\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 $$\sum_{i j =1}^n \frac{\partial}{\partial x_i} \Big( a_{ij}(x) \frac{\partial}{\partial x_j} \Big)u + f(x,u , \nabla u) + c(x) u =0\,,$$ with Dirichlet boundary conditions on bounded and unbounded domains. In this article, the above 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 $$\label{e1.1} \ell u := \sum_{i j =1}^n \frac{\partial}{\partial x_i} \Big( a_{ij}(x) \frac{\partial}{\partial x_j} \Big)u + f(x,u , \nabla u) + c(x) u =0$$ 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 \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} and if $u\neq 0$, then \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} \begin{lemma} \label{lem2.1} Assume {\rm (H1)--(H3)} hold. Let $u$ and $v$ be 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 $$\label{e2.7} 0= \int_G \Big[ v^2 a\Big(\nabla[\frac uv ], \nabla [\frac uv] \Big)+ u^2 \frac{ f(x,v,\nabla v)}v \Big]dx$$ 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 $$\label{e2.8b} 0=\int_G \Big\{u^2 a\Big( \nabla[\frac vu ], \nabla [\frac vu] \Big) - vf(x,v,\nabla v)\Big\}dx$$ 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 $$\label{iii} tg_1(x,t)\leq 0\quad\text{for all }(x,t)\in \mathbb{R}^n \times \mathbb{R},$$ $\overrightarrow{ B} = ( b_1(x) , b_2(x) , \dots ,b_n(x) )$ 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 $$\label{e2.8} L u := \sum_{i j =1}^n \frac{\partial}{\partial x_i} \Big( a_{ij}(x) \frac{\partial}{\partial x_j} \Big)u + c(x) u =0 \quad \text{in } \mathbb{R}^n$$ 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 $$\label{e2.9} \big( p(r)y' \big)' + q(r)y =0 \quad \text{in } \mathbb{R}_+ \,.$$ 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 $$\label{e3.1} \ell u := \sum_{i j =1}^n \frac{\partial}{\partial x_i} \Big( a_{ij}(x) \frac{\partial}{\partial x_j} \Big)u + f(x,u,\nabla u) + c(x)u=0 \quad \text{in } G,$$ 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. 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