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

\AtBeginDocument{{\noindent\small 
\emph{Electronic Journal of Differential Equations}, 
Vol. 2009(2009), No. 108, pp. 1--6.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or
http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2009 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2009/108\hfil Positive solutions]
{Positive solutions for semi-linear  elliptic
 equations in exterior domains}

\author[H. M\^aagli, S. Turki, N. Zeddini\hfil EJDE-2009/108\hfilneg]
{Habib M\^aagli, Sameh Turki, Noureddine Zeddini} % in alphabetical order

\address{Habib M\^aagli \newline
D\'epartement de Math\'ematiques, Facult\'e des Sciences de Tunis,
Campus Universitaire, 2092 Tunis, Tunisia}
\email{habib.maagli@fst.rnu.tn}

\address{Sameh Turki \newline
D\'epartement de Math\'ematiques, Facult\'e des Sciences de Tunis,
Campus Universitaire, 2092 Tunis, Tunisia}
\email{sameh.turki@ipein.rnu.tn}

\address{Noureddine Zeddini \newline
D\'epartement de Math\'ematiques, Facult\'e des Sciences de Tunis,
Campus Universitaire, 2092 Tunis, Tunisia}
\email{noureddine.zeddini@ipein.rnu.tn}

\thanks{Submitted August 12, 2009. Published September 10, 2009.}
\subjclass[2000]{34A12, 35J60} 
\keywords{Positive solutions; nonlinear elliptic equations; exterior domain}

\begin{abstract}
 We prove the existence of a solution, decaying to zero at
 infinity, for the  second order differential
 equation
 $$
 \frac{1}{A(t)}(A(t)u'(t))'+\phi(t)+f(t,u(t))=0,\quad t\in (a,\infty).
 $$
 Then we give a simple proof, under some sufficient conditions which
 unify and generalize most of those given in the bibliography,  for
 the existence of a positive solution for the semilinear second order
 elliptic equation
 $$
 \Delta u+\varphi(x,u)+g( |x|) x.\nabla u =0,
 $$
 in an exterior domain of the Euclidean space ${\mathbb{R}}^{n},n\geq 3$.
\end{abstract}

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

\section{Introduction}

The semilinear elliptic equation
\begin{equation} \label{1}
\Delta u+\varphi(x,u)+g( |x|) x.\nabla u=0,\quad  x\in G_\delta =
\{x\in {\mathbb{R}}^{n}  : |x|> \delta >0\},
\end{equation}
constitutes the object of numerous investigations in the last few
years (see \cite{A1,C2,C3,C4,E1,E2,E3,N1,W1}). The function
$\varphi$ is nonnegative and locally H\"{o}lder continuous in
$G_\delta\times {\mathbb{R}}$ for which there exist two continuous
functions  $q: [\delta, \infty)\to [0, \infty)$ and $\omega: [0,
\infty)\to [0,\infty)$ such that
$$
0 \leq \varphi(x,t)\leq q(|x|)\omega(t),\quad t\in [0, \infty),\; x
\in G_\delta\,.
$$
So far, the optimal sufficient condition stated to ensure the
existence of a positive solution, decaying to zero at infinity, for
\eqref{1} in some $G_B$ with $B>\delta$ is
\begin{equation}\label{1'}
\int_\delta^\infty r\,\left[q(r)+g^-(r)\right]\,dr<\infty \,,
\end{equation}
where $g^-(r)=\max(-g(r),0)$ for $r\geq \delta$.

To apply the method of sub-solutions and super-solutions developed
in \cite{N1} and other works, the scaling function $ |x|= r=
\beta(s)=(\frac{s}{n-2})^{1/(n-2)}$ plays a capital role in  finding
a radial super-solution for \eqref{1} of the form $u(x)= h(|x|)=
h(r)$, where $h$ is chosen so that $y(s)= s h( \beta(s))$ satisfies
a nonlinear differential equation
\begin{equation} \label{2}
y''(s)+ G( s, y(s), y'(s))=0\,\quad s\geq s_0= (n-2)\delta^{n-2}.
\end{equation}
As a sub-solution of  \eqref{1} we understand any function $\omega
\in C^2(G_B) \cap C(\overline{G_B})$ such that $\Delta \omega (x)
+\varphi(x,\omega (x))+g(|x|)x\,\cdot \nabla \omega (x) \geq0$ in
$G_B$. For the super-solution, the sign of the inequality should be
reversed.

Our aim in this paper is twofold. Firstly, we study in section 2 the
existence of solutions, having a nonnegative limit at infinity, for
the problem
\begin{equation}\label{3}
\frac{1}{A(t)}(A(t)u'(t))' +\phi(t)+ f(t,u(t))= 0,\quad  t\in
(a,\infty),
\end{equation}
where $A$ and $f$ satisfy some hypothesis stated in the next
section. Secondly, in section 3, we omit the scaling function
$\beta$ defined before and we give a simple proof for the existence
of positive solutions, decaying to zero at infinity, in some $G_B$,
$B>\delta$  for the semi-linear elliptic equation \eqref{1}. This
will be done under  sufficient conditions given by the hypotheses
(A3)-(A4) below, which improve and generalize \eqref{1'}. More
precisely we will prove the existence of a positive solution to
\eqref{1} even when $\int_\delta^\infty r\,g^-(r)\,dr=\infty$.

\section{Positive solutions of second-order ODEs}

In this section, we are concerned with the existence of positive
solutions for the problem
\begin{equation}\label{5}
\begin{gathered}
\frac{1}{A(t)}(A(t)u'(t))' + \phi(t)+ f(t,u(t))= 0, \quad
\text{for } t \geq a>1 \\
Au'(a) = -\alpha  \leq 0\,,\quad \lim_{t\to \infty}u(t) = \lambda
\geq 0 \,,\quad \text{with } \alpha +\lambda>0\,,
\end{gathered}
\end{equation}
where $A$ is a positive and differentiable function  on
$[1,\infty)$, $\phi$ is a nonnegative continuous function on
$[1,\infty)$ and $f:[1,\infty )\times [0,\infty)\to [0,\infty)$ is
continuous such that $f(x,0)=0$.

In the sequel we suppose that $ \int_1^\infty
\frac{1}{A(t)}\,dt<\infty$ and we denote by
$$
G(t)=A(t)\Big( \int_t^{\infty}\frac{1}{A(s)}\,ds\Big)
$$
for $ t\geq 1$.  The following hypotheses satisfied by
 $A$, $\phi$ and  $f$  throughout this section:
\begin{itemize}
\item[(A1)] $ \int_1^\infty G(t) \phi(t)\,dt<\infty$;

\item[(A2)] For each $c>0$,
there exists a continuous function $ k: [1,\infty)\to [0,\infty) $
such that
$$
|f(t,u)-f(t,v)|\leq k(t)|u-v|\quad \text{for any } (t,u,v)\in
[1,\infty) \times [0,c]\times [0,c]
$$
and $\int_1^{\infty}G(t)\,k(t)\,dt<\infty$.
\end{itemize}
Our first existence result is the following.

\begin{theorem}\label{thm1}
Let $\alpha\geq 0$ and $\lambda \geq 0$ with $\alpha + \lambda>0$.
Under the hypotheses {\rm (A1)-(A2)}, there exists $a>1$ such that
 \eqref{5} has a unique positive solution
$u \in C^1([a, \infty), {\mathbb{R}})$.
\end{theorem}

\begin{proof}
 Let
$$
c>M:= \lambda+ \alpha  \int_1^\infty \frac{1}{A(t)}\,dt +
\int_1^\infty G(t) \phi (t)\,dt.
$$
 From  (A2), there exists a $k$ such that
$|f(s,u)-f(s,v)|\leq k(s)|u-v|\, \text{ for  any } (s,u,v)\in
[1,\infty) \times [0,c]\times [0,c]$ and
$\int_1^{\infty}G(t)\,k(t)\,dt<\infty$.
  Let $a>1$ such that
$$
\int_a^\infty G(t)k(t)\,dt < 1-\frac{M}{c}:=\sigma.
$$
We denote by $ C_b([a,\infty), \mathbb{R})$ the set of continuous
bounded real valued functions on $[a, \infty)$ and by
$$
\Gamma: =
 \{u\in C_b([a, \infty),{\mathbb{R}}): \lambda \leq  u \leq c\}\,.
$$
Then  $\Gamma$ endowed with the  supremum norm is a Banach space. To
apply a fixed point argument, we define the operator $T$ on $\Gamma$
by
\begin{equation}\label{6}
Tu(r)=\lambda+\alpha\int_r^\infty \frac{1}{A(t)}dt+\int_r^\infty
\frac{1}{A(t)}\Big(\int _a^t A(s) [\phi(s)+ f(s,u(s))]ds\Big)dt.
\end{equation}
First, we claim that $T (\Gamma) \subset \Gamma$. Indeed, from (A2)
and Fubini theorem, we get that for each $u\in \Gamma$ any $r\geq
a$,
\begin{align*}
\lambda \leq Tu(r) &\leq  \lambda+\alpha \int_a^\infty
\frac{1}{A(t)}dt +\,\int_a^\infty \frac{1}{A(t)}\Big[\int_a^t
A(s)\big(\phi(s)+ck(s)\big)\,ds\Big]\,dt \\
&\leq \lambda+ \alpha \int_a^\infty \frac{1}{A(t)}dt + \int_a^\infty
G(s) \phi(s)\,ds +c \, \int_a^\infty G(s)k(s)\,ds \leq  c.
\end{align*}
Now, we have to show that $T$ is a contraction on $(\Gamma,
\|.\|_\infty)$. Indeed,  let $u, v\in \Gamma$
 and $r \in [a,\infty)$. Then by the assumption (A2) and Fubini
theorem we have
\begin{align*}
|Tu(r)-Tv(r)| &\leq  \int_r^\infty \frac{1}{A(t)}\Big(\int_a^t
A(s)k(s)|u(s)-v(s)|\,ds\Big)dt \\
&\leq {\|u-v\|_\infty}\int_a^\infty A(s)k(s)\Big(\int_s^\infty
\frac{1}{A(t)}\,dt\Big) ds,
\end{align*}
which implies that $\|Tu-Tv\|_{\infty}\leq  \sigma
\,{\|u-v\|_\infty}$. Thus, by the Banach fixed point theorem, there
exists a unique point $u\in(\Gamma,\|.\|_\infty)$ such that $T u =
u$. It is easy to verify that $u$ is  the unique solution in
$C^1([a,\infty),{\mathbb{R}})$ for \eqref{5}. This completes the
proof.
\end{proof}


It is worth pointing out that for any given $u(a)\geq 0$ and
$u'(a)\leq 0$, the corresponding solution to the equation is unique
and defined for all times (that is, blowup is not possible), see
\cite{B1,C1,H1}. Also and under more restrictive conditions, the
asymptotic behavior of the solutions have been studied, see
\cite{L1}.

\begin{example}\label{exa21} \rm
Let $\sigma>0$ and $\theta : [1,\infty)\to {\mathbb{R}}$ be a
continuous function such that $ \lim_{t\to \infty}\theta (t)=0$. Let
$A(t)=t^{\sigma+1}\exp\big( \int_1^t\frac{\theta(s)}{s}\,ds\big)$.
Then  $ \lim_{t \to \infty}  \frac{t\,A'(t)}{A(t)}=\sigma +1>1$. So
$ \int_1^\infty \frac{1}{A(s)}ds<\infty$ and we have $ \int_t^\infty
\frac{1}{A(s)}\,ds\sim
 \frac{t}{\sigma \, A(t)}$ as $t \to \infty$.
Consequently $G(t)\sim  \frac{t}{\sigma}$ as $t \to \infty$.

 Let $q, \rho$ be respectively two
nontrivial nonnegative continuous function on $[1,\infty)$ and
$[0,\infty)$ such that $ \int_1^\infty t\,q(t)dt<\infty$ and put
$f(t,u)=q(t) \int_0^u\rho(s)ds$. Then for each nonnegative
continuous function $\phi$ on $[1,\infty)$ satisfying $
\int_1^\infty t\,\phi (t) dt<\infty$, there exists $a>1$ such that
\eqref{5} has a unique positive solution $u\in
C^1([a,\infty),{\mathbb{R}})$.
\end{example}

\section{Applications to elliptic equations}

In this section, we are concerned with the nonlinear second order
elliptic equation \eqref{1} in an exterior domain $G_\delta=\{x\in
{\mathbb{R}}^n : |x|> \delta\}$, where $n\geq 3$ and $\delta \geq
0$. We  prove, under some assumptions on the functions $\varphi, g$,
that  \eqref{1} has a positive solution in $G_B$ for $B\geq \delta$
decaying to zero as $|x|$ tends to infinity. More precisely, we omit
the function $\beta$ defined in section 1 and we apply the result in
section 2 to give a simple proof for the existence of positive
solution, decaying to zero, for \eqref{1} in $G_B$ with $B$ large
enough.

To this aim, we consider two continuous functions $\varphi$ and $g$
satisfying
\begin{itemize}
\item[(A3)] $\varphi \in C( G_\delta \times {\mathbb{R}},{\mathbb{R}}_+)$
and there exists a nonnegative continuous function $f$  on
$[\delta,\infty)\times \mathbb{R}$ such that $f(t,0)=0$ and a
nonnegative continuous function $\phi$ on $[\delta,\infty)$ such
that $0\leq \varphi(x,u)\leq f(|x|,u)+\phi(|x|)$. Moreover for each
$c>0$, there exists a nontrivial nonnegative continuous function $k$
defined on $[\delta,\infty)$ such that,
$$
|f(t,u)-f(t,v)|\leq k(t)|u-v|,\quad \forall u,v\in {[0,c]},\;
 \forall t \geq \delta ;
$$

\item[(A4)]
$$
\int_\delta^\infty [k(t)+\phi(t)] A(t)\Big(\int_t^\infty
\frac{1}{A(r)}dr\Big)dt<\infty,
$$
 where $A(t)=t^{n-1} \exp \big(- \int_\delta^t\xi\, g^-(\xi)d\xi \big)$
and $g^-=\max(-g,0)$.

\end{itemize}

In the particular case  when $ \int_\delta^\infty r\,
g^-(r)\,dr<\infty$, hypothesis (A4) reduces to $ \int_\delta^\infty
t\,[k(t)+\phi(t)]\,dt<\infty$. So
 hypothesis (A4) is weaker than the condition
\eqref{1'} given in the introduction where $\phi=0$.

Next, we recall the following two lemmas needed to achieve the proof
of our second main result.

\begin{lemma}[\cite{N1}] \label{lem1}
If for some $B\geq \delta$, there exists a nonnegative sub-solution
$w$ and a nonnegative super-solution $v$ to \eqref{1} in $G_B$, such
that $w(x)\leq v(x)$ for all $x\in \overline{G_B}$, then \eqref{1}
has a solution $u$ in $G_B$, such that $w\leq u\leq v$ in
$\overline{G_B}$ and $u=v$ on $S_B=\{x\in {\mathbb{R}}^n/ |x|=B\}$.
\end{lemma}

\begin{lemma}[{\cite[Theorem 3.5]{G1}}] \label{lem2}
Let $\pounds$ be a uniformly elliptic operator on a domain $\Omega$.
Let $u\in C^2(\Omega)$ such that $\pounds u \geq 0$ in $\Omega$. If
there exists $x_0 \in \Omega$ satisfying $ \sup_{{x\in \Omega}}\,
u(x)= u(x_0)$, then $u$ is constant in all $\Omega$.
\end{lemma}

Now, we give our main result in this section.

\begin{theorem} \label{thm3.4}
 Let $\delta>0$ and assume  {\rm (A3)-(A4)}.
 Then  \eqref{1} has a positive solution
$u$ in $G_B$ for some $B\geq \delta$, such that $ \lim_{x \to
\infty}u(x)=0$.
\end{theorem}

\begin{proof} We will apply Lemma \ref{lem1}. Clearly  the trivial
function $w=0$ is a sub-solution of \eqref{1} in $G_\delta$. Next,
we try to find a positive radial super-solution $y(r)=y(|x|)$ for
\eqref{1} with $ \lim_{r\to \infty} y(r)=0$. Taking into account
(A3), it suffices to find a function $y$ such that
\begin{gather*}
y''+[\frac{n-1}{r}+rg(r)]y'+f(r,y) +\phi(r) \leq 0 \quad
 \text{for } r>B>\delta \\
\lim_{r\to \infty}y(r)=0.
\end{gather*}
Now, taking into account of Theorem \ref{thm1}, it suffices to find
$B>\delta$ and  a solution for the problem
\begin{align*}
y''+[\frac{n-1}{r}-r g^-(r)]y'+f(r,y)+\phi(r)=0,\, \quad r>B \\
 y'(r)<0 ,\quad r>B, \quad \lim_{r\to \infty}y(r)=0.
\end{align*}
Or equivalently,
\begin{equation}\label{8}
\begin{gathered}
 \frac{1}{A(r)}(A(r)y'(r))'+f(r,y)+\phi(r)=0,\,\quad r>B \\
y'(r)<0 ,\quad r>B, \quad \lim_{r\to \infty} y(r)=0\,,
\end{gathered}
\end{equation}

 where
$$
A(r)= \,r^{n-1}\exp\Big(- \int_\delta^r \xi g^-(\xi)d\xi\Big).
$$
So it follows from hypotheses  (A3)-(A4) and Theorem \ref{thm1} that
there exists $B>\delta$ such that
 \eqref{8} has a positive solution $y(r)$ on $[B,\infty)$.
Obviously $y$ is a super-solution for \eqref{1} in $G_B$. Hence, by
Lemma \ref{lem1},  problem \eqref{1} has a solution $u$ in $G_B$
such that $0\leq u(x) \leq y(|x|)$ in $G_B$ and  $u= y> 0 $ on
$S_B$.

 Next, we prove that the solution $u$ is positive in $G_B$.
Suppose that there exists $x_0\in G_B$ such that $u(x_0)=0$. Then,
the uniformly  elliptic operator $\pounds u:=\Delta u+
g(|x|)x.\nabla u $ satisfies
  $\pounds (-u)\geq \varphi(x,u) \geq 0$
in  $G_B$ and $ \sup_{{x\in G_B}}( -u(x))= -u(x_0)=0$. Hence by
Lemma \ref{lem2} we obtain $u=0$ in $G_B$. From the continuity of
$u$ in $\overline{G_B}$, this contradicts the fact that $u>0$ on
$S_B$ and shows that $u(x)>0$, for all $x\in G_B$.
\end{proof}

\begin{example} \label{exa3.1} \rm
In the sequel, we define by $\mathop{\rm Log}_0 t=t$ and
$\mathop{\rm Log}_mt=\mathop{\rm Log}(\mathop{\rm Log}_{m-1}t)$ for
$m\in {\mathbb{N}}^{\star}$ and $t$ large enough. Let $\delta_m>0$
such that $\mathop{\rm Log}_{m}(\delta_m)=1$ and let $g$ be a
continuous function on $[\delta_m,\infty)$ such that
\begin{equation}\label{eq3.2}
g^-(r)=\max(-g(r),0)= \frac{\gamma}{r\,
 \prod_{k=0}^m \mathop{\rm Log}_k(r)},
\end{equation}
where  $ \gamma >0$ if $m \in {\mathbb{N}}^{\star}$ and $0< \gamma
<n-2$ if $m=0$.
 Then $t\,g^-(t)=\gamma  \frac{d}{dt}( \mathop{\rm Log}_{m+1}t)$ and so
$$
\exp\Big( \int_{\delta_{m}}^ts\,g^-(s)\,ds\Big)= (\mathop{\rm
Log}\nolimits _mt)^\gamma.
$$
Thus,   $ \int_{\delta_m}^\infty r\,g^-(r)\,dr=\infty$ and
 (A4) is satisfied if and only if
$$
 \int_{\delta_m}^\infty t [k(t)+\phi(t)]dt<\infty.
$$
Indeed, this follows from Example \ref{exa21} with
$\theta(s)=-s^2\,g^-(s),\sigma=n-2$ if $m \in {\mathbb{N}}^{\star}$
and $\theta=0,\sigma=n-2-\gamma$ if $m=0$.

Now, using this fact we deduce that if $g$ is a function  where
$g^-$ is given by \eqref{eq3.2}, if  $\phi$ and $k$  are two
nonnegative continuous functions on $[\delta_m, \infty)$ satisfying
$ \int_{\delta_m}^\infty t\,[ k(t)+\phi(t)]\,dt<\infty$ and if
$0\leq \varphi( x,u)\leq k(|x|) u^\alpha + \phi(|x|)$ for
$\alpha\geq 1$, then there exists $B>\delta_m$ such that
 \eqref{1} has a positive solution $u$ on $G_B$ decaying
to zero at infinity.
\end{example}

\subsection*{Acknowledgements}
The authors want to thank the anonymous referee for his/her careful
reading of the original manuscript and the helpful suggestions.

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\end{document}