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\AtBeginDocument{{\noindent\small
2005-Oujda International Conference on Nonlinear Analysis.
\newline {\em Electronic Journal of Differential Equations},
Conference 14, 2006, pp. 163--172.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or
http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2006 Texas State University - San Marcos.}
\vspace{9mm}}
\setcounter{page}{163}


\begin{document}
\title[\hfilneg EJDE/Conf/14 \hfil Existence of two nontrivial solutions]
{Existence of two nontrivial solutions for semilinear elliptic problems}

\author[A. R. El Amrouss, F. Moradi, M. Moussaoui \hfil EJDE/Conf/14 \hfilneg]
{Abdel R. El Amrouss, Fouzia Moradi, Mimoun Moussaoui}

\address{Abdel R. El Amrouss \newline 
University Mohamed 1er, Faculty of sciences\\
Department of Mathematics, Oujda, Morocco}
\email{amrouss@sciences.univ-oujda.ac.ma}

\address{Fouzia Moradi\newline
University Mohamed 1er, Faculty of sciences\\
Department of Mathematics, Oujda, Morocco}
\email{foumoradi@yahoo.fr}

\address{Mimoun Moussaoui \newline
University Mohamed 1er, Faculty of sciences\\
Department of Mathematics, Oujda, Morocco}
\email{moussaoui@sciences.univ-oujda.ac.ma}

\date{}

\thanks{Published September 20, 2006.}
\subjclass[2000]{58E05, 35J65, 56J20}
\keywords{Variational elliptic problem; resonance; critical group; 
\hfill\break\indent Morse theory; minimax method}

\begin{abstract}
 This paper concerns the existence of multiple nontrivial solutions for  some
 nonlinear problems. The first nontrivial solution is found using a  minimax
 method, and the second by computing the Leray-Schauder index  and the
 critical group near 0.
\end{abstract}

\maketitle\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{lemma}[theorem]{Lemma}

\section{Introduction}

We consider the Dirichlet problem
\begin{equation}
\begin{gathered} -\Delta u = \lambda_k u + f(u) \quad\mbox{in }\Omega \\ u =
0 \quad \mbox{on }\partial\Omega, \end{gathered}  \label{e1}
\end{equation}
where $\Omega $ is a bounded domain in $\mathbb{R}^{n}$, and $f:\Omega
\times \mathbb{R}\to\mathbb{R}$ is a nonlinear function satisfying the
Carath\'{e}odory conditions, and $0<\lambda _{1}<\lambda _{2}\leq \dots
\lambda _{k}\leq \dots $ is the sequence of eigenvalues of the problem
\begin{gather*}
-\Delta u=\lambda u\quad \mbox{in }\Omega , \\
u=0\quad \mbox{on }\partial \Omega.
\end{gather*}
Let us denote by $E(\lambda _{j})$ the $\lambda _{j}$-eigenspace and by $%
F(s) $ the primitive $\int_{0}^{s}f(t)\,dt$ .

There are several works studying the problem
\begin{equation}
\begin{gathered} -\Delta u=\lambda _{k}u+f(x,u)+h\quad \mbox{in }\Omega , \\
u=0\quad \mbox{on }\partial \Omega \,. \end{gathered}  \label{e2}
\end{equation}
where $h\in L^{2}(\Omega )$; see for example \cite{C-O,D-G,Do,El2,E-M}. We
write
\begin{gather*}
l_{\pm }(x)=\liminf_{s\to\pm \infty }\frac{f(x,s)}{s},\quad k_{\pm
}(x)=\limsup_{s\to\pm \infty }\frac{f(x,s)}{s}, \\
L_{\pm }(x)=\liminf_{s\to\pm \infty } \frac{2F(x,s)}{s^{2}},\quad K_{\pm
}(x) =\limsup_{s\to\pm \infty }\frac{2F(x,s)}{s^{2}}\,.
\end{gather*}
In \cite{Do}, the solvability of (\ref{e2}) for every $h\in L^{2}(\Omega)$,
is ensured when
\begin{equation*}
0<\upsilon _{k}\leq l_{\pm }(x)\leq k_{\pm }(x)\leq \upsilon _{k+1}<\lambda
_{k+1}-\lambda _{k},
\end{equation*}
where $\upsilon _{k}$ and $\upsilon _{k+1}$ are constants.

However, in the autonomous case $f(x,s)=f(s)$, De Figuerido and Gossez \cite
{D-G} introduced a density condition that requires $\frac{f(s)}{s} $ to be
between $0$ and $\alpha =\lambda _{k+1}-\lambda _{k}$ as $s\to\pm \infty $,
and showed the existence of solution for any $h$. Next in \cite{C-O}, Costa
and Oliveira proved an existence result for (\ref{e2}) under the following
conditions:
\begin{gather}
0\leq l_{\pm }(x)\leq k_{\pm }(x)\leq \lambda _{k+1}-\lambda _{k} \quad
\text{uniformly for a.e } x\in \Omega,  \label{e3} \\
0\preceq L_{\pm }(x)\leq K_{\pm }(x)\preceq \lambda _{k+1}-\lambda _{k}
\quad \text{uniformly for a.e } x\in \Omega .  \label{e4}
\end{gather}
Here the relation $a(x)\preceq b(x)$ indicates that $a(x)\leq b(x)$  on $%
\Omega $, with strict inequality holding on subset of positive measure.

Later in \cite{E-M}, the authors proved an existence result in situation $%
L_{\pm }(x)=0$ for a.e $x\in \Omega $ and $K_{\pm }(x)=\lambda
_{k+1}-\lambda _{k}$ for a.e $x\in \Omega $. They replaced (\ref{e4}) by
classical resonance conditions of Ahmad-Lazer-Paul on two sides of (\ref{e4}%
) and showed that (\ref{e2}) is solvable. More recently, in \cite{El2}, the
author interested to study the existence of two nontrivial solutions in the
case $k=1$ and under other weaker conditions cited above.

The aim of this paper is to generalize the above result for $k\geq 1$. We
assume the following assumptions:

\begin{itemize}
\item[(F0)]  $|f'(s)|\leq c(|s|^{p}+1)$, $s\in \mathbb{R}$, 
$p<\frac{4}{n-2}$ if $n\geq 3$ and no restriction if $n=1,2$.

\item[(F1)]  $sf(s)\geq 0$ for $|s|\geq r>0$ and
\begin{equation*}
\limsup_{s\to \pm \infty }\frac{f(s)}{s}\leq \lambda _{k+1}-\lambda
_{k}=\alpha .
\end{equation*}

\item[(F2)]  $\lim_{\Vert v\Vert \rightarrow \infty ,v\in E(\lambda
_{k})}\int F(v(x))dx=+\infty $.

\item[(F3)]  There exists $\eta \in \mathbb{R}$, $0<\eta <\alpha $, such
that
\begin{equation*}
\liminf_{n\rightarrow +\infty }\frac{\mu (G_{n})}{n}>0
\end{equation*}
where $G_{n}=\{s\in ]-n,n[,s\neq 0$, and $\frac{f(s)}{s}\leq \alpha -\eta \}$
and $\mu $ denotes the Lebesgue measure on $\mathbb{R}$.

\item[(F4)]  $f'(0)+\lambda _{k}<\lambda _{1}$
\end{itemize}

\begin{theorem} \label{thm1.1}
Let $f$ be $ C^{1}$ function, with $f(0)=0$, that satisfies the
conditions (F0)-(F4). Then  \eqref{e1} has at least two
nontrivial solutions.
\end{theorem}

This paper is organized as follows: In section 2, we give some technical
lemmas and some results of critical groups. The proof of our result is
carried out in section 3.

\section{Preliminaries Lemmas}

Let us consider the functional defined on $H_{0}^{1}(\Omega )$ by
\begin{equation*}
\Phi (u)=\frac{1}{2}\int_{\Omega }|\nabla u|^{2}dx-\frac{1}{2}\lambda
_{k}\int u^{2}dx-\int F(u)dx.
\end{equation*}
where $H_{0}^{1}(\Omega )$ is the usual Sobolev space obtained through the
completion of $C_{c}^{\infty }(\Omega )$ with respect to the norm induced by
the inner product
\begin{equation*}
\langle u,v\rangle =\int_{\Omega }\nabla u\nabla vdx,\quad u,v\in
H_{0}^{1}(\Omega ).
\end{equation*}
It is well known that under a linear growth condition on $f$, the functional
$\Phi $ is well defined on $H_{0}^{1}(\Omega )$, weakly lower
semi-continuous and $\Phi \in C^{1}( H_{0}^{1},\mathbb{R})$, with
\begin{equation*}
\langle \Phi '(u),v\rangle =\int_{\Omega }\nabla u\nabla vdx-\lambda
_{k}\int uvdx-\int f(u)vdx,\quad \mbox{for }u,v\in H_{0}^{1}(\Omega ).
\end{equation*}
Consequently, the weak solutions of the problem \eqref{e1} are the critical
points of the functional $\Phi $. Moreover, under the condition(F0), $\Phi $
is a $C^{2}$ functional with the second derivative given by
\begin{equation*}
\Phi ^{\prime\prime}(u) v.w=\int \nabla v\nabla wdx-\lambda _{k}\int
vwdx-\int f'(u)vwdx,
\end{equation*}
for $u,v,w\in H_{0}^{1}(\Omega )$.

Since we are going to apply the variational characterization of the
eigenvalues, we shall decompose the space $H_{0}^{1}(\Omega )$ as $%
E=E_{-}\oplus E_{k}\oplus E_{k+1}\oplus E_{+}$, where $E_{-}$ is the
subspace spanned by the $\lambda _{j}$- eigenfunctions with $j<k$ and $E_{j}$
is the eigenspace generated by the $\lambda _{j}$-eigenfunctions and $E_{+}$
is the orthogonal complement of $E_{-}\oplus E_{k}\oplus E_{k+1}$ in $%
H_{0}^{1}(\Omega )$ and we shall decompose for any $u\in H_{0}^{1}(\Omega )$
as following $u=u^{-}+u^{k}+u^{+}$ where $u^{-}\in E_{-}$, $u^{k}\in E_{k}$,
$u^{k+1}\in E_{k+1}$ and $u^{+}\in E_{+}$. We can verify easily that
\begin{gather}
\int |\nabla u|^{2}\,dx-\lambda _{i}\int |u|^{2}\,dx\geq \delta _{i}\Vert
u\Vert ^{2}\quad \forall u\in \oplus _{j\geq i+1}E_{j}  \label{e5} \\
\int |\nabla u|^{2}\,dx-\lambda _{i}\int |u|^{2}\,dx\leq -\delta _{i}\Vert
u\Vert ^{2}\quad \forall u\in \oplus _{j\leq i}E_{j},  \label{e6}
\end{gather}
where $\delta _{i}=\min \{1-\frac{\lambda _{i}}{\lambda _{i+1}},\frac{
\lambda _{i}}{\lambda _{i-1}}-1\}$.

\subsection{A compactness condition}

To apply minimax methods for finding critical points of $\Phi $, we need to
verify that $\Phi $ satisfies a compacteness condition of the Palais-Smail
type which was introduced by Cerami \cite{Ce}, and recently was generalized
by the first author in \cite{El}.

\noindent\textbf{Definition.} Let $E$ be a real Banach space and $\Phi \in
C^{1}(E,\mathbb{R})$. \newline
(i) A sequence $(u_{n})$ is said to be a $(C)_{c}$ sequence, at the level $%
c\in \mathbb{R}$, if there is a sequence $\epsilon _{n}\to0$, such that
\begin{gather}
\Phi (u_{n})\to c  \label{e7} \\
\| u_{n}\| \langle \Phi '(u_{n}),v\rangle _{H_{0}^{1},H^{-1}}\leq {%
\epsilon _{n}}\| v\| \quad \forall v\in H_{0}^{1}.  \label{e8}
\end{gather}
(ii) A functional $\Phi \in C^{1}(E,\mathbb{R})$, is said to satisfy a
condition $(C)_{c}$, at the level $c\in \mathbb{R}$, if every $(C)_{c}$
sequence $(u_{n})$, possesses a convergent subsequence.

Now, we present some technical lemmas.

\begin{lemma} \label{lm2.1}
Let $(u_n)\subset H_0^1(\Omega)$ and $(p_n) \subset L^\infty(\Omega)$
be sequences, and let A a nonnegative constant  such that
$$
0 \leq p_n(x) \leq A \quad\mbox{a.e.  in  $\Omega$  and   for   all }
  n \in \mathbb{N}
$$
and
$p_n\rightharpoonup 0 $ in the weak* topology of $L^\infty$, as
$n\to\infty$.
Then, there are subsequences $(u_n),(p_n)$ satisfying the above conditions,
and there is a positive integer $n_0$ such that for all $ n\geq n_0$,
\begin{equation}\label{e9}
\int p_n u_n((u_n^- + u_n^k) - (u_n^{k+1} + u_n^+)) \,dx \geq
\frac{-\delta_k}{2} \|u_n^+ + u_n^{k+1}\|^2.
\end{equation}
\end{lemma}

\begin{proof}
Since $p_{n}\geq 0$ a.e. in $\Omega $, we see that
\begin{equation}  \label{e10}
\begin{aligned}
&\int p_n u_n((u_n^-+u_n^k)-(u_n^{k+1}+u_n^+))\\
&\geq -\int p_n(u_n^+ + u_n^{k+1})^2\,dx \\
&\geq -\Big[\int p_n\big(\frac{u_n^+
+u_n^{k+1}}{\|u_n^++u_n^{k+1}\|}\big)^2 \,dx
\Big]\|u_n^++u_n^{k+1}\|^2. \end{aligned}
\end{equation}
Moreover, by the compact imbedding of $H_{0}^{1}(\Omega )$ into $
L^{2}(\Omega )$ and $p_{n}\rightharpoonup 0$ in the weak* topology of $
L^{\infty }$, when $n\to\infty $, then there are subsequences $
(u_{n}),(p_{n})$ such that
\begin{equation*}
\int p_{n}\Big(\frac{u_{n}^{+}+u_{n}^{k+1}}{\| u_{n}^{+}+u_{n}^{k+1}\| }\Big)
^{2}\,dx\to0.
\end{equation*}
Therefore, there exists $n_{0}\in \mathbb{N}$ such that for $n\geq n_{0}$ we
have
\begin{equation}  \label{e11}
\int p_{n}\Big(\frac{u_{n}^{+}+u_{n}^{k+1}}{\| u_{n}^{+}+u_{n}^{k+1}\| }\Big)
^{2}\,dx\leq  \frac{\delta _{k}}{2}.
\end{equation}
Combining inequalities (\ref{e10}) and (\ref{e11}), we get inequality (\ref
{e9}).
\end{proof}

\begin{lemma} \label{lm2.2}
Let $(u_n)\subset H_0^1(\Omega)$ be a $(C)$ sequence.
If
$$
f_n(x)=\frac{f(u_n(x))}{u_n(x)} \chi_{[|u_n(x)| \geq r_{\epsilon}]}
\rightharpoonup 0
$$
in the weak* topology of $L^\infty$, as
$n\to\infty$. Then, there is subsequence $(u_n)$ such that
$(\|u_n^- + (u_n^+ + u_n^{k+1})\|)_n$
  is  uniformly  bounded  in  $n$.
\end{lemma}

\begin{proof}
Since $(u_{n})_{n}\subset H_{0}^{1}$ be a (C) sequence, \eqref{e7} and
\eqref{e8} are satisfied. Now, we prove that the sequence $(\|
u_{n}^{-}+u_{n}^{+}+u_{n}^{k+1}\| )_{n}$ is uniformly bounded in $n$.
Take $v=(u_{n}^{-}+u_{n}^{k})-(u_{n}^{+}+u_{n}^{k+1})$ in (\ref{e8}),
$p_{n}(x)=f_{n}(x)$, and
\begin{align*}
\Lambda =\Big\{& -\int |\nabla u_{n}^{-}|^{2}+\lambda _{k}\int
|u_{n}^{-}|^{2}\,dx+\int |\nabla (u_{n}^{+}+u_{n}^{k+1})|^{2} \\
& -\lambda _{k}\int |u_{n}^{+}+u_{n}^{k+1}|^{2}\,dx+\int
p_{n}u_{n}((u_{n}^{-}+u_{n}^{k})-(u_{n}^{k+1}+u_{n}^{+}))\,dx\Big\} \\
\Gamma =\Big\{& \epsilon _{n}+\int_{|u_{n}(x)|\leq r_{\epsilon
}}|f(u_{n}(x)||(u_{n}^{+}+u_{n}^{k+1})-(u_{n}^{-}+u_{n}^{k})|\,dx\Big\}.
\end{align*}
Then $\Lambda \leq \Gamma $. By the Poincar\'{e} inequality, from (\ref{e5}
), (\ref{e6}), (\ref{e9}), and $\Lambda \leq \Gamma $, it follows that there
exists constants $A_{\epsilon }$ and $B_{\epsilon }$ such that
\begin{equation*}
\frac{\delta _{k}}{2}\| u_{n}^{-}+(u_{n}^{+}+u_{n}^{k+1})\| ^{2}\leq
\epsilon _{n}+A_{\epsilon }\| u_{n}^{-}+(u_{n}^{+}+u_{n}^{k+1})\|
+B_{\epsilon }.
\end{equation*}
This gives that $(\|u_{n}^{-}+(u_{n}^{+}+u_{n}^{k+1})\| )_{n}$ is uniformly
bounded in $n$.
\end{proof}

\begin{lemma} \label{lm2.3}
Let $(u_n)\subset H_0^1(\Omega)$ such that $\|u_n^- + (u_n^+ +
u_n^{k+1})\|$ is uniformly bounded in $n$ and there exists $A$
such that if $A \leq \Phi(u_n)$, then
$$
\int F(\frac{u_n^k}{2}) \,dx \leq M.
$$
\end{lemma}

\begin{proof}
 From $A\leq \Phi (u_{n})$, and Poincar\'{e} inequality, we have
\begin{equation}
\int F(\frac{u_{n}^{k}}{2})dx\leq -A+\int [ F(\frac{u_{n}^{k}}{2}
)-F(u_{n})]dx+\frac{1}{2}\Vert u_{n}^{-}+u_{n}^{+}+u_{n}^{k+1}\Vert
^{2}.  \label{e12}
\end{equation}
Since  $f\in C^{1}\left( \overline{\Omega },\mathbb{R}\right) $ satisfy
(F1), there exists two functions  $\gamma ,h:\Omega \to\mathbb{R}$
 such  that
\begin{equation*}
f(t)=t\gamma \left( t\right) +h\left( t\right)
\end{equation*}
with  $0\leq \gamma \left( t\right) =\frac{f(t)}{t}\chi [ |
t|\geq r]\leq \lambda _{k+1}-\lambda _{k}$  and  $
h(t)=f(t)\chi [ |t|<r]$.
However, by the mean value theorem, we get
\begin{equation}
\begin{aligned} \int [F(\frac{u_{n}^{k}}{2})-F(u_{n})]dx
&=\int_{\Omega}\int_{0}^{1}f(t\frac{u_{n}^{k}}{2}+(1-t)u_{n})dt(
\frac{u_{n}^{k}}{2}-u_{n})dx \\&=\int_{\Omega
}\int_{0}^{1}h(t\frac{u_{n}^{k}}{2}+(1-t)u_{n})dt(
\frac{u_{n}^{k}}{2}-u_{n})dx \\
&\quad + \int_{\Omega }\int_{0}^{1}\gamma
(t\frac{u_{n}^{k}}{2}+(1-t)u_{n})[t(\frac{u_{n}^{k}}{2}-u_{n})^{2}+(
\frac{u_{n}^{k}}{2}-u_{n})u_{n}] \end{aligned}  \label{e13}
\end{equation}
Set  $t_{1}=\min \{t\in [ 0,1]:\int_{0}^{1}h(t\frac{
u_{n}^{k}}{2}+(1-t)u_{n})\neq 0\}$ and
$t_{2}=\max \{t\in [ 0,1]: \int_{0}^{1}h(t\frac{u_{n}^{k}}{2
}+(1-t)u_{n})\neq 0\}$. It is clear that
\begin{equation}
(t_{2}-t_{1})|\frac{u_{n}^{k}}{2}-u_{n}|\leq 2r.  \label{e14}
\end{equation}
So that using (\ref{e13}),(\ref{e14}) and the Poincar\'{e} inequality, and
an elementary  inequality
\begin{equation*}
(\frac{a}{2}-b)^{2}+(\frac{a}{2}-b)b\leq (a-b)^{2}.
\end{equation*}
We have
\begin{equation}  \label{e15}
\begin{aligned}
&\int [ F(\frac{u_{n}^{k}}{2})-F(u_{n})]dx \\
&\leq \int_{\Omega }\int_{t_{1}}^{t_{2}}h(t\frac{u_{n}^{k}}{2}
 +(1-t)u_{n})dt\big( \frac{u_{n}^{k}}{2}-u_{n}\big) dx
+\frac{\lambda _{k+1}-\lambda
_{k}}{4\lambda _{1}}\| u_{n}^{-}+u_{n}^{+}+u_{n}^{k+1}\| ^{2} \\
&\leq 2r\sup_{|s|\leq r} |f(s)|\mathop{\rm meas}(\Omega)
 +\frac{\lambda _{k+1}-\lambda _{k}}{4\lambda _{1}}\|
u_{n}^{-}+u_{n}^{+}+u_{n}^{k+1}\| ^{2}.
 \end{aligned}
\end{equation}
 From (\ref{e12}) and (\ref{e15}), there exists $M>0$ such that
\begin{equation*}
\int F(\frac{u_{n}^{k}}{2})\,dx\leq M.
\end{equation*}
\end{proof}

\subsection{Critical groups}

Let $H$ be a Hilbert space and $\Phi \in C^{1}(H,\mathbb{R})$ satisfying the
Palais-Smaile condition or the Cerami condition. Set $\Phi ^{c}=\{u\in H\mid
\Phi (u)\leq c\}$ and denote by $H_{q}(X,Y)$ the q-th relative singular
homology group with integer coefficient. The critical groups of $\Phi $ at
an isolated critical point u with $\Phi (u)=c$ are defined by
\begin{equation*}
C_{q}(\Phi ,u)=H_{q}(\Phi ^{c}\cap U,\Phi ^{c}\cap U\setminus \{u\});\quad
q\in Z.
\end{equation*}
where $U$ is a closed neighborhood of $u$.

Let $K=\{u\in H\mid \Phi '(u)=0\}$ be the set of critical points of 
$\Phi $ and $a<\inf_{K}\Phi $. The critical groups of $\Phi $ at infinity are
defined by
\begin{equation*}
C_{q}(\Phi ,\infty )=H_{q}(H,\Phi ^{a}); \quad q\in Z
\end{equation*}
We will use the notation $\deg (\Phi ',U,0)$ for the Leray-Schauder
degree of $\Phi $ with respect to the set $U$ and the value $0$. Denote also
by $\mathop{\rm index}(\Phi ',u)$ the Leray-Schauder index of 
$\Phi'$ at critical point $u$. Recall that this quantity is defined as 
$\lim_{r\to0}\deg (\Phi ',B_{r}(u),0)$, if this limit exists, where 
$B_{r}(u)$ is the ball of radius $r$ centered at $u$.

\begin{proposition}[\cite{Ch}] \label{prop2.1}
If  $u$ is  a mountain  pass  point  of  $\Phi $,  then
\begin{equation*}
C_{q}(\Phi ,u)=\delta _{q,1}Z.
\end{equation*}
\end{proposition}

\begin{proposition}[\cite{B-L}] \label{prop2.2}
 Assume  that $ H=H^{+}\oplus H^{-}$, $\Phi $  is
bounded from below on $H^{+}$ and $\Phi (u)\to-\infty $  as
$\|u\| \to\infty $ with  $u\in H^{-}$. Then
\begin{equation*}
C_{\mu }(\Phi ,\infty )\neq 0,\quad \text{with  }\mu =\dim
H^{-}<\infty .
\end{equation*}
\end{proposition}

\section{Proof of Theorem \ref{thm1.1}}

First, we prove that $\Phi $ satisfies the Cerami condition.

\begin{lemma} \label{lm3.1}
Under the assumptions  (F0)--(F3), $\Phi$ satisfies the $(C)_c$
condition on $ H_0^1(\Omega)$, for all $c \in \mathbb{R}$.
\end{lemma}

\begin{proof}
Let $(u_{n})_{n}\subset H_{0}^{1}$ be a $(C)_{c}$ sequence, i.e
\begin{gather}
\Phi (u_{n})\to c  \label{e16} \\
\Vert u_{n}\Vert \langle \Phi '(u_{n}),v\rangle
_{H_{0}^{1},H^{-1}}\leq {\epsilon _{n}}\Vert v\Vert \quad \forall v\in
H_{0}^{1},  \label{e17}
\end{gather}
where $\epsilon _{n}\to0$. It clearly suffices to show that
$(u_{n})_{n}$ remains bounded in $H_{0}^{1}$. Assume by contradiction.
Defining $z_{n}=\frac{u_{n}}{\Vert u_{n}\Vert }$, we have
$\Vert z_{n}\Vert=1$ and, passing if necessary to a subsequence,
we may assume that $z_{n}\rightharpoonup z$ weakly in $H_{0}^{1}$,
$z_{n}\to z$ strongly in $L^{2}(\Omega )$ and $z_{n}(x)\to z(x)$ a.e. in
$\Omega $. By the
 linear growth of  $f$ , the sequence
$\big( \frac{f(u_{n}(x) )}{\|u_{n}\| }\big) _{n}$  remains  bounded  in
$L^{2}$, then for a subsequence, we have
\begin{equation*}
\frac{f(u_{n}(x) )}{\|u_{n}\| }\rightharpoonup \zeta
\quad \text{in }L^{2}.
\end{equation*}
and by standard arguments based on assumptions F0),F1), $\zeta $ can be
written as
$ \zeta (x)=m(x)z(x)$,
where $ m$  satisfies (see \cite{C-O}).
\begin{equation*}
0\leq m(x)\leq \lambda _{k+1}-\lambda _{k}\quad \text{a.e. in   }\Omega .
\end{equation*}

However, divide (\ref{e17}) by  $\|u_{n}\| ^{2}$  and goes to
the limit we obtain
\begin{equation*}
\frac{\langle \Phi '(u_{n}),v\rangle }{\|u_{n}\| }=\int
\nabla z_{n}\nabla v-\lambda _{k}\int z_{n}v-\int \frac{f(u_{n})}{\|
u_{n}\| }vdx\to0
\end{equation*}
for every  $v\in H_{0}^{1}$. On the other hand, since  $z_{n}$  converges
to  $z$  weakly in $ H_{0}^{1}$,  strongly in $L^{2}$ and
$\frac{f( u_{n}(x)) }{\|u_{n}\| }$
converges weakly in $L^{2}$ to  $\zeta $, we deduce
\begin{equation}
\frac{\langle \Phi '(u_{n}),v\rangle }{\|u_{n}\| }
\to\int \nabla z\nabla v-\lambda _{k}\int zv-\int \zeta vdx=0\quad
\forall v\in H_{0}^{1}(\Omega ).  \label{e18}
\end{equation}

\subsection*{Claim:}
We will prove that $z_{n}\to z$  strongly in  $H_{0}^{1}$.
Indeed, taking $ v=z$ in (\ref{e18}) we have
\begin{equation}
\|z\| ^{2}=\lambda _{k}\int z^{2}+\int m(x)z^{2}.  \label{e19}
\end{equation}
On the other hand, by (\ref{e17}) it results
\begin{equation}
\frac{\langle \Phi '(u_{n}),u_{n}\rangle }{\|u_{n}\|
^{2}}\to1-\lambda _{k}\int z^{2}-\int m(x)z^{2}=0.  \label{e20}
\end{equation}
 From (\ref{e19}) and (\ref{e20}), it follows $\|z\| =1$. Since
$z_{n}\rightharpoonup z$, $\|z_{n}\| \to\|z\| $ and
$H_{0}^{1}( \Omega ) $  is convex uniformly
space the claim follows.
So that, $z$  is a nontrivial solution of problem
\begin{equation}
\begin{gathered} -\Delta z= ( \lambda_k +m(x))z \quad\mbox{in }\Omega\\z =
0 \quad\mbox{on }\partial\Omega. \end{gathered}  \label{e21}
\end{equation}
We now distinguish three cases:  i) $ \lambda _{k}<m(x)+\lambda _{k}$
and $   m(x)+\lambda _{k}<\lambda _{k+1}$ on subset of positive measure;
(ii) $  m(x)+\lambda _{k}\equiv \lambda _{k}$;  (iii) $  m(x)+\lambda
_{k}\equiv \lambda _{k+1}$.

\subsection*{Case i:}
We have $z$  is a nontrivial solution of problem (\ref{e21}), then 1 is an
eigenvalue of  this problem. On the other hand, by strict monotonicity $
\lambda _{k}\left( \lambda _{k}+m(x)\right) <1$  and $\lambda _{k+1}\left(
\lambda _{k}+m(x)\right) >1$ , which gives a contradiction.

\subsection*{Case ii:}
By (F1), for $ \varepsilon >0$, there exists a constant
$r_{\varepsilon }>r$  such that
\begin{equation}
0\leq \frac{f(s)}{s}\leq \lambda _{k+1}-\lambda _{k}+\varepsilon \quad
\forall |s|\geq r_{\varepsilon }  \label{e22}
\end{equation}
Put  $f_{n}(x)=\frac{f\big( u_{n}(x) \big) }{u_{n}(x)}\chi
\{|u_{n}(x)|\geq r_{\varepsilon }\}$, which
remains bounded in $L^{\infty }$, passing if necessary to a subsequence,
$f_{n}\to l$  in the weak* topology of $L^{\infty }$.
By  (\ref{e22}), the  $L^{\infty }$-function  $l$  satisfies
\begin{equation*}
0\leq l(x) \leq \lambda _{k+1}-\lambda _{k}+\varepsilon\quad
\text{a.e.in }\Omega
\end{equation*}
Multiply  $f_{n}$ by $z_{n}^{2}$, integrate on $\Omega $ and going
to the limit, to have
\begin{equation*}
\int f_{n}z_{n}^{2}dx=\int \frac{f\left( u_{n}(x) \right) }{
\|u_{n}\| }z_{n}\to\int m(x)z^{2}dx=\int
l(x)z^{2}dx=0.
\end{equation*}
By the unique continuation Property  of  $\Delta $  and  $l\geq 0$, we
deduce that $  l\equiv 0$   a.e.in  $\Omega $. Then, by
lemma \ref{lm2.2} and lemma \ref{lm2.3} there exists  $ M>0$   such that
\begin{equation*}
\int F(\frac{u_{n}^{k}}{2})dx\leq M\text{.}
\end{equation*}
This is a contradiction with assumption (F2) and
$\|u_{n}^{k}\| \to+\infty $.

\subsection*{Case iii:}
If $ m(x)\equiv \lambda _{k+1}-\lambda _{k}$.
Dividing (\ref{e16})  by $ \|u_{n}\| ^{2}$, we obtain
\begin{equation*}
\frac{\Phi ( u_{n}) }{\|u_{n}\| ^{2}}=\frac{1}{2}
\|z_{n}\| ^{2}-\frac{\lambda _{k}}{2}\int z_{n}^{2}-\int \frac{
F(u_{n}(x))}{\|u_{n}\| ^{2}}dx\to0,\quad \text{as }
n\to\infty .
\end{equation*}
However, it results that
\begin{equation*}
\underset{n\to+\infty }{\lim }\int \frac{F(u_{n}(x))}{\|
u_{n}\| ^{2}}dx=\frac{1}{2}\alpha \int z^{2}dx.
\end{equation*}
 Applying  Fatou's lemma, we have
\begin{equation*}
\int_{z>0}( \alpha -K_{+}) z^{2}dx+\int_{z<0}\left( \alpha
-K_{-}\right) z^{2}dx\leq 0.
\end{equation*}
This is a contradiction with assumption (F3), since (F3) is equivalent to
 $K_{\pm }=\limsup_{s\to\pm \infty }\frac{2F(s)}{s^{2}}<\alpha $.
(see \cite{G-O}). The proof of lemma is complete.
\end{proof}

\begin{lemma} \label{lm3.2}
Under the hypothesis of Theorem \ref{thm1.1}, the functional
$\Phi$ has the following properties:
\begin{itemize}
\item[(i)]  $\Phi (w)\to+\infty $, as $\|w\|
\to+\infty $, $w\in W^{+}=E_{k+1}\oplus E_{+}$.
 \item[(ii)]$\Phi (v)\to-\infty $, as $\|v\| \to
+\infty $, $w\in W^{-}=E_{k}\oplus E_{-}$.
\end{itemize}
\end{lemma}

\begin{proof}
(i) $\Phi $ is coercive on $W^{+}$. Indeed, the assumption (F3) is
equivalent to  $K_{\pm }=\limsup_{s\to\pm \infty }
\frac{2F(s)}{s^{2}}<\alpha $. Thus, there exists an $B_{\varepsilon }\geq 0$
 such that
\begin{equation*}
F(s)\leq \frac{\alpha }{2}s^{2}-\varepsilon s^{2}+B_{\varepsilon } \quad
\forall s\in \mathbb{R}.
\end{equation*}
Hence, for every $ w\in W^{+}$, we obtain
\begin{align*}
\Phi (w)& =\frac{1}{2}\Vert w\Vert ^{2}-\frac{\lambda _{k}}{2}\int
w^{2}-\int F(w)\,dx \\
& \geq \frac{\lambda _{k+1}-\lambda _{k}}{2\lambda _{k+1}}\|w\|
^{2}-\frac{\alpha -2\varepsilon }{2}\int w^{2}-B_{\varepsilon }|\Omega
|\\
& \geq \frac{\varepsilon }{\lambda _{k+1}}\Vert w\Vert ^{2}-B_{\varepsilon
}|\Omega |.
\end{align*}
However, $\Phi (w)\to+\infty $, as $\|w\| \to +\infty $.

\noindent (ii) Assume by contradiction that there exists a constant $B>0$  and
a sequence $(v_{n})\subset V$ with $\Vert v_{n}\Vert \to\infty $
such that
\begin{equation*}
B\leq \Phi (v_{n})\leq -\delta \Vert v_{n}^{-}\Vert ^{2}.
\end{equation*}
Therefore, by lemma \ref{lm2.3},  since $\Vert v_{n}^{-}\Vert $  is
bounded, there exists $M>0$ such that
\begin{equation*}
\int F(\frac{v_{n}^{k}}{2})\,dx\leq M
\end{equation*}
which contradicts (F2).
 \end{proof}

\begin{lemma} \label{lm3.3}
Under the condition (F4), the functional $\Phi$ has the following
properties:
\begin{itemize}
\item[(i)]  There is an $R > 0$ and $\beta >0$ such that $\Phi \geq
\beta$ on $\partial B_R(0)$.
\item[(ii)] $C_{q}(\Phi ,0)=\delta
_{q,0}Z$
\end{itemize}
\end{lemma}

\begin{proof}
(i) We start by proving the first assertion. On one hand, it is easy to see
that if $\lambda _{k}+f'(0)\leq 0$  we have
\begin{equation*}
\Phi ''(0)u.u\geq \Vert u\Vert ^{2}.
\end{equation*}
On the other hand, where  $\lambda _{k}+f'(0)>0$, the
Poincar\'{e}'s inequality gives that
\[
\Phi ''(0)u.u
=\Vert u\Vert ^{2}-\lambda _{k}\int u^{2}-\int f'(0)u^{2}dx
\geq \big( 1-\frac{\lambda _{k}+f'(0)}{\lambda _{1}}\big)
\Vert u\Vert ^{2}
\]
Put   $\gamma =1-\frac{\lambda _{k}+f'(0)}{\lambda _{1}}$
and  by (F4), we have  $\gamma >0$  and
\begin{equation*}
\Phi ''(0)u.u\geq \gamma \Vert u\Vert ^{2}.
\end{equation*}
Taylor's formula implies
\[
\Phi (u) =\frac{1}{2}\Phi ''(0)u.u+o(\Vert u\Vert ^{2})
\geq \big( \frac{\gamma }{2}+\frac{o(\Vert u\Vert ^{2})}{\Vert u\Vert ^{2}
}\big) \Vert u\Vert ^{2}
\]
with    $\frac{o(\Vert u\Vert ^{2})}{\Vert u\Vert ^{2}}\to0$,
as  $\Vert u\Vert \to0$. Consequently, the assertion (i) follows.

\noindent (ii) Since $u=0$ is a local mininum of $\Phi $,  we have
\begin{equation*}
C_{q}(\Phi ,0)=\delta _{q,0}Z.
\end{equation*}
\end{proof}

\begin{lemma} \label{lm3.4}
The functional $\Phi $  has at least one critical point
$u_{0}$, such that
$$ C_{q}(\Phi ,u_{0})=\delta _{q,1}Z.
$$
\end{lemma}

\begin{proof}
According to (ii) of Lemma \ref{lm3.2}, $ \Phi $  is anti-coercive on
$W^{-}$ we can find an  $e\in H_{0}^{1}$  such that
$\|e\| \geq M>R$  and$ \Phi (e)\leq 0$. So by mountain pass theorem,
there  exists a critical point $ u_{0}$  of mountain pass type, such that
\begin{equation*}
 C_{1}(\Phi ,u_{0})\neq 0.
\end{equation*}
By proposition \ref{prop2.1}, it results that
$C_{q}(\Phi ,u_{0})=\delta _{q,1}Z$.
The proof of lemma is complete.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm1.1}]
For this proof we distinguish two cases.
\subsection*{Case 1:}
If  $k=1$, we assume that $ \{0,u_{0}\}$  is the critical
set of $\Phi $  and let $R>0$, such that  $\{0,u_{0}\}
\subset B_{R}(0)$. By the Riesz representation theorem we can write
\begin{equation*}
\langle \Phi '(u),v\rangle =\langle u,v\rangle -\langle Nu,v\rangle
,\quad \mbox{for  all }u,v\in H_{0}^{1}(\Omega )
\end{equation*}
where $ \langle u,v\rangle =\int_{\Omega }\nabla u\nabla v$  and $\langle
Nu,v\rangle =\int [\lambda _{1}u+f(u)]v\,dx$.  So that, $\Phi '=I-N $
and By the Sobolev embedding theorem, $N$   is compact. We see
that $\Phi '$ has the form Identity-compact, so that Leary-Shauder
techniques are applicable
\begin{equation}
\begin{aligned}
\deg (\Phi ',B_{R}(0),0)
&= \mathop{\rm index}(\Phi ',0)+index(\Phi ',u_{0}) \\
&=\sum_{q=0}^{\infty }( -1)^{q}\dim C_{q}\left( \Phi ,0\right)
+\sum_{q=0}^{\infty }(-1)^{q}\dim C_{q}\left( \Phi ,u_{0}\right) \\
&=1-1=0 \end{aligned}  \label{e23}
\end{equation}
In a similar way we can define a compact map
$T:H_{0}^{1}(\Omega )\to H_{0}^{1}(\Omega )$ by
\begin{equation*}
\langle Tu,v\rangle =\int ( \lambda _{1}+\mu ) uv\,dx
\end{equation*}
where $0<\mu <\lambda _{2}-\lambda _{1}$. Now we claim that there is a
priori bound in  $H_{0}^{1}(\Omega )$  for all possible solutions of
the family of equations (see \cite{Nk})
\begin{gather*}
-\Delta u-\lambda _{1}u=(1-t)\mu u+tf(u)\quad \mbox{in }\Omega \\
u=0\quad \mbox{on }\partial \Omega.
\end{gather*}
The homotopy invariance of Leray-Schauder  degree  implies
\begin{equation*}
\deg (\Phi ',B_{R}(0),0)=\deg (I-T,B_{R}(0),0)=-1.
\end{equation*}
This contradicts (\ref{e23}).

\subsection*{Case 2:}
If  $k\geq 2$, by Lemma \ref{lm3.1}, the functional $\Phi $ satisfies
the  condition (C). Since  $\Phi $  is weakly lower semi continuous and
coercive on  $W^{+}$, $\Phi $  is bounded from below on  $W^{+}$.
Moreover, by  (ii) of Lemma \ref{lm3.2},  $\Phi $  is anti-coercive on
$W^{-}$, thus we can apply the proposition \ref{prop2.2} and we conclude
that
\begin{equation*}
 C_{\mu }(\Phi ,\infty )\neq 0
\end{equation*}
where  $\mu =\dim W^{-}\geq k\geq 2$. It follows from the Morse inequality
that  $\Phi $  has a critical point $u_{1}$ with
\begin{equation*}
 C_{\mu }(\Phi ,u_{1})\neq 0.
\end{equation*}
Since  $\mu \neq 1$  and  $\mu \neq 0$, then the problem \eqref{e1}  has
at least two nontrivial  solutions. The proof of theorem is complete.
\end{proof}

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\end{document}