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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. 149--153.\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}{149}

\begin{document}

\title[\hfilneg EJDE/Conf/14 \hfil Multiplicity results]
{Multiplicity results for nonlinear elliptic equations}

\author[S. Benmouloud, M. Khiddi, M. Sbai \hfil EJDE/Conf/14 \hfilneg]
{Samira Benmouloud, Mostafa Khiddi, Simohammed Sbai}  

\address{Samira Benmouloud \newline
E.G.A.L, D\'ept. Maths, Fac. Sciences,
Universit\'e Ibn Tofail, BP. 133, K\'enitra, Maroc}
\email{ben.sam@netcourrier.com}

\address{Mostafa Khiddi \newline
E.G.A.L, D\'ept. Maths, Fac. Sciences, Universit\'e Ibn Tofail, BP.
133, K\'enitra, Maroc} 

\address{Simohammed Sbai \newline
E.G.A.L, D\'ept. Maths, Fac. Sciences,
Universit\'e Ibn Tofail, BP. 133, K\'enitra, Maroc}
\email{sbaisimo@netcourrier.com}

\date{}
\thanks{Published September 20, 2006.}
\subjclass[2000]{35J20, 35J65}
\keywords{Semilinear elliptic equations; critical Sobolev exponent}

\begin{abstract}
 Let $\Omega$  be a bounded domain in $\mathbb{R}^{N}$,  $N\geq 3$,
 and $p=\frac{2N}{N-2}$ the limiting Sobolev exponent.
 We show that for $f\in H^1_0(\Omega)^\ast$, satisfying  suitable
 conditions, the nonlinear elliptic problem
 \begin{gather*}
 -\Delta u =|u |^{ p-2 }u +f \quad \mbox{in } \Omega   \\
 u=0 \quad \mbox{on } \partial\Omega
 \end{gather*}
 has at least  three solutions  in $H_{0}^{1}(\Omega)$.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}


\section{Introduction}

It is well known  \cite[Theorems 1 and 2]{t1} that for $f\neq0$
and $\|f\|$ sufficiently small, the problem
\begin{equation} \label{e1}
\begin{gathered}
-\Delta u =|u |^{ p-2 }u +f \quad \mbox{on } \Omega   \\
u=0 \quad \mbox{on } \partial\Omega
\end{gathered}
\end{equation}
has at least two distinct solutions  $\textbf{u}_0$ and
$\textbf{u}_1$ which are critical points of the functional
$$
I (u) = \frac{1}{2}\int_{\Omega } |\nabla u|^{2}
-\frac{1}{p}\int_{\Omega }|u |^{p} -\int_{\Omega }fu,
$$
 such that
 $I(\textbf{u}_1)>I(\textbf{u}_0)$.
In this note we suppose $f\geq0$ and satisfies
\begin{equation}
\|f\|<\frac{\alpha}{N}S^{\frac{N}{4}}, \label{e*}
\end{equation}
where
$$
\frac{1}{2}<\alpha <(\frac{N-2}{N+2})^{\frac{N+2}{4}},
\quad\text{and}\quad
S=\inf_{u\in H^1_0(\Omega)\|u\|_p=1}\|\nabla u\|^2_2,
$$
which corresponds to the best constant for the Sobolev embedding
 $ H^1_0(\Omega)\hookrightarrow L^p(\Omega)$.
 We determine a special $\omega_\varepsilon$, from the extremal functions
 for the Sobolev inequality in  $\mathbb{R}^N$, and consider
  $\Gamma$ the class of continuous paths joining
  $0$  to $\omega_\varepsilon$.

\begin{proposition} \label{prop1}
Let
$$
c=\inf_{\gamma\in\Gamma} \sup_{t\in[0,1]} I(\gamma(t)) .
$$
Then there is a
sequence $(u_j) \subset H^1_0(\Omega)$  such that
\begin{gather*}
I(u_{j})\to c ,\\
I'(u_{j}) \to  0 \quad \text{in }
(H_{0}^{1}(\Omega))^\ast,\\
I(\textbf{u}_0)<I(\textbf{u}_1)<c.
\end{gather*}
\end{proposition}

Let  $\textbf{u}$ denotes the weak limit in $H^1_0(\Omega) $ of (a
subsequence of) $(u_n)$, our principal result is as follows.

\begin{theorem} \label{thm1}
Let $f\in H^1_0(\Omega)^\ast$, $f\geq 0$ satisfies
\eqref{e*}. Then either
\begin{enumerate}
    \item $I(\textbf{u})=c$ and Problem \eqref{e1}  has at least
three solutions.     Or
     \item $I(\textbf{u})\leq c-\frac{1}{N}S^{N/2}$.
\end{enumerate}
\end{theorem}

 Note that the existence results of biharmonic analogue of
Problem \eqref{e1} have been studied in  [2], so a result similar to
that of Theorem \ref{thm1} may be established for the bilaplacian operator.

\section{The proof of   Proposition \ref{prop1}}

We start with a variant of the mountain  pass theorem of
Ambrosetti-Rabinowitz  without the Palais-Smale condition


\begin{theorem} \label{thm2}
Let  $E$ be a real Banach space and $I\in
C^1(E,\mathbb{R})$. Suppose there exists a neighborhood  $U$
of  $0$ in $E$ and a constant  $\rho>0$ such that
\begin{enumerate}
    \item[(H1)]  $I(u)\geq \rho$, for all $u\in \partial U$.

    \item[(H2)] $I(0)<\rho$ and,  $I(v)<\rho$  for some
$ v\in E\setminus U$.
\end{enumerate}
Let
 $$
 c=\inf_{\gamma\in\Gamma} \max_{t\in[0,1]} I(\gamma(t)),
$$
where
$$
\Gamma=\{ \gamma :[0,1]\to E,\text{ is  continuous, }\gamma(0)=0,\;
\gamma(1)=v\}.
$$
Then there is a sequence $(u_n)$ in $E$ such that
\begin{gather*}
 I(u_{n}) \to  c ,\\
I'(u_{n}) \to  0 \quad\text{in }E^*.
\end{gather*}
\end{theorem}

On $H^1_0(\Omega)$ we define a variational functional
  $I:H^1_0(\Omega)\to \mathbb{R}$ for problem \eqref{e1}, by
$$
I (u) = \frac{1}{2}\|\nabla u\|^{2}_{2} -\frac{1}{p}\|\ u \|^{p}_{p}
-\int_{\Omega }fu.
$$
 Clearly $I$ is $C^1$ on $E$ and  $I(0)=0$. We shall verify the assumptions
of Theorem \ref{thm2}

\subsection*{Verification of (H1)}
Let  $r\in ]0,\alpha S^{N/4}[  $ and $u\in H^1_0(\Omega))$ be  such
that $\|\nabla u\|_2=r $. We have
$$
I (u)  \geq \frac{1}{2}r^{2} -\frac{1}{p}r^{p}S^{-p/2}-\|f\|r.
$$
Letting $r\to \alpha S^{N/4}$, we obtain
 $$
 I(u)\geq \frac{1}{2}\alpha^{2}S^{N/2}
-\frac{1}{p}\alpha^{p}S^{N/2}-\frac{1}{4N}\alpha^{2}S^{N/2}.
 $$
Set
  $$
\rho=\frac{\alpha^{p}S^{N/2}}{2N},
  $$
hence
   $I(u)>\rho$ for all $u \in\partial B(0,r)$.

\subsection*{Verification of (H2)}

Assume  $0\in\Omega$ and let $\phi\in \mathcal{C}^\infty_0(\Omega)$
be a fixed function such that $\phi\equiv1$ for $x$ in some
neighborhood of $0$. For $\varepsilon>0$, define
$$
u_{\epsilon}(x)=\frac{\phi(x)}{(\epsilon+|x|^{2})^{\frac{N-2}{2}}},\quad
v_{\epsilon}(x)=\frac{u_{\epsilon(x)}}{\|u_{\epsilon}\|_{p}}.
$$
 Hence, from  \cite{b3},
\begin{equation}
\|\nabla v_{\epsilon}\|_{2}^{2}=S + O(\epsilon^{\frac{N-2}{2}}).
\label{e2.1}
\end{equation}
 For every $\mu\neq0$,  \cite[Lemma 2.1]{t1}, gives a real  $t^+>0$
such that
\begin{equation}
t^{+}>(\frac{\|\nabla\mu v_{\epsilon}\|_{2}^{2}}{(p-1)\|\mu
v_{\epsilon}\|_{p}^{p}})^{\frac{1}{p-2}}=
\frac{1}{\mu}(\frac{N-2}{N+2})^{\frac{N-2}{4}}\|\nabla
v_{\epsilon}\|_{2}^{\frac{N-2}{2}}\label{e2.2}
\end{equation}
and
\begin{equation}
t^{+}<\frac{1}{\mu}\|\nabla v_{\epsilon}\|_{2}^{\frac{N-2}{2}}.
\label{e2.3}
\end{equation}
Set  $\omega_{\epsilon}= t^{+}\mu v_{\epsilon} $. We have
$$
\|\nabla\omega_{\epsilon}\|_{2}=t^{+}\mu \|\nabla
v_{\epsilon}\|_{2}>(\frac{N-2}{N+2})^{\frac{N-2}{4}}\|\nabla
v_{\epsilon}\|_{2}^{\frac{N}{2}}>(\frac{N-2}{N+2})^{\frac{N-2}{4}}
S^{\frac{N}{4}}>\alpha S^{\frac{N}{4}}>r.
$$
On the other hand,  from  \eqref{e2.2} and  \eqref{e2.3},
we get
\begin{align*}
I(\omega_{\epsilon})
&< \frac{1}{2}(t^{+})^{2}\|\nabla
\omega_{\epsilon} \|^{2}_{2} -\frac{1}{p}(t^{+})^{p}\\
&< \frac{1}{2\mu^{2}}\|\nabla v_{\epsilon}
\|^{N}_{2}-\frac{1}{\mu^{p}}\frac{1}{p}(\frac{N-2}{N+2})^{\frac{p(N-2)}{4}}
\|\nabla v_{\epsilon} \|^{N}_{2}\,.
\end{align*}
Using  \eqref{e2.1}, we deduce
$$
I(\omega_{\epsilon}) < (\frac{1}{2\mu^{2}}-\frac{1}{\mu^{p}}
\frac{N-2}{N+2}(\frac{N-2}{N+2})^{\frac{N}{2}})(S +
O(\epsilon^{\frac{N-2}{2}}))^{N/2}
 <  \frac{\epsilon_{0}^{p}S^{N/2}}{2N},
$$
for  $\mu $ large enough.
Then
$c\geq\rho > I(\omega_\epsilon)$.
Recall that $\omega_\epsilon\in \Lambda^-$ (\cite[Lemma 2.1]{t1} with
$$
\Lambda^-=\{ u\in H^1_0(\Omega) / <I'(u),u>=0, \| \nabla
u\|^2_2-(p-1)\|u\|^p_p<0\},
$$
and that  $\inf_{\Lambda^-}I$ is attained by $\textbf{u}_1$
\cite[Theorem 2]{t1}. We conclude that
 $$
 c\geq\rho > I(\omega_\epsilon)\geq I(\textbf{u}_1)  >I(\textbf{u}_0).
 $$

\section{Proof of the Theorem \ref{thm1}}
Applying  Proposition \ref{prop1} we obtain a sequence  $(u_j)\subset
 H^1_0(\Omega)$ such that
\begin{gather}
I(u_j) \to c, \label{e3.1} \\
I'(u_j) \to  0 \quad\text{in }H^1_0(\Omega)^\ast. \label{e3.2}
\end{gather}

This implies that $\|\nabla u_j \|_2$ is uniformly bounded. Hence
for a subsequence of $u_j$, still denoted by $u_j$, we can find
$\textbf{u}\in H^1_0(\Omega)$ such that
\begin{gather*}
u_j \to \textbf{u}\quad\text{weakly  in }H^1_0(\Omega),\\
 u_j \to \textbf{u} \quad\text{strongly  in } L^q, \; q<p,\\
u_j \to \textbf{u} \quad\text{a.e.  on }\Omega.
\end{gather*}
 From \eqref{e3.2}, we deduce that $\textbf{u}$ is a  (weak) solution of
Problem \eqref{e1}. In particular $\textbf{u}$ satisfies
\begin{equation}
\|\textbf{u}\|^2_2-\|\textbf{u}\|^p_p=\int f\textbf{u} \label{e3.3}
\end{equation}
Let  $u_j=\textbf{u}+v_j$, where $v_j\to 0$ weakly in
$H^1_0(\Omega)$ and  $v_j\to 0$ a.e on $\Omega$. We have
 $$
\|\nabla u_{j}\|^{2}_{2}=\|\nabla \textbf{u}\|^{2}_{2}+\|\nabla
v_{j}\|^{2}_{2}+\circ(1).
 $$
and by \eqref{e3.1},
$$
I(\textbf{u}) + \frac{1}{2}\|\nabla
v_{j}\|_{2}^{2}-\frac{1}{p}\|v_{j}\|_{p}^{p}=c + o(1),
$$
thanks to Brezis-Lieb Lemma \cite{b4}.
By  \eqref{e3.2} and  \eqref{e3.3},
$\|\nabla v_{j}\|_{2}^{2}-\| v_{j}\|_{p}^{p}=o(1)$,
which gives
$$
I(\textbf{u}) + \frac{1}{N}\|\nabla v_{j}\|_{2}^{2}=c + o(1).
$$
Set $l= \lim_{j\to +\infty}\|\nabla
v_{j}\|_{2}^{2} $, then  $ \lim_{j\to
+\infty}\| v_{j}\|_{p}^{p}=l $. Using  Sobolev inequality one see
that $l\geq S l ^{2/p}$.
Then $l=0$, or  $l\geq S^{\frac{N}{2}}$.
We get, either
$$
I(\textbf{u})=c,
$$
and since
$$
I(\textbf{u})>I(\textbf{u}_1)>I(\textbf{u}_0),
$$
$\textbf{u}$ is a solution of Problem \eqref{e1} distinct from
$\textbf{u}_o$ and $\textbf{u}_1$,
or
$$
I(\textbf{u})\leq c-\frac{1}{N}S^{\frac{N}{2}}.
$$

\begin{remark} \label{rmk1} \rm
One can show that  $c<\frac{1}{N}S^{\frac{N}{2}}$, consequently
$I(\textbf{u})<0$ in the second case
\end{remark}

\section{Semilinear biharmonic equation}

In \cite{b1}, Benmouloud considered the  problem
\begin{gather*}
\Delta^2 u =|u |^{ p-2 }u +f \quad \mbox{in } \Omega   \\
\Delta u=u=0 \quad \mbox{on } \partial\Omega
\end{gather*}
where $\Omega$ is a bonded domain in $\mathbb{R}^N$, $N\geq5$
 $p=\frac{2N}{N-4}$ and $\Delta^2$ denotes the biharmonic operator.
She proved that for $f\in H^{-1}$  subject to a suitable condition,
this problem has at least two distinct solutions in
$H^2(\Omega)\cap H^1_0(\Omega)$.
The existence of on solution follows from the mountain-pass
theorem, with Palais-Smale condition,
 and a second is obtained by a constrained minimization (see also \cite{b2}).

It follows from this study that an  analog result of Theorem \ref{thm1} may
be established by a similar argument with suitable smallness
 condition on $f$.


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\bibitem{b1} S. Benmouloud, \emph{Existence de solutions pour un problème
biharmonuque non homogène avec exposant critique de Sobolev.} Bull.
Belg. Math. Soc. 8 (2001), 555-565

\bibitem{b2} S. Benmouloud, M. Sbai; \emph{A perturbed biharmonic
minimization problem with critical exponent} \`a paraitre dans le
volume 7 de Math-Recherche et applications.

\bibitem{b3} H. Brezis, L. Niremberg; \emph{Positive Solutions of NonLinear
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\bibitem{b4} H. Brezis, E. Lieb; \emph{A relation between pointwise
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\bibitem{t1} G. Tarantello, \emph{On nonhomogeneous elliptic equations
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\end{thebibliography}




\end{document}