\documentclass[reqno]{amsart}

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2004(2004), No. 49, pp. 1--10.\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 2004 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}

\title[\hfilneg EJDE-2004/49\hfil Multiple solutions]
{Multiple solutions for inhomogeneous nonlinear
elliptic problems arising in astrophyiscs}

\author[Marco Calahorrano \& Hermann Mena\hfil EJDE-2004/49\hfilneg]
{Marco Calahorrano \& Hermann Mena}  % in alphabetical order

\address{Marco Calahorrano \hfill\break
Escuela Polit\'{e}cnica Nacional, Departamento de Matem\'{a}tica,
Apartado 17-01-2759, Quito, Ecuador}
\email{calahor@server.epn.edu.ec}

\address{Hermann Mena \hfill\break
Escuela Polit\'{e}cnica Nacional, Departamento de Matem\'{a}tica,
Apartado 17-01-2759, Quito, Ecuador}
\email{hmena@server.epn.edu.ec}

\date{}
\thanks{Submitted May 15, 2003. Published April 6, 2004.}
\subjclass[2000]{35J65, 85A30, 35J20}
\keywords{Solar flares, variational methods, inhomogeneous nonlinear 
\hfill\break\indent
elliptic problems}


\begin{abstract}
 Using variational methods we prove the existence and multiplicity
 of solutions for some  nonlinear inhomogeneous elliptic
 problems on a bounded domain in $\mathbb{R}^n$, with $n\geq 2$ and
 a smooth boundary, and when the domain is $\mathbb{R}_+^n$.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{thm}{Theorem}[section]
\newtheorem{lem}[thm]{Lemma}
\newtheorem{prop}[thm]{Proposition}
\newtheorem{rem}[thm]{Remark}

\newcommand{\norm}[1]{\|#1\|}
\newcommand{\normw}[1]{\|#1\|_{W^{1,2}_0(\Omega)}}

\section{Introduction}

In this paper we study the boundary-value problem
\begin{equation} \label{eq:*}
\begin{gathered}
-\Delta u+c(x)u=\lambda f(u)\quad\mbox{in } \Omega \\
 u=h(x)\quad \mbox{on }\partial\Omega
\end{gathered}
\end{equation}
when $\Omega$ is a bounded domain in $\mathbb{R}^{n}$, with $n\geq
2$ and smooth boundary $\partial\Omega$, and when the domain is $
\mathbb{R}_{+}^n :=\mathbb{R}^{n-1}\times \mathbb{R}_{+}$ with
$\mathbb{R}_{+}=\{y\in \mathbb{R}: y>0\}$. The function
$f:]-\infty,+\infty[\to \mathbb{R}$ is assumed to satisfy the
following conditions:
\begin{itemize}
\item[(f1)] There exists $s_{0}>0$ such that $f(s)>0$ for all
$s\in ]0,s_{0}[$.
\item[(f2)] $f(s)=0$ for $s\leq0$ or $s\geq s_{0}$.
\item[(f3)] $f(s)\leq as^{\sigma}$, $a$ is a positive constant and
$1<\sigma<\frac{n+2}{n-2}$
 if $n>2$ or $\sigma>1$ if $n=2$.
\item[(f4)] There exists $l>0$ such that $|f(s_{1})-f(s_{2})|\leq
 l|s_{1}-s_{2}|$, for all $s_{1}$, $s_{2}$ $\in \mathbb{R}$.
\end{itemize}
The function $h$ is a non-negative bounded, smooth, $h\neq 0$,
$\min h<s_{0}$ and $c\geq0$, and $c\in L^{\infty}(\Omega)\bigcap
C(\overline{\Omega})$.

Note that problem \eqref{eq:*} is equivalent to
\begin{equation}\label{eq:*1}
\begin{gathered}
-\Delta \omega+c(x)\omega=\lambda f(\omega+\tau)\quad\mbox{in }\Omega \\
 \omega=0\quad \mbox{on }\partial\Omega\,,
\end{gathered}
\end{equation}
where $\omega=u-\tau$ and $\tau$ is a solution of
\begin{equation} \label{eq:**}
\begin{gathered}
-\Delta \tau+c(x)\tau=0 \quad \mbox{in } \Omega \\
 \tau=h(x)\quad \mbox{on }\partial\Omega\,.
\end{gathered}
\end{equation}

We will study \eqref{eq:*1} instead of \eqref{eq:*}.
In section 2 using variational techniques we will find an interval
$\Lambda \subset \mathbb{R}_{+}$ such that for all $\lambda\in\Lambda$
there exist at least three positive solutions of \eqref{eq:*1},
for $\|\tau\|_{L^{\sigma+1}(\Omega)}$ small enough. This result is
better than the one obtained by  Calahorrano and Dobarro in  \cite{cd}.

In section 3, we will study the problem \eqref{eq:*1} for $\inf
c(x)>0$ and $\Omega$ big enough, by this we mean that there exists
$x_{0}\in\Omega$ such that the Euclidean ball with center $x_{0}$
and radius R is contained  in $\Omega$, with R large enough. In
this case, we will eliminate the restrictions on $\tau$, obtaining
similar results.

Problem \eqref{eq:*} is a generalization of an astrophysical
gravity-free model of solar flares in the half plane
$\mathbb{R}^{2}_{+}$, given in  \cite{h1},  \cite{h2} and  \cite{h3},
namely:
\begin{equation} \label{eq:hw}
\begin{gathered}
-\Delta u=\lambda f(u)\quad \mathbb{R}^{2}_{+} \\
 u(x,0)=h(x)\quad \forall x\in \mathbb{R}
 \end{gathered}
\end{equation}
besides the above mentioned conditions for $f$ and $h$, the
authors are interested in finding a positive range of $\lambda 's$
in which there is multiplicity of solutions for \eqref{eq:hw}, see
\cite{h1,h2,h3} for a detail description.

In section 4, a related problem is reviewed
\begin{equation}\label{eq:Rn}
\begin{gathered}
-\Delta \omega+c(x)\omega=\lambda f(\omega+\tau)\quad \mbox{in }\mathbb{R}_+^n \\
 \omega(x,0)=0\quad \forall x\in \mathbb{R}^{n-1}
\end{gathered}
\end{equation}
and we prove the existence of solutions of \eqref{eq:Rn} as limit of a
special family of solutions of
\begin{equation}\label{eq:Dr}
\begin{gathered}
-\Delta \omega+c(x)\omega=\lambda f(\omega+\tau)\quad \mbox{in }D_{R}\\
 \omega=0\quad \mbox{on }\partial D_{R}
 \end{gathered}
\end{equation}
where
\[
D_{R}=\{(x_{1},\dots,x_{n})\in\ \mathbb{R}_+^n: \sum^{n}_{i=1}x^{2}_{i}<R^{2}\}
\]
and $R$ is large enough.
Besides these solutions are absolute minima of the natural
associated functional for small $\lambda 's$ and local but not
global minima for large $\lambda 's$.

\section{Variational Method}

Similarly to section 1, let $\tau$ be the solution of
\begin{equation} \label{eq:**2}
\begin{gathered}
-\Delta \tau+c(x)\tau =0\quad\mbox{in }\Omega\\
 \tau =h(x)\quad \mbox{on } \partial\Omega\,.
\end{gathered}
\end{equation}
Problem \eqref{eq:*} is equivalent to
\begin{equation}\label{eq:*2}
\begin{gathered}
-\Delta \omega+c(x)\omega =\lambda f(\omega+\tau)\quad \mbox{in }\Omega \\
 \omega =0\quad \mbox{on } \partial\Omega
\end{gathered}
\end{equation}
where $\omega=u-\tau$. Therefore, we are studying (\ref{eq:*2}) instead
of \eqref{eq:*}.

Since $f\geq 0$, then any solution of (\ref{eq:*2}) is positive by
the maximum principle, furthermore $\omega=0$ is solution of
(\ref{eq:*2}) if and
only if $\lambda=0$.
On the other hand $\tau$ achieves its maximum and minimum on the
boundary, i.e.
$\inf_{\partial\Omega}\tau\leq\tau(x)\leq\sup_{\partial\Omega}\tau$.

Let $H^{1}_{0}(\Omega)$ be the usual Sobolev space, with
$\|u\|^{2}=\int_{\Omega}|\nabla u|^{2}dx$.
We define for all $\lambda\geq 0$ and for all non-negative
function $\tau$ such that
$\|\tau\|_{L^{\sigma+1}(\Omega)}\equiv\Gamma<\infty$ the $C^{1}$
functional, \cite{am},
$\Phi_{\lambda,\tau}:H^{1}_{0}(\Omega)\to R $,
\[
\Phi_{\lambda,\tau}(u)=\frac{1}{2}\int_{\Omega}[c(x)u^{2}+|\nabla
u |^{2}]dx-\lambda \int_{\Omega}F(u+\tau)dx
\]
where, $F(s)=\int^{s}_{0}f(t)dt$.

If $u\in H^{1}_{0}(\Omega)$, $\Phi'_{\lambda,\tau}(u)=0$
($\Phi'$ is the gradient of $\Phi$) then $u$ is a weak and, by
regularity strong
solution of (\ref{eq:*2}).

Since $f$ is bounded, it is easy to prove that
$\Phi_{\lambda,\tau}$ is coercive and verifies the Palais-Smale
condition for all $\lambda$ non negative (using methods like in
the case c=0, \cite{r}). Then $\Phi_{\lambda,\tau}$ attains its
global infimum on a function $u_{\lambda,\tau}\in
H^{1}_{0}(\Omega)$ for all $\lambda$ non negative.


\begin{thm} \label{thm1}
Let us assume (f1)--(f4). For all $\Gamma>0$
small enough there exists an interval
$]\underline{\lambda},\overline{\lambda}(\Gamma)[$ with
$\underline{\lambda}>0$ such that for all
$\lambda\in]\underline{\lambda},\overline{\lambda}(\Gamma)[$ the
problem \eqref{eq:*2} has at least three positive solutions. Moreover
$\overline{\lambda}(\Gamma)\to +\infty$  as
$\Gamma\to 0$.
\end{thm}

To prove Theorem \ref{thm1}, we will use arguments as those in \cite{cd},
for which the following lemmas are necessary.

\begin{lem} \label{lm2}
There exists $\omega_{0}\geq0$, $\omega_{0}\neq 0$ and
$\underline{\lambda}>0 $ such that for all $\lambda>\underline{\lambda}$
and for all $\tau\geq 0$, $\Phi_{\lambda,\tau}(\omega_{0})<0$
\end{lem}

\begin{proof}. Let $B_{r}(x_{0})$ denote an euclidean ball with
center at $x_{0}$ and radius $r$. Let $x_{0}\in \Omega$ and $R>0$
such that $B_{R}(x_{0})\subset\Omega$. Then for all $0<\delta<R$,
$B_{\rho}(x_{0})\subset B_{R}(x_{0})$, where $\rho=R-\delta$.
Now, we define
\[
\omega_{\delta,R}(x) = \begin{cases}
s_{0}  & \text{if } |x-x_{0}|\leq\rho\\
\frac{s_{0}}{\delta}(R-|x-x_{0}|) & \text{if }\rho\leq|x-x_{0}|\leq R\\
0 & \text{if } |x-x_{0}|\geq R
\end{cases}
\]
So, using  the H\"{o}lder and Poincar\'{e} inequalities
\begin{align*}
\Phi_{\lambda,\tau}(\omega_{\delta,R})&=\frac{1}{2}\|\omega_{\delta,R}\|^{2}+\frac{1}{2}\int_{\Omega}c(x)(\omega_{\delta,R})^{2}dx
- \lambda\int_{\Omega}F(\omega_{\delta,R}+\tau)dx\\
&\leq\frac{1}{2}\|\omega_{\delta,R}\|^{2}+\frac{\|c\|_{L^{\infty}}}{2}\int_{B_{R}(x_{0})}(\omega_{\delta,R})^{2}dx
- \lambda\int_{B_{\rho}(x_{0})}F(s_{0}+\tau)dx\\
&\leq\frac{1}{2}\|\omega_{\delta,R}\|^{2}+\frac{\|c\|_{L^{\infty}}}{2}\Bigr(\frac{|B_{R}(x_{0})|}{\omega_{n}}\Bigl)^{\frac{2}{n}}\|\omega_{\delta,R}\|^{2} - \lambda F(s_{0})\int_{B_{\rho}(x_{0})}dx\\
&=\frac{s^{2}_{0}(1+\|c\|_{L^{\infty}}R^{2})}{2\delta^{2}}\int_{B_{R}(x_{0})-B_{\rho}(x_{0})}dx
- \lambda F(s_{0})\int_{B_{\rho}(x_{0})}dx\\
&=\frac{s^{2}_{0}(1+\|c\|_{L^{\infty}}R^{2})(R^{n}-(R-\delta)^{n})\omega_{n}}{2\delta^{2}}-\lambda
F(s_{0})(R-\delta)^{n}\omega_{n}
\end{align*}
where $\omega_{n}$ denotes the volume of the unit ball in $R^{n}$.
Let
\[
\underline{\lambda}(\delta)\equiv\frac{s^{2}_{0}(1+\|c\|_{L^{\infty}}R^{2})
(R^{n}-(R-\delta)^{n})}{2F(s_{0})\delta^{2}(R-\delta)^{n}}
\]
If $\delta=tR$, $0<t<1$, results in
\[
\underline{\lambda}(\delta)=\frac{s^{2}_{0}(1
+\|c\|_{L^{\infty}}R^{2})}{2F(s_{0})R^{2}}\Big(\frac{1-(1-t)^{n}}{t^{2}(1-t)^{n}}\Big).
\]
then
$\Phi_{\lambda,\tau}(\omega_{\delta,R})<0$ for all $\lambda>\underline{\lambda}(\delta)>0$,
and for all $\tau\geq0$.
Let
\[
\psi(t)\equiv\frac{1-(1-t)^{n}}{t^{2}(1-t)^{n}}
\]
and let $t_{1}\in ]0,1[$ such that
$\psi(t_{1})=\min_{]0,1[}\psi(t)$. If $\delta_{1}=t_{1}R$,
$\omega_{o}=\omega_{\delta_{1},R}$ and
$\underline{\lambda}=\underline{\lambda}(\delta_{1})$, then there
results
\[
\Phi_{\lambda,\tau}(\omega_{0})<0 \quad \forall \lambda>\underline{\lambda}>0 \quad
and \quad \forall \tau\geq0
\]
Moreover,
\[
\|\omega_{0}\|=s_{0}\Bigl(\omega_{n}\Bigr)^{1/2}R^{\frac{n-2}{2}}\Bigl(\frac{1-(1-t_{1})^{n}}{t^{2}_{1}}\Bigr)^{1/2}
\]
\end{proof}

\begin{lem} \label{lm3}
There exists a constant K=K($a,\sigma,\Omega$) such that for all
$\lambda<\overline{\lambda}(\Gamma)$ and $\|u\|=\Gamma$, $\Phi_{\lambda,\tau}(u)>0$
where $\overline{\lambda}\equiv K\Gamma^{1-\sigma}$.
\end{lem}

\begin{proof}  From (f3),
\[
\int_{\Omega}F(u+\tau)dx=\int_{\Omega}\int^{u+\tau}_{0}f(t)dt\,dx\leq
\int_{\Omega}\frac{a(u+\tau)^{\sigma+1}}{\sigma+1}\,dx
\]
then, using the Sobolev immersion and Poincar\'{e} inequalities\\
\begin{align*}
\Phi_{\lambda,\tau}(u)&= \frac{1}{2}\|u\|^{2}+\frac{1}{2}\int_{\Omega}c(x)u^{2}dx
- \lambda\int_{\Omega}F(u+\tau)dx\\
&\geq \frac{1}{2}\|u\|^{2}
- \lambda\int_{\Omega}\frac{a(u+\tau)^{\sigma+1}}{\sigma+1}dx\\
&\geq \frac{1}{2}\|u\|^{2}
- \lambda\Bigl(\frac{a}{\sigma+1}\Bigr)(\|u\|_{L^{\sigma+1}(\Omega)}
+\|\tau\|_{L^{\sigma+1}(\Omega)})^{\sigma+1}\\
&\geq \frac{1}{2}\|u\|^{2}
- \lambda\Bigl(\frac{a}{\sigma+1}\Bigr)(C(\Omega)\|u\|+\Gamma)^{\sigma+1},
\end{align*}
where $C(\Omega)$ is a constant depending on $\Omega$.
Setting
\[
K=\frac{\sigma+1}{2a(C(\Omega)+1)^{\sigma+1}}
\]
it follows that for all $\lambda<\overline{\lambda}(\Gamma)\equiv
K\Gamma^{1-\sigma}$, $\Phi_{\lambda,\tau}(u)>0$.
\end{proof}


\begin{rem} \label{rmk4} \rm
(i) Since $\overline{\lambda}(\Gamma)=K\Gamma^{1-\sigma}$ it
follows
$\overline{\lambda}\to +\infty$ as $\Gamma\to 0$.\\
(ii) $\Phi_{\lambda,\tau}(0)$ and $\Phi_{\lambda,\tau}'(0)(v)$ are negative for all $\lambda>0$
and  $v\geq0$, $v \neq 0$.
\end{rem}

\begin{lem} \label{lm5}
For all $0<\lambda<\overline{\lambda}(\Gamma)$ there exists
$\overline{u}\in H^{1}_{0}(\Omega)$ with $\|\overline{u}\|<\Gamma$
such that $\Phi_{\lambda,\tau}(\overline{u})<0$ and $\Phi_{\lambda,\tau}'(\overline{u})=0$.
\end{lem}

\begin{proof} Using Lemma \ref{lm3} we prove that $\Phi_{\lambda,\tau}(u)>0$, for
$0<\lambda<\overline{\lambda}(\Gamma)$ and $u$ such that
$\|u\|=\Gamma$. Moreover $\Phi_{\lambda,\tau}(0)<0$ y $\Phi_{\lambda,\tau}'(0)(v)\neq0$. Keeping
in mind that the solution of
\[
\begin{gathered}
\frac{d\alpha}{dt}=W(\alpha(t))\\
\alpha(0)=0
\end{gathered}
\]
where $W=-V$, $V$ pseudo-gradient vector field for $\Phi_{\lambda,\tau}$ in the set of
regular points of $\Phi_{\lambda,\tau}$, with
$0<\lambda<\overline{\lambda}$.

Since $\Phi_{\lambda,\tau}$ verifies the Palais-Smale condition and is bounded from
below, using  \cite[Theorem 5.4]{p} we have that
\begin{enumerate}
\item $\alpha:[0,+\infty[\to H^{1}_{0}(\Omega)$ is
continuous.
\item $\Phi_{\lambda,\tau}(\alpha(t))$ is strictly decreasing.
\item $\alpha(t)\to \overline{u}$ as $t\to
+\infty$, $\Phi_{\lambda,\tau}'(\overline{u})=0$.
\end{enumerate}
then, $\overline{u}$ satisfies the required conditions.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm1}]  Let $\omega_{0}$ and
$\underline{\lambda}$ be defined in Lemma \ref{lm2}. Using Lemma \ref{lm3} for
$\Gamma<\|\omega_{0}\|$, there exists
$\overline{\lambda}(\Gamma)>0$ such that $\Phi_{\lambda,\tau}(u)>0$ for all
$\lambda<\overline{\lambda}$ and $\|u\|=\Gamma$. But since
$\underline{\lambda}$ is independent of $\Gamma$, using Remark \ref{rmk4}
$\underline{\lambda}<\overline{\lambda}(\Gamma)$ for $\Gamma$
small enough.

Now we claim that for $\Gamma$ small enough there exists
$\widehat{u}\in H^{1}_{0}(\Omega)$, $\|\widehat{u}\|>\Gamma$ such
that for all
$\underline{\lambda}<\lambda<\overline{\lambda}(\Gamma)$
$\Phi_{\lambda,\tau}(\widehat{u})<0$ and $\Phi_{\lambda,\tau}'(\widehat{u})=0$.
Indeed, we remember that for all
$\underline{\lambda}<\lambda<\overline{\lambda}(\Gamma)$ lemmas 3
and 2 are verified. Keeping in mind that the solution of
\[
\begin{gathered}
\frac{d\beta}{dt}=W(\beta(t))\\
\beta(0)=\omega_{0}
\end{gathered}
\]
Using similar arguments as those in Lemma \ref{lm5} we find the critical point
$\widehat{u}$ with $\|\widehat{u}\|>\Gamma$.
Let
\[
c\equiv\inf_{\delta\in\Theta}\sup_{u\in\delta}\Phi_{\lambda,\tau}(u)
\]
where $\Theta$ is the set paths
\[
\Theta=\{\gamma\in
C([0,1],H^{1}_{0}(\Omega)):\gamma(0)=\overline{u},
\gamma(1)=\omega_{0}\}
\]
we are able to apply the Mountain Pass Theorem of
Ambrosetti-Rabinowitz \cite{ar}. Then $c$ is achieved in
$H^{1}_{0}(\Omega)$ at a
function $\widetilde{u}$.
Finally using Lemma \ref{lm5} we prove Theorem \ref{thm1}.
\end{proof}

\begin{rem} \label{rmk6} \rm
(i) If we define $\mu\in R_{-}$,
\[
\mu\equiv\min_{0\leq
t\leq\Gamma}\frac{1}{2}t^{2}-\lambda\frac{a}{\sigma+1}(C(\Omega)t+\Gamma)^{\sigma+1}
\]
it is easy to prove
\[
\Phi_{\lambda,\tau}(\widehat{u})<\mu\leq\Phi_{\lambda,\tau}(\overline{u})<0<\Phi_{\lambda,\tau}(\widetilde{u})
\]
(ii) Unlike  \cite{h1},  \cite{h2},  \cite{h3} and  \cite{cd},
where the size of $\|\tau\|_{L^{\infty}(\Omega)}$ is relevant, in
our approach the condition
$\Gamma\equiv\|\tau\|_{L^{\sigma+1}(\Omega)}$ small is of primary
importance. Note, that $\Gamma$ small does not say anything about
$\|\tau\|_{L^{\infty}(\Omega)}$.
\end{rem}


\section{$\Omega$ big enough}

Now we study problem (\ref{eq:*2}) for $\inf c(x)>0$ and
$\Omega\subset\mathbb{R}^{n}$ ($n\geq3$) big enough. By big enough we
mean that there exists $x_{0}\in\Omega$ such that the euclidean
ball with center $x_{0}$ and radius R is contained in
$\Omega$, with R large enough.

Let $W^{1,2}_0(\Omega)$ be the usual Sobolev space, with $\normw
u^{2}=\int_{\Omega}[u^{2}+|\nabla u|^{2}]dx$ and $\Gamma\equiv
\norm \tau_{L^{2}(\Omega)}$. If $\inf c(x)>0$, then
\begin{equation} \label{e3.1}
\normw u^{2}\leq \frac{1}{m}\int_{\Omega}[c(x)u^{2}+|\nabla u|^{2}]dx
\end{equation}
where $m\equiv \min$\{$\inf c(x),1$\}.

As was seen in section 2 we find an interval $\Lambda' \subset
\mathbb{R}_{+}$ such that for all $\lambda\in\Lambda'$ there exists
at least three positive solutions of (\ref{eq:*2}) and we
eliminate
the restrictions on $\tau$ .
Consequently we obtain:

\begin{thm} \label{thm7}
Let us assume (f1)--(f4). For all $\Gamma>0$ and R large
enough there exists an interval
$]\underline{\lambda}(R),\overline{\lambda}[$ with
$\underline{\lambda}(R)>0$ such that for all
$\lambda\in]\underline{\lambda},\overline{\lambda}[$ the problem
(\ref{eq:*2}) has at least three positive solutions.
\end{thm}

To prove this theorem, we need to redefine
$\underline{\lambda}$ and $\overline{\lambda}$.
Therefore, let
\[
\omega_{\delta,R}(x) = \begin{cases}
\dfrac{s_{0}}{\delta^{1/4}} & \text{if } |x-x_{0}|\leq\rho\\
\dfrac{s_{0}}{\delta^{5/4}} (R-|x-x_{0}|) & \text{if } \rho\leq|x-x_{0}|\leq R\\
0 & \text{if } |x-x_{0}|\geq R
\end{cases}
\]
If we define $\omega_{o}=\omega_{\delta_{1},R}$ where
$\delta_{1}=t_{1}R$ and $t_{1}\in ]0,1[$ such that
$\psi(t_{1})=\min_{]0,1[}\psi(t)$,
$\psi(t)\equiv\frac{1-(1-t)^{n}}{t^{\frac{5}{2}}(1-t)^{n}}$; then
with a similar development to Lemma \ref{lm2}, we obtain
\[
\Phi_{\lambda,\tau}(\omega_{0})<0 \quad \forall \lambda>\underline{\lambda}>0 \quad
and \quad \forall \tau\geq0
\]
where
\[
\underline{\lambda}(R)=\frac{s^{2}_{0}(1+\|c\|_{L^{\infty}}R^{2})}{2F
(\tau(x_{0}))R^{\frac{5}{2}}}\Bigl(\frac{1-(1-t_{1})^{n}}{t^{\frac{5}{2}}_{1}
(1-t_{1})^{n}}\Bigr).
\]
On the other hand, using the modification, to $n\geq3$
\begin{equation} \label{e3.2}
\|\nabla\omega_{0}\|_{L^{2}(\Omega)}=s_{0}\Bigl(\omega_{n}\Bigr)^{1/2}
R^{\frac{2n-5}{4}}\Bigl(\frac{1-(1-t_{1})^{n}}{t_{1}^{\frac{5}{2}}}\Bigr)^{1/2}
\to\infty
\end{equation}
as $R\to \infty$.
Since
\[
0\leq \lim_{s\to
0^{+}}\frac{2F(s)}{s^{2}}\leq\lim_{s\to
0^{+}}\frac{f(s)}{s}=0
\]
for (f3) and since $F$ is bounded, we define
\begin{equation} \label{e3.3}
\frac{b}{2}\equiv\sup_{s>0}\frac{F(s)}{s^{2}}<+\infty
\end{equation}

\begin{lem} \label{lm8}
For all $\lambda<\overline{\lambda}$ and $\normw u=\Gamma$,
$\Phi_{\lambda,\tau}(u)>0$.
\end{lem}

\begin{proof} Using  \eqref{e3.1} and \eqref{e3.3}
\begin{align*}
\Phi_{\lambda,\tau}(u)&= \frac{1}{2}\int_{\Omega}[c(x)u^{2}+|\nabla u|^{2}]dx
- \lambda\int_{\Omega}F(u+\tau)dx\\
&\geq \frac{m}{2}\normw u^{2}
- \frac{\lambda b}{2}\int_{\Omega}(u+\tau)^{2}dx\\
&\geq \frac{m}{2}\normw u^{2} - \frac{\lambda b}{2}(\norm
u_{L^{2}(\Omega)}+\norm \tau _{L^{2}(\Omega)})^{2}\\
&> \frac{m}{2}\normw u^{2} - \frac{\lambda b}{2}(\normw u+\norm \tau
_{L^{2}(\Omega)})^{2}
\end{align*}
So, when we define
$\overline{\lambda}\equiv m/4b$,
then for all $\lambda<\overline{\lambda}$, $\Phi_{\lambda,\tau}(u)>0$.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm7}]
Let $\omega_{0}$ and $\underline{\lambda}(R)$ be as above, using Lemma \ref{lm8} there
exists $\overline{\lambda}>0$ such that $\Phi_{\lambda,\tau}(u)>0$ for all
$\lambda<\overline{\lambda}$ and $\normw u=\Gamma$. From the
$\underline{\lambda}$, $\overline{\lambda}$ definition and \eqref{e3.2}
to R large enough $\underline{\lambda}<\overline{\lambda}$ and
$\|\omega_{0}\|_{W^{1,2}_{0}(\Omega)}>\Gamma$. Finally using a
similar development to Theorem \ref{thm1}, Theorem \ref{thm7} is proven.
\end{proof}

\begin{rem} \label{rmk9} \rm
For $n=2$ Theorem \ref{thm7} is false.
\end{rem}

\section{The problem in $R^{n}_{+}$}

Let $W^{1,2}_0(\mathbb{R}_+^n)$ and $V^{1,2}_{c,0}(\mathbb{R}_+^n)$ be the completion of
$C^{\infty}_{0}(\mathbb{R}^{n}_{+})$ in
$(\|.\|^{2}_{2}+\|\nabla(.)\|^{2}_{2})^{1/2}$ and
$(\|c.\|^{2}_{2}+\|\nabla(.)\|^{2}_{2})^{1/2}$ respectively,
where $\|.\|_{2}$ is the usual
$L^{2}$ norm  for the respective domain.
If $\inf c(x)>0$, then by \eqref{e3.1},
\[
W^{1,2}_0(\mathbb{R}_+^n)\sim V^{1,2}_{c,0}(\mathbb{R}_+^n)
\]
We define for all $\lambda\geq 0$ and for all non-negative
function $\tau$ such that $\|\tau\|_{L^{\sigma+1}(\mathbb{R}_+^n)}<\infty$,
the functional $\Phi_{\lambda,\tau,\infty}:W^{1,2}_0(\mathbb{R}_+^n)\to \mathbb{R}$
\[
\Phi_{\lambda,\tau,\infty}(u)=\frac{1}{2}\int_{\mathbb{R}_+^n}[c(x)u^{2}+|\nabla u|^{2}]dx -
\lambda\int_{\mathbb{R}_+^n}F(u+\tau)dx
\]
where $F(s)=\int^{t}_{0}f(t)dt$.

The function $\Phi_{\lambda,\tau,\infty}$ is well-defined; even more if
$u\in W^{1,2}_0(\mathbb{R}_+^n)$, using (f3) and
Sobolev immersion we obtain
\begin{align*}
0\leq \int_{\mathbb{R}_+^n}F(u+\tau)
&\leq
\frac{a}{\sigma+1}\int_{\mathbb{R}_+^n}(u+\tau)^{\sigma+1}\\
&\leq \frac{a}{\sigma+1}(\norm
u_{L^{\sigma+1}(\mathbb{R}_+^n)}+\|\tau\|_{L^{\sigma+1}(\mathbb{R}_+^n)})^{\sigma+1}\\
&\leq \frac{a}{\sigma+1}(C_{s}\norm
u_{W^{1,2}_0(\mathbb{R}_+^n)}+\|\tau\|_{L^{\sigma+1}(\mathbb{R}_+^n)})^{\sigma+1}
\end{align*}
where $C_{s}$ is the usual Sobolev immersion constant.
Then using \eqref{e3.1}
\begin{equation} \label{e4.1}
\Phi_{\lambda,\tau,\infty}(u)\geq  \frac{m}{2}\norm u^{2}_{W^{1,2}_0(\mathbb{R}_+^n)}
-\lambda\frac{a}{\sigma+1}(C_{s}\norm u_{W^{1,2}_0(\mathbb{R}_+^n)}+\|\tau\|_{L^{\sigma+1}
(\mathbb{R}_+^n)})^{\sigma+1}
\end{equation}
It is easy to verify that $\Phi_{\lambda,\tau,\infty}$ is a $C^{1}$ functional, so if
$u\in W^{1,2}_0(\mathbb{R}_+^n)$ is a critical point of $\Phi_{\lambda,\tau,\infty}$ then $u$ is a weak solution
and by
regularity, so classical solution of \eqref{eq:Rn}.


\begin{prop} \label{prop10}
(i) Let $m$ be as above then for all
$\lambda<\frac{m}{b}$, $\Phi_{\lambda,\tau,\infty}$ is coercive and bounded from below.\\
(ii) For all $\lambda<\frac{\inf c(x)}{l}$, \eqref{eq:Rn} has at most one
solution in $W^{1,2}_0(\mathbb{R}_+^n)$.
\end{prop}

\begin{proof} (i) Using \eqref{e3.1} and \eqref{e3.3}
\begin{align*}
\Phi_{\lambda,\tau,\infty}(u)&\geq \frac{m}{2}\norm u^{2}_{W^{1,2}_0(\mathbb{R}_+^n)}
- \frac{\lambda b}{2}\int_{\mathbb{R}_+^n}(u+\tau)^{2}\\
&> \frac{m}{2}\norm u^{2}_{W^{1,2}_0(\mathbb{R}_+^n)} - \frac{\lambda b}{2}(\norm
u_{W^{1,2}_0(\mathbb{R}_+^n)}+\norm \tau _{L^{2}(\mathbb{R}_+^n)})^{2}\\
&= \Bigl(\frac{m-\lambda b }{2}\Bigr)\|u\|^{2}_{W^{1,2}_0(\mathbb{R}_+^n)}-\lambda
b\|u\|_{W^{1,2}_0(\mathbb{R}_+^n)}\|\tau\|_{L^{2}(\mathbb{R}_+^n)}-\frac{\lambda
b}{2}\|\tau\|^{2}_{L^{2}(\mathbb{R}_+^n)}
\end{align*}
so, (i) is proven.

\noindent (ii) The uniqueness is proved as in \cite{aa}.
Indeed: if $u_{1}$ and $u_{2}$ are two  solutions
of \eqref{eq:Rn} then,
\[
\inf
c(x)\int_{\mathbb{R}_+^n}(u_{1}-u_{2})^{2}dx\leq\int_{\mathbb{R}_+^n}[c(x)(u_{1}-u_{2})^{2}+|\nabla
(u_{1}-u_{2})|^{2}]dx\leq\lambda l\int_{\mathbb{R}_+^n}(u_{1}-u_{2})^{2}dx
\]
\end{proof}

Now we consider problem \eqref{eq:Dr} and we define
$\Phi_{\lambda,\tau,R}:W^{1,2}_{0}(D_{R})\to \mathbb{R}$ in the
same way that $\Phi_{\lambda,\tau,\infty}$. It can be verified that, if $R'\geq R$,
then
\[
W^{1,2}_{0}(D_{R})\subset W^{1,2}_{0}(D_{R'})\subset W^{1,2}_0(\mathbb{R}_+^n)
\]
in addition for all $u\in W^{1,2}_{0}(D_{R})$,
$\Phi_{\lambda,\tau,\infty}(u)\leq\Phi_{\lambda,\tau,R'}(u)\leq\Phi_{\lambda,\tau,R}(u)$,
more precisely
\begin{equation} \label{e4.2}
\Phi_{\lambda,\tau,R'}(u)=\Phi_{\lambda,\tau,R}(u)-\lambda\int_{D_{R'}-D_{R}}F(\tau)dx
\end{equation}

\begin{rem} \label{rmk11} \rm
There exists a positive constant $C=C(a,\sigma,C_{s},m)$ such that
for all
$\lambda<\overline{\overline{\lambda}}(\|\tau\|_{L^{\sigma+1}(\mathbb{R}_+^n)})$
and for all $u$:
$\|u\|_{W^{1,2}_0(\mathbb{R}_+^n)}=\|\tau\|_{L^{\sigma+1}(\mathbb{R}_+^n)}$,
$\Phi_{\lambda,\tau,\infty}(u)$ $>0$, where
$\overline{\overline{\lambda}}(\|\tau\|_{L^{\sigma+1}(\mathbb{R}_+^n)})\equiv
C\|\tau\|^{1-\sigma}_{L^{\sigma+1}(\mathbb{R}_+^n)}$. In fact,
applying \eqref{e4.1} and taking
\[
C\equiv\frac{(\sigma+1)m}{2a}[C_{s}+1]^{-\sigma-1}
\]the result is obvious.
Furthermore for \eqref{e4.2}
\[
\Phi_{\lambda,\tau,R}(u)>0 \quad\forall u\in
W^{1,2}_{0}(D_{R})\quad \norm
u_{W^{1,2}_{0}(D_{R})}=\|\tau\|_{L^{\sigma+1}(\mathbb{R}_+^n)}
\]
then as in Lemma \ref{lm5}, for
$\lambda<\overline{\overline{\lambda}}$ there exists
$\overline{u}_{R}\in W^{1,2}_{0}(D_{R})$ with
$\|\overline{u}_{R}\|_{W^{1,2}_{0}(D_{R})}<\|\tau\|_{L^{\sigma+1}(\mathbb{R}_+^n)}$
such that $\Phi_{\lambda,\tau,R}(\overline{u}_{R})<0$ and
$\Phi'_{\lambda,\tau,R}(\overline{u}_{R})=0$.
\end{rem}


Now we will prove a sufficient condition to approximate solutions
of \eqref{eq:Rn} with solutions of \eqref{eq:Dr} with $R$ large enough.

\begin{lem} \label{lm12}
Let $f$ and $\tau$ be as above and $\lambda\in R_{+}$. Suppose
$(R_{n})_{n}$ is a sequence $\mathbb{R}_{+}$ such that
$R_{n}\to +\infty$ and $(u_{n})_{n}$ is a sequence of
positive solutions of \eqref{eq:Dr} with $R_{n}$ instead of R, such that
for all n,  $u_{n}\in W^{1,2}_{0}(D_{R_{n}})$ and $(u_{n})_{n}$ is
bounded in $W^{1,2}_0(\mathbb{R}_+^n)$, i.e. there exists $\Gamma'>0$ such that for
all n, $\|u_{n}\|_{L^{2}(D_{R_{n}})}+\|\nabla
u_{n}\|_{L^{2}(D_{R_{n}})}<\Gamma'$. Then, there exists a
subsequence (called again $(u_{n})_{n})$) and a function $u\in W^{1,2}_0(\mathbb{R}_+^n)$
such that $u_{n}\to u$ weakly in $W^{1,2}_0(\mathbb{R}_+^n)$ and $u$ is a
classical solution of \eqref{eq:Rn}.
\end{lem}

\begin{proof} Using the Calder\'{o}n-Zygmund
 inequality for all $n$ \cite[theorems 9.9 and 9.11]{gt},
$u_{n}\in W^{1,2}_{0}(D_{R_{n}})\bigcap H^{2,p}(D_{R_{n}})$.
($H^{2,p}(D_{R_{n}})$ denotes the usual
Sobolev space  \\ $W^{2,p}(D_{R_{n}})$). Fixed $R'>0$, for any
$\Omega'\subset\subset D_{R'}$,
\[
\|u_{n}\|_{H^{2,p}(\Omega')}\leq
C(\|u_{n}\|_{L^{p}(D_{R'})}+\|\lambda
f(u_{n}+\tau)\|_{L^{p}(D_{R'})})
\]
for all $n$ such that  $R_{n}>R'$. The constant $C$ depends on
$D_{R'}$, $n$, $p$ and $\Omega'$. Since $(u_{n})$ is bounded
in $W^{1,2}_0(\mathbb{R}_+^n)$, using Sobolev immersion and Poincar\'{e} inequality
\[
\|u_{n}\|_{H^{2,p}(\Omega')}\leq C(C_{1}\Gamma'+\lambda \sup
f |D_{R'}|^{\frac{1}{p}})
\]
for $p$ such that
\begin{gather*}
1<p<\frac{2n}{n-2}  \quad \textrm{if } n\geq3\\
1<p \quad \textrm{if } n=2
\end{gather*}
and for all n such that $R_{n}>R'$.
 From this and the Sobolev embedding theorem for $\Omega'$,
there exists a subsequence $(u_{n})_{n}$ such that if n=2,3
$u_{n}\to u $ in $C^{1,\alpha}(\overline{\Omega'})$ and
if $n\geq 4$ and $1<p<\min\bigl(\frac{n}{2},\frac{2n}{n-2}\bigr)$
is fixed, $u_{n}\to u $ in $L^{q}(\Omega')$, $1\leq
q<\frac{np}{n-2p}$.
Since $\Omega'$ is an arbitrary and relatively compact such
that $\Omega'\subset\subset D_{R_{n}}$ and $R_{n}\to
+\infty$, we obtain that the above convergence  are in
$C^{1,\alpha}_{\rm loc}(\mathbb{R}_+^n)$ and $L^{q}_{\rm loc}(\mathbb{R}_+^n)$, respectively. In
particular
\begin{equation} \label{e4.3}
u_{n}\to u \quad\mbox{in }  L^{1}_{\rm loc}(\mathbb{R}_+^n)
\end{equation}
On the other hand, since $(u_{n})_{n}$ is bounded in $W^{1,2}_0(\mathbb{R}_+^n)$, and
reflexivity
\begin{equation} \label{e4.4}
u_{n}\to u \quad \mbox{weakly in } W^{1,2}_0(\mathbb{R}_+^n)
\end{equation}
then using Sobolev immersion
\begin{equation} \label{e4.5}
u_{n}\to u \quad \mbox{weakly in } \quad L^{p}(\mathbb{R}_+^n)
\end{equation}
where
\begin{gather*}
2\leq p<\frac{2n}{n-2}  \quad \text{if } n\geq3\\
2\leq p \quad \text{if }n=2
\end{gather*}
By \eqref{e4.4}, if we prove that for all $v\in C^{\infty}_{0}(\mathbb{R}_+^n)$
\[
\int_{\mathbb{R}_+^n}f(u_{n}+\tau)vdx\to \int_{\mathbb{R}_+^n}f(u+\tau)vdx
\]
our lemma will follow. Based on this and for fixed $v\in
C^{\infty}_{0}(\mathbb{R}_+^n)$, we consider the function
\[
w=\frac{f(u+\tau)}{u+\tau}v
\]
It is easy to see that $w\in L^{p'}(\mathbb{R}_+^n)$, where $p'$ is such
that $\frac{1}{p}+\frac{1}{p'}=1$. Now
\begin{equation} \label{e4.6}
\begin{aligned}
&\int_{\mathbb{R}_+^n}f(u_{n}+\tau)vdx \\
&=  \int_{\mathbb{R}_+^n}\Bigl[f(u_{n}+\tau)-(u_{n}+\tau)\frac{f(u+\tau)}{u+\tau}\Bigr]vdx
+ \int_{\mathbb{R}_+^n}(u_{n}+\tau)w\,dx
\end{aligned}
\end{equation}
By \eqref{e4.5}, the last term of the right hand side of \eqref{e4.6} tends to
$\int_{\mathbb{R}_+^n}f(u+\tau)v$. On the other hand, by (f4)
\begin{equation} \label{e4.7}
\Bigl|\int_{\mathbb{R}_+^n}\Bigl[f(u_{n}+\tau)-(u_{n}+\tau)\frac{f(u+\tau)}{u+\tau}\Bigr]vdx
\Bigr|\leq 2l\int_{\mathop{\rm supp} (v)}|u-u_{n}||v|dx
\end{equation}
so by \eqref{e4.3}, the first term of the second member in \eqref{e4.6} tends to 0.
\end{proof}

\begin{thm} \label{thm13}
Let $\Gamma$, $f$, $\tau$ and $\overline{\overline{\lambda}}$ be as
above. Then, for all $\lambda$,
$0<\lambda<\overline{\overline{\lambda}}$ the local minima
$\overline{u}_{R}$ of $\Phi_{\lambda,\tau,R}$ obtained in Remark \ref{rmk11},
approximate the local minima of $\Phi_{\lambda,\tau,\infty}$ on
the ball $B_{\Gamma}$ of center 0 and radius $\Gamma$ in $W^{1,2}_0(\mathbb{R}_+^n)$.
As consequence
$\nu_{\infty}\equiv\inf_{B_{\Gamma}}\Phi_{\lambda,\tau,\infty}$,
is a minimum and by Proposition \ref{prop10} it is the unique, if $\lambda$
is small enough (i.e. $0<\lambda<\frac{\inf c(x)}{l}$).
\end{thm}

\begin{proof} Using the Lemma \ref{lm12}, we only need to prove that
$\Phi_{\lambda,\tau,R}(\overline{u}_{R})\to \nu_{\infty}$
as $R\to\infty$. Because of this we consider $(u_{R})_{R}$
in $C^{\infty}_{0}(\mathbb{R}_+^n)$ such that $u_{R}\in W^{1,2}_{0}(D_{R})$
and $\Phi_{\lambda,\tau,\infty}(u_{R})\to\nu_{\infty}$ as $R\to\infty$.
Then
\[
\nu_{\infty}\leq\Phi_{\lambda,\tau,R}(\overline{u}_{R})
\leq\Phi_{\lambda,\tau,R}(u_{R})=\Phi_{\lambda,\tau,\infty}(u_{R})-\lambda\int_{\mathbb{R}_+^n-D_{R}}F(\tau)dx
\]
by \eqref{e4.2}, $\lambda\int_{\mathbb{R}_+^n-D_{R}}F(\tau)dx\to 0$ as $R\to\infty$.
\end{proof}

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\end{document}