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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. 249--254.\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}{249}

\begin{document}

\title[\hfilneg EJDE/Conf/14 \hfil Positive solutions in the semi-positone case]
{Existence of non-negative solutions for nonlinear equations in the
semi-positone case}

\author[N. Yebari,  A. Zertiti \hfil EJDE/Conf/14 \hfilneg]
{Naji Yebari, Abderrahim Zertiti}  % in alphabetical order

\address{Naji Yebari \newline
D\'epartement de Math\'ematiques\\
Universit\'e Abdelmalek Essaadi\\
B. P. 2121, Tetouan, Maroc}
\email{nyebari@hotmail.com, nyebari@fst.ac.ma}

\address{Abderrahim Zertiti \newline
D\'epartement de Math\'ematiques\\
Universit\'e Abdelmalek Essaadi\\
B. P. 2121, Tetouan, Maroc} 
\email{zertiti@fst.ac.ma}

\date{}
\thanks{Published September 20, 2006.}
\subjclass[2000]{35J65, 35J25}
\keywords{Superlinear semi-positone problems; variational methods;
\hfill\break\indent fibring method}

\begin{abstract}
 Using the fibring method we prove the existence of non-negative
 solution of the $p$-Laplacian boundary value problem
 $-\Delta_pu=\lambda f(u)$, for any  $\lambda >0$ on any regular
 bounded domain of  $\mathbb{R}^N$, in the special case
 $f(t)=t^q-1$.
\end{abstract}

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


\section{Introduction and main results}

In this paper we are interested in finding nonnegative
solutions to the equation
\begin{equation}
\begin{gathered}
-\Delta _pu=\lambda f(u) \quad\text{in }\Omega , \\
u=0 \quad\text{on } \partial \Omega ,
\end{gathered} \label{e1.1}
\end{equation}
for some specific $f$ in the  non positone case ($f(0)<0$), under
assumptions stated below.

Here $\Omega $ is a connected and bounded subset of $\mathbb{R}^N$ with
boundary $\partial \Omega $ in $C^{1,\alpha }$. We set
\[
\Delta _pu=\mathop{\rm div}(| \nabla u| ^{p-2}\nabla u).
\]
When $p=2$, this type of problem in the  nonpositone case can be studied via
the shooting method. Existence of a radially symmetric nonnegative solution
for $\lambda >0$ sufficiently small have been obtained in
\cite{a1,b1} and
nonexistence of such a solution for $\lambda >0$ large have been established
in  \cite{a1,c1}, in the framework of the semi positone case and $f$ is
superlinear. Observe that, since $f(0)<0$, the constant $0$ is an upper
solution of \eqref{e1.1} and as a consequence it is not possible, in general, to
apply the usual techniques (for example: the method of upper and lower
solutions, etc.) and we shall work in the framework of the so-called
fibration method introduced by  Pohozaev in \cite{p1}, and then
developed in \cite{p2,p3,p4}. We shall assume that $f$
has the form
\begin{equation}
f(t)=t^q-1,\quad \text{with }q>p-1  \label{e1.2}
\end{equation}
To avoid the noncompactness problem we shall always assume that the
problem is subcritical, in the sense of the critical exponent
for $\Omega $,
\begin{equation}
p^{\star }=\begin{cases}
\frac{Np}{N-p} & \text{if }1<p<N, \\
+\infty  & \text{if }p\geq N.
\end{cases} \label{e1.3}
\end{equation}
Let
\begin{equation}
u=\theta v, \quad
\theta =\lambda ^{-(\frac 1{q-p+1})}, \quad
\mu =\lambda ^{\frac q{q-p+1}}>0 \,.  \label{e1.4}
\end{equation}
and
\begin{equation}
\begin{gathered}
-\Delta _pu=u^q-\mu \quad\text{in } \Omega , \\
u>0 \quad\text{in }\Omega , \\
u=0 \quad\text{on } \partial \Omega
\end{gathered}  \label{e1.5}
\end{equation}
It can be seen that \eqref{e1.1}, \eqref{e1.2}, \eqref{e1.4} and
\eqref{e1.5} are equivalent.
Also let  $p$ and $q$ satisfy
\begin{equation}
0<p-1<q<p^{*}-1  \label{e1.6}
\end{equation}
where $p^{*}$ is given by \eqref{e1.3}. Concerning $\mu $, we shall
assume its positivity.

By a solution of \eqref{e1.5}, we mean a
$W_0^{1,p}(\Omega )\cap L^\infty (\Omega )$
function which is a critical point of the functional
\[
E(v)=-\frac 1p\int_\Omega | \nabla v| ^pdx
+\frac 1{q+1}\int_\Omega | v| ^{q+1}dx-\mu \int_\Omega | v(x)| dx
\]
and therefore satisfies
\[
\int_\Omega (| \nabla u| ^{p-2}\nabla u.\nabla \varphi
-(u^q-\mu )\varphi )dx=0\text{ }
\]
for every $\varphi \in W_0^{1,p}(\Omega )\cap L^\infty (\Omega )$.

Our main result is as follows.

\begin{theorem} \label{thm1}
Let assumptions \eqref{e1.3} and \eqref{e1.6} be satisfied. Then there
exists a nontrivial nonnegative solution $u\in W_0^{1,p}(\Omega )\cap
L^\infty (\Omega )$ of problem \eqref{e1.1} for any $\lambda >0$.
Moreover, $u\in C^{1,\alpha }(\overline{\Omega })$ for some
$\alpha >0$.
\end{theorem}

\section{Proof of the main theorem}

The proof is based on the fibering method and is divided into five stages.

\noindent\textbf{Step 1:}
We introduce the Euler functional associated with \eqref{e1.5} as
follows
\[
E(u)=-\frac 1p\int_\Omega | \nabla u| ^pdx+\frac
1{q+1}\int_\Omega | u| ^{q+1}dx-\mu \int_\Omega
| u(x)| dx
\]
According to the fibering method, we set
\begin{equation}
u(x)=rv(x),  \label{e2.1}
\end{equation}
where $r\in \mathbb{R}^{+}$ and $v\in W_0^{1,p}(\Omega )$.
Then we obtain
\begin{equation}
\widetilde{E}(r,v)=E(r,v)=-\frac{| r| ^p}
p\int_\Omega | \nabla v| ^pdx+\frac{| r| ^{q+1}
}{q+1}\int_\Omega | v| ^{q+1}dx-\mu r\int_\Omega
| v(x)| dx  \label{e2.2}
\end{equation}
We introduce the fibering functional
\begin{equation}
\int_\Omega | \nabla v| ^pdx=1  \label{e2.3}
\end{equation}
Under condition \eqref{e2.3} the functional $\widetilde{E}$
takes the form
\begin{equation}
\widetilde{E}(r,v)=-\frac{r^p}p+\frac{r^{q+1}}{q+1}\int_\Omega
| v| ^{q+1}dx-\mu r\int_\Omega | v(x)| dx
\label{e2.4}
\end{equation}
The bifurcation equation is
\begin{equation}
0=\frac{\partial \widetilde{E}}{\partial r}=-r^{p-1}+r^q\int_
\Omega | v| ^{q+1}dx-\mu \int_\Omega | v(x)| dx
\label{e2.5}
\end{equation}
which gives
\begin{equation}
-r^p+r^{q+1}\int_\Omega | v| ^{q+1}dx-\mu
r\int_\Omega | v(x)| dx=0 . \label{e2.6}
\end{equation}
Let set
\begin{equation}
\widetilde{E}(v)=E(r(v)v)  \label{e2.7}
\end{equation}

\noindent\textbf{Step 2:}
Let us consider the variational problem
\begin{equation}
M_0=\sup \big\{ \widetilde{E}(v);\;v\in W_0^{1,p}(\Omega
)/\int_\Omega | \nabla v| ^pdx=1\big\}.   \label{e2.8}
\end{equation}
It follows that
\begin{equation} \label{e2.9}
\widetilde{E}(v)
= \min_{r\geq 0}\widetilde{E}(r,v)\\
=\min_{r\geq 0}\{-\frac{r^p}p +\frac{r^{q+1}}{q+1}\int_\Omega
| v|^{q+1}dx-\mu r\int_\Omega | v(x)| dx\}<0,
\end{equation}
as a matter of fact, \eqref{e2.6} gives
$$
-\frac{r^p(v)}p=-\frac{r^{q+1}(v)}p\int_\Omega |
v| ^{q+1}dx+\mu \frac{r(v)}p\int_\Omega |v(x)| dx,
$$
On the other hand,
\begin{align*}
\widetilde{E}(v)
&= E(r(v)v)\\
&=-\frac{r^{q+1}(v)}p\int_\Omega | v| ^{q+1}dx
  +\mu \frac{r(v)}p\int_\Omega |v(x)| dx\\
&\quad +\frac{r^{q+1}(v)}p\int_\Omega | v| ^{q+1}dx
  -\mu r(v)\int_\Omega | v(x)| dx,
\end{align*}
which gives
\begin{equation}
\widetilde{E}(v)=\frac{(p-q-1)}{(q+1)p}r^{q+1}(v)\int_\Omega
| v| ^{q+1}dx-\mu r(v)(1-\frac 1p)\int_\Omega |
v(x)| dx  \label{e2.10}
\end{equation}
By \eqref{e1.6}, $\widetilde{E}(v)<0$.

Let us prove the following Lemma.

\begin{lemma} \label{lem2.1}
The sequence maximizing  problem \eqref{e2.8} is bounded in
$W_0^{1,p}(\Omega )$.
\end{lemma}

\begin{proof}
Let $(v_n)$ be a maximizing sequence for \eqref{e2.8}. We set
\begin{equation}
v_n(x)=c_n+\overline{v}_n(x)  \label{e2.11}
\end{equation}
with
\begin{equation}
\int_\Omega \overline{v}_n(x) dx=0\, . \label{e2.12}
\end{equation}
Since
\begin{equation}
\int_\Omega | \nabla v_n| ^pdx=\int_\Omega |
\nabla \overline{v}_n| ^pdx=1  \label{e2.13}
\end{equation}
and by the Sobolev embedding theorems (the Poincare-Wirtinger inequality),
the sequence $(\overline{v}_n)$ is bounded in $W^{1,p}(\Omega )$.
 From the bifurcation equation \eqref{e2.5}, we obtain
\begin{equation}
r_n^p=r_n^{q+1}\int_\Omega | c_n+\overline{v}_n|
^{q+1}dx-\mu r_n\int_\Omega | c_n+\overline{v}_n| dx.
\label{e2.14}
\end{equation}
Let us assume that
\begin{equation}
c_n\to +\infty , \quad \text{as }n\to +\infty\,.  \label{e2.15}
\end{equation}
Then
\begin{equation}
\int_\Omega | 1+\frac{\overline{v}_n}{c_n}|
^{q+1}dx=\frac 1{c_n^{q+1}r_n^{q-p+1}}+\frac \mu
{c_n^qr_n^q}\int_\Omega | 1+\frac{\overline{v}_n}{c_n}
| dx\,.  \label{e2.16}
\end{equation}
By embedding results, there exists $C>0$ such that
\[
\| \overline{v}_n\| _{W^{1,p}(\Omega )}\leq C, \quad\forall
n\in \mathbb{N}
\]
Using \eqref{e2.15} and since by assumption \eqref{e1.6} the space
$W^{1,p}(\Omega )$ is compactly embedded in $L^{q+1}(\Omega )$. We
may assume that $(\overline{v}_n)$ converges strongly in latter
space. Then from \eqref{e2.16} we have
\begin{equation}
\int_\Omega | 1+\frac{\overline{v}_n}{c_n}|
^{q+1}dx\to | \Omega | >0,\quad \text{as }n\to +\infty.
\label{e2.18}
\end{equation}
The proof is complete.
\end{proof}

Hence, we can assume that the sequence $(v_n)$ converges weakly in
$W_0^{1,p}(\Omega )$. By assumption \eqref{e1.6}, it follows that
$v_n\to \overline{v}$ in $L^{q+1}(\Omega )$.
This implies that
\[
\| \nabla v_0\| _p\leq \liminf_{n\to +\infty}\| \nabla v_n\| _p\,.
\]
Since
\[
\| \nabla v_n\| _p^p=\int_\Omega | \nabla v_n| ^pdx=1,
\]
we obtain
\begin{equation}
0\leq \| \nabla v_0\| _p^p=\int_\Omega | \nabla
v_0| ^pdx\leq 1.  \label{e2.19}
\end{equation}
Now we shall prove the equality
\begin{equation}
\int_\Omega | \nabla v_0| ^pdx=1.  \label{e2.20}
\end{equation}
We assume the contrary; i.e, that
\begin{equation}
\int_\Omega | \nabla v_0| ^pdx<1.  \label{e2.21}
\end{equation}
Note that
\begin{equation}
0<\int_\Omega | \nabla v_0| ^pdx \,. \label{e2.22}
\end{equation}
Otherwise, if $\int_\Omega | \nabla v_0| ^pdx=0$,
$v_0=c_0$ is a constant, and from \eqref{e2.8}, we have for all
$\epsilon >0$ there exist $n_0\in \mathbb{N}$ such that for all
$n\geq n_0$ we have
\[
M_0-\epsilon < \widetilde{E}(v_n)<M_0.
\]
Let $\theta \in ] 0,1[$. Then
\begin{equation}
\widetilde{E}(\theta v_n)-\epsilon \leq M_0-\epsilon <\widetilde{E}(v_n)<M_0
 \label{e2.23}
\end{equation}
by \eqref{e2.10}. We recall that
\[
\widetilde{E}(v_n)=\frac{(p-q-1)}{(q+1)p}r_n^{q+1}\int_\Omega
| v_n| ^{q+1}dx-\mu r_n(\frac{p-1}p)\int_\Omega |
v_n(x)| dx\,.
\]
Using \eqref{e2.23}, we see that
\[
(1-\theta )\mu r_n(\frac{p-1}p)\int_\Omega | v_n(x)|
dx<\epsilon \;\text{for all }n\geq n_0.
\]
Then we have
\[
r_n\int_\Omega | v_n(x)| dx \to r_0\int_\Omega | v_0(x)| dx=0\
\]
as $n\to +\infty$ and $v_0=c_0=0$ which gives $\widetilde{E}(v_0)
=0$. This contradicts $M_0<0$.

Due to \eqref{e2.21} and \eqref{e2.22}, there exists $\theta >1$
(i.e., $\theta ^p=1/\int_\Omega | \nabla v_0(x)| ^pdx>1)$ such
that $v_{*}=\theta v_0$ satisfies
\[
\int_\Omega | \nabla v_{*}(x)| ^pdx=1
\]
and
\begin{align*}
\widetilde{E}(v_{*})
&=\widetilde{E}(\theta v_0)=\min _{r\geq
0}\big\{ -\frac{r^p}p+\frac{r^{q+1}}{q+1}\theta ^{q+1}\int_\Omega
| v_0| ^{q+1}dx-\mu r\theta \int_\Omega |v_0(x)| dx\big\} \\
&= \min_{\rho \geq 0}\big\{ -\frac{\rho ^p}{
p\theta ^p}+\frac{\rho ^{q+1}}{q+1}\int_\Omega | v_0|
^{q+1}dx-\mu \rho \int_\Omega | v_0(x)| dx\big\} \\
&> \min_{\rho \geq 0}\big\{ -\frac{\rho ^p}p+\frac{\rho ^{q+1}}{q+1}
\int_\Omega | v_0| ^{q+1}dx-\mu \rho \int_\Omega
| v_0(x)| dx\big\}.
\end{align*}
Thus,
\[
\widetilde{E}(v_{*})>\;\widetilde{E}(v_0).
\]
This inequality contradicts the definition of \eqref{e2.8}. Thus, we have
obtained a solution to the variational problem.
\smallskip

\noindent\textbf{Step 3:}
\[
\widetilde{E}(v_0)=\sup \big\{ \widetilde{E}(v);\;v\in
W_0^{1,p}(\Omega )/\int_\Omega | \nabla v| ^pdx=1\big\}
\]
The fibering method implies  $r=r_0=r(v_0)$ where $r_0>0$ and
\begin{align*}
&-\frac{r_0^p}p+\frac{r_0^{q+1}}{q+1}\int_\Omega | v_0|
^{q+1}dx-\mu r_0\int_\Omega | v_0(x)|\, dx
\\
&=\min_{r\geq 0}\big\{ -\frac{r^p}p+\frac{r^{q+1}}{q+1}\int_\Omega | v|
^{q+1}dx-\mu r\int_\Omega | v(x)| dx\big\}
\end{align*}

To complete the proof, we must show that the equation \eqref{e1.5} is
verified. We can assume that $v_0$ is nonnegative by replacing $v_n$ by
$| v_n| $. Moreover, there exists a Lagrange multiplier
$\sigma $ such that
\begin{equation}
\widetilde{E}'(v_0).h=\sigma \Big(\int_\Omega |
\nabla (.)| ^pdx\Big)'(v_0).h\quad \forall  h\in W_0^{1,p}(\Omega
).  \label{e2.24}
\end{equation}
 From the above equation, and by taking $v_0$ as test function, we have
\[
r_0\big\{ \int_\Omega ((r_0v_0)^q-\mu )v_0dx\big\} =p\sigma
\int_\Omega | \nabla (v_0)| ^pdx=p\sigma \,.
\]
By \eqref{e2.6} we obtain $\sigma =\frac{r_0^p}p>0$. Then we can
write
\[
\widetilde{E}'(v_0)=p\sigma (-\Delta _pv_0)
\]
which is equivalent to
\[
-\Delta _p(r_0v_0)=\;(r_0v_0)^q-\mu \,.
\]
Then if we set $u=r_0v_0\geq 0$, we can see that $u$ is a solution of
problem \eqref{e1.5}. \smallskip

\noindent \textbf{Step 4:}
For $u\geq 0$, we have $\widetilde{E}(v_0)<0$, thus the solution
$u\geq 0$ is non trivial.

\noindent\textbf{Step 5:}
We have obtained the nonnegative nontrivial solution $u $ to
problem \eqref{e1.5}. A standard bootstrap argument (see  Drabek \cite{d1})
shows that $u\in L^\infty (\Omega )$. Then the asserted regularity
of $u\in C_{\rm Loc}^{1,\alpha }(\Omega )$ follows by Tolksdorf \cite{t1}.
Thus the theorem is proved.

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\end{thebibliography}

\end{document}