\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2011 (2011), No. 152, 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 2011 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2011/152\hfil Nonexistence of radial positive solutions] {Nonexistence of radial positive solutions for a nonpositone system in an annulus} \author[S. Hakimi \hfil EJDE-2011/152\hfilneg] {Said Hakimi} \address{Said Hakimi \newline Universit\'e Abdelmalek Essaadi\\ Facult\'e des sciences \\ D\'epartement de Math\'ematiques \\ BP 2121, T\'etouan, Morocco} \email{h\_saidhakimi@yahoo.fr} \thanks{Submitted May 2, 2011. Published November 10, 2011.} \subjclass[2000]{35J25, 34B18} \keywords{Nonpositone problem; radial positive solutions} \begin{abstract} In this article we study the nonexistence of radial positive solutions for a nonpositone system in an annulus by using energy analysis and comparison methods. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \allowdisplaybreaks \section{Introduction} We study the nonexistence of radial positive solutions for the system $$\begin{gathered} -\Delta u(x)=\lambda f(v(x)),\quad x\in \Omega \\ -\Delta v(x)=\mu g(u(x)),\quad x\in \Omega \\ u(x)=v(x)=0,\quad x\in \partial \Omega, \end{gathered} \label{eq1}$$ where $\lambda$, $\mu \geq \varepsilon _0>0$, $\Omega$ is an annulus in $\mathbb{R}^N$: $\Omega =C(0,R,\widehat{R}) =\{x\in \mathbb{R}^N: R<|x| <\widehat{R}\}$, ($0\sigma$. \end{theorem} \noindent\textbf{Remark.} Existence result for positive solutions with superlinearities satisfying (C1), $\lambda=\mu$ and $\lambda$ small can be found in \cite{h1, h2}. Existence results, for the single equation case can be found in \cite{a1,c1,h4}, and non-existence results in \cite{a1,b1,h5}. To prove Theorem \ref{thm2.1}, we need the next three lemmas. Here we use ideas adapted from Hai, Oruganti and Shivaji \cite{h3}. \begin{lemma} \label{lem2.2} There exists a positive constant $C$ such that for $\lambda \mu$ large, \begin{equation*} u(R_0)+v(R_0)\leq C, \end{equation*} where $R_0=(R+\widehat{R})/2$. \end{lemma} \begin{proof} Multiplying the first equation in \eqref{eq2} by a positive eigenfunction, say $\phi$ corresponding to $\lambda _1$, and using (C1) we obtain \begin{equation*} -\int_R^{\widehat{R}}(r^{N-1}u')'\phi dr \geq\int_R^{\widehat{R}}\lambda (a_1v-b_1)\phi r^{N-1}dr; \end{equation*} that is, $$\int_R^{\widehat{R}} \lambda_1 u r^{N-1}\phi dr \geq\int_R^{\widehat{R}}\lambda (a_1v-b_1)\phi r^{N-1}dr. \label{eq3}$$ Similarly, using the second equation in \eqref{eq2} and (C2), we obtain $$\int_R^{\widehat{R}} \lambda_1 v r^{N-1}\phi dr \geq\int_R^{\widehat{R}}\mu (a_2u-b_2)\phi r^{N-1}dr. \label{eq4}$$ Combining \eqref{eq3} and \eqref{eq4}, we obtain $$\int_R^{\widehat{R}}[ \lambda _1-\lambda \mu \frac{a_1a_2}{\lambda _1}] v\Phi r^{N-1}dr\geq \int_R^{\widehat{R}}\mu [ -\lambda \frac{a_2b_1}{\lambda _1}-b_2] \Phi r^{N-1}dr.$$ Now, if $\lambda \mu a_1a_2/2 \geq \lambda _1^2$, then $$\int_R^{\widehat{R}}\mu [ -\lambda a_2b_1-b_2\lambda _1] \Phi r^{N-1}dr\leq \int_R^{\widehat{R}}-\frac{\lambda \mu }2a_1a_2v\Phi r^{N-1}dr;$$ that is, $$\int_R^{\widehat{R}}\frac{a_1a_2}2v\Phi r^{N-1}dr\leq \int_R^{\widehat{R}}[ a_2b_1+\frac{b_2\lambda _1}{\varepsilon _0}] \Phi r^{N-1}dr, \label{eq5}$$ (because $\lambda \geq \varepsilon _0$). Similarly $$\int_R^{\widehat{R}}\frac{a_1a_2}2u\Phi r^{N-1}dr\leq \int_R^{\widehat{R}}[ a_1b_2+\frac{b_1\lambda _1}{\varepsilon _0}] \Phi r^{N-1}dr. \label{eq6}$$ Adding \eqref{eq5} and \eqref{eq6}, we obtain the inequality $$\int_R^{\widehat{R}}(u+v)\Phi r^{N-1}dr\leq \frac{2}{a_1a_2}\int_R^{\widehat{R}}[ a_1b_2+\frac{b_1\lambda _1}{\varepsilon _0}+a_2b_1+\frac{b_2\lambda _1}{\varepsilon _0}] \Phi r^{N-1}dr.\\$$ Then \begin{align*} (u+v)(R_0)\int_{\overline{t}}^{R_0}\Phi r^{N-1}dr &\leq \int_{\overline{t}}^{R_0}(u+v)\Phi r^{N-1}dr\\ &\leq \int_{R}^{\widehat {R}}(u+v)\Phi r^{N-1}dr\\ &\leq \frac{2}{a_1a_2}\int_R^{\widehat{R}}[ a_1b_2+\frac{b_1\lambda _1}{\varepsilon _0}+a_2b_1+\frac{b_2\lambda _1}{\varepsilon _0}] \Phi r^{N-1}dr, \end{align*} where $\overline{t}=\max (\overline{t}_1,\overline{t}_2)$ with $\overline{t}_1$ and $\overline{t}_2$ are such that $\overline{t}_1=\max \{ r\in (R,\widehat{R}) :u'(r)=0\},\quad \overline{t}_2=\max \{ r\in (R,\widehat{R}): v'(r)=0\}.$ The proof is complete. \end{proof} We remark that $\overline{t}_i\leq R_0$, for $i=1,2$, was shown in \cite{g1}. Now, assume that there exists $z\geq 0$ ($z\not\equiv 0$) on $\overline{I}$ where $I=(\alpha,\beta)$, and a constant $\gamma$ such that $$-(r^{N-1}z')'\geq \gamma r^{N-1}z\,,\quad r\in I. \label{eq7}$$ Let $\lambda _1=\lambda _1(I)>0$ denote the principal eigenvalue of $$\begin{gathered} -(r^{N-1}\Psi')'=\lambda r^{N-1}\Psi,\quad r\in (\alpha,\beta)\\ \Psi(\alpha)=0=\Psi (\beta), \end{gathered} \label{eq8}$$ where $0<\alpha<\beta\leq 1$. \begin{lemma} \label{lem2.3} Let \eqref{eq7} hold. Then $\gamma \leq \lambda _1(I)$. \end{lemma} \begin{proof} Multiplying \eqref{eq7} by $\Psi$ ($\Phi>0$), an eigenfunction corresponding to the principal eigenvalue $\lambda _1(I)$, and integrating by parts (twice) we obtain $$\int_\alpha^\beta[ \gamma -\lambda _1(I)] r^{N-1}z\Psi dr\leq \beta^{N-1}\Psi '(\beta)z(\beta)-\alpha^{N-1}\Psi'(\alpha)z(\alpha). \label{eq9}$$ However, $\Psi'(\beta)<0$ and $\Psi'(\alpha)>0$; hence the right-hand side of \eqref{eq9} is less than or equal to zero. Then $\gamma\leq\lambda _1(I)$, and the proof is complete. \end{proof} Now, we define $R_1=R_0+\frac{\widehat{R}-R_0}3, \quad R_2=R_0+\frac{2(\widehat{R}-R_0)}3.$ \begin{lemma}\label{lem2.4} For $\lambda \mu$ sufficiently large, $u(R_2)\leq \beta _2$ or $v(R_2)\leq \beta _1$, where $\beta _1$ and $\beta _2$ are the unique positive zeros of $f$ and $g$ respectively. \end{lemma} \begin{proof} We argue by contradiction. Suppose that $u(R_2)>\beta _2$ and $v(R_2)>\beta _1$. \noindent\textbf{Case 1:} $u(R_1)>\rho _2$ or $v(R_1)>\rho _1$, where $\rho _1=\frac{\beta_1 +\theta_1}2$ and $\rho _2=\frac{\beta_2 +\theta_2}2$ ($\theta_1$ and $\theta_2$ are the unique zeros of $F$ and $G$ respectively where $F(x)=\int_0^xf(t)dt$ and $G(x)=\int_0^xg(t)dt$). If $u(R_1)>\rho _2$ then $-(r^{N-1}v')' =\mu r^{N-1}g(u) \geq \varepsilon _0r^{N-1}g(\rho _2)\quad\text{in }J=(R_0,R_1)$ and $v(r)\geq \beta _1$ on $\bar{J}$. Let $\omega$ be the unique solution of \begin{gather*} -(r^{N-1}\omega')' = \varepsilon _0r^{N-1}g(\rho _2)\quad \text{in }J\\ \omega =\beta _1\quad \text{in }\partial J. \end{gather*} Then by comparison arguments, $v(r)\geq \omega (r)=\varepsilon _0g(\rho _2)\omega _0(r)+\beta _1$ on $\bar{J}$, where $\omega _0$ is the unique (positive) solution of \begin{gather*} -(r^{N-1}\omega _0')' =r^{N-1}\quad\text{in }J \\ \omega _0 = 0\quad \text{on }\partial J. \end{gather*} In particular, there exists $\overline{\beta }_1>\beta _1$ (we choose $\overline{\beta }_1$ such that $f(\overline{\beta }_1)\neq 0$) such that $v(R_0+\frac{2(R_1-R_0)}3) \geq \omega (R_0+\frac{2(R_1-R_0)}3) \geq \overline{\beta }_1$ in $J^{*}=(R_0+\frac{R_1-R_0}3,R_0+\frac{2(R_1-R_0)}3)$. Then \begin{align*} -(r^{N-1}(u-\beta _2)')' &=\lambda r^{N-1}f(v)\\ &\geq \lambda r^{N-1}f(\overline{\beta }_1)\\ &\geq (\frac{\lambda f(\overline{\beta }_1)}C) r^{N-1}(u-\beta _2)\quad \text{on }J^{*}, \end{align*} (where $C$ is as in Lemma \ref{lem2.2}). Since $u-\beta _2>0$ on $\bar{J}^*$, it follows that $$\frac{\lambda f(\overline{\beta }_1)}C\leq \lambda _1(J^{*}), \label{eq10}$$ where $\lambda _1(J^{*})$ is the principal value of \eqref{eq8} (with $(\alpha,\beta)=J^{*}$). Next we consider \begin{align*} (r^{N-1}(v-\beta _1)')' &= \mu r^{N-1}g(u)\\ &\geq \mu r^{N-1}g(\rho _2)\\ &\geq (\frac{\mu g(\rho _2)}C) r^{N-1}(v-\beta _1)\quad \text{on }J. \end{align*} Since $v-\beta _1>0$ on $\bar{J}$, it follows that $$\frac{\mu g(\rho _2)}C\leq \lambda _1(J), \label{eq11}$$ where $\lambda _1(J)$ is the principal value of \eqref{eq8} (with $(\alpha,\beta)=J$). Combining \eqref{eq10} and \eqref{eq11}, we obtain $$\frac{\lambda \mu f(\overline{\beta }_1)g(\rho _2)}{C^2}\leq \lambda _1(J^{*})\lambda _1(J),$$ But $f(\overline{\beta }_1)$, $g(\rho _2)$ and $C$ are fixed positive constants. This is a contradiction for $\lambda \mu$ large. A similar contradiction can be reached for the case $v(R_1)>\rho _1$. \noindent\textbf{Case 2:} $u(R_1)\leq \rho _2$ and $v(R_1)\leq \rho _1$. Then $\beta _20$ on $\bar{J}_4$, we have $$\frac{\lambda \widetilde{K}_1}{\beta _2}\leq \lambda _1(J_4), \label{eq15}$$ where $\widetilde{K}_1=-f(\beta _1/2)$ and $\lambda_1(J_4)$ is the principal eigenvalue of \eqref{eq8} (with $(\alpha,\beta)=J_4$). Similarly, there exists $r_2\in (r_1,\widehat{R})$ (independent of $\lambda \mu$) such that $$v(r_2)<\frac{\beta _1}2.$$ Hence \begin{gather*} -(r^{N-1}u')' = \mu r^{N-1}f(v)\leq 0\quad \text{on } J_5=(r_2,\widehat{R})\\ u(r_2) \leq C,\quad u(\widehat{R})=0, \end{gather*} then, by a comparison argument we obtain $$u(r)\leq \omega _1(r)=\frac C{\int_{r_2}^{\widehat{R}}s^{1-N}ds}\int_r^{\widehat{R}}s^{1-N}ds;$$ thus \begin{gather*} -(r^{N-1}\omega _1')'=0,\quad \text{on }J_5,\\ \omega_1(r_2)=C,\quad \omega _1(\widehat{R})=0. \end{gather*} Arguing as before there exists $r_3\in (r_2,\widehat{R})$ (independent of $\lambda \mu$) such that $$u(r_3)\leq \omega _1(r_3)\leq \frac{\beta _2}20 on \bar{J}_6, it follows that $$\frac{\mu \widetilde{K}_2}{\beta _1}\leq \lambda_1(J_6),\label{eq16}$$ where \widetilde{K}_2=-g(\beta _1/2) and \lambda_1(J_6) is the principal eigenvalue of \eqref{eq8} (with (\alpha,\beta)=J_6). Combining \eqref{eq15} and \eqref{eq16}, we obtain$$ \frac{\lambda \mu \widetilde{K}_1\widetilde{K}_2}{\beta _1\beta _2}\leq \lambda _1(J_4)\lambda _1(J_6),  which is a contradiction to $\lambda \mu$ being large. A similar contradiction can be reached for the case $v(R_2)\leq \beta_1$. Hence Theorem \ref{thm2.1} is proven. \end{proof} \begin{thebibliography}{00} \bibitem{a1} D. Arcoya and A. Zertiti; \emph{Existence and non-existence of radially symmetric non-negative solutions for a class of semi-positone problems in annulus}, Rendiconti di Mathematica, serie VII, Volume 14, Roma (1994), 625-646. \bibitem{b1} K .J. Brown, A. Castro and R. Shivaji; \emph{Non-existence of radially symmetric non-negative solutions for a class of semi-positone problems}, Diff. and Int. Equations,2. (1989), 541-545. \bibitem{c1} A. Castro and R. Shivaji; \emph{Nonnegative solutions for a class of radially symmetric nonpositone problems}, Proc. AMS, 106(3) (1989), pp. 735-740. \bibitem{g1} B. Gidas, W. M. 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