\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 $$\label{e1.1} \ell u := \sum_{i j =1}^n \frac{\partial}{\partial x_i} \Big( a_{ij}(x) \frac{\partial}{\partial x_j} \Big)u + f(x,u) + c(x) u \,.$$ 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 $w\neq 0$, \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} 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)}u - \frac{f(x,w)}w \big\} \,; \end{aligned} also for $\lambda \neq 0$ if $\ell u=0$, then $$\label{e2.1iii} \ell (\lambda u) = f(x, \lambda u) - \lambda f(x,u) \,.$$ \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 $$\label{e2.2} \ell v = \sum_{i j =1}^n D_i\big( a_{ij}(x) D_j \big)v + c(x) v =0 \quad \text{in } G \,; \quad v\big|_{\partial G}=0,$$ then \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{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 \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{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 \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} 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 $$\label{e2.7} 0= \int_G \Big[ v^2 a\Big( \nabla[\frac uv ], \nabla [\frac uv] \Big)+ u^2 \frac{f(x,v)}v \Big]dx$$ 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 $$\label{e2.8} L u := \sum_{i j =1}^n \frac{\partial}{\partial x_i} \Big( a_{ij}(x) \frac{\partial}{\partial x_j} \Big)u + c(x) u =0 \quad \text{in } \mathbb{R}^n$$ 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 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 $$\label{e3.1i} L u := \sum_{i j =1}^n \frac{\partial}{\partial x_i} \Big( a_{ij}(x) \frac{\partial}{\partial x_j} \Big)u + c(x)u=0 \quad \text{in } G$$ with either $$\label{e3.1ii} u\big|_{\partial G} =0 \,; \quad u>0 \quad \text{in }G$$ or $$\label{e3.1iii} \nabla u|_{\partial G}=0 \,; \quad u>0 \quad \text{in } G .$$ 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 $$\label{3.2i} \ell u := \sum_{i j =1}^n \frac{\partial}{\partial x_i} \Big( a_{ij}(x) \frac{\partial}{\partial x_j} \Big)u + f(x,u) +c(x)u=0 \quad \text{in } G$$ with either $$\label{3.2ii} u\big|_{\partial G} =0 \,; \quad u>0 \quad \text{in }G$$ or $$\label{3.2iii} \nabla u|_{\partial G}=0 \,; \quad u>0 \text{ in } G .$$ (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 $G_v:=\{ x\in G : u(x)1$ such that for all $(\lambda , x , t) \in (\lambda_0 , \infty)\times G\times (0, \infty)$, $$\label{e3.5} \lambda f(x,t) - f(x,\lambda t) >0 \,.$$ 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}. \begin{thebibliography}{00} \bibitem{j1} J. Jaros, T. Kusano \& N. Yosida; Picone-type Inequalities for Nonlinear Elliptic Equations and their Applications \emph{J. of Inequal. \& Appl.} (2001), vol. 6, 387-404 . \bibitem{k1} K. Kreith; \emph{Piconne's identity and generalizations}, Rend. Mat., Vol. 8 (1975), 251-261. \bibitem{k2} T. Kusano, J. Jaros, N. Yoshida; \emph{A Picone-type identity and Sturmian comparison and oscillation theorems for a class of half-linear partial differential equations of second order}, Nonlinear Analysis, Vol. 40 (2000), 381-395. \bibitem{o1} M. Otani; \emph{Existence and nonexistence of nontrivial solutions of some nonlinear degenerate elliptic equations}, J. Functional Anal., Vol. 76 (1988), 140-159. \bibitem{p1} M. Picone; \emph{Sui valori eccezionali di un parametro da cui dipende una equazione differenziale lineare ordinaria del secondo ordine}, Ann. Scuola Norm. Pisa, Vol. 11 (1910), 1-141. \bibitem{s1} S. Sakaguchi; \emph{Concavity properties of solutions to some degenerate quasilinear elliptic Dirichlet problems}, Ann. Scuola Norm. Sup. Pisa (1987), 404-421. \bibitem{s2} C. A. Swanson; \emph{ A dichotomy of PDE Sturmian theory}, SIAM Reviews vol. 20, no. 2 (1978), 285-300. \bibitem{t1} Tadi\'e; \emph{Comparison Results for Quasilinear Elliptic equations via Picone-type Identity: Part I: Quasilinear Cases}, in print in Nonlinear Analysis (10;1016/J.na.2008.11073) \bibitem{t2} Tadi\'e; \emph{Uniqueness results for decaying solutions of semilinear $P$-Laplacian}, Int. J. Appl. Math., vol. 2, no. 10 (2000), 1143-1152. \bibitem{t3} Tadi\'e; \emph{On Uniqueness Conditions for Decreasing Solutions of Semilinear Elliptic Equations }, Zeitschrift Anal. und ihre Anwendungen vol. 18, no. 3 (1999), 517-523. \bibitem{t4} Tadi\'e; \emph{Uniqueness results for some boundary value elliptic problems via convexity }, Int. J. Diff. Equ. Appl., vol. 2, no. 1 (2001), 47-53. \bibitem{t5} Tadi\'e; \emph{Sturmian comparison results for quasilinear elliptic equations in $\mathbb{R}^n$}, Electronic J. of Differential Equations vol. 2007, no. 26 (2007), 1-8. \end{thebibliography} \end{document}