\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2014 (2014), No. 228, pp. 1--12.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2014 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2014/228\hfil Quasilinear equations of $N$-biharmonic type]
{Solutions to quasilinear equations of $N$-biharmonic type with degenerate coercivity}

\author[S. Aouaoui \hfil EJDE-2014/228\hfilneg]
{Sami Aouaoui}  % in alphabetical order

\address{Sami Aouaoui \newline
Institut Sup\'erieur des Math\'ematiques Appliqu\'ees et
de l'Informatique de Kairouan  \\
Avenue Assad Iben Fourat, 3100 Kairouan, Tunisie \newline
Phone +216 77 226 575; Fax +216 77 226 575}
\email{sami\_aouaoui@yahoo.fr}

\thanks{Submitted June 22, 2014. Published October 27, 2014.}
\subjclass[2000]{35D30, 35J20, 35J61, 58E05}
\keywords{Variational method; $N$-biharmonic operator; mountain pass theorem;
\hfill\break\indent  minimax level; iterative scheme; critical point}

\begin{abstract}
 In this article we show the existence of multiple solutions for 
 quasilinear equations in divergence form with degenerate coercivity.
 Our strategy is to combine a variational method and an iterative technique
 to obtain the solutions.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{definition}[theorem]{Definition}
\allowdisplaybreaks

\section{Introduction and statement of main results}

In this article, we study the  quasilinear equation
\begin{equation} \label{eP}
 - \operatorname{div}\big(a(x,u) | \nabla u|^{N-2} \nabla u\big)
+ V(x) |u|^{N-2} u + \Delta^2_N u=  f(x,u) +h(x),\quad x \in \mathbb{R}^N,
\end{equation}
where $ N \geq 2$, $\Delta^2_N u = \Delta (| \Delta u|^{N-2} \Delta u)$, and
$ h \in L^{N'}( \mathbb{R}^N)$, $N'= \frac{N}{N-1}$, $h \neq 0$ and $ h \geq 0$.
Concerning the functions $V$, $f$ and $ a$, we have the following assumptions:
 \begin{itemize}

\item[(V1)] $V: \mathbb{R}^N \to \mathbb{R} $ is a continuous function such that
 $$
V(x) \geq V_0 > 0 ,\quad \forall x \in \mathbb{R}^N,
$$
where $ V_0 $ is a positive constant.

\item[(V2)]  For every $ M > 0$,
 $\operatorname{meas}(\{x \in \mathbb{R}^N,\ V(x) \leq M\}) < + \infty$,
where ``meas'' denotes the Lebesgue measure in $ \mathbb{R}^N$.

\item[(H1)] $f : \mathbb{R}^N \times \mathbb{R} \to \mathbb{R} $
is a Carath\'eodory function. We assume that for every positive real number
$ k > 0$, there exist two positive constants $ \alpha_k > N-1 $ and
$ C_k > 0 $ such that
$$
|f(x,s)| \leq C_k |s|^{\alpha_k}, \quad \text{a.e. $x \in \mathbb{R}^N$ and for all
$s \in \mathbb{R}$ with $|s| \leq k$}.
$$

\item[(H2)] There exists $ \nu > N $ such that
$$
 \nu F(x,s) \leq f(x,s) s,\quad \forall (x,s) \in \mathbb{R}^N \times \mathbb{R},
 \text{ where } F(x,s) = \int_0^s f(x,t) dt.
$$

\item[(H3)]  There exist two real numbers $ A > 0 $ and $ p > N$, such that
$$
F(x, s) \geq A s^p,\quad \text{a.e. }x \in \mathbb{R}^N,\; \forall s \geq 0.
$$

\item[(H4)] There exist two positive constants $ \beta_0 $ and $ \beta_1 $ such that
$$
|f(x,s_1) -f(x,s_2)| \leq \beta_0 z^{\beta_1} |s_1 -s_2|^{N-1},
$$
a.e $ x \in \mathbb{R}^N$, $\forall z \in [0,1]$ and $\forall s_1,s_2 \in [-z,z]$.

\item[(H5)] $ a: \mathbb{R}^N \times \mathbb{R} \to \mathbb{R} $ is a continuous
function satisfying the following property: for every $ k > 0$, there exist
$ 0 < a_k < a'_k < + \infty $ such that
$$
a_k \leq a(x,s) \leq a'_k,\quad \forall x \in \mathbb{R}^N\ \text{ and }
 |s| \leq k.
$$

\item[(H6)] There exists a constant $ L > 0 $ such that
$$
|a(x,s_1) -a(x,s_2)| \leq L |s_1 -s_2|^{N-1},\quad \text{a.e. }x \in \mathbb{R}^N,\;
 \forall s_1,s_2 \in [-1,1].
$$
\end{itemize}

\subsection*{Examples}
  When $N = 2$, for $ f$, we can choose:
\begin{itemize}
\item[(1)] $ f(x,s) = \lambda |s|^{\alpha-1} s$, $\lambda > 0$, $\alpha > 1$.
\item[(2)] $ f(x,s) = \lambda |s|^{\alpha-1} s + |s|^{\beta-1} s (e^{p_0 s^2} -1)$,
$ \lambda > 0$, $\alpha > 1$, $\beta > 1 $ and $ p_0 > 0$.
\end{itemize}
For $ a$, we can choose:
\begin{itemize}
\item[(1)] $ a(x,s) = 1 + |s|^{\sigma-1} s,\ \sigma > 1$.
\item[(2)] $ a(x,s) = \frac{1}{1+ s^2}$.
\end{itemize}

Many articles about  problems similar to \eqref{eP}, having a divergence
part of the form $ -\operatorname{div}(A(x,u) | \nabla u|^{p-2} \nabla u) $
with degenerate coercivity, have been published.  Among them, 
the following model is of special interest:
$$
-\operatorname{div}( \frac{| \nabla u|^{p-2} \nabla u}{(1 + |u|)^q})
= f\ \text{in}\ \Omega,
$$
where $ \Omega $ is some open (bounded in the majority of cases) domain of
 $ \mathbb{R}^N$, $N \geq 2$, $q > 0$, $p > 1 $ and $ f $ is   datum
satisfying some summability condition.
See for example \cite{a4,b1,b2,b3,b4,b5} and references therein.
We want to mention also the model
$$
-\operatorname{div}(A(x,u)| \nabla u|^{p(x)-2} \nabla u) + |u|^{p(x)-2} u
= Z(x, u ,  \nabla u),\quad \text{in } \mathbb{R}^N,\; N \geq 3,
$$
where $ p $ is some bounded and Lipschitz continuous function.
This model was studied in \cite{a6} in the very special framework of the generalized
Sobolev space with variable exponents. In the previously cited works,
the authors use approximations in order to overcome the lack of coercivity.
Then, establish a priori estimates on the sequence of approximative solutions, 
and then use the passage to the limit to finally obtain a weak
solution for the initial equation.

 In this article, we develop a new method to deal with such kind of problems.
The main idea in this new method is inspired by the work \cite{f1}.
In  \cite{f1},   de Figueiredo,  Girardi and Matzeu considered the
 semilinear elliptic  equation
\begin{equation} \label{e1}
\begin{gathered}
- \Delta u  = f(x,u, \nabla u)\quad \text{in } \Omega, \\
 u = 0\quad \text{on } \partial \Omega,
\end{gathered}
\end{equation}
where $ \Omega $ is a bounded smooth domain of $ \mathbb{R}^N$, $N \geq 3$.
Because the dependence of the nonlinearity on the gradient of the solution,
 \eqref{e1} is non-variational and a direct attack to it using critical point
theory is not possible. The new approach by de Figueiredo,  Girardi and  Matzeu
consists of associating with \eqref{e1} a family of semilinear elliptic problems
 with no dependence on the gradient. Namely, for each $ w \in H_0^1( \Omega)$,
they considered the problem
\begin{equation} \label{e2}
\begin{gathered}
- \Delta u  = f(x,u, \nabla w)\quad \text{in } \Omega, \\
 u = 0\quad \text{on } \partial \Omega.
\end{gathered}
\end{equation}
Thus, the authors have ``frozen'' the gradient term. Problem \eqref{e2} is of
variational nature and could be treated by variational method.
 We have to mention that this idea was used in some later works dealing also
with nonlinear problems involving nonlinearities with dependence on the gradient.
 We can cite \cite{f2,g1,g2,l6,m1,s1}.

 In this article, we try to use this idea to discuss a completely different
kind of problem. In fact, in the problem \eqref{eP} and in contrast with \eqref{e1}
(and similar equations), the nonvariational nature is not due to the dependence
of the right-hand term on the gradient of the solution but it is in reality
due to the presence of the coefficient $ a(x,u) $ in the divergence part.
Hence, we will try to ``freeze'' the term $ a(x,u)$. The ``associated''
problem will be variational and consequently could be treated using the
critical point theory. An iterative scheme will be performed in order to
obtain weak solutions for the initial problem \eqref{eP}.
This method allows us to obtain a multiplicity result which, knowing that in
the majority of cases the classical nonvariational methods give the existence
of one solution, could be seen as an interesting result.

 The existence of the $ N-$biharmonic operator, $ \Delta^2_{N}$, is remarkable.
The importance of studying fourth-order equations lies in the fact that they
can describe some physical phenomena as the deformations of an elastic beam
in equilibrium state (see \cite{l4,w2}).
Laser and  McKenna \cite{l3} pointed out that this type of nonlinearity
provides a model to study travelling waves in suspension bridges.
For this reason, there is a wide literature that deals with existence
and multiplicity of solutions for nonlinear fourth-order elliptic problems
in bounded and unbounded domains. See for example
\cite{h1,k1,l5,m2}  and references therein.
On the other hand, the study of nonlinear equations involving the
$N$-Laplacian operator, $ N \geq 2$, which is a borderline case for the
Sobolev embedding, could be considered as one of the most interesting topics
of research during last decades. A special interest has been given to equation
 of $ N$-Laplacian type containing nonlinear terms which have a subcritical
or critical exponential growth. See \cite{a1,a2,a3,c1,f3,l1,l2,o1,o2,o3,o4,t1,w1}
and references therein.
Here, we highlight the fact that in the present work we deal with a more
general type of nonlinearity which includes the case of exponential growth.

The appropriate space in which the problem \eqref{eP} will be studied is the
 subspace of $ W^{2,N}( \mathbb{R}^N)$,
$$
 E= \Big\{u \in W^{2,N}( \mathbb{R}^N):
 \int_{\mathbb{R}^N} V(x) |u|^N \,dx < + \infty\Big\},
$$
which is a Banach reflexive space equipped with the norm
$$
\|u\| = \Big(\int_{\mathbb{R}^N} (| \nabla u|^N + V(x) |u|^N + | \Delta u|^N)\,dx
\Big)^{1/N}.
$$
In view of (V1), we clearly have
$$
E \hookrightarrow W^{2,N}( \mathbb{R}^N) \hookrightarrow
 L^{q}( \mathbb{R}^N),\quad \forall N \leq q \leq +\infty.
$$
Also there exists a positive constant $ \delta_0 > 0 $ such that
\begin{equation}
|u|_{L^{\infty}( \mathbb{R}^N)} \leq \delta_0 \|u\|,\quad \forall u \in E.
\label{e1.1}
\end{equation}
Furthermore, since (V2) holds, we obtain (see \cite{o2})
the compactness of the embedding
$$
E \hookrightarrow L^p( \mathbb{R}^N),\quad \text{for all } p \geq N.
$$
This compact embedding will be crucial in the proof of our multiplicity result.

 \begin{definition} \label{def1.1}\rm
 A function $ u \in E$ is said to be a weak solution of the problem \eqref{eP}
if it satisfies
\begin{align*}
&\int_{ \mathbb{R}^N}  a(x,u) |\nabla u |^{N-2} \nabla u \nabla v \,dx
+ \int_{\mathbb{R}^N}  V(x)|u|^{N-2} uv \,dx + \int_{\mathbb{R}^N} | \Delta u|^{N-2}
\Delta u \Delta v \,dx \\
& =  \int_{\mathbb{R}^N} f(x,u) v \,dx + \int_{\mathbb{R}^N} hv \,dx,\quad
 \forall v \in E.
\end{align*}
\end{definition}
The main result in the present paper is given by the following theorem.

 \begin{theorem} \label{thm1.1}
 Assume that {\rm (V1), (V2), (H1)--(H6)} hold.
Then, there exist $ A_0 > 0 $ and $ d_0 > 0 $ with the following property:
 if $ A > A_0$, and $ |h|_{L^{N'}( \mathbb{R}^N)} < d_0$, then problem
\eqref{eP} admits at least two nontrivial weak solutions.
\end{theorem}

\section{Proof of main resutls}

The proof of Theorem \ref{thm1.1} will be divided into several steps.
First, for $ w \in E$, we introduce the functional $ I_w $ defined on $ E $ by
$$
I_{w}(u)= \int_{\mathbb{R}^N} \frac{a(x,w)| \nabla u|^N + V(x) |u|^N
+ | \Delta u|^N}{N} \,dx  -  \int_{ \mathbb{R}^N} F(x,u) \,dx
-\int_{\mathbb{R}^N} h u \,dx.
$$

 \begin{lemma} \label{lem2.1}
Assume that {\rm (V1), (V2), (H1),  (H5)} hold.
Then, there exist $ 0 < \rho < \frac{1}{\delta_0}$, $\mu > 0 $, and $ d > 0 $
independent of $ w $ such that
$$
I_{w}(u) \geq \mu,\quad \text{for } \|u\| = \rho,
$$
provided that $  \|w\| \leq \frac{1}{\delta_0} $ and
$ |h|_{L^{N'}( \mathbb{R}^N)} < d$.
\end{lemma}

 \begin{proof}
 For $ \|w\| \leq \frac{1}{\delta_0}$, by \eqref{e1.1} it yields
 $ |w|_{L^{\infty}( \mathbb{R}^N)} \leq 1 $ and by (H5) we can assert
that there exist $ 0 < a_1 < a'_1 < + \infty $ such that
\begin{equation}
a_1 \leq a(x,w(x)) \leq a'_1,\quad \forall x \in \mathbb{R}^N. \label{e2.1}
\end{equation}
For $ \|u\| \leq 1/\delta_0$, then by \eqref{e1.1} it yields
$$
|u(x)| \leq 1,\ \text{a.e.}\ x \in \mathbb{R}^N.
$$
 By (H1), we get the existence of two constants $ \alpha > N-1 $ and $ c_1 > 0 $
such that
\begin{equation}
|f(x,u(x))| \leq c_1 |u(x)|^{\alpha},\ \text{a.e}\ x \in \mathbb{R}^N. \label{e2.2}
\end{equation}
This implies
$$
\int_{\mathbb{R}^N} F(x,u) \,dx \leq c_2 \|u\|^{\alpha +1}.
$$
This inequality and \eqref{e2.1} give
 $$
 I_{w}(u) \geq \min\{1,\ a_1\} \frac{\|u\|^N}{N}
- c_2 \|u\|^{\alpha +1} - |h|_{L^{N'}( \mathbb{R}^N)} \|u\|.
$$
Since $ \alpha +1 > N$, then one can easily find
$ 0 < \rho < \min \{1,\ \frac{1}{\delta_0}\} $ small enough such that
 $$
\min\{1, a_1\}\frac{\rho^N}{N}- c_2 \rho^{1 + \alpha} \geq \min\{1, a_1\}
\frac{\rho^N}{2N}.
$$
It follows that
$$
I_{w}(u) \geq \min\{1, a_1\}\frac{ \rho^N}{2N} - |h|_{L^{N'}( \mathbb{R}^N)} \rho,
\quad \text{for } \|u\| = \rho.
$$
We complete the proof of Lemma \ref{lem2.1} by taking
$ d = \min\{1,a_1\} \frac{\rho^{N-1}}{4N} $ and
$ \mu =  \min\{1,a_1\} \frac{\rho^{N}}{4N}$.
\end{proof}

 \begin{lemma} \label{lem2.2}
 Assume that {\rm (V1), (V2), (H1), (H3), (H5)} hold.
Then, there exists $ \vartheta \in E $ independent of $ w $ such that
$ \| \vartheta\| > \rho $ and $ I_{w}( \vartheta) < 0 $ for all
$ w \in E $ with $ \|w\| \leq \frac{1}{\delta_0}$.
\end{lemma}

\begin{proof}
Let $ \varphi \in C_0^{\infty}( \mathbb{R}^N) $  be such that $ \varphi \neq 0 $
and $ \varphi \geq 0$. For $ t > 0$, we have
\begin{align*}
I_{w}(t \varphi)  
&\leq \max\{1, a'_1\}\frac{t^N}{N} \| \varphi\|^N
- \int_{\mathbb{R}^N} F(x,t \varphi) \,dx\\
&\leq \max\{1, a'_1\}\frac{t^N}{N} \| \varphi\|^N -  A t^p
| \varphi |_{L^p( \mathbb{R}^N)}^p.
\end{align*}
Since $ p > N$, we have
$$
\max\{1,a'_1\}\frac{t^N}{N} \| \varphi\|^N - A t^p |
\varphi |_{L^p( \mathbb{R}^N)}^p \to - \infty,\quad \text{as } t \to + \infty.
$$
Thus, we can choose $ \vartheta = t_0 \varphi $ where $ t_0 $ is large enough
such that $ \|t_{0} \varphi\| > 1 > \rho$.
This completes the proof.
\end{proof}

Now, by the Mountain Pass Theorem without the Palais-Smale condition 
(see \cite{a5,w3}),
there exists a sequence $ (u_{n,w}) \subset E $ such that $ I'_{w}(u_{n,w}) \to 0 $
and $I_{w}(u_{n,w}) \to c_{w} = \inf_{ \gamma \in \Gamma}
\sup_{0 \leq t \leq 1} I_{w}( \gamma(t))$, where
$$
\Gamma = \{ \gamma \in C([0,1],\ E),\; \gamma(0) =0,\; \gamma(1)
= t_{0} \varphi = \vartheta \}. 
$$


\begin{lemma} \label{lem2.3}
 Assume that {\rm (V1), (V2), (H1)--(H3), (H5)} hold.
Let $ w \in E $ with $ \|w\| \leq \frac{1}{\delta_0}$. Then, for every
$ 0 < \eta < \frac{1}{\delta_0}$, there exist $ A_{\eta} > 0 $ and
$ d_{\eta} > 0  $  such that:
if $ A > A_{\eta}$, and $ |h|_{L^{N'}( \mathbb{R}^N)} < d_{\eta} $
then the functional $ I_{w} $ admits a nontrivial critical point
$ u_{w } \in E  $ such that $ 0 < \mu \leq I_{w}(u_{w}) = c_{w}$,
where $ \mu $ is given by Lemma \ref{lem2.1}.
Moreover, $ \|u_{w}\|  \leq  \eta$.
\end{lemma}

 \begin{proof}
We have
$$
I_{w}(u_{n,w}) - \frac{1}{ \nu} \langle I'_{w}(u_{n,w}),\ u_{n,w}
\rangle \leq c_{w} + o_n(1) (1 + \|u_{n,w}\|).
$$
Using (H2) and \eqref{e2.1}, we have
\begin{equation}
\min\{1,a_1\}( \frac{1}{N} - \frac{1}{ \nu}) \|u_{n,w}\|^N \leq c_{w} + o_n(1)
(1 + \|u_{n,w}\|) + |h|_{L^{N'}( \mathbb{R}^N)} \|u_{n,w}\|. \label{e2.3}
\end{equation}
Then, $ (u_{n,w}) $ is a bounded sequence in $ E$. Now, by Young's inequality,
there exists $ c_3 > 0 $ such that
$$
|h|_{L^{N'}( \mathbb{R}^N)} \|u_{n,w}\| \leq \frac{\min\{1,a_1\}}{2}
( \frac{1}{N} -\frac{1}{\nu}) \|u_{n,w}\|^N +c_3 |h|_{L^{N'}( \mathbb{R}^N)}^{N'}.
 $$
Putting this inequality in \eqref{e2.3}, we obtain
$$
\frac{\min\{1,a_1\}}{2}( \frac{1}{N} - \frac{1}{ \nu}) \|u_{n,w}\|^N
\leq c_{w} + o_n(1) (1 + \|u_{n,w}\|) + c_3 |h|_{L^{N'}( \mathbb{R}^N)}^{N'}.
$$
By passing to the upper limit, we obtain
\begin{equation}
 \limsup_{n \to +\infty} \|u_{n,w}\|^N
\leq \frac{2c_{w}}{\min\{1,a_1\}(\frac{1}{N} - \frac{1}{ \nu})}
+ c_4 |h|_{L^{N'}( \mathbb{R}^N)}^{N'}. \label{e2.4}
\end{equation}
Now, observe that by the even definition of $ c_{w}$, we have
$$
c_{w}  \leq \max_{t \geq 0} I_{w}(t \varphi)
\leq \max_{t \geq 0} \Big(\frac{\max\{1,a'_1\}t^N \| \varphi\|^N}{N}
-A t^p | \varphi|_{L^p( \mathbb{R}^N)}^p\Big).
$$
It is clear that the function
$$
K(t) = \frac{\max\{1,a'_1\}t^N \| \varphi\|^N}{N}
-A t^p | \varphi|_{L^p( \mathbb{R}^N)}^p
$$
defined on $ [0, +\infty[ $ attains its maximum at
$$
t_{\rm max} = (\frac{\max\{1,a'_1\} \| \varphi\|^N}{A p |
\varphi|_{L^p( \mathbb{R}^N)}^p})^{ \frac{1}{p-N}}.
$$
Thus,
$$
\max_{t \geq 0} K(t) = \max\{1,a'_1\} \| \varphi\|^N (\frac{1}{N}- \frac{1}{p})
\Big( \frac{\max\{1,a'_1\} \| \varphi\|^N}{ p A | \varphi|_{L^p( \mathbb{R}^N)}^p}
\Big)^{ \frac{N}{p-N}}.
$$
Hence,
\begin{equation}
 c_{w} \leq \max\{1,a'_1\} \| \varphi\|^N (\frac{1}{N}- \frac{1}{p})
\Big( \frac{ \max\{1,a'_1\} \| \varphi\|^N}{ p A | \varphi
|_{L^p( \mathbb{R}^N)}^p}\Big)^{ \frac{N}{p-N}}. \label{e2.5}
\end{equation}
Denote
$$
\Sigma(A) = \max\{1,a'_1\} \| \varphi\|^N (\frac{1}{N}
- \frac{1}{p})( \frac{ \max\{1,a'_1\} \| \varphi\|^N}
{ p A | \varphi|_{L^p( \mathbb{R}^N)}^p})^{ \frac{N}{p-N}}.
$$
Fix $ 0 < \eta <  \frac{1}{\delta_0}$.
It is clear that there exists $ A_{\eta} > 0 $ large enough such that
$$
\Sigma(A) \leq \frac{\min\{1,a_1\}}{4} ( \frac{1}{N} - \frac{1}{\nu}) \eta^N,
$$
provided that $ A > A_{\eta}$. On the other hand, we can choose
$ |h|_{L^{N'}( \mathbb{R}^N)} $ small enough such that
$$
c_4  |h|_{L^{N'}( \mathbb{R}^N)}^{N'} \leq \frac{ \eta^N}{2}.
$$
Hence, by \eqref{e2.4} and \eqref{e2.5} we deduce that
$$
\limsup_{n \to +\infty} \|u_{n,w}\|^N \leq  \eta^N.
$$
It follows, that there exists $ n_0 > 1 $ large enough such that
$$
\|u_{n,w}\| \leq \Big( \frac{2 \Sigma(A)}{\min\{1,a_1\}( \frac{1}{N}
-\frac{1}{\nu})} +c_4 |h|_{L^{N'}( \mathbb{R}^N)}^{N'} \big)^{1/N}
\leq \eta < \frac{1}{\delta_0},\quad \forall n \geq n_0.
$$
Up to a subsequence, $ (u_{n,w}) $ is weakly convergent to some point $ u_w $
in $ E$. We claim that, up to a subsequence, $ (u_{n,w}) $ is strongly
convergent to $ u_{w} $ in $ E$. First, observe that by \eqref{e2.2} we have
$$
\int_{\mathbb{R}^N} |f(x,u_{n,w})|^{N'} \,dx
\leq c_5\int_{\mathbb{R}^N} |u_{n,w}|^{ \alpha N'} \,dx.
$$
Thus, we get the boundedness of the sequence $ (f(\cdot, u_{n,w})) $
in $ L^{N'}( \mathbb{R}^N)$.
This fact together with the compact embedding
$ E \hookrightarrow\hookrightarrow L^N( \mathbb{R}^N) $ imply
\begin{equation}
 \int_{ \mathbb{R}^N} |f(x,u_{n,w})(u_{n,w} -u_{w})| \,dx \to 0,\quad n \to + \infty.
 \label{e2.6}
\end{equation}
Using \eqref{e2.6} and the weak convergence of $ (u_{n,w}) $ to $ u_w $ in $ E$,
we obtain
\begin{align*}
&\int_{\mathbb{R}^N} a(x,w) (| \nabla u_{n,w}|^{N-2} \nabla u_{n,w}
- | \nabla u_{w}|^{N-2} \nabla u_{w}) \nabla (u_{n,w} -u_w) \,dx \\
& + \int_{\mathbb{R}^N} V(x) (|u_{n,w}|^{N-2} u_{n,w}
- |u_w|^{N-2} u_w)(u_{n,w} -u_w) \,dx \\
& + \int_{\mathbb{R}^N} (| \Delta u_{n,w}|^{N-2} \Delta u_{n,w}
- | \Delta u_{w}|^{N-2} \Delta u_{w}) \Delta (u_{n,w} -u_w) \,dx \\
& \to 0,\quad \text{as } n \to + \infty.
\end{align*}
Recalling the standard inequality
\begin{equation}
(|x|^{N-2} x - |y|^{N-2} y)(x-y) \geq 2^{-N} |x-y|^N,\quad
 \forall x,y \in \mathbb{R}^r,\; \forall r \geq 1, \label{e2.7}
\end{equation}
 we can deduce that, up to a subsequence, $ (u_{n,w}) $ is strongly convergent
to $ u_{w} $ in  $ E$.
Consequently, $ u_{w} $ is a critical point of $ I_{w} $ and
$ I_{w}(u_{w}) = c_{w} \geq \mu > 0$. Moreover, taking into account that
 $$
\|u_{n,w}\| \leq \eta,\quad  \forall n \geq n_0,
$$
and passing to the limit as $ n \to +\infty$, we obtain
$\|u_{w}\| \leq \eta$.
\end{proof}

 \begin{lemma} \label{lem2.4}
 Assume that {\rm (V1)--(V2), (H1), (H5)} hold.
 Let $ w \in E $ be such that $ \|w\| \leq \frac{1}{\delta_0}$.
Then, the functional $ I_{w} $ admits a nontrivial weak solution $ U_{w} \in E $
such that $ I_{w}(U_{w}) \leq -\sigma < 0 $ and $ \|U_{w}\| \leq \rho$, where
$ \rho $ is given by Lemma \ref{lem2.1} and $ \sigma $ is some positive constant
independent of $ w$.
\end{lemma}

\begin{proof}
 Let $ \varphi $ be the function introduced and used in Lemma \ref{lem2.2}.
Clearly, we can choose  $ 0 \leq \varphi(x) \leq 1$ for all $x \in \mathbb{R}^N$.
For $ 0 < t < 1$, we have
\begin{equation}
I_{w}(t \varphi) \leq \max\{1,a'_1\} \frac{t^N}{N} \| \varphi\|^N
-  \int_{\mathbb{R}^N} F(x,t\varphi) \,dx
 - t \int_{\mathbb{R}^N} h \varphi \,dx. \label{e2.8}
\end{equation}
By (H1), we can easily obtain
$$
\lim_{t \to 0^+} \int_{\mathbb{R}^N} \frac{F(x,t \varphi)}{t} \,dx =0.
$$
Moreover, since
$$
\int_{\mathbb{R}^N} h \varphi \,dx > 0,
$$
by \eqref{e2.8} one can easily find
$ 0 < t_{1} < \inf(1, \frac{\rho}{\| \varphi\|})$ small enough and
independent of $ w $, and $ \sigma > 0 $ also independent of $ w $
such that
$$
I_{w}(t_1 \varphi) \leq -\sigma < 0.
$$
Now, denote
$$
\theta_{w} = \inf\{I_{w}(u),\ \|u\| \leq \rho\}.
$$
 In view of Lemma \ref{lem2.1} and by  the Ekeland's variational principle (see \cite{e1}),
there exists a sequence $ (U_{n,w}) \subset E $ such that
$$
 \|U_{n,w}\| \leq \rho,\ I_{w}(U_{n,w}) \to \theta_{w},\quad \text{and}
\quad I'_{w}(U_{n,w}) \to 0.
$$
Up to a subsequence, $ (U_{n,w}) $ is weakly convergent to some point
$ U_w $ in $E$. Observe that $ \rho < \frac{1}{\delta_0} $ and arguing
as for \eqref{e2.6}, we can prove that
$$
\int_{\mathbb{R}^N} f(x,U_{n,w})(U_{n,w} - U_{w}) \,dx \to 0,\quad
 \text{as } n \to +\infty.
$$
Proceeding exactly as for the sequence $ (u_{n,w})$, we can easily
show that, up to a subsequence, $ (U_{n,w}) $ is strongly
convergent to $ U_{w} $ in $ E$. Therefore, the point
 $ U_{w} $ is a critical point of $ I_{w} $ satisfying
$$
I_{w}(U_{w}) = \theta_{w} \leq -\sigma < 0,\quad \text{and}\quad
 \|U_{w}\| \leq \rho.
$$
This completes the proof. \end{proof}

\subsection*{Proof of Theorem \ref{thm1.1} completed}
 To conclude the proof, an iterative scheme will be performed.
Let $ 0 < \eta  < 1/\delta_0 $ and fix $ u_0 \in E $ such that
$ \|u_0\| \leq \eta$. By Lemma \ref{lem2.3}, under the condition $ A > A_{\eta}$,
and $ |h|_{L^{N'}( \mathbb{R}^N)} < d_{\eta}$, the functional $ I_{u_0} $
admits a nontrivial critical point $ u_{1} \in E  $ such that
$$
 I_{u_{0}} (u_{1}) \geq \mu > 0,\quad \|u_{1}\| \leq \eta.
$$
Similarly, the functional $ I_{u_{1}} $ admits a critical point $ u_2 $ such that
$$
I_{u_{1}}(u_2) \geq \mu  > 0,\quad \|u_2\| \leq \eta.
$$
This way, we construct a sequence $ (u_{n}) \subset E  $ such that
$$
\|u_{n}\| \leq  \eta,\quad I_{u_{n-1}}(u_{n}) \geq \mu > 0,
$$
and $ u_{n} $ is a critical point of the functional $ I_{u_{n-1}}$.
Thus, we have
\begin{equation}
  \begin{aligned}
& \int_{ \mathbb{R}^N} a(x,u_{n-1})| \nabla u_{n}|^{N-2} \nabla u_{n} \nabla v \,dx \\
& + \int_{ \mathbb{R}^N} V(x) |u_{n}|^{N-2} u_{n} v \,dx
  + \int_{\mathbb{R}^N} | \Delta u_{n}|^{N-2} \Delta u_{n} \Delta v \,dx \\
& =  \int_{ \mathbb{R}^N} f(x,u_{n}) v \,dx + \int_{\mathbb{R}^N} hv \,dx,\quad
 \forall v \in E.
\end{aligned} \label{e2.9}
\end{equation}
Similarly, we have
\begin{equation} \begin{aligned}
 & \int_{ \mathbb{R}^N} a(x,u_{n})| \nabla u_{n+1}|^{N-2} \nabla u_{n+1}
\nabla v \,dx \\
& + \int_{ \mathbb{R}^N} V(x) |u_{n+1}|^{N-2} u_{n+1} v \,dx
  + \int_{\mathbb{R}^N} | \Delta u_{n+1}|^{N-2} \Delta u_{n+1} \Delta v \,dx \\
& =  \int_{ \mathbb{R}^N} f(x,u_{n+1}) v \,dx + \int_{\mathbb{R}^N} hv \,dx,\quad
  \forall v \in E.
\end{aligned} \label{e2.10}
\end{equation}
Taking $ v = u_{n+1} -u_n $ as test function in \eqref{e2.9} and \eqref{e2.10},
and subtracting one equation from the other, we obtain
\begin{equation} \begin{aligned}
& \int_{\mathbb{R}^N} a(x,u_{n})
 \Big(| \nabla u_{n+1}|^{N-2} \nabla u_{n+1} -| \nabla u_{n}|^{N-2} \nabla u_{n}\Big)
 \nabla (u_{n+1} - u_{n}) \,dx\\
& + \int_{\mathbb{R}^N} (a(x,u_{n}) -a(x,u_{n-1})) | \nabla u_{n}|^{N-2}
 \nabla u_{n} \nabla (u_{n+1} -u_{n}) \,dx \\
& + \int_{\mathbb{R}^N} V(x) \Big(|u_{n+1}|^{N-2} u_{n+1} - |u_{n}|^{N-2} u_{n}\Big)
 (u_{n+1}  -u_{n}) \,dx\\
& + \int_{\mathbb{R}^N} \Big(| \Delta u_{n+1}|^{N-2} \Delta u_{n+1}
- | \Delta u_{n}|^{N-2} \Delta u_{n}\Big) \Delta (u_{n+1} -u_{n}) \,dx\\
& =  \int_{\mathbb{R}^N} (f(x,u_{n+1}) -f(x,u_{n})) (u_{n+1} -u_{n}) \,dx.
\end{aligned} \label{e2.11}
\end{equation}
Since $ \|u_n\| \leq \eta$, for all $n \geq1$, it follows by \eqref{e1.1} that
$$
|u_n|_{L^{\infty}( \mathbb{R}^N)},\ |u_{n+1}|_{L^{\infty}( \mathbb{R}^N)}
\leq \delta_0 \eta < 1,\quad \forall n \geq 0.
$$
By $ (H_4)$, it yields
$$
|f(x,u_{n+1}(x)) -f(x,u_n(x))| \leq \beta_0 (\delta_0 \eta)^{\beta_1} |u_{n+1}(x)
-u_n(x)|^{N-1},
$$
a.e. $x \in \mathbb{R}^N$, for all $n \geq 0$.
Consequently
\begin{equation}
\int_{\mathbb{R}^N} (f(x,u_{n+1}) -f(x,u_n))(u_{n+1} -u_n) \,dx
\leq \beta_0 (\delta_0 \eta)^{\beta_1} \int_{\mathbb{R}^N} |u_{n+1} -u_n|^N \,dx.
 \label{e2.12}
\end{equation}
 If we take $ \eta $ small enough such that
$$
\beta_0 (\delta_0 \eta)^{\beta_1} \leq \frac{V_0 \min\{1,a_1\} 2^{-N}}{4},
$$
then from \eqref{e2.12} we infer
\begin{equation}
\begin{aligned}
& \int_{\mathbb{R}^N} (f(x,u_{n+1}) -f(x,u_n))(u_{n+1} -u_n) \,dx \\
& \leq \frac{ \min\{1,a_1\} 2^{-N}}{4} \int_{\mathbb{R}^N} V(x) |u_{n+1} -u_n|^N \,dx \\
& \leq \frac{ \min\{1,a_1\} 2^{-N}}{4} \|u_{n+1} -u_n\|^N.
\end{aligned} \label{e2.13}
\end{equation}
On the other hand, by Young's inequality we have
 \begin{align*}
& \int_{\mathbb{R}^N} |a(x,u_n) -a(x,u_{n-1})|\ | \nabla u_n|^{N-1} | \nabla (u_{n+1}
-u_n)| \,dx \\
& \leq \frac{\min\{1,a_1\} 2^{-N}}{4} \int_{\mathbb{R}^N}
 | \nabla (u_{n+1} -u_n)|^N \,dx \\
&\quad +c_6 \int_{\mathbb{R}^N} |a(x,u_n) -a(x,u_{n-1})|^{N'} | \nabla u_n|^N \,dx,
 \end{align*}
and by (H6) it follows that
\begin{equation} \begin{aligned}
&  \int_{\mathbb{R}^N} |a(x,u_n) -a(x,u_{n-1})|\ | \nabla u_n|^{N-1} |
 \nabla (u_{n+1} -u_n)| \,dx \\
&  \leq \frac{\min\{1,a_1\} 2^{-N}}{4} \|u_{n+1} -u_n\|^N
 + c_6 L^{N'} |u_n -u_{n-1}|_{L^{\infty}( \mathbb{R}^N)}^N \|u_n\|^N \\
& \leq \frac{\min\{1,a_1\} 2^{-N}}{4} \|u_{n+1} -u_n\|^N
 + (c_6 L^{N'} \delta_0^N \eta^N) \|u_n -u_{n-1}\|^N.
\end{aligned} \label{e2.14}
\end{equation}
Using  \eqref{e2.7}, \eqref{e2.11}, \eqref{e2.13} and \eqref{e2.14},
 we obtain
\begin{equation}
\frac{\min\{1,a_1\}}{ 2^{N+1}} \|u_{n+1} -u_n\|^N
\leq (c_6 L^{N'} \delta_0^N \eta^N) \|u_n -u_{n-1}\|^N. \label{e2.15}
\end{equation}
Set
$$
\Gamma(\eta) = \Big(\frac{c_6 L^{N'} \delta_0^N \eta^N 2^{N+1}}{\min\{1,a_1\}}
\Big)^{1/N}.
$$
By \eqref{e2.15}, it yields
\begin{equation}
\|u_{n+1} -u_n\| \leq \Gamma(\eta) \|u_n -u_{n-1}\|,\quad \forall n \geq 1.
 \label{e2.16}
\end{equation}
Clearly, we can choose $ \eta $ small enough such that $ \Gamma( \eta) < 1$.
 Therefore, by \eqref{e2.16} $ (u_n) $ is
a Cauchy sequence and by consequence it is strongly convergent to some point
 $ u \in E$.
Passing to the limit as $ n \to +\infty $ in \eqref{e2.9}, we conclude that
$ u $ satisfies
\begin{align*}
&\int_{ \mathbb{R}^N} a(x,u)| \nabla u|^{N-2} \nabla u \nabla v \,dx
+ \int_{ \mathbb{R}^N} V(x) |u|^{N-2} uv \,dx
+ \int_{\mathbb{R}^N} | \Delta u|^{N-2} \Delta u \Delta v \,dx \\
& =  \int_{ \mathbb{R}^N} f(x,u) v \,dx + \int_{\mathbb{R}^N} hv \,dx,\quad
 \forall v \in E.
\end{align*}
According to Definition 1.1, this means that $ u $ is a weak solution
of problem \eqref{eP}. On the other hand, we have
$ I_{u_{n-1}}(u_{n}) \geq \mu > 0$, for all $n \geq 2$.
Hence,
\begin{align*}
& \int_{ \mathbb{R}^N} \frac{a(x,u_{n-1})| \nabla u_{n}|^N + V(x)|u_{n}|^N
+ | \Delta u_n|^N}{N} \,dx \\
&-  \int_{ \mathbb{R}^N} F(x,u_{n}) \,dx -\int_{\mathbb{R}^N} h u_{n} \,dx
\geq \mu > 0.
\end{align*}
Passing to the limit as $ n \to + \infty$, it follows
\begin{align*}
\Psi(u)  &= \int_{ \mathbb{R}^N} \frac{a(x,u)| \nabla u|^N + V(x) |u|^N
+ | \Delta u|^N}{N} \,dx  \\
&\quad -  \int_{ \mathbb{R}^N} F(x,u) \,dx
- \int_{\mathbb{R}^N} hu \,dx \geq \mu > 0.
\end{align*}
Now, using Lemma \ref{lem2.4} it is immediate that an iterative scheme
 could be performed to construct a sequence $ (U_{n}) \subset E $ such that,
for all $ n \geq 1$,
$$
\|U_{n}\| \leq \rho < \frac{1}{\delta_0},\quad
I_{U_{n-1}}(U_{n}) \leq -\sigma < 0,
 $$
 and $ U_{n} $ is a critical point of the functional $ I_{U_{n-1}}$.
Moreover, using the same arguments as for the sequence $ (u_n)$,
we can easily prove that the sequence $ (U_{n}) $
is strongly convergent to some point $ U \in E $ which is a weak
solution of problem \eqref{eP}. Furthermore, we have
\begin{align*}
  \Psi(U)  &= \int_{ \mathbb{R}^N} \frac{a(x,U)| \nabla U|^N + V(x) |U|^N
 + | \Delta U|^N}{N} \,dx   \\
&\quad - \int_{ \mathbb{R}^N} F(x,U) \,dx - \int_{\mathbb{R}^N} hU \,dx
\leq -\sigma < 0.
\end{align*}
Hence, $ u \neq U$. This completes the proof of Theorem \ref{thm1.1}.

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\end{document}