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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2016 (2016), No. 217, pp. 1--15.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2016 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2016/217\hfil Multiplicity of solutions]
{Multiplicity of solutions for equations involving a nonlocal term
and the biharmonic operator}

\author[G. M. Figueiredo, R. G. Nascimento \hfil EJDE-2016/217\hfilneg]
{Giovany M. Figueiredo, R\'ubia G. Nascimento}

\address{Giovany M. Figueiredo \newline
Universidade Federal do Par\'a,
Faculdade de Matem\'atica,
CEP: 66075-110 Bel\'em - Pa, Brazil}
\email{giovany@ufpa.br}

\address{R\'ubia G. Nascimento \newline
Universidade Federal do Par\'a,
Faculdade de Matem\'atica,
CEP: 66075-110 Bel\'em - Pa, Brazil}
\email{rubia@ufpa.br}

\thanks{Submitted April 20, 2016. Published August 16, 2016.}
\subjclass[2010]{34B15, 34B18. 35J35, 35G30}
\keywords{Beam equation; Berger equation; critical exponent;
\hfill\break\indent variational methods}

\begin{abstract}
 In this work we study the existence and multiplicity result of solutions
 to the equation
 \begin{gather*}
 \Delta^{2}u-M\Big(\int_{\Omega}|\nabla u|^{2} \,dx\Big)\Delta u
 = \lambda |u|^{q-2}u+  |u|^{2^{**}}u \quad\text{in }\Omega, \\
 u=\Delta u=0 \quad\text{on }\partial\Omega,
 \end{gather*}
 where $\Omega$ is a bounded smooth domain of $\mathbb{R}^{N}$,
 $N\geq 5$,  $1 < q<2$ or $2<q<2^{**}$, $M:\mathbb{R}^{+}\to\mathbb{R}^{+}$
 is a continuous function. Since there is a competition between the
 function $M$ and the critical exponent, we need to make a truncation on
 the function $M$. This truncation allows to define an auxiliary problem.
 We show that, for $\lambda$ large,  exists one solution and for
 $\lambda$ small there are infinitely many solutions for the auxiliary problem.
 Here we use arguments due to Brezis-Niremberg \cite{BrezNiremb} to show
 the existence result and genus theory due to Krasnolselskii \cite{kras} to
 show the multiplicity result.  Using the size of $\lambda$, we show that
 each solution of the auxiliary problem is a solution of the original problem.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{corollary}[theorem]{Corollary}
\allowdisplaybreaks

\section{Introduction}

In this work we deal with questions of existence and multiplicity
of solutions to an equation involving a nonlocal term and biharmonic operator.
More precisely we study the equation
\begin{equation} \label{ePl}
\begin{gathered}
 \Delta^{2}u-M\Big(\int_{\Omega}|\nabla u|^{2} \,dx\Big)\Delta u
 =\lambda |u|^{q-2}u+  |u|^{2^{**}-2}u \quad\text{in } \Omega, \\
u=\Delta u=0 \quad\text{on }\partial\Omega,
\end{gathered}
\end{equation}
where $\Omega\subset\mathbb{R}^{N}$ is a bounded smooth
domain, $1 < q<2$ or $2<q<2^{**}$ and $M:\mathbb{R}^{+}\to
\mathbb{R}^{+}$ is a continuous function that satisfies conditions
which will be stated later. Here $2^{**}=\frac{2N}{N-4}$ with
$N\geq 5$ and $\Delta^{2}$ is the biharmonic operator; that is,
$$
\Delta^{2}u= \sum^{N}_{i=1} \frac{\partial^{4}}{\partial
x_{i}^{4}}u+ \sum^{N}_{i\neq j} \frac{\partial^{4}}{\partial x_{i}^{2}x_{j}^{2}}u.
$$

Our study was strongly motivated by extensible beam equation type or
of a stationary Berger plate equation, as can be seen below.

In 1950, Woinowsky-Krieger \cite{Woinowsky} studied the equation
\begin{equation}
\begin{gathered}
 \frac{\partial^{2}u}{\partial t^{2}}+
 \frac{EI}{\rho} \frac{\partial^{4}u}{\partial
x^{4}}-\Big( \frac{H}{\rho}+ \frac{EA}{2\rho
L}\int_0^{L}| \frac{\partial u}{\partial x}|^{2} \,dx \Big)
 \frac{\partial^{2}u}{\partial x^{2}} =0,
\end{gathered} \label{eWK}
\end{equation}
where $L$ is the length of the beam in the rest position, $E$ is the
Young modulus of the material, $I$ is the cross-sectional moment of
inertia, $\rho$ is the mass density, $H$ is the tension in the rest
position and $A$ is the cross-sectional area. This model was
proposed to modify the theory of the dynamic Euler-Bernoulli beam,
assuming a nonlinear dependence of the axial strain on the
deformation of the gradient. Owing to its importance in engineering,
physics and material mechanics, since  such model was proposed, this
class of problems has been studied. These studies are focused on the
properties of its solutions, as can be seen in \cite{Ball,Ball1,Dickey,Medeiros}
and references therein. More recent references with important details about the
physical motivation of \eqref{eWK} can be seen in
\cite{Ashfaque,Kang,Ma,Yang}.

In 1955, Berger \cite{Berger} studied the equation
\begin{equation}
\begin{gathered}
 \frac{\partial^{2}u}{\partial t^{2}} + \Delta ^{2}u +
\Big( Q +  \int_{\Omega}|\nabla u|^{2} dx
\Big)\Delta u = f(u,u_{t},x),
\end{gathered} \label{eB}
\end{equation}
which is called the Berger plate model \cite{Chueshov}, as a
simplification of the von Karman plate equation which describes
large deflection of plate, where the parameter $Q$ describes in-plane
forces applied to the plate and the function $f$ represents
transverse loads which may depend on the displacement $u$ and the
velocity $u_{t}$.

Problem \eqref{ePl} is a generalization of the stationary
problem associated with problem \eqref{eWK} in dimension one or problem
\eqref{eB} in dimension two. Before stating our main results, we need the
following hypotheses on the function $M:\mathbb{R}^{+}\to \mathbb{R}^{+}$:
The function $M$ is continuous, increasing and there exists $0< m_0$  such
that
\begin{equation}
 M(t)\geq  m_0=M(0), \quad\text{for all } t \in \mathbb{R}^{+}. \label{eM}
\end{equation}
A typical example of a function satisfying this condition is
$$
M(t)=m_0 + bt
$$
with $b\geq 0$ and for all $t\geq 0$, which is the one considered for
 \eqref{eWK}  by Woinowsky-Krieger  \cite{Woinowsky} and  for
\eqref{eB}  by Berger in \cite{Berger}.
 However, our hypotheses about the function $M$ include other functions,
such as
$M(t) = m_0 + ln(1 + t)$,
$ M(t)=m_0+bt+ \sum_{i=1}^{k}b_{i}t^{d_{i}}$ with
$b_{i}\geq 0$ and $d_{i}\in (0,1)$ for all $i\in \{1,2,\ldots, k\}$
or $M(t)=\exp t$.

The  first result is related to the case $1<q<2$ with a small
positive parameter $\lambda$.


\begin{theorem}\label{Teorema1}
If $1<q<2$ and \eqref{eM} hold, then exists a positive constant
$\lambda^{*}$ such that \eqref{ePl} has infinitely many
solutions, for all $\lambda \in (0,\lambda^{*})$. Moreover, if
$u_{\lambda}$ is one of these solutions, then
$u_{\lambda} \in C^{4,\alpha}(\Omega)\cap C^{3}(\overline{\Omega})$
with $\alpha \in (0,1)$ and
$$
 \lim_{\lambda\to 0}\|u_{\lambda}\|=0.
$$
\end{theorem}

The  second result is related with the case $2< q< 2^{**}$ with
$\lambda$ large.

\begin{theorem}\label{Teorema3}
If $2< q< 2^{**}$ and \eqref{eM} hold, then exists a positive constant
$\lambda^{**}$ such that \eqref{ePl} has a nontrivial
solution $u_{\lambda}$, for all $\lambda \in (\lambda^{**},+\infty)$. Moreover,
$u_{\lambda} \in C^{4,\alpha}(\Omega)\cap
C^{3}(\overline{\Omega})$ with $\alpha \in (0,1)$ and
$$
 \lim_{\lambda\to +\infty}\|u_{\lambda}\|=0.
$$
\end{theorem}


Problem \eqref{ePl} with the function $M$ constant and
subcritical growth was exhaustively studied, as can be seen in
\cite{Alves,Bonano,Cabada,Grossinho,Liu,Ye,Zhou} and references therein.
On the other hand, there are only a few works dedicated to equations
modeling  stationary beam equations or Berger plate equation;
that is, problems involving a function $M$ depending on the gradient
of the solution of problem. In this direction, we mention the papers
\cite{Cabada,Ma1,Ma2,Wang,Wang1,Wang2,Zelati}.
The difficulty that arises in the
study of this class of problems is the growth of the operator
$\widehat{M}(\|u\|^{2})=m_0\|u\|^{2}+ \frac{b}{2}\|u\|^{4}$, where
$M(t)= \int^{t}_0M(s) \,ds$ and  $m_0,b >0$. This
requires us to impose a 4-superlinear growth on  the nonlinearity
$f$; that is, $f(x,t)=t^{p}$ with $p \in (3,
2^{**}=\frac{2N}{N-4})$. But $2^{**}=\frac{2N}{N-4}\to 2$ as
$N\to +\infty$. To circumvent this difficulty, it is common
to fix $N \leq 4$ because, in this case, $2^{**}=\infty$ or $M$
bounded or make a truncation on function $M$. In \cite{Ma1} the
author shows some existence results using the Ekeland variational
principle and also discuss a numerical example considering a general
function $M$ and $N=1$.  In \cite{Ma2} the author gives a necessary
and sufficient condition for the existence of solutions when the
nonlinearity  is increasing considering again   a general function
$M$ and $N=1$.  In \cite{Wang} the authors show the existence of
nontrivial solution using the Mountain Pass Theorem considering the
function $M$ bounded and $N\geq 1$. In \cite{Wang1} the authors show
the existence of nontrivial solution using an iterative scheme of
Mountain Pass "approximated" solutions considering the case
$M(t)=\lambda(a+bt)$, $N\geq 1$, $a,b>0$ and $\lambda>0$ small. The
paper \cite{Wang2}  is a version of  \cite{Wang1}  in
$\mathbb{R}^{N}$.  In \cite{Zelati} the authors analyze from both
the physical and the analytical viewpoints problem \eqref{ePl}
with $N=1$ and $M(t)=\gamma +t$. In this article, the author
consider two cases namely: $\gamma>0$ and $\gamma<0$. In
\cite{Cabada1} the author consider a version of problem
\eqref{ePl} in $\mathbb{R}^{N}$ with a general version of $M$
and $N\geq 5$.

In this article, we complement the results found in  \cite{Ma1,Ma2,Wang,Wang1}
 in the following sense:
\begin{itemize}
\item[(i)]   Unlike of \cite{Ma1}, \cite{Ma2},
\cite{Wang} and \cite{Wang1}, we overcome the difficult of
competition between the operator and the critical exponent  without
consider $N\leq 4$ or $M$ bounded or $M(t)=a+bt$ with $a,b$ small.
In our work we use a truncation on function $M$ and we use the size
of lambda to show that the solution of truncated problem is a
solution of original problem. Of course, the estimates on the
operator of the truncated problem was adapted from \cite{Wang}.

\item[(ii)] Moreover, we  study the asymptotic behavior of
solution of problem \eqref{ePl} when $\lambda\to
\infty$. This study was not observed in the articles above.

\end{itemize}
Unfortunately we do not have information on the case $1<q<2$ and
$\lambda$ large or on the case $2<q<2^{**}$ and $\lambda$ small.

In the proof of theorem \ref{Teorema1} we use an argument that can
be found in \cite{Bernis} and the proof of Theorem \ref{Teorema3} we
use an argument that can be found in \cite{GP}, for example. But,
due to the presence of the function $M$ and its truncation, some
estimates more refined are necessary, such as in Lemmas
\ref{comportamentoassimtotico100} and \ref{limitacao}.

In recent years, problems involving biharmonic or polyharmonic
operators have  received a special attention, in particular problems
where the nonlinearity has a critical growth. In this interesting
book  \cite{GGS1}, the reader can find a lot of results involving
this class of operator and an excellent bibliography about this
subject. In addition to this book, we would like to cite the
papers \cite{BWW,Bernis,G,GGS2,Ge1,Ge2,Pucci,yao}
and references therein.

The plan of this article is as follows.
In Section 2, we define the truncated problem.
In section 3, we recall some properties of genus
theory, we prove some  technical lemmas on truncated problem and we
prove the Theorem \ref{Teorema1}. The proof of Theorem
\ref{Teorema3} is made in section 4.


\section{Auxiliary problem and variational framework}

Since  intend to work with $N \geq 5$, we use a truncation
argument. Here we are assuming, without loss of generality, that $M$
is unbounded. Otherwise, the truncation of the function $M$ is not
necessary. We make a truncation on the function $M$ for the case
$1<q<2$ and another truncation on function $M$ for the case
$2<q<2^{**}$ as follows:

From \eqref{eM}, there exists $t_0>0$ such that $ m_0 < M(t_0) <
 \frac{2^{**}}{2}m_0$ for the case $1<q<2$   and
$m_0< M(t_0) <  \frac{q}{2}m_0$ for the case
$2<q<2^{**}$. We set
\begin{equation}\label{pedidodoreferee}
M_0(t):= \begin{cases}
M(t), &  \text{if }  0 \leq t \leq t_0, \\
M(t_0) & \text{if }  t \geq t_0.
\end{cases}
\end{equation}
From \eqref{eM} we obtain
\begin{gather}\label{vaidarcerto}
 M_0(t) \leq  \frac{2^{**}}{2}m_0 \quad \text{in the case } 1<q<2, \\
\label{vaidarcerto1}
M_0(t) \leq  \frac{q}{2}m_0 \quad \text{in the case }  2<q<2^{**}.
\end{gather}

The proofs of  Theorems \ref{Teorema1} and  \ref{Teorema3} are based
on a careful study of solutions of the  auxiliary problem
\begin{equation}
\begin{gathered}
\Delta^{2} u-M_0\Big( \int_{\Omega}|\nabla u|^{2} dx \Big) \Delta u
 = \lambda |u|^{q-2}u+|u|^{2^{**}-2}u \quad\text{in } \Omega,  \\
u=\Delta u=0 \quad\text{on } \partial\Omega,
\end{gathered}
 \label{eTl}
\end{equation}
where $N$ and $\lambda$ are as in the introduction.

We say that $u \in H:=H^{2}(\Omega)\cap H^{1}_0(\Omega)$ is a
weak solution of  problem \eqref{eTl} if $u$ satisfies
\begin{align*}
&\int_{\Omega}\Delta u \Delta \phi \,dx +
M_0\Big(\int_{\Omega}|\nabla u|^{2} dx
\Big)\int_{\Omega} \nabla u\nabla \phi dx \\
&= \lambda\int_{\Omega}|u|^{q-2}u\phi
 \,dx+\int_{\Omega}|u|^{2^{**}-2}u\phi  \,dx,
\end{align*}
for all $\phi \in  H$.

Note that $H$ is a Hilbert space with the norm
$$
\|u\|^{2}=  \int_{\Omega}|\Delta u|^{2} \,dx
 + \int_{\Omega}|\nabla u|^{2} dx
$$
and we will look for solutions of  \eqref{eTl} by finding
critical points of the $C^1$-functional $I_{\lambda}: H\to
\mathbb{R}$ given by
$$
I_{\lambda}(u) = \frac{1}{2}\int_{\Omega}|\Delta u|^{2} \,dx
+ \frac{1}{2}\widehat{M_0}\Big(\int_{\Omega}|\nabla u|^{2}
dx \Big)  - \frac{\lambda}{q}\int_{\Omega}|u|^{q} \,dx
-\frac{1}{2^{**}}\int_{\Omega}|u|^{2^{**}} \,dx ,
$$
where $\widehat{M_0}(t)= \int^{t}_0M_0(s) \,ds$.
Note that
\begin{align*}
I_{\lambda}'(u)\phi
&=  \int_{\Omega}\Delta u \Delta
\phi \,dx + M_0\Big(\int_{\Omega}|\nabla u|^{2} dx
\Big) \int_{\Omega} \nabla u\nabla \phi dx\\
&\quad - \lambda \int_{\Omega}|u|^{q-2}u\phi
 \,dx- \int_{\Omega}|u|^{2^{**}-2}u\phi  \,dx,
\end{align*}
for all $\phi \in  H$. Hence critical points of  $I_{\lambda}$
are weak solutions for \eqref{eTl}.


To use variational methods, we first derive some results
related to the Palais-Smale  compactness condition.

We say that a sequence $(u_{n})\subset  H$ is a Palais-Smale
sequence for the functional $I_{\lambda}$ if
\begin{equation}\label{***}
I_{\lambda}(u_{n})\to c_{\lambda} \quad \text{and} \quad
\|I_{\lambda}'(u_{n})\|\to 0 \quad\text{in }H'.
\end{equation}

If \eqref{***} implies the existence of a subsequence $(u_{n_{j}})
\subset (u_{n})$ which converges in $H$, we say that $I_{\lambda}$
satisfies the Palais-Smale condition. If this strongly convergent
subsequence exists only for some $c_{\lambda}$ values, we say that
$I_{\lambda}$ satisfies a local Palais-Smale condition.



\section{Case $1<q<2$ }

We start by considering some basic notions on the Krasnoselskii
genus that we will use in the proof of Theorem \ref{Teorema1}.

\subsection{Genus theory}

Let $E$ be a real Banach space. Let us denote by $\mathfrak{A}$ the
class of all closed subsets  $A\subset E\setminus \{0\}$ that are
symmetric with respect to the origin, that is, $u\in A$ implies
$-u\in A$.

\begin{definition}\label{genero} \rm
Let $A\in \mathfrak{A}$. The Krasnoselskii genus $\gamma(A)$ of $A$
is defined as being the least positive integer $k$ such that exists
an odd mapping $\phi \in C(A,\mathbb{R}^{k})$ such that $\phi(x)\neq 0$
for all $x\in A$. If such  a $k$ does not exist we set
$\gamma(A)=\infty$. Furthermore, by definition,
$\gamma(\emptyset)=0$.
\end{definition}

In the sequel we will establish only the properties of the genus
that will be used in this work. More information on this
subject may be found in the references  \cite{Ambrosetti,Castro,Davi,kras}.

\begin{theorem}
Let $E={\mathbb{R}}^{N}$ and let $\partial\Omega$ be the boundary of an open,
symmetric and bounded subset $\Omega \subset {\mathbb{R}}^{N}$ with $0
\in \Omega$. Then $\gamma(\partial\Omega)=N$.
\end{theorem}

\begin{corollary}\label{esfera}
$\gamma(S^{N-1})=N$.
\end{corollary}

\begin{proposition}\label{paracompletar}
If $K \in \mathfrak{A}$ and $\gamma(K) \geq 2$, then
$K$ has infinitely many points.
\end{proposition}

\subsection{Proof of Theorem \ref{Teorema1}}

The genus theory requires that the functional $I_{\lambda}$ be
bounded below. Since this not occur, it is necessary to make other
truncation. The plan of the proof is to show that the set of
critical points of the truncated functional  is compact, symmetric,
does not contain the zero and has genus  more than $2$. Thus, by
Proposition \ref{paracompletar}, this functional has infinitely many
critical points. With the size of lambda, we show that each critical
point of the truncated functional  is a solution of the auxiliary
problem and solution of the original problem.

Here we adapt arguments from \cite{GP}. We make a
truncation in the functional $I_{\lambda}$ as follows:
From \eqref{eM} and Sobolev's embedding, we obtain
\begin{align*}
I_{\lambda}(u) \geq \frac{k_0}{2}\|u\|^{2} -\frac{\lambda}{q
S_{q}^{q/2}}\|u\|^{q} -
\frac{1}{2^{**}S^{2^{**}/2}}\|u\|^{2^{**}}=g(\|u\|^{2}),
\end{align*}
where $k_0=\min\{1,m_0\}$,
 \begin{gather*}
S_{q}:=\inf \big\{ \|u\|^{2} : u \in H  \text{ and }
\int_{\Omega} |u|^{q}\,dx=1\big\}, \\
S:=\inf \big\{ \|u\|^{2} : u \in H  \text{ and }
\int_{\Omega} |u|^{2^{**}}\,dx=1\big\},
\end{gather*}
and
\begin{equation} \label{gabriel1}
g(t)=\frac{k_0}{2} t - \frac{\lambda}{q S_{q}^{q/2}}t^{q/2} -
 \frac{1}{2^{**}S^{2^{**}/2}}t^{2^{**}/2}.
\end{equation}
Hence, there exists $\tau_1>0$ such that, if
$\lambda \in (0,\tau_1)$, then $g$ attains its positive maximum.

Denoting by $R_0(\lambda) < R_1(\lambda)$  the only roots of  $g$. We
have  the following result.

\begin{lemma}\label{comportamentoassimtotico100}
\begin{equation}\label{comportamentoassimtotico1}
R_0(\lambda)\to 0 \quad \text{as } \lambda \to 0.
\end{equation}
\end{lemma}

\begin{proof}
From $g(R_0(\lambda))=0$ and $g'(R_0(\lambda))>0$, we have
\begin{gather}\label{lambda1}
CR_0(\lambda)= \frac{\lambda}{q
S_{q}^{q/2}}R_0(\lambda)^{q/2}+
 \frac{1}{2^{**}S^{2^{**}/2}}R_0(\lambda)^{2^{**}/2}, \\
\label{lambda2}
C> \frac{\lambda}{2
S_{q}^{q/2}}R_0(\lambda)^{q-2/2}+
 \frac{1}{2S^{2^{**}/2}}R_0(\lambda)^{2^{**}-2/2},
\end{gather}
for all $\lambda\in (0,\tau_1)$. From, \eqref{lambda1} we
conclude that $R_0(\lambda)$ is bounded. Suppose that
$R_0(\lambda)\to \widetilde{R}>0$ as $\lambda\to 0$. Then
\begin{gather}
C= \frac{1}{2^{**}S^{2^{**}/2}}\widetilde{R}^{2^{**}-2/2}, \\
\label{lambda2b}
C\geq  \frac{1}{2S^{2^{**}/2}}\widetilde{R}^{2^{**}-2/2},
\end{gather}
which is a contradiction, because $2^{**}>2$. Therefore
$R_0(\lambda)\to 0$ as $\lambda\to 0$.
\end{proof}

We consider $\tau_1$ such that $R_0\leq M(t_0)$ and we make
the following truncation on the functional $I_{\lambda}$:

 Take $\phi \in C^{\infty}_0([0,+\infty))$, $0\leq \phi(t)\leq 1$, for all
$ t\in [0,+\infty)$, such that $\phi(t)=1$
if $t\in [0,R_0]$ and $\phi(t)=0$ if $t\in [R_1, +\infty)$. Now,
we consider the truncated functional
$$
J_{\lambda}(u)
= \frac{1}{2}\int_{\Omega}|\Delta u|^{2} \,dx
+ \frac{1}{2}\widehat{M_0}\Big(\int_{\Omega}|\nabla u|^{2}
dx \Big)  -
 \frac{\lambda}{q} \int_{\Omega}|u|^{q} \,dx
- \phi(\|u\|^{2}) \frac{1}{2^{**}} \int_{\Omega}
|u|^{2^{**}} \,dx.
$$
Note that $J_{\lambda} \in C^{1}(H,\mathbb{R})$ and, as in
\eqref{gabriel1}, $J_{\lambda}(u)\geq  \overline{g}(\|u\|^{2})$,
where
$$
\overline{g}(t)= \frac{k_0}{2} t - \frac{\lambda}{q
S_{q}^{q/2}}t^{q/2} -
\phi(t) \frac{1}{2^{**}S^{2^{**}/2}}t^{2^{**}/2}.
$$

Note that if $\|u\|^{2}\leq R_0$ then
$J_{\lambda}(u)=I_{\lambda}(u)$ and if $\|u\|^{2}\geq R_1$, then
\[
J_{\lambda}(u)= \frac{1}{2}\int_{\Omega}|\Delta u|^{2} \,dx
+ \frac{1}{2}\widehat{M_0}\Big(\int_{\Omega}|\nabla u|^{2}
dx \Big)-
 \frac{\lambda}{q} \int_{\Omega}|u|^{q} \,dx.
\]
Thus, we conclude that the functional $J_{\lambda}$ is coercive and,
hence, $J_{\lambda}$ is bounded below.

Now, we show that $J_{\lambda}$ satisfies the local Palais-Smale
condition. For this, we need the following technical result, which
is an analogous of \cite[Lemma 3.3 ]{Bernis}.
 Here,  $\lambda_1$ is the first eigenvalue  of the problem
\begin{equation}
\begin{gathered}
\Delta^{2}{u}=\lambda u, \quad \text{in }  \Omega \\
u=\Delta u=0, \quad \text{on } \partial \Omega.
\end{gathered} \label{eEP}
\end{equation}

\begin{lemma}\label{nivelbaixo}
Let $(u_{n}) \subset H$ be  a bounded sequence such that
$$
I_{\lambda}(u_{n})\to c_{\lambda} \quad  \text{and} \quad
I_{\lambda}'(u_{n})\to 0 \quad  \text{as }  n\to \infty.
$$
If
\[
c_{\lambda}
<\frac{2}{N}S^{N/4}-\lambda^{\frac{2}{(2-q)}}
\bigl(\frac{1}{q}-\frac{1}{2^{**}}\bigl)
|\Omega|^{\frac{(2-q)}{2}}\big[q\bigl(\frac{1}{q}-\frac{1}{2^{**}}\bigl)
|\Omega|^{\frac{(2-q)}{2}}\frac{N}{4}\lambda^{2}_1\big]^{\frac{q}{(2-q)}},
\]
 then we have that, up to a subsequence, $(u_{n})$ is strongly
convergent in  $H$.
\end{lemma}

\begin{proof}
Taking a subsequence, we suppose that
\begin{gather*}
|\Delta u_n|^2 \rightharpoonup |\Delta u|^2 + \mu, \quad
|\nabla u_n|^2 \rightharpoonup |\nabla u|^2 + \gamma, \\
|u_n|^{2^{**}} \rightharpoonup |u|^{2^{**}} + \nu
\quad \text{(weak$^*$-sense of measures).}
\end{gather*}

Using the concentration compactness-principle by Lions 
\cite[Lemma 2.1]{Lio2}, we obtain at most countable index set
$\Lambda$, sequences $(x_i) \subset \mathbb{R}^N$,
$(\mu_i), (\gamma_i), (\nu_i), \subset [0,\infty)$, such that
\begin{equation}
\nu  =  \sum_{i \in \Lambda}\nu_{i}\delta_{x_{i}},\quad
\mu\geq \sum_{i \in \Lambda}\mu_{i}\delta_{x_{i}}, \quad
\gamma\geq \sum_{i \in
\Lambda}\gamma_{i}\delta_{x_{i}}, \quad
S\nu_{i}^{2/2^{**}}\leq \mu_{i},
 \label{lema_infinito_eq11}
\end{equation}
for all $i \in\Lambda$, where $\delta_{x_i}$ is the Dirac mass at
$x_i \in \mathbb{R}^{N}$.

Now we claim that $\Lambda=\emptyset$. Arguing by contradiction,
assume that $\Lambda\neq\emptyset$ and fix $i \in \Lambda$. Consider
$\psi \in C_0^{\infty}(\Omega,[0,1])$ such that $\psi \equiv 1$ on
$B_1(0)$, $\psi \equiv 0$ on $\Omega \setminus B_2(0)$ and
$|\nabla \psi|_{\infty} \leq 2$. Defining
$\psi_{\varrho}(x) := \psi((x-x_i)/\varrho)$ where  $\varrho>0$, we have that
$(\psi_{\varrho}u_n)$ is bounded. Thus
$I_{\lambda}'(u_n)(\psi_{\varrho}u_n) \to 0$; that is,
\begin{align*}
&\int_{\Omega}u_{n} \Delta u_n  \Delta \psi_{\varrho} \,dx
 + \int_{\Omega} \psi_{\varrho}|\Delta u_n|^2 \,dx
 + 2 \int_{\Omega} \Delta u_n  \nabla \psi_{\varrho}\nabla u_n \,dx\\
& + M_0\Big( \int_{\Omega}|\nabla u_{n}|^{2} \,dx \Big)
 \int_{\Omega}u_{n} \nabla u_n \nabla \psi_{\varrho} \,dx
 + M_0\Big( \int_{\Omega}|\nabla u_{n}|^{2} \,dx \Big)
 \int_{\Omega} \psi_{\varrho}|\nabla u_n|^2 \,dx \\
&= \lambda \int_{\Omega}|u_n|^{q}\psi_{\varrho} \,dx
 +  \int_{\Omega} \psi_{\varrho}|u_n|^{2^{**}} \,dx + o_n(1).
\end{align*}
Since the support of $\psi_{\varrho}$ is contained in
$B_{2\varrho}(x_{i})$, we obtain
$$
\Big| \int_{\Omega}u_{n}\Delta u_{n} \Delta
\psi_{\varrho} \,dx \Big|\leq \int_{B_{2\rho}(x_{i})} |\Delta
u_{n}||u_{n} \Delta \psi_{\varrho}| \,dx.
$$

By H\"older inequality and the fact that the sequence $(u_{n})$ is
bounded in $H$ we have
\begin{align*}
\big| \int_{\Omega}u_{n}\Delta u_{n} \Delta
\psi_{\varrho} \,dx \big|
&\leq C \Big( \int_{B_{2\varrho}(x_{i})}|u_{n}
\Delta \psi_{\varrho}|^{2} \,dx\Big)^{1/2} \\
&\leq C \Big(\int_{B_{2\varrho}(x_{i})}|u_{n}|^{2}
|\Delta \psi_{\varrho}|^{2} \,dx\Big)^{1/2}.
\end{align*}
By the Dominated Convergence Theorem
$ \int_{B_{2\varrho}(x_{i})} |u_{n}\Delta
\psi_{\varrho}|^{2} \,dx \to 0$ as $n \to +\infty$ and $\varrho \to
0$. Thus, we obtain
\begin{equation*}
 \lim_{\varrho\to
0}\big[ \lim_{n\to \infty} \int_{\Omega}
u_{n}\Delta u_n  \Delta \psi_{\varrho} \,dx \big]=0.
\end{equation*}
Using the same reasoning we obtain
\begin{gather*}
 \lim_{\varrho\to 0}\big[ \lim_{n\to \infty} \int_{\Omega}
u_{n}\nabla u_n  \nabla \psi_{\varrho} \,dx \big]=0, \\
\lim_{\varrho\to 0}\big[ \lim_{n\to \infty} \int_{\Omega}
 \Delta u_n  \nabla \psi_{\varrho}\nabla u_n \,dx \big]=0, \\
 \lim_{\varrho\to 0}\lim_{n\to \infty}
\big[ \int_{\Omega}\psi_{\varrho}|u_n|^{q} \,dx\big]=0.
\end{gather*}
Since $0<m_0\leq M_0(t) \leq M(t_0)$, for all $t \in \mathbb{R}$, we obtain
$$
 \lim_{\varrho\to 0}\lim_{n\to
\infty}\big[M_0(\|u_{n}\|^{2}) \int_{\Omega}u_{n} \nabla u_n
\nabla \psi_{\varrho} \,dx \big]=0.
$$
Thus, we have
$$
\int_{\Omega} \psi_{\varrho}\mathrm{d}\mu \leq \int_{\Omega}
\psi_{\varrho}\mathrm{d}\mu+m_0\int_{\Omega}
\psi_{\varrho}\mathrm{d}\gamma \leq \int_{\Omega}
\psi_{\varrho}\mathrm{d}\nu+o_{\varrho}(1).
$$
Letting $\varrho \to 0$ and using standard theory of Radon measures,
we conclude that $\mu_i \leq \nu_i$. It follows from
\eqref{lema_infinito_eq11} that
$$
\mu_{i}\geq S\nu_{i}^{2/2^{**}}\geq S\mu _{i}^{2/2^{**}},
$$
where we conclude that
\begin{equation}\label{lemafinitoeq22}
\mu_{i}\geq S^{N/4}.
\end{equation}

Now we shall prove that the above inequality cannot occur, and
therefore the set $\Lambda$ is empty. Indeed, arguing by
contradiction, let us suppose that $\mu_i\geq S^{N/4}$, for some
$i\in \Lambda$. Thus,
\[
c_{\lambda} = I_{\lambda}(u_n) -  \frac{1}{2^{**}}
I_{\lambda}'(u_n)u_n + o_n(1).
\]
Since $M_0(t)\leq  \frac{2^{**}}{2}m_0$ for all $t
\in \mathbb{R}$, we have
\begin{align*}
c_{\lambda} \geq \frac{2}{N} \int_{\Omega}|\Delta u_{n}|^{2} \,dx-
\lambda
(\frac{1}{q}-\frac{1}{2^{**}}) \int_{\Omega} |u_n|^{q} \,dx.
\end{align*}
Letting $n\to\infty$, we obtain
\begin{align*}
c_{\lambda} \geq  \frac{2}{N}\mu_{i}
+\frac{2}{N} \int_{\Omega}|\Delta u|^{2} \,dx- \lambda
(\frac{1}{q}-\frac{1}{2^{**}}) \int_{\Omega} |u|^{q} \,dx.
\end{align*}
Hence,
\begin{align*}
c_{\lambda} \geq  \frac{2}{N}S^{N/4}
+\frac{2}{N}\frac{1}{\lambda^{2}_1} \int_{\Omega}|
u|^{2} \,dx- \lambda
(\frac{1}{q}-\frac{1}{2^{**}}) \int_{\Omega} |u|^{q} \,dx.
\end{align*}
By H\"older's inequality
\begin{align*}
c_{\lambda} \geq  \frac{2}{N}S^{N/4}
+\frac{2}{N}\frac{1}{\lambda^{2}_1} \int_{\Omega}|
u|^{2} \,dx- \lambda
(\frac{1}{q}-\frac{1}{2^{**}})|\Omega|^{\frac{(2-q)}{2}}\Big( \int_{\Omega}
|u|^{2} \,dx\Big)^{q/2}.
\end{align*}
Note that
$$
f(t)= \frac{2}{N}\frac{1}{\lambda^{2}_1}t^{2}-\lambda
(\frac{1}{q}-\frac{1}{2^{**}})|\Omega|^{\frac{(2-q)}{2}}t^{q}
$$
is a continuous function that attains its absolute minimum, for
$t>0$, at the point
$$
\alpha_0=\Big[q\lambda\bigl(\frac{1}{q}-\frac{1}{2^{**}}\bigl)|\Omega|^{(2-q)/2}\frac{N}{4}\lambda^{2}_1\Big]^{\frac{1}{(2-q)}}.
$$
Hence,
\begin{align*}
c_{\lambda} \geq  \frac{2}{N}S^{N/4}
+\frac{2}{N}\frac{1}{\lambda^{2}_1}\alpha^{2}_0- \lambda
(\frac{1}{q}-\frac{1}{2^{**}})|\Omega|^{\frac{(2-q)}{2}}\alpha_0^{q}.
\end{align*}
So
\begin{align*}
c_{\lambda} \geq  \frac{2}{N}S^{N/4} - \lambda
(\frac{1}{q}-\frac{1}{2^{**}})|\Omega|^{\frac{(2-q)}{2}}\alpha_0^{q}.
\end{align*}
Thus, we conclude that
\begin{align*}
c_{\lambda} \geq  \frac{2}{N}S^{N/4} - \lambda^{\frac{2}{(2-q)}}
(\frac{1}{q}-\frac{1}{2^{**}})|\Omega|^{\frac{(2-q)}{2}}
\Big[q\bigl(\frac{1}{q}-
\frac{1}{2^{**}}\bigl)|\Omega|^{\frac{(2-q)}{2}}\frac{N}{4}
\lambda^{2}_1\Big]^{\frac{q}{(2-q)}},
\end{align*}
which is a contradiction. Thus $\Lambda$ is empty and it follows
that $u_n \to u$ in $L^{2^{**}}(\Omega)$. Thus, up to a subsequence,
$$
 \lim_{n\to\infty}\Big[ \int_{\Omega}|\Delta
u_{n}|^{2} \,dx + M_0\bigl( \int_{\Omega}|\nabla
u_{n}|^{2} \,dx\bigl) \int_{\Omega}|\nabla u_{n}|^{2} \,dx\Big]
=\lambda \int_{\Omega}|u|^{q} \,dx+ \int_{\Omega}|u|^{2^{**}} \,dx.
$$
Moreover, since $u_{n}\rightharpoonup u$ in $H$ and
$M_0\bigl( \int_{\Omega}|\nabla u_{n}|^{2} \,dx\bigl)\to \beta$,
for some $\beta \geq 0$, we have
$$
\big[ \int_{\Omega}|\Delta u|^{2} \,dx +
\beta \int_{\Omega}|\nabla u|^{2} \,dx\big]
=\lambda \int_{\Omega}|u|^{q} \,dx+ \int_{\Omega}|u|^{2^{**}} \,dx.
$$
We claim that
$$
 \lim_{n\to\infty}\|u_{n}\|^{2}=\|u\|^{2}
$$
because, otherwise, we have
$$
 \limsup_{n\to\infty} \int_{\Omega}|\Delta
u_{n}|^{2} \,dx<  \int_{\Omega}|\Delta u|^{2} \,dx
$$
or
$$
 \limsup_{n\to\infty} \int_{\Omega}|\nabla
u_{n}|^{2} \,dx\ <  \int_{\Omega}|\nabla
u|^{2} \,dx.
$$
The second inequality implies
$$
 \limsup_{n\to\infty}M_0\bigl( \int_{\Omega}|\nabla
u_{n}|^{2} \,dx\bigl) \int_{\Omega}|\nabla
u_{n}|^{2} \,dx\ <\beta \int_{\Omega}|\nabla
u|^{2} \,dx.
$$
Thus, in either of these two cases, we have
\begin{align*}
&\lambda \int_{\Omega}|u|^{q} \,dx+ \int_{\Omega}|u|^{2^{**}} \,dx \\
&=  \limsup_{n\to\infty}\big[ \int_{\Omega}|\Delta
u_{n}|^{2} \,dx + M_0\bigl( \int_{\Omega}|\nabla
u_{n}|^{2} \,dx\bigl) \int_{\Omega}|\nabla u_{n}|^{2} \,dx\big]\\
&<  \int_{\Omega}|\Delta u|^{2} \,dx +
\beta \int_{\Omega}|\nabla u|^{2} \,dx \\
& =\lambda \int_{\Omega}|u|^{q} \,dx+ \int_{\Omega}|u|^{2^{**}} \,dx,
\end{align*}
which is a contradiction.  Hence,
$\|u_{n}-u\|^{2} =o_{n}(1)$.
\end{proof}


By Lemma \ref{nivelbaixo}  we conclude that, there exists
$\tau_{2}>0$ such that, for all $\lambda \in (0,\tau_{2})$ we obtain
\begin{align*}
\frac{2}{N}S^{N/4}-\lambda^{\frac{2}{(2-q)}}\bigl(\frac{1}{q}-\frac{1}{2^{**}}\bigl)
|\Omega|^{\frac{(2-q)}{2}}\Big[q\bigl(\frac{1}{q}-\frac{1}{2^{**}}\bigl)|\Omega|^{\frac{(2-q)}{2}}\frac{N}{4}\lambda^{2}_1\Big]^{\frac{q}{(2-q)}}
>0
\end{align*}
and, hence, if $(u_{n})$ is a bounded sequence  such that
$I_{\lambda}(u_{n})\to c$,  $I_{\lambda}'(u_{n})\to 0$ with $c<0$,
then $(u_{n})$ has a subsequence convergent.

\begin{lemma}\label{faltava}
If $J_{\lambda}(u) <0$, then $\|u\|^{2}< R_0\leq M(t_0)$ and
$J_{\lambda}(v)=I_{\lambda}(v)$, for all $v$ in a small enough
neighborhood of $u$. Moreover, $J_{\lambda}$ satisfies a local
Palais-Smale condition for $c_{\lambda}<0$.
\end{lemma}

\begin{proof}
Since $\lambda\in (0,\tau_1)$ and $J_{\lambda}(u) <0$, then by definition
of $\overline{g}$, we obtain
$\overline{g}(\|u\|^{2})\leq J_{\lambda}(u) <0$.
Consequently, $J_{\lambda}(u)=I_{\lambda}(u)$. Hence, we conclude  $\|u\|^{2} <
R_0\leq M(t_0)$. Moreover, since $J_{\lambda}$ is a continuous
functional, we derive $J_{\lambda}(v)=I_{\lambda}(v)$, for
all $v \in B_{R_0/2}(0)$. Besides, if $(u_{n})$ is a sequence such
that $J_{\lambda}(u_{n})\to c_{\lambda}<0$ and
$J_{\lambda}'(u_{n})\to 0$, for $n$ sufficiently large,
$I_{\lambda}(u_{n})=J_{\lambda}(u_{n})\to c_{\lambda}<0$ and
$I_{\lambda}'(u_{n})=J_{\lambda}'(u_{n})\to 0$. Since
$J_{\lambda}$ is coercive, we obtain that $(u_{n})$ is bounded in $H$.
From Lemma \ref{nivelbaixo}, for all $\lambda \in (0,\tau_{2})$, we
obtain
\[
c_{\lambda}<0
<\frac{2}{N}S^{N/4}-\lambda^{\frac{2}{(2-q)}}\bigl(\frac{1}{q}-\frac{1}{2^{**}}\bigl)
|\Omega|^{\frac{(2-q)}{2}}\Big[q\bigl(\frac{1}{q}-\frac{1}{2^{**}}\bigl)
|\Omega|^{\frac{(2-q)}{2}}\frac{N}{4}\lambda^{2}_1\Big]^{\frac{q}{(2-q)}}
\]
and, hence, up to a subsequence, $(u_{n})$ is strongly convergent in
$H$.
\end{proof}


Now, we construct an appropriate mini-max sequence of negative
critical values for the functional $J_{\lambda}$.

\begin{lemma}\label{minimax}
Given $k \in \mathbb{N}$, there exists $\epsilon = \epsilon(k)>0$
such that
$$
\gamma(J_{\lambda}^{-\epsilon}) \geq k,
$$
where $J_{\lambda}^{-\epsilon}=\{u \in H: J_{\lambda}(u) \leq -\epsilon\}$ and
$\gamma$ was given in definition 3.1.
\end{lemma}

\begin{proof}
Fix $k \in \mathbb{N}$, let $X_{k}$ be a
$k$-dimensional subspace of $H$. Thus, there exists $C_k>0$ such that
$$
-C(k)\|u\|^{q}\geq -  \int_{\Omega}|u|^{q} \,dx,
$$
for all $u \in X_{k}$.
We now use the inequality above and \eqref{vaidarcerto} to conclude
that
\begin{align*}
J_{\lambda}(u)\leq \frac{2^{**}}{4}\|
u\|^{2}-\frac{C(k)}{q}\|u\|^{q}= \| u \|^{q}\Big(\frac{k_1}{2}\|
u\|^{2-q}-\frac{C(k)}{q}\Big).
\end{align*}
Considering $R>0$ sufficiently small, there exists
$\epsilon=\epsilon(R)>0$ such that
$$
J_{\lambda}(u)<-\epsilon < 0,
$$
for all $u\in {\mathcal{S}_R}=\{u\in X_k; \| u \|=R \}$. Since $X_k$
and $\mathbb{R}^k$ are isomorphic and $\mathcal{S}_R$ and $S^{k-1}$
are homeomorphic, where $S^{k-1}$ is the sphere of $\mathbb{R}^k$.
Then we conclude from Corollary \ref{esfera} that
$\gamma(\mathcal{S}_R)=\gamma(S^{k-1})=k$. Moreover, since
${\mathcal{S}_R} \subset J_{\lambda}^{-\epsilon}$ and
$J_{\lambda}^{-\epsilon}$ is symmetric and closed,  we have
$$
k= \gamma ({\mathcal{S}_R})\leq \gamma( J_{\lambda}^{-\epsilon}).
$$
\end{proof}

Now for each $k \in \mathbb{N}$, we define the sets
\begin{gather*}
\Gamma_{k}=\{C \subset H\backslash\{0\}: C \text{ is closed }, C=-C
\text{ and }  \gamma(C) \geq k\}, \\
K_{c}=\{u \in H\backslash\{0\}: J_{\lambda}'(u)=0  \text{ and }
J_{\lambda}(u)=c\}
\end{gather*}
and the number
$$
c_{k}= \inf_{C\in \Gamma_{k}} \sup_{u \in
C}J_{\lambda}(u).
$$


\begin{lemma}\label{minimax1}
For each $k \in \mathbb{N}$, the number $c_{k}$ is negative.
\end{lemma}

\begin{proof}
From Lemma \ref{minimax}, for each $k\in
\mathbb{N}$ there exists $\epsilon >0$ such that
$\gamma(J_{\lambda}^{-\epsilon}) \geq k$. Moreover,
$ 0 \notin J_{\lambda}^{-\epsilon}$ and
$J_{\lambda}^{-\epsilon}\in \Gamma_{k}$. On the other hand
$$
 \sup_{u\in J_{\lambda}^{-\epsilon}}J_{\lambda}(u)\leq
-\epsilon.
$$
Hence,
$$
-\infty < c_{k}= \inf_{C\in
\Gamma_{k}} \sup_{u \in C}J_{\lambda}(u) \leq
 \sup_{u\in J_{\lambda}^{-\epsilon}}J_{\lambda}(u) \leq
-\epsilon <0.
$$
\end{proof}

The next Lemma allows us to prove the existence of critical points
of $J_{\lambda}$.

\begin{lemma}\label{minimax2}
If $c=c_{k}=c_{k+1}=\dots =c_{k+r}$ for some $r \in \mathbb{N}$, then
there exists $\lambda^{*}>0$ such that
$$
\gamma(K_{c})\geq r+1,
$$
for $\lambda \in ( 0, \lambda^{*})$.
\end{lemma}

\begin{proof}
Since $c=c_{k}=c_{k+1}=\dots =c_{k+r} <0$, for
$\lambda^{*}=\min\{\tau_1,\tau_{2}\}$ and for all
$\lambda \in (0,\lambda^{*})$, from Lemma \ref{nivelbaixo} and Lemma
\ref{minimax1}, we obtain that $K_{c}$ is a compact set. Moreover,
$K_{c}= - K_{c}$. If $\gamma(K_{c})\leq r$, there exists a closed
and symmetric neighborhood $U$ of $ K_{c}$ such that
$\gamma(U)= \gamma(K_{c}) \leq r$. Note that we can choose
$U\subset J_{\lambda}^{0}$ because $c<0$. By the deformation lemma
\cite{benci} we have an odd homeomorphism $ \eta: H\to H$
such that $\eta(J_{\lambda}^{c+\delta}-U)\subset
J_{\lambda}^{c-\delta}$ for some $\delta > 0$ with $0<\delta < -c$.
Thus, $J_{\lambda}^{c+\delta}\subset J_{\lambda}^{0}$ and by
definition of $c=c_{k+r}$, there exists $A \in \Gamma_{k+r}$ such
that $ \sup_{u \in A} < c+\delta$, that is,
$A \subset J_{\lambda}^{c+\delta}$ and
\begin{equation}\label{estrela1}
\eta(A-U) \subset \eta ( J_{\lambda}^{c+\delta}-U)\subset
J_{\lambda}^{c-\delta}.
\end{equation}
But $\gamma(\overline{A-U})\geq \gamma(A)-\gamma(U) \geq k$ and
$\gamma(\eta(\overline{A-U}))\geq  \gamma(\overline{A-U})\geq k$.
Then $\eta(\overline{A-U}) \in \Gamma_{k}$ and this contradicts
\eqref{estrela1}. Hence, the lemma is proved.
\end{proof}

\begin{remark}\label{Teorema22} \rm
If $-\infty< c_1 < c_{2} < \dots < c_{k}< \dots <0$ with
$c_{i}\neq c_{j}$, since each $c_{k}$ is a critical value of $J_{\lambda}$,
then we obtain infinitely many critical points of $J_{\lambda}$ and,
hence problem \eqref{eTl} has infinitely many solutions.

On the other hand, if there are two constants $c_{k}=c_{k+r}$, then
$c=c_{k}=c_{k+1}=\dots =c_{k+r}$ and from Lemma \ref{minimax2}, there
exists $\lambda^{*}>0$ such that
$$
\gamma(K_{c})\geq r+1 \geq 2
$$
for all $\lambda \in (0,\lambda^{*})$. From Proposition
\ref{paracompletar}, $K_{c}$ has infinitely many points, that is,
problem \eqref{eTl} has infinitely many solutions.
\end{remark}

\begin{proof}[Proof of Theorem \ref{Teorema1}]
 Let $\lambda^{*}$ be as in Lemma \ref{minimax2} and, for
$\lambda< \lambda^{*}$, let $u_{\lambda}$ be the nontrivial solution of problem
\eqref{eTl} found in remark \ref{Teorema22}. Thus
$J_{\lambda}(u_{\lambda})=I_{\lambda}(u_{\lambda}) <0$. Hence,
\begin{equation}\label{provaassimp1}
 \int_{\Omega}|\nabla u_{\lambda}|^{2} \,dx\leq
\|u_{\lambda}\|^{2} \leq R_0 \leq t_0.
\end{equation}
By the definition of $M_0$ we obtain
$$
M_0\Big( \int_{\Omega}|\nabla u_{\lambda}|^{2} \,dx\Big)
=M\Big( \int_{\Omega}|\nabla u_{\lambda}|^{2} \,dx\Big),
$$
which implies that $u_{\lambda} $ is a solution of
\eqref{ePl}. Moreover, from \eqref{provaassimp1} and
\eqref{comportamentoassimtotico1}, we conclude
$$
 \lim_{\lambda \to 0}\|u_{\lambda}\|=0.
$$
Since for each solution $u_{\lambda}$ we have that
$M(\|u_{\lambda}\|^{2})\geq m_0>0$ is a positive number, then the
regularity of these solutions is a consequence of
\cite[Theorem 2.1]{Bernis}.
\end{proof}


\section{Case $2<q< 2^{**}$}

In this section, we adapt for our study some ideas from
\cite{jmaa}. In the sequel, we prove that the functional
$I_{\lambda}$ has the Mountain Pass Geometry. This fact is proved in
the next lemmas:


\begin{lemma}\label{geometria1}
Assume that condition \eqref{eM} holds. There exist positive numbers
$\rho$ and $\alpha $ such that
$$
I_{\lambda}(u)\geq \alpha>0, \quad \forall u\in H : \|u\|=\rho.
$$
\end{lemma}

\begin{proof}  From \eqref{eM}, we have
$$
I_{\lambda}(u) \geq  \frac{k_0}{2}\|u\|^{2} -
\frac{\lambda}{q}\int_{\Omega}|u|^{q}\,dx -
\frac{1}{2^{**}}\int_{\Omega}|u|^{2^{**}}\,dx,
$$
where $k_0=\min\{1, m_0\}$. So, using Sobolev's Embedding Theorem,
there exists a positive constant $C>0$ such that
$$
I_{\lambda}(u) \geq   C\|u\|^{2} -\lambda C\|u\|^{q} -
C\|u\|^{2^{*}}.
$$
Since $2< q< 2^{**}$, the result follows by choosing $\rho>0$ small
enough.
\end{proof}

\begin{lemma}\label{segundageometria1}
For all $\lambda>0$, there exists $e\in H$ with
 $I_{\lambda}(e)<0$ and $\| e\| >\rho$, where $\rho$ was given
 in Lemma \ref{geometria1}.
\end{lemma}

\begin{proof}
 Fix $v_0\in C^{\infty}_0(\Omega) \setminus \{0\}$ with
$v_0 \geq 0$ in $\Omega$ and $\|v_0\|=1$.
Using \eqref{vaidarcerto1}   we obtain
$$
I_{\lambda}(tv_0)
\leq \frac{1}{2}\max\{1, \frac{m_0q}{2}\} t^{2}
 - \frac{t^{2^{**}}}{2^{**}}\int_{\Omega}|v_0|^{2^{**}}\,dx.
$$
Since $2< q<2^{**}$,  the result follows by considering
$e=\overline{t}v_0$ for some $\overline{t}>0$ large enough.
\end{proof}

Using a version of the Mountain Pass Theorem due to Ambrosetti and
Rabinowitz \cite{Ambrosetti}, without (PS) condition  (see
\cite[p.12]{Willem}), there exists a sequence $(u_{n})\subset H$
satisfying
$$
I_{\lambda}(u_{n})\to c_{\lambda} \quad\text{and} \quad I_{\lambda}'(u_{n})\to 0,
$$
where
\begin{gather*}
c_{\lambda} = \inf_{\gamma \in \Gamma} \max_{t \in [0,1]}
I_{\lambda}(\gamma(t))>0, \\
\Gamma := \{ \gamma \in C([0,1],H) : \gamma(0)=0,\; I_{\lambda}(\gamma(1)) < 0\}.
\end{gather*}

Next, we shall prove an estimate for $c_{\lambda}$.

\begin{lemma}\label{nivel}
If  condition \eqref{eM} holds, then
$ \lim_{\lambda\to \infty}c_{\lambda}=0$.
\end{lemma}

\begin{proof}
Since the functional $I_{\lambda}$ has the
Mountain Pass geometry, it follows that there exists $t_{\lambda}>0$
satisfying $I_{\lambda}(t_{\lambda}v_0)=\max_{t\geq 0}I_{\lambda}(tv_0)$,
 where $v_0$ is the function given by Lemma
\ref{segundageometria1}, that  does not depend of $\lambda$.
Hence, from \eqref{vaidarcerto1} we obtain
\begin{equation}\label{casos}
t_{\lambda}^{2} \frac{1}{2}\max\{1, \frac{m_0q}{2}\} \geq \lambda
t_{\lambda}^{q}\int_{\Omega}|v_0|^{q}\,dx
+t_{\lambda}^{2^{**}}\int_{\Omega}|v_0|^{2^{**}} \,dx\geq
t_{\lambda}^{2^{**}}\int_{\Omega}|v_0|^{2^{**}} \,dx,
\end{equation}
which implies that $(t_{\lambda})$ is bounded. Thus, there exists a
sequence $\lambda_{n} \to +\infty$ and $\beta_0 \geq 0$ such that
$ t_{\lambda_{n}}\to \beta_0$ as $n \to +\infty$.
Consequently, exists $D>0$ such that
\[
t_{\lambda_{n}}^{2} \frac{1}{2}\max\{1, \frac{m_0q}{2}\}\leq D \quad
\forall n \in \mathbb{N},
\]
and so
\[
t^{q}_{\lambda_{n}} \lambda_{n}\int_{\Omega}|v_0|^{q} \,dx
+t_{\lambda_{n}}^{2^{**}}\int_{\Omega}|v_0|^{2^{**}} \leq D
\quad \forall n \in \mathbb{N}.
\]
If $\beta_0>0$, the above inequality leads to
\begin{align*}
 \lim_{n\to
\infty}\lambda_{n}t^{q}_{\lambda_{n}}\int_{\Omega}|v_0|^{q} \,dx
+t_{\lambda_{n}}^{2^{**}}\int_{\Omega}|v_0|^{2^{**}}
=+\infty,
\end{align*}
which is a contradiction. Thus, we conclude that $\beta_0=0$. Now,
let us consider the path $\gamma_{*}(t)=te$ for $t \in [0,1]$,
 to get the  estimate
$$
0<c_{\lambda}\leq \max_{t \in
[0,1]}I(\gamma_{*}(t))=I(t_{\lambda}v_0) \leq
Ct_{\lambda}^{2},
$$
for some positive $C$. In this way,
$ \lim_{\lambda\to \infty}c_{\lambda}=0$.
\end{proof}

\begin{lemma}\label{limitacao}
Let $(u_{n}) \subset H$  be a sequence such that
$$
I_{\lambda}(u_{n}) \to c_{\lambda} \quad \text{and} \quad
I_{\lambda}'(u_{n}) \to 0.
$$
Then
$$
\|u_{n}\|^{2}\leq t_0, \quad
\text{for all $n\in \mathbb{N}$ where $t_0$ is given in
\eqref{pedidodoreferee}}.
$$
\end{lemma}

\begin{proof} Assuming, by contradiction, that, up to a
subsequence that $\|u_{n}\|^{2}> t_0$. Thus, from
\eqref{vaidarcerto1} we obtain
\[
c_{\lambda} =I_{\lambda}(u_{n})-
\frac{1}{q}I_{\lambda}'(u_{n})u_{n} + o_{n}(1)\geq
\frac{1}{2}\widehat{M}_0(\|u_{n}\|^{2})
-\frac{1}{q}M(t_0)\|u_{n}\|^{2}+o_{n}(1).
\]
Thus
\begin{equation}\label{comportamentoassimtotico2}
c_{\lambda} \geq
\bigl(\frac{1}{2}m_0-\frac{1}{q}M(t_0)\bigl)\|u_{n}\|^{2}
 + o_{n}(1).
\end{equation}
Since $m_0 < M(t_0) < \frac{q}{2}m_0$, we obtain
$$
c_{\lambda}\geq \bigl(\frac{1}{2}m_0-\frac{1}{q}M(t_0)\bigl)t_0.
$$
 But this last inequality is in contradiction with  Lemma
\ref{nivel}. Hence $(u_{n})$ is bounded in $H$ by constant
$\sqrt{t_0}$.
\end{proof}


\begin{proof}[Proof of Theorem \ref{Teorema3}]
 From Lemma \ref{nivel} we have
$ \lim_{\lambda\to +\infty}c_{\lambda} =0$.
Therefore, there exists $\lambda^{**}>0$ such that
\begin{equation}\label{ref4}
c_{\lambda}< \frac{2}{N}S^{\frac{N}{4}},
\end{equation}
for all $\lambda\geq \lambda^{**}$.  Now, fix $\lambda\geq
\lambda^{**}$ and let us to show that \eqref{eTl} admits
a positive solution. From Lemmas \ref{geometria1} and
\ref{segundageometria1}, there exists a bounded sequence $(u_{n})
\subset H$ satisfying
$$
I_{\lambda}(u_{n}) \to c_{\lambda} \quad \text{and} \quad
I_{\lambda}'(u_{n}) \to 0.
$$
Arguing as in Lemma \ref{nivelbaixo} we conclude that
$u_{n}\to u_{\lambda}$ in $L^{2^{**}}(\Omega)$. This
convergence implies that $u_{n}\to u_{\lambda}$ in $H$.
Thus, $u_{\lambda}$ is a solution of \eqref{eTl}. Moreover, by
Lemma \ref{limitacao}, $u_{\lambda}$ is a solution of Problem
\eqref{ePl} and from \eqref{comportamentoassimtotico2} and Lemma
\ref{nivel} we obtain
$$
 \lim_{\lambda\to + \infty}\|u_{\lambda}\|=0.
$$
Since  for each solution $u_{\lambda}$ we have that
$M(\|u_{\lambda}\|^{2})\geq m_0>0$ is a positive number, then the
regularity of these solutions is a consequence of \cite[Theorem
2.1]{Bernis}.
\end{proof}


\subsection*{Acknowledgments}
Giovany M. Figueiredo was supported by PROCAD/CASA\-DINHO: 552101/2011-7,
 CNPq/PQ  301242/2011-9 and CNPQ/CSF  200237/2012-8
R\'ubia G. Nascimento was supported by PROCAD/CASADINHO: 552101/2011-7.

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\end{document}