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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2010(2010), No. 51, pp. 1--5.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2010 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2010/51\hfil Oscillation criteria]
{Oscillation criteria for semilinear elliptic equations with
a damping term in $\mathbb{R}^n$}

\author[Tadie\hfil EJDE-2010/51\hfilneg]
{Tadie}

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

\thanks{Submitted September 10, 2009. Published April 9, 2010.}
\subjclass[2000]{35J60, 35J70}
\keywords{Picone's identity; semilinear elliptic equations}

\begin{abstract}
We use a method based on Picone-type identities to find
oscillation conditions for the equation
$$
\sum_{i j =1}^n \frac{\partial}{\partial x_i}
\Big( a_{ij}(x) \frac{\partial}{\partial x_j} \Big)u +
f(x,u , \nabla u)  + c(x) u \,,
$$
with Dirichlet boundary conditions on bounded and unbounded domains.
This method substitudes the traditional Riccati techniques
\cite{m1,xu} used for unbounded domains.
\end{abstract}

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

\section{Introduction}

We consider semilinear  Dirichlet problems associated with the
elliptic equation
\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 , \nabla u)  + c(x) u
\end{equation}
in a smooth, open and bounded (or unbounded) domain
$G\subset \mathbb{R}^n$, $n \geq 3$.

Oscillation conditions for \eqref{e1.1} when $f$  does not depend
on $ \nabla u $  are shown in \cite{t2}. Inspired by those results,
we find conditions on $f$, $a_{ij}$
and $c$ for \eqref{e1.1} to be oscillatory in $ \mathbb{R}^n$.
We recall that \eqref{e1.1} is said to be oscillatory in
$ \mathbb{R}^n $ if  for all $R>0$,
any of its (classical) solutions (extended to the whole space) has
a simple zero in $ \Omega_R:=\{ x\in \mathbb{R}^n : \|x\|>R \}$.

In this article, we use the notation:
\begin{gather*}
D_i \{ . \} := \frac{\partial }{\partial x_i } \{. \} :=\{.\}_{,i}\,;\\
a(Y,W) :=\sum_{i,j=1}^n a_{ij}Y^i W^j,\quad
\text{for } Y , W \in \mathbb{R}^n ,\; a\in M_{n\times n} \,,
\end{gather*}
where $M_{n \times n}$ denotes the space of $n\times n$-matrices.
The function $f(x,u,\nabla u)$ plays the role of the damping
term in \eqref{e1.1}.
We use the hypotheses:
\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}\times \mathbb{R}^n ; \mathbb{R})$
is non constant;
$ \mathbb{R}_+ := (0 , \infty) $  and
$\bar{ \mathbb{R}}_+:=[0 , \infty)$.
The (classical) solutions  for \eqref{e1.1} are those which
belong to the space $C^1(\overline{G})\cap C^2(G)$.

\item[(H3)] Teh function $f$  satisfies:
for each $t\in \mathbb{R}$, $\xi \in \mathbb{R}^n$,
\begin{itemize}
\item[(i)] $tf(x,t,\xi)>0 $  or
\item[ii)] $tf(x,t,\xi)<0$
\end{itemize}
for all $x \in G$.
\end{itemize}

Oscillatory solutions will be extended to the whole space, if
they were expressed only in a bounded set $G$.
When the domain is the whole space $\mathbb{R}^n$, Hypotheses (H1)--(H3)
need to hold outside $G$, for the oscillatory results to be true.

\section{Preliminaries}

For (smooth) functions $ u $ and $ 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 $
satisfies the property that if $ w\neq 0 $, then
\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,\nabla w)}w - \frac{f(x,u,\nabla 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,\nabla u)}u - \frac{f(x,w,\nabla w)}w \big\} \,.
\end{aligned}
\end{equation}

\begin{lemma} \label{lem2.1}
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,\nabla 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  \text{or} \quad v\big|_{\partial G}=0 .
\end{gather}
Then  as in \eqref{e2.1i},
\begin{gather*}
\sum_{i,j=1}^n D_i \big\{ v a_{ij}(x)D_j v  - \frac{v^2}u \; a_{ij}D_j u
\big\}
  = u^2 a\Big( \nabla[\frac vu ], \nabla [\frac vu] \Big)  - vf(x,v,\nabla v)\\
\text{if $u\neq 0$  in $G$  and if $v\neq 0$  in  $G$}\\
\sum_{i,j=1}^n D_i \Big\{ u a_{ij}(x)D_j u  - \frac{u^2}v
\; a_{ij}D_j v \Big\}
= v^2 a\Big(\nabla[\frac uv ] , \nabla[ \frac uv] \Big)
+ u^2 \frac{f(x,v,\nabla v)}v \,.
\end{gather*}
Then the two solutions cannot be simultaneously non zero throughout
$G$. Consequently
\begin{itemize}
\item[(i)] there is no  non negligible  domain $\Omega \subset G $
in which the solutions $  u  $ and $v$  satisfy $ uv>0$ and
$u|_{\partial \Omega}=v|_{\partial \Omega}=0$;

\item[(ii)]  in between two  consecutive zeroes of each one lies
one zero of the other.
\end{itemize}
\end{lemma}

\begin{proof}
Assume that in $G$  two solutions $u$ and $v$ are of the
same sign and have  value zero on $\partial G$.
Assume that  (H3i) holds. Then 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,\nabla v)}v \Big]dx
\end{equation}
which cannot hold as the second member is strictly positive.
Assume that (H3ii) holds . Then integration over $G$ of \eqref{e2.1ii}
with $v$ replacing $w$  gives
\begin{equation}\label{e2.8b}
0=\int_G \Big\{u^2 a\Big( \nabla[\frac vu ],
\nabla [\frac vu] \Big)  - vf(x,v,\nabla v)\Big\}dx
\end{equation}
and  we get the same conclusion as the second member of the equation
is strictly positive.
\end{proof}

\begin{remark} \label{rmk2.2} \rm
  Among the admissible functions $f$ we have:

\subsection*{(A1)} Define $f(x,u,\nabla u):= g_1(x,u) + g_2(u,\nabla u)$
for all $t \neq 0$, $\xi \in \mathbb{R}^n$, $x\in \mathbb{R}^n $.
In the case (H3i), $ tg_1( x,t) $ and $tg_2(t,\xi) $ are strictly
positive functions.
In the case (H3ii),  $ tg_1( x,t) $ and $tg_2(t,\xi) $ are strictly
negative  functions.
In either case
$$
\int_G u^2  \frac{ f(x,v,\nabla v)}v \,dx \geq 0\,.
$$
\subsection*{(A2)} Define
$f(x,u,\nabla u) := g_1(x,u) + \overrightarrow{ B}.\nabla \zeta(u)$,
where
\begin{equation} \label{iii}
tg_1(x,t)\leq 0\quad\text{for all }(x,t)\in \mathbb{R}^n \times \mathbb{R},
\end{equation}
$\overrightarrow{ B} = ( b_1(x) , b_2(x) , \dots ,b_n(x) ) $ is
a vector field, $u \nabla \zeta(u) \equiv \nabla \psi(u) $ for some
$\psi \in C^1(\mathbb{R}) $  which keeps the same sign
in $\mathbb{R}$ and either
\begin{gather}
\frac{\partial b_i}{\partial x_i} \geq 0 \quad
\text{for $i= 1, 2 , \dots ,n $,  if  $\psi$ is a non negative function},
\label{iv}\\
\frac{\partial b_i}{\partial x_i} \leq 0 \quad
\text{for $i= 1, 2 , \dots ,n $,  if  $\psi$ is a non positive function}.
\label{v}
\end{gather}
Simple calculations show that anyone of the two conditions
\eqref{iv} or \eqref{v} leads to
$$
\int_G \{ -uf(x,u,\nabla u)\} dx \geq 0
$$
and \eqref{e2.8b} applies.

The condition (A2) applies for example to the perturbed Schrodinger
equation (see \cite{m1})
$$
\Delta u +  \langle \overrightarrow{b}(x) , \nabla u  \rangle
+ c(x)u =0\,.
$$
\end{remark}

\subsection{Oscillation criteria}

\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
$\Omega_R:=\{ x\in \mathbb{R}^n : |x|> R \}$.
A solution of \eqref{e1.1} will be said to be oscillatory if its
extension over $\mathbb{R}^n $ is oscillatory.
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 $r_0>0$, define
\[
P(t) := \int_{r_0}^t \frac{dr}{p(r)} \quad \text{if }
\lim_{t\to\infty} p(t)=\infty
\]
and
\[
\Pi(t) := \int_{r_0}^t \frac{dr}{p(r)} \quad \text{if }
\lim_{t\to\infty}  p(t)<\infty.
\]
From \cite[Lemma 3.1 and Theorem 3.1]{k2}, we have the
following result  (see also  \cite{t2}).

\begin{lemma} \label{lem2.3}
Let $r_0>0$,
\begin{itemize}
\item[(i)]
$  \int_{r_0}^\infty q(r)dr =\infty$ or
\[
\int_{r_0}^\infty q(r)dr <\infty \quad \text{and}\quad
\lim \inf_{r\nearrow \infty}
\big\{ P(r)\int_r^\infty q(s)ds \big\} >\frac 14
\]

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

The above lemma also holds when
$A(r) $ and $ C(r) $ are  replaced, respectively, by
\[
\overline{A}(r):=\frac 1{\omega_n  r^{n-1}} \int_{|x|=r}  \max_{I_n}
\{a_{ij}(x)\} ds \quad \rm{and}\quad
  \overline{C}(r):=\frac 1{\omega_n  r^{n-1} } \int_{|x|=r} c(x)ds \,,
\]
where $\omega_n $ denotes the area of the unit sphere
in $\mathbb{R}^n$. (\cite{t2})

\section{Main result}

From  Lemma \ref{lem2.3} and the preceding  results, we have the de
following theorem.

\begin{theorem} \label{thm3.1}
Consider in a bounded and regular domain $G\subset \mathbb{R}^n $
the equation
\begin{equation} \label{e3.1}
\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,\nabla u) +  c(x)u=0
\quad \text{in } G,
\end{equation}
where {\rm (H1), (H2)} hold in the whole space $\mathbb{R}^n$.
If in addition
\begin{itemize}
\item[(a)] either {\rm (H3)} holds in $\mathbb{R}^n$  and
the  functions $a_{ij} $ and $c$ satisfy (i) or (ii)
of Lemma \ref{lem2.3},  or
\item[(b)] \eqref{iii}--\eqref{v} hold
\end{itemize}
then  \eqref{e3.1} is oscillatory in $\mathbb{R}^n$.
\end{theorem}

\begin{proof}
From Lemma \ref{lem2.3}, conditions (i) and (ii) imply that \eqref{e2.8}
is oscillatory.
From Lemma \ref{lem2.1} and  Remark \ref{rmk2.2}, if \eqref{e2.8}
is oscillatory, so is \eqref{e3.1}.
\end{proof}

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\end{document}
      
