alpha_{\beta}}{4}\log(e\sqrt[4]{n})\bigg{)}^{\frac{1}{\gamma}}}&\mbox{
if }\frac{1}{\sqrt[4]{n}}\leq|x|\leq\frac{1}{2}\\\ &\zeta_{n}&\mbox{ if
}\frac{1}{2}\leq|x|\leq 1\end{array}\right.$

where $\zeta_{n}\in C^{\infty}_{0}(B)$ is such that
$\zeta_{n}(\frac{1}{2})=\frac{1}{\big{(}\frac{\alpha_{\beta}}{16}\log(e^{4}n)\big{)}^{\frac{1}{\gamma}}}\big{(}\log
2e\big{)}^{1-\beta}$, $\displaystyle\frac{\partial\zeta_{n}}{\partial
x}(\frac{1}{2})=\frac{-2(1-\beta)}{\bigg{(}\frac{\alpha_{\beta}}{4}\log(e\sqrt[4]{n})\bigg{)}^{\frac{1}{\gamma}}}\big{(}\log(2e)\big{)}^{-\beta}$
$\displaystyle\zeta_{n}=\frac{\partial\zeta_{n}}{\partial
x}=0~{}~{}\mbox{on}~{}~{}\partial B$ and the functions $\xi_{n}$,
$\nabla\xi_{n}$, $\Delta\xi_{n}$ are all $\displaystyle
o\bigg{(}\frac{1}{\log(e\sqrt[4]{n})}\bigg{)}$.
Let $v_{n}(x)=\displaystyle\frac{w_{n}}{\|w_{n}\|}$. We have,
$v_{n}\in\mathbf{W}$ , $\|v_{n}\|^{2}=1.$

We compute $\Delta w_{n}(x)$, we get

$\Delta
w_{n}(x)=\displaystyle\left\\{\begin{array}[]{rclll}\displaystyle\frac{-(1-\beta)(4-2\beta)|x|^{-2\beta}}{\big{(}\frac{\alpha_{\beta}}{4n}\big{)}^{\frac{1}{\gamma}}\big{(}\log(e\sqrt[4]{n})\big{)}^{\frac{\gamma-1}{\gamma}}}\mbox{
}\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }&\mbox{ if
}0\leq|x|\leq\frac{1}{\sqrt[4]{n}}\\\ \\\
\displaystyle\frac{-(1-\beta)\bigg{(}\log(\frac{e}{|x|})\bigg{)}^{-\beta}\bigg{(}2+\beta\big{(}\log\frac{e}{|x|}\big{)}^{-1}\bigg{)}}{\bigg{(}\frac{\alpha_{\beta}}{4}\log(e\sqrt[4]{n})\bigg{)}^{\frac{1}{\gamma}}}&\mbox{
if }\frac{1}{\sqrt[4]{n}}\leq|x|\leq\frac{1}{2}\\\ \\\
\triangle\zeta_{n}\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{
}\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{
}\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{
}\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }&\mbox{ if }\frac{1}{2}\leq|x|\leq
1\end{array}\right.$

So,

$\|w_{n}\|^{2}=\underbrace{2\pi^{2}\int^{\frac{1}{\sqrt[4]{n}}}_{0}r^{3}|\Delta
w_{n}(x)|^{2}\big{(}\log\frac{e}{r}\big{)}^{\beta}dr}_{I_{1}}+\underbrace{2\pi^{2}\int^{\frac{1}{2}}_{\frac{1}{\sqrt[4]{n}}}r^{3}|\Delta
w_{n}(x)|^{2}\big{(}\log\frac{e}{r}\big{)}^{\beta}dr}_{I_{2}}+\underbrace{2\pi^{2}\int^{1}_{\frac{1}{2}}r^{3}|\Delta
w_{n}(x)|^{2}\big{(}\log\frac{e}{r}\big{)}^{\beta}dr}_{I_{3}}$

we have,

$\begin{array}[]{rclll}\displaystyle I_{1}&=&\displaystyle
2\pi^{2}\frac{(1-\beta)^{2}(4-2\beta)^{2}}{\big{(}\frac{\alpha_{\beta}}{4n}\big{)}^{\frac{2}{\gamma}}\big{(}\log(e\sqrt[4]{n})\big{)}^{\frac{2(\gamma-1)}{\gamma}}}\int^{\frac{1}{\sqrt[4]{n}}}_{0}r^{3-4\beta}\big{(}\log\frac{e}{r}\big{)}^{\beta}dr\\\
&=&\displaystyle
2\pi^{2}\frac{(1-\beta)^{2}(4-2\beta)^{2}}{\big{(}\frac{\alpha_{\beta}}{4n}\big{)}^{\frac{2}{\gamma}}\big{(}\log(e\sqrt[4]{n})\big{)}^{\frac{2(\gamma-1)}{\gamma}}}\left[\frac{r^{4-4\beta}}{4-4\beta}(\log\frac{e}{r}\big{)}^{\beta}\right]^{\frac{1}{\sqrt[4]{n}}}_{0}\\\
&+&2\pi^{2}\frac{\beta(1-\beta)^{2}(4-2\beta)^{2}}{\big{(}\frac{\alpha_{\beta}}{4n}\big{)}^{\frac{2}{\gamma}}\big{(}\log(e\sqrt[4]{n})\big{)}^{\frac{2(\gamma-1)}{\gamma}}}\displaystyle\int^{\frac{1}{\sqrt[4]{n}}}_{0}\frac{r^{4-4\beta}}{4-4\beta}\big{(}\log\frac{e}{r}\big{)}^{\beta-1}dr\\\
&=&\displaystyle o\big{(}\frac{1}{\log e\sqrt[4]{n}}\big{)}\cdot\\\
\end{array}$

Also,

$\begin{array}[]{rclll}\displaystyle I_{2}&=&\displaystyle
2\pi^{2}\frac{(1-\beta)^{2}}{\big{(}\frac{\alpha_{\beta}}{4}\big{)}^{\frac{2}{\gamma}}\big{(}\log(e\sqrt[4]{n})\big{)}^{\frac{2}{\gamma}}}\int_{\frac{1}{\sqrt[4]{n}}}^{\frac{1}{\frac{1}{2}}}\frac{1}{r}\big{(}\log\frac{e}{r}\big{)}^{-\beta}\big{(}2+\beta\big{(}\log\frac{e}{r}\big{)}^{-1}\big{)}^{2}dr\\\
&=&\displaystyle-2\pi^{2}\frac{(1-\beta)^{2}}{\big{(}\frac{\alpha_{\beta}}{4}\big{)}^{\frac{2}{\gamma}}\big{(}\log(e\sqrt[4]{n})\big{)}^{\frac{2}{\gamma}}}\left[\frac{\beta^{2}}{-1-\beta}\big{(}\log\frac{e}{r}\big{)}^{-\beta-1}+4\big{(}\log\frac{e}{r}\big{)}^{-\beta}+\frac{4}{1-\beta}\big{(}\log\frac{e}{r}\big{)}^{1-\beta}\right]^{\frac{1}{2}}_{\frac{1}{\sqrt[4]{n}}}\\\
&=&\displaystyle 1+o\big{(}\frac{1}{(\log
e\sqrt[4]{n})^{\frac{2}{\gamma}}}\big{)}\cdot\\\ \end{array}$

and $I_{3}=\displaystyle o\big{(}\frac{1}{(\log
e\sqrt[4]{n})^{\frac{2}{\gamma}}}\big{)}.$ Then
$\|w_{n}\|^{2}=1+o\big{(}\frac{1}{(\log
e\sqrt[4]{n})^{\frac{2}{\gamma}}}\big{)}$. Also,
for $0\leq|x|\leq\frac{1}{\sqrt[4]{n}}$, $\displaystyle
v^{\gamma}_{n}(x)\geq\displaystyle\bigg{(}\frac{\log(e\sqrt[4]{n})}{\alpha_{\beta}}\bigg{)}+o(1)\cdot$

###  4.2 Min-Max level estimate

We are going to the desired estimate.

######  Lemma 5.

For the sequence $(v_{n})$ identified by (4.1), there exists $n\geq 1$ such
that

$\max_{t\geq
0}\mathcal{J}(tv_{n})<\displaystyle\frac{1}{2}(\frac{\alpha_{\beta}}{\alpha_{0}})^{\frac{2}{\gamma}}\cdot$
(4.3)

###### Proof.

By contradiction, suppose that for all $n\geq 1$,

$\max_{t\geq
0}\mathcal{J}(tv_{n})\geq\displaystyle\frac{1}{2}(\frac{\alpha_{\beta}}{\alpha_{0}})^{\frac{2}{\gamma}}\cdot$

Therefore, for any $n\geq 1$, there exists $t_{n}>0$ such that

$\max_{t\geq
0}\mathcal{J}(tv_{n})=\mathcal{J}(t_{n}v_{n})\geq\displaystyle\frac{1}{2}(\frac{\alpha_{\beta}}{\alpha_{0}})^{\frac{2}{\gamma}}$

and so,

$\frac{1}{2}t_{n}^{2}-\int_{B}F(x,t_{n}v_{n})dx\geq\displaystyle\frac{1}{2}(\frac{\alpha_{\beta}}{\alpha_{0}})^{\frac{2}{\gamma}}\cdot$

Then, by using $(H_{1})$

$t_{n}^{2}\geq\displaystyle(\frac{\alpha_{\beta}}{\alpha_{0}})^{\frac{2}{\gamma}}\cdot$
(4.4)

On the other hand,

$\frac{d}{dt}\mathcal{J}(tv_{n})\big{|}_{t=t_{n}}=t_{n}-\int_{B}f(x,t_{n}v_{n})v_{n}dx=0,$

that is

$t_{n}^{2}=\int_{B}f(x,t_{n}v_{n})t_{n}v_{n}dx.$ (4.5)

Now, we claim that the sequence $(t_{n})$ is bounded in $(0,+\infty)$.
Indeed, it follows from $(H_{5})$ that for all $\varepsilon>0$, there exists
$t_{\varepsilon}>0$ such that

$f(x,t)t\geq(\gamma_{0}-\varepsilon)e^{\alpha_{0}t^{\gamma}}~{}~{}\forall|t|\geq
t_{\varepsilon},~{}~{}\mbox{uniformly in}~{}~{}x\in B.$ (4.6)

Using (4.4) and (4.5), we get

$t^{2}_{n}=\int_{B}f(x,t_{n}v_{n})t_{n}v_{n}dx\geq\int_{0\leq|x|\leq\frac{1}{\sqrt[4]{n}}}f(x,t_{n}v_{n})t_{n}v_{n}dx\cdot$

Since

$\frac{t_{n}}{\|w_{n}\|}\big{(}\frac{\log
e\sqrt[4]{n}}{\alpha_{\beta}}\big{)}^{\frac{1}{\gamma}}\rightarrow\infty~{}~{}\mbox{as}~{}~{}n\rightarrow+\infty,$

then it follows from (4.6) that for all $\varepsilon>0$, there exists $n_{0}$
such that for all $n\geq n_{0}$

$t^{2}_{n}\geq\displaystyle(\gamma_{0}-\varepsilon)\int_{0\leq|x|\leq\frac{1}{\sqrt[4]{n}}}e^{\alpha_{0}t^{\gamma}_{n}v^{\gamma}_{n}}dx$
$t^{2}_{n}\geq\displaystyle
2\pi^{2}(\gamma_{0}-\varepsilon)\int_{0}^{\frac{1}{\sqrt[4]{n}}}r^{3}e^{\alpha_{0}t^{\gamma}_{n}(\displaystyle\big{(}\frac{\log(e\sqrt[4]{n})}{\alpha_{\beta}}\big{)}+o(1)}dr.$
(4.7)

Hence,

$1\geq\displaystyle 2\pi^{2}(\gamma_{0}-\varepsilon)~{}~{}\displaystyle
e^{\alpha_{0}t^{\gamma}_{n}(\displaystyle\big{(}\frac{\log(e\sqrt[4]{n})}{\alpha_{\beta}}\big{)}+o(1))-3\log
n-2\log t_{n}}.$

Therefore $(t_{n})$ is bounded. Also, we have from the formula (4.5) that

$\displaystyle\lim_{n\rightarrow+\infty}t_{n}^{2}\geq\displaystyle(\frac{\alpha_{\beta}}{\alpha_{0}})^{\frac{2}{\gamma}}\cdot$

Now, suppose that

$\displaystyle\lim_{n\rightarrow+\infty}t_{n}^{2}>\displaystyle(\frac{\alpha_{\beta}}{\alpha_{0}})^{\frac{2}{\gamma}},$

then for $n$ large enough, there exists some $\delta>0$ such that
$t_{n}^{\gamma}\geq\frac{\alpha_{\beta}}{\alpha_{0}}+\delta$. Consequently the
right hand side of (4.7) tends to infinity and this contradicts the boudness
of $(t_{n})$. Since $(t_{n})$ is bounded, we get

$\displaystyle\lim_{n\rightarrow+\infty}t_{n}^{2}=\displaystyle(\frac{\alpha_{\beta}}{\alpha_{0}})^{\frac{2}{\gamma}}\cdot$
(4.8)

Let

$\mathcal{A}_{n}=\\{x\in B|t_{n}v_{n}\geq
t_{\varepsilon}\\}~{}~{}\mbox{and}~{}~{}\mathcal{C}_{n}=B\setminus\mathcal{A}_{n},$
$\begin{array}[]{rclll}t_{n}^{2}&=&\displaystyle\int_{B}f(x,t_{n}v_{n})t_{n}v_{n}dx=\int_{\mathcal{A}_{n}}f(x,t_{n}v_{n})t_{n}v_{n}dx+\int_{\mathcal{C}_{n}}f(x,t_{n}v_{n})t_{n}v_{n}$$\\\
&\geq&\displaystyle(\gamma_{0}-\varepsilon)\int_{\mathcal{A}_{n}}e^{\alpha_{0}t_{n}^{\gamma}v_{n}^{\gamma}}dx+\int_{\mathcal{C}_{n}}f(x,t_{n}v_{n})t_{n}v_{n}dx\\\
&=&\displaystyle(\gamma_{0}-\varepsilon)\int_{B}e^{\alpha_{0}t_{n}^{\gamma}v_{n}^{\gamma}}dx-(\gamma_{0}-\varepsilon)\int_{\mathcal{C}_{n}}e^{\alpha_{0}t_{n}^{\gamma}v_{n}^{\gamma}}dx\\\
&+&\displaystyle\int_{\mathcal{C}_{n}}f(x,t_{n}v_{n})t_{n}v_{n}dx.\end{array}$

Since $v_{n}\rightarrow 0~{}~{}\mbox{a.e in }~{}~{}B$,
$\chi_{\mathcal{C}_{n}}\rightarrow 1~{}~{}\mbox{a.e in}~{}~{}B$, therefore
using the dominated convergence theorem, we get

$\displaystyle\int_{\mathcal{C}_{n}}f(x,t_{n}v_{n})t_{n}v_{n}dx\rightarrow
0~{}~{}\mbox{and}~{}~{}\int_{\mathcal{C}_{n}}e^{\alpha_{0}t_{n}^{\gamma}v_{n}^{\gamma}}dx\rightarrow\frac{\pi^{2}}{2}\cdot$

Then,

$\
\lim_{n\rightarrow+\infty}t_{n}^{2}=\displaystyle(\frac{\alpha_{\beta}}{\alpha_{0}})^{\frac{2}{\gamma}}\geq(\gamma_{0}-\varepsilon)\lim_{n\rightarrow+\infty}\int_{B}e^{\alpha_{0}t_{n}^{\gamma}v_{n}^{\gamma}}dx-(\gamma_{0}-\varepsilon)\frac{\pi^{2}}{2}\cdot$

On the other hand,

$\int_{B}e^{\alpha_{0}t_{n}^{\gamma}v_{n}^{\gamma}}dx\geq\int_{\frac{1}{\sqrt[4]{n}}\leq|x|\leq\frac{1}{2}}e^{\alpha_{0}t_{n}^{\gamma}v_{n}^{\gamma}}dx+\int_{\mathcal{C}_{n}}e^{\alpha_{0}t_{n}^{\gamma}v_{n}^{\gamma}}dx\cdot$

Then, using (4.4)

$\lim_{n\rightarrow+\infty}t_{n}^{2}\geq\lim_{n\rightarrow+\infty}\displaystyle(\gamma_{0}-\varepsilon)\int_{B}e^{\alpha_{0}t_{n}^{\gamma}v_{n}^{\gamma}}dx\geq\displaystyle\lim_{n\rightarrow+\infty}(\gamma_{0}-\varepsilon)2\pi^{2}\int^{\frac{1}{2}}_{\frac{1}{\sqrt[4]{n}}}r^{3}e^{\frac{4\big{(}\log\frac{e}{r}\big{)}^{2}}{\log(e\sqrt[4]{n})\|w_{n}\|^{\gamma}}}dr$

and, making the change of variable

$s=\frac{4\log\frac{e}{r}}{\log(e\sqrt[4]{n})\|w_{n}\|^{\gamma}},$

we get

$\begin{array}[]{rclll}\displaystyle\lim_{n\rightarrow+\infty}t_{n}^{2}&\geq&\displaystyle\lim_{n\rightarrow+\infty}\displaystyle(\gamma_{0}-\varepsilon)\int_{B}e^{\alpha_{0}t_{n}^{\gamma}v_{n}^{\gamma}}dx\\\
\\\
&\geq&\displaystyle\lim_{n\rightarrow+\infty}2\pi^{2}(\gamma_{0}-\varepsilon)\frac{\|w_{n}\|^{\gamma}\log(e\sqrt[4]{n})}{4}e^{4}\int^{\frac{4}{\|w_{n}\|^{\gamma}}}_{\frac{4\log
2e}{\|w_{n}\|^{\gamma}\log(e\sqrt[4]{n})}}\displaystyle
e^{\frac{\|w_{n}\|^{\gamma}\log(e\sqrt[4]{n})}{4}~{}(s^{2}-4s)}ds\\\ \\\
&\geq&\displaystyle\lim_{n\rightarrow+\infty}2\pi^{2}(\gamma_{0}-\varepsilon)\frac{\|w_{n}\|^{\gamma}\log(e\sqrt[4]{n})}{4}e^{4}\int^{\frac{4}{\|w_{n}\|^{\gamma}}}_{\frac{4\log
2e}{\|w_{n}\|^{\gamma}\log(e\sqrt[4]{n})}}e^{-\frac{\|w_{n}\|^{\gamma}\log(e\sqrt[4]{n})}{4}~{}4s}ds\\\
&=&\displaystyle\lim_{n\rightarrow+\infty}(\gamma_{0}-\varepsilon)\frac{\pi^{2}}{2}e^{4}(-e^{-4\log
e\sqrt[4]{n}}+e^{-4\log(2e)}\big{)}\\\
&=&\displaystyle(\gamma_{0}-\varepsilon)\frac{\pi^{2}e^{4(1-\log
2e)}}{2}=\displaystyle(\gamma_{0}-\varepsilon)\frac{\pi^{2}}{32}\cdot\end{array}$

It follows that

$\displaystyle(\frac{\alpha_{\beta}}{\alpha_{0}})^{\frac{2}{\gamma}}\geq(\gamma_{0}-\varepsilon)\frac{\pi^{2}}{32}$

for all $\varepsilon>0$. So,

$\gamma_{0}\leq\frac{1024(1-\beta)}{\alpha^{1-\beta}_{0}},$

which is in contradiction with the condition $(H_{5})$.
Now by Proposition 3.1, the functional $\mathcal{J}$ satisfies the $(PS)$
condition at a level
$c<\displaystyle\frac{1}{2}(\frac{\alpha_{\beta}}{\alpha_{0}})^{\frac{2}{\gamma}}$
. So, by Lemma 3 and Lemma 4, we deduce that the functional $\mathcal{J}$ has
a nonzero critical point $u$ in $\mathbf{W}$. From maximum principle, the
solution $u$ of the problem (1.1) is positive. The Theorem 1.3 is proved. ∎

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