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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2009(2009), No. 67, pp. 1--7.\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-2009/67\hfil Comparison results]
{Comparison results for semilinear 
elliptic equations via Picone-type identities}

\author[Tadie\hfil EJDE-2009/67\hfilneg]
{Tadie}

\address{Tadie \newline
Mathematics Institut \\
Universitetsparken 5 \\
 2100  Copenhagen, Denmark}
\email{tad@math.ku.dk}

\thanks{Submitted November 14, 2008. Published May 14, 2009.}
\thanks{Dedicated to my late son Nkayum Tadie Abissi
(+ 11/03/07) and to my cousin \hfill\break\indent
Tagne David Pierre (+ 01/11/08); requiescate in pacem.}
\subjclass[2000]{35J60, 35J70}
\keywords{Picone's identity; semilinear elliptic equations}

\begin{abstract}
 By means of a Picone's type identity, we prove
 uniqueness and oscillation of solutions to
 an elliptic semilinear equation with Dirichlet boundary
 conditions.
\end{abstract}

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

\section{Introduction}


The aim of this work is to provide some comparison and uniqueness
results for semilinear  Dirichlet problems in a smooth,
open and bounded domain $G\subset \mathbb{R}^n$,  $n \geq 3$.
The problems are related to  the
elliptic operators
\begin{equation} \label{e1.1}
\ell u := \sum_{i j =1}^n \frac{\partial}{\partial x_i}
\Big( a_{ij}(x) \frac{\partial}{\partial x_j} \Big)u +
 f(x,u)  + c(x) u \,.
\end{equation}
The notation in this article is as follows:
\begin{gather*}
D_i \{ . \} := \frac{\partial }{\partial x_i } \{. \} :=\{.\}_{,i} \,; \\
 \forall  Y , W \in \mathbb{R}^n \text{ and }
 a\in M_{n\times n},
 \quad a(Y,W) :=\sum_{i,j=1}^n a_{ij}Y^i W^j ,
\end{gather*}
where $M_{n \times n}$ denotes the space of
$n\times n$-matrices.
The  hypotheses on the coefficients are:
\begin{itemize}
\item[(H1)]  The functions  $ a_{ij} \in C^1( \overline{G}; \mathbb{R}_+)$
are symmetric and continuous with
$$
\sum_{i,j=1}^n a_{ij}(x)\xi_i \xi_j \geq 0 \quad
\forall ( x , \xi)\in G \times  \mathbb{R}^n  \quad
( >0  \text{ if } \xi \neq 0) .
$$

\item[(H2)] The function $ c \in C( \overline{G}; \mathbb{R})$;
$ f\in C(\mathbb{R}^n \times \mathbb{R}; \mathbb{R})$
is non constant;  $ \mathbb{R}_+ := (0 , \infty) $  and
$\bar{ \mathbb{R}}_+:=[0 , \infty)$.
The (classical) solutions  for \eqref{e1.1} 
belong to the space $C^1(\overline{G})\cap C^2(G)$.
\end{itemize}

\section{Preliminaries}

For the (smooth) functions $ u , w  $, as in \cite{j1},
from the expressions
$ D_i \{ u a_{ij}D_ju - (u^2/w) a_{ij}D_jw\} $  and  $ u\ell u $
we have that if $ w\neq 0 $,
\begin{equation}  \label{e2.1i}
\begin{aligned}
 &\sum_{i,j=1}^n D_i \big\{ u a_{ij}(x)D_j u  - \frac{u^2}w \; a_{ij}D_j w
\big\}  \\
  &= w^2 a\Big( \nabla[\frac uw ], \nabla [\frac uw] \Big)
+ u\ell u - \frac{u^2}w \ell w + u^2
 \big\{ \frac{f(x,w)}w - \frac{f(x,u)}u \big\}
\end{aligned}
\end{equation}
and if $ u\neq 0 $, then
\begin{equation}  \label{e2.1ii}
\begin{aligned}
& \sum_{i,j=1}^n D_i \Big\{ w a_{ij}(x)D_j w  - \frac{w^2}u \; a_{ij}D_j u
\Big\}  \\
&= u^2 a\Big(\nabla[\frac wu ] , \nabla[ \frac wu] \Big)
+  w\ell w -   \frac{w^2}u \ell u +
w^2 \big\{ \frac{f(x,u)}u - \frac{f(x,w)}w \big\} \,;
\end{aligned}
\end{equation} 
 also for $\lambda \neq 0 $ if $ \ell u=0 $, then
\begin{equation} \label{e2.1iii}
 \ell (\lambda u) =  f(x, \lambda u) - \lambda f(x,u) \,.
\end{equation}

\begin{remark} \label{rmk2.0} \rm
 Most of the results  will be established by the means of integrating
over $G$ (which is a regular domain) allowing the integration by
parts along  its boundary $ \partial G $;
this in cases like the left side of say, \eqref{e2.1i} , \eqref{e2.1ii}
 and many other cases makes the
 left side of the integral to be zero when $ u|_{\partial G} =0 $.
\end{remark}

\begin{lemma} \label{lem2.1}
If $ u_1 $ and $ w_1 $ are classical  solutions of
\begin{equation} \label{e2.2}
 \ell v =  \sum_{i j =1}^n D_i\big( a_{ij}(x) D_j \big)v
+ c(x) v =0 \quad \text{in }  G \,; \quad v\big|_{\partial G}=0,
\end{equation}
then
 \begin{equation} \label{e2.3}
\begin{aligned}
\sum_{i.j=1}^n   D_i \big\{ u_1 a_{ij}D_j u_1 - \frac{u_1^2}{w_1}
a_{ij}D_j w_1  \big\}
&=w_1^2 \sum_{i.j=1}^n  a_{ij} D_i [\frac{u_1}{w_1}]
D_j [\frac{u_1}{w_1}]\\
&= w_1^2  a( \nabla  [\frac{u_1}{w_1}] ,\nabla [\frac{u_1}{w_1}] ) \,.
\end{aligned}
\end{equation}
\end{lemma}

The proof of the above lemma follows from the
identities \eqref{e2.1i}-\eqref{e2.1ii} where $f\equiv 0$.


\begin{lemma} \label{lem2.2}
If $ u ,  v \in C^2 $ with $ v\neq 0 $ then
\begin{equation} \label{e2.4}
\begin{aligned}
 &v^2 a( \nabla[\frac uv ] , \nabla[\frac uv] )
+ \sum_{i,j=1}^n D_i \Big( \frac{u^2}v a_{ij}D_jv \Big) \\
 &= a(\nabla u,\nabla u) + u^2 \frac{\ell v}v - c(x)u^2
 - \frac{u^2 f(x,v)}v \,.
\end{aligned}
\end{equation}
\end{lemma}

\begin{proof}
As in \cite{s1}, for all $u,v \in C^2 $ with $  v\neq 0 $,
\[
 D_i \big\{ a_{ij}\; \frac{u^2}v \; D_jv \big\}
 = \frac{2u}v a_{ij} D_iu D_j v -  \big(\frac uv \big)^2 a_{ij} D_iv D_i v
+ \frac{u^2}v  D_i(a_{ij}v_j)
\]
and
\begin{equation} \label{e2.5}
\begin{aligned}
&v^2 a_{ij} D_i\Big( \frac uv \Big)D_j\Big( \frac uv \Big) \\
&=a_{ij} D_iu D_ju  - \frac uv  a_{ij} ( D_iuD_jv + D_juD_iv \; ) +
 \Big(\frac uv \Big)^2 a_{ij} D_iv D_i v\,;
\end{aligned}
\end{equation}
thus
\begin{align*}
&\sum_{i,j=1}^n \Big\{ v^2 a_{ij} D_i\big(\frac uv \big) D_j
\big(\frac uv \big) + D_i \Big( \frac{u^2}v a_{ij}D_jv \Big) \Big\}\\
&= v^2 a( \nabla[ \frac uv ], \nabla[\frac uv ] )
 + \sum_{i,j=1}^n D_i \Big( \frac{u^2}v a_{ij}D_jv \Big)  \\
&=\sum_{i,j=1}^n a_{ij}D_iu D_ju
 + \frac{u^2}v \sum_{i,j=1}^n D_i( a_{ij}D_j v ) \\
&:= a(\nabla u,\nabla u) + u^2 \frac{\ell v}v - c(x)u^2
- \frac{u^2 f(x,v)}v \,.
\end{align*}
Then \eqref{e2.4} follows.
\end{proof}

To ensure that solutions can be extended in the whole $\mathbb{R}^n $
  we set the hypothesis
\begin{itemize}
\item[(H3)]  for all $x\in \mathbb{R}^n$  and
 all $t\in \mathbb{ R} \setminus \{0\}$, it holds $tf(x,t)>0 $.
\end{itemize}

\begin{lemma} \label{lem2.3}
Assume {\rm (H1)--(H3)} hold.
Let $u$ and $v$ be respectively  solutions of
\begin{gather} \label{e2.6i}
  \ell v := \sum_{i j =1}^n \frac{\partial}{\partial x_i}
 \Big( a_{ij}(x)  \frac{\partial}{\partial x_j} \Big)v
 + c(x) v + f(x,v) =0 \quad \text{in } G ; \\
\label{e2.6ii}
  L u := \sum_{i j =1}^n \frac{\partial}{\partial x_i}
 \Big( a_{ij}(x) \frac{\partial}{\partial x_j} \Big)u
 + c(x) u =0 \quad \text{in } G; \\
\label{e2.6iii}
 u\big|_{\partial G} =0 \,; \quad u>0
\text{ in $ G$ and $v>0$ somewhere in $G$}.
\end{gather}
Then $v$ has a zero inside $G$. The same conclusion holds
in the case where the inequalities are reverse in \eqref{e2.6iii}.
Consequently any component of the support of $u$ or  that of $-u$
contains a zero of and vise versa.
\end{lemma}

\begin{proof}
Assume that $ v>0 $ in $G$. The integration over $G$ of \eqref{e2.1i}
where $v$  replaces $w$ , gives
\begin{equation} \label{e2.7}
 0= \int_G \Big[ v^2 a\Big( \nabla[\frac uv ], \nabla [\frac uv] \Big)+
 u^2  \frac{f(x,v)}v \Big]dx
\end{equation}
which cannot hold as the second member is strictly positive.
If the inequalities in \eqref{e2.6iii} are reverse we get the
same conclusion  by applying the result to $-u$ and $-v$.
\end{proof}

\subsection{Oscillatory solutions}

\subsection*{Definition} % 2.4
A function $u$  is said to be oscillatory  in $\mathbb{R}^n $ if
for all $R>0$, $u $ has a simple zero in
$\Omega_R:=\{ x\in \mathbb{R}^n : |x|> R \}$.
Equation \eqref{e1.1} is said to be oscillatory if it has oscillatory
solutions.

For the equation
\begin{equation} \label{e2.8}
 L u := \sum_{i j =1}^n \frac{\partial}{\partial x_i}
 \Big( a_{ij}(x) \frac{\partial}{\partial x_j} \Big)u
 + c(x) u =0 \quad \text{in } \mathbb{R}^n
\end{equation}
and for $r>0$ and $ I_n:=\{ (i,j) : i,j \in 1,2,\dots  n \,.\}$, define
\begin{gather*}
A(r):= \max_{\{I_n : |x|=r \}}\{ a_{ij}(x)\} \,, \quad
C(r):=\min_{|x|=r} c(x)\,, \\
p(r):=r^{n-1}A(r) \,, \quad
q(r):= r^{n-1} C(r)\end{gather*}
 and the associated  equation
\begin{equation} \label{e2.9}
 \big( p(r)y' \big)' + q(r)y =0 \quad \text{in } \mathbb{R}_+  \,.
\end{equation}
For some $r_0>0$, define
\[
P(t) := \int_{r_0}^t \frac{dr}{p(r)} \quad \text{if }
 \lim_{\infty} p(t)=\infty
\]
and
\[
\Pi(t) := \int_{r_0}^t \frac{dr}{p(r)} \quad \text{if }
 \lim_{\infty}  p(t)<\infty.
\]
From \cite[Lemma 3.1 and Theorem 3.1]{k2}, we have the
following result.

\begin{lemma} \label{lem2.4}
Let $r_0>0$,
\begin{itemize}
\item[(i)]
$  \int_{r_0}^\infty q(r)dr =\infty$ or
\[
\int_{r_0}^\infty q(r)dr <\infty \quad \text{and}\quad
\lim_{r\nearrow \infty} \inf
\big\{ P(r)\int_r^\infty q(s)ds \big\} >\frac 14
\]

\item[(ii)]
$ \Pi$ is bounded and $\int_{r_0}^\infty \Pi(r)^2 q(r)dr =\infty$,
or
\[
\int_{r_0}^\infty \Pi(r)^2 q(r)dr <\infty \quad \text{and}\quad
 \lim_{r\nearrow \infty} \inf
\big\{ \frac 1{\Pi(r)} \int_r^\infty \Pi(s)^2 q(s)ds \big\} > \frac 14
\]
\end{itemize}
If either (i) or (ii) holds,
then \eqref{e2.9}  is oscillatory, and so is  \eqref{e2.8}.
\end{lemma}

 From \cite[Remark 3.3]{k2}, Lemma 2.4 also holds when
$A(r) $ and $ C(r) $ are  replaced, respectively, by
\begin{gather*}
\overline{a}(r):=\frac 1{\omega_n  r^{n-1}} \int_{|x|=r}  \max_{I_n}
\{a_{ij}(x)\} ds,\\
  \overline{C}(r):=\frac 1{\omega_n  r^{n-1} } \int_{|x|=r} c(x)ds
\end{gather*}
where $\omega_n $ denotes the area of the unit sphere
in $\mathbb{R}^n$.

\section{Main results}

\begin{theorem} \label{thm3.1}
Consider the problem
 \begin{equation} \label{e3.1i}
 L u := \sum_{i j =1}^n \frac{\partial}{\partial x_i}  \Big( a_{ij}(x)
 \frac{\partial}{\partial x_j} \Big)u  + c(x)u=0 \quad \text{in } G
\end{equation}
with either
\begin{equation} \label{e3.1ii}
 u\big|_{\partial G} =0 \,; \quad  u>0 \quad \text{in }G
\end{equation}
or
\begin{equation} \label{e3.1iii}
 \nabla u|_{\partial G}=0 \,; \quad u>0 \quad \text{in } G .
\end{equation}
Under the hypotheses {\rm (H1)-(H2)}, any  two solutions $u$ and $v$
 of the problem \eqref{e3.1i}, \eqref{e3.1ii} or the problem
\eqref{e3.1i}, \eqref{e3.1iii}
  must  satisfy $ u= k v$ for some constant $k\in \mathbb{R}$.
\end{theorem}

\begin{proof}
If $u$ and $v$ are two such solutions then  after integrating both
sides of \eqref{e2.3}, we get the right side
 strictly positive while the left one is zero  (see Remark \ref{rmk2.0}.
This is absurd unless $\nabla[\frac uv]\equiv 0$ in $G$.
\end{proof}

\begin{theorem} \label{thm3.2}
Assume that {\rm (H1)-(H2)} hold.
For the problem
\begin{equation} \label{3.2i}
 \ell u := \sum_{i j =1}^n \frac{\partial}{\partial x_i}
 \Big( a_{ij}(x) \frac{\partial}{\partial x_j} \Big)u
 + f(x,u) +c(x)u=0 \quad \text{in } G
\end{equation}
with either
\begin{equation} \label{3.2ii}
  u\big|_{\partial G} =0 \,; \quad  u>0 \quad \text{in }G
\end{equation}
or
\begin{equation} \label{3.2iii}
 \nabla u|_{\partial G}=0 \,; \quad u>0  \text{ in }  G .
\end{equation}
(1) If $f(x,t)$ or $ \frac {f(x,t)}t $ is decreasing in $t>0$ for
any $x\in G$ then  any of the problems
\eqref{e3.1i}, \eqref{e3.1ii}; or
\eqref{e3.1i}, \eqref{e3.1iii} of \eqref{e1.1}  has at most one
positive classical solution.

\noindent(2)  Moreover if $ t \mapsto \frac {f(x,t)}t $ is monotone
in $ t>0 $ uniformly for $x\in G$ then any two solutions $u$ and $v$
 of \eqref{e1.1} must intersect in the sense that  each of the sets
$G_u:=\{ x\in G : u(x)>v(x)\} $ and
$ G_v:=\{ x\in G : u(x)<v(x) \} $ has  a non zero measure.
\end{theorem}

\begin{proof}
Let $u$ and $v$ be two such solutions.

(1)   From \eqref{e2.1i}-\eqref{e2.1ii}
\begin{gather*}
 0=\int_G v^2 a(\nabla[\frac uv] , \nabla[\frac uv])
  + u^2 \big\{ \frac {f(x,v)}v - \frac{f(x,u)}u \big\} dx
\\
0=\int_G u^2 a(\nabla[\frac vu] , \nabla[\frac vu]) - v^2
\big\{ \frac {f(x,v)}v - \frac{f(x,u)}u \big\} dx
\end{gather*} \\
whence
\begin{equation} \label{e3.3}
 0= \int_G\Big[  v^2 a(\nabla[\frac uv] , \nabla[\frac uv])
+ u^2 a(\nabla[\frac vu] , \nabla[\frac vu])
+ \{ u^2 - v^2 \}\big\{ \frac{f(x,v)}v - \frac{f(x,u)}u \big\} \Big] dx
\end{equation}
 and the conclusion follows from the fact that in any of the cases,
the left hand side of \eqref{e3.3} is zero and the right strictly
positive.

(2)  From \eqref{e2.1i}-\eqref{e2.1ii}, with
$ X(x):=\frac{f(x,v)}v - \frac{f(x,u)}u $
\begin{align*}
0 &=\int_G \Big\{ v^2 a(\nabla[\frac uv] , \nabla[\frac uv])
 + u^2 X(x) \Big\}dx\\
 &=\int_G \Big\{ u^2 a(\nabla[\frac vu] , \nabla[\frac vu])
 - v^2 X(x) \Big\}dx
\end{align*}
whence
\begin{equation} \label{e3.4}
0=\int_G\Big[ v^2 a(\nabla[\frac uv] , \nabla[\frac uv])
+  u^2 a(\nabla[\frac vu] , \nabla[\frac vu]) +
\{ u^2 - v^2\} X(x) \Big] dx \,.
\end{equation}
If $ t \mapsto \frac{f(x,t)}t $ is increasing  and $ u-v $ does
not change sign in $G$ then \eqref{e3.4}
 cannot hold as its second member would be strictly positive.
Thus to have two distinct solutions in this case
 none of  $G_u$ and $G_v$ must have zero measure.
\end{proof}

\begin{theorem} \label{thm3.3}
Assume that there is $ \lambda_0>1 $ such that
for all $(\lambda , x , t) \in (\lambda_0 ,
\infty)\times G\times (0, \infty)$,
\begin{equation} \label{e3.5}
  \lambda f(x,t) - f(x,\lambda t) >0 \,.
\end{equation}
Then if  for all $x\in G$, $t\mapsto \frac{f(x,t)}t $ is strictly
 increasing in $t>0$, \eqref{e1.1} has at most one positive solution.
\end{theorem}

\begin{proof}
Let $u$ and $v$ be two distinct solutions; for
$G_u :=\{ x\in G : u(x)>v(x) \}$, we have
$\nabla\{ u-v\}|_{\partial G_u} \not\equiv 0 $; otherwise
from  \eqref{e2.1ii},
\[
 0=\int_{G_u} \Big[ u^2 a(\nabla[\frac vu] , \nabla[\frac vu])
+ v^2 \{\frac{f(x,u)}u - \frac{f(x,v)}v \} \Big]dx
\]
which would not hold as the second member would be strictly positive.

Let $ W \in C(G) $ be defined by
$ W(x):= (u\vee v)(x):=\max\{ u(x) , \; v(x) \}$.
Then  $W$  is a  weak subsolution of \eqref{e1.1}.
We chose $ \lambda_0>1 $ such that
for all $(x, \lambda)\in G \times (\lambda_0 , \infty)$
$W(x) < \lambda u (x):=V(x)$. By \eqref{e3.5},
 $V$ is a supersolution for \eqref{e1.1} and there is a solution $w$,
say, such that $ W\leq w \leq V $ in $G$ , by the super-sub-solutions
method. This conflicts with the fact that any two solutions of
\eqref{e1.1} must intersect by Theorem 3.2. In fact
such $w$ would not intersect $u$ nor $v$ in the sense of
Theorem 3.2.
\end{proof}

\begin{theorem} \label{thm3.4}
Assume that {\rm (H1)--(H3)} hold in the whole $\mathbb{R}^n$.
If in addition
 (i) and (ii) of the Lemma 2.4  hold,  then
$$
\ell u := \sum_{i j =1}^n \frac{\partial}{\partial x_i}
 \Big( a_{ij}(x) \frac{\partial}{\partial x_j} \Big)u
+ f(x,u) +c(x)u=0
$$
is oscillatory in $\mathbb{R}^n$.
\end{theorem}

The proof of the above theorem is a  mere application of
 Lemmas \ref{lem2.3} and \ref{lem2.4}.


\begin{theorem}[Wirtinger-type inequalities] \label{thm3.5}
Assume that {\rm (H1)--(H2)} hold.
Let $v$ be a classical solution of \eqref{e1.1} and $u$ be a
function in $C^1(\overline{G})$
such that $ u\big|_{\partial G}=0 $. Then
\[
\int_G v^2 a( \nabla[\frac uv ],\nabla[ \frac uv] ) \, dx
 \leq \int_G a(\nabla u,\nabla u) dx
\]
and
\[
 \int_G \big\{ c(x) u^2 + \frac {u^2}v f(x,v) \big\} dx
 \leq \int_G a(\nabla u,\nabla u)\, dx \,.
\]
\end{theorem}

The proof of the above theorem  follows from the integration
over $G $ of both sides of \eqref{e2.4}.


\subsection*{Concluding remarks}
Some of these results can be extended to more general quasilinear
equations including  the $p$-Laplacian equations; see \cite{t1}.

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\end{document}