\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2016 (2016), No. 326, pp. 1--5.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu} \thanks{\copyright 2016 Texas State University.} \vspace{8mm}} \begin{document} \title[\hfilneg EJDE-2016/326\hfil Nonexistence of stable solutions] {Nonexistence of stable solutions to $p$-Laplace equations with exponential nonlinearities} \author[P. Le \hfil EJDE-2016/326\hfilneg] {Phuong Le} \address{Phuong Le \newline Department of Economic Mathematics, Banking University of Ho Chi Minh City, Vietnam} \email{phuongl@buh.edu.vn} \thanks{Submitted November 3, 2016. Published December 22, 2016.} \subjclass[2010]{35A01, 35B06, 35B35, 35J92} \keywords{$p$-Laplace equations; stable solutions; exponential nonlinearity; \hfill\break\indent nonexistence} \begin{abstract} In this note we prove the nonexistence of stable solutions to the $p$-Laplace equation $-\Delta_p u = e^u$ on the entire Euclidean space $\mathbb{R}^N$, where $p>2$ and $N < \frac{p(p+3)}{p-1}$. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \newtheorem{definition}[theorem]{Definition} \allowdisplaybreaks \section{Introduction and statement of main results} We consider the $p$-Laplace equation \begin{equation}\label{problem:main} -\Delta_p u = e^u \quad \text{in } \mathbb{R}^N, \end{equation} where $p>2$ and $\Delta_p u = \operatorname{div}(|\nabla u|^{p-2}\nabla u)$ is the usual $p$-Laplace operator. We recall that $u\in C^1(\mathbb{R}^N)$ is said to be a weak solution of \eqref{problem:main} if \begin{equation}\label{definition:solution} \int_{\mathbb{R}^N} |\nabla u|^{p-2} ( \nabla u, \nabla\varphi ) dx = \int_{\mathbb{R}^N} e^u\varphi dx \end{equation} for every $\varphi \in C_c^1(\mathbb{R}^N)$. This article concern the stable solutions of \eqref{problem:main} in the following sense. \begin{definition}\label{def1.1} \rm A weak solution $u$ of \eqref{problem:main} is stable if \begin{equation*} \int_{\mathbb{R}^N} |\nabla u|^{p-2} |\nabla\varphi|^2 dx + (p-2) \int_{\mathbb{R}^N} |\nabla u|^{p-4} (\nabla u, \nabla\varphi)^2 dx - \int_{\mathbb{R}^N} e^u\varphi^2 dx \ge 0 \end{equation*} for every $\varphi \in C_c^1(\mathbb{R}^N)$. \end{definition} Note that the above expression is nothing but the second that the variation of the energy functional associated with \eqref{problem:main} is non-negative. Thus, if $u \in C^1(\mathbb{R}^N)$ is a local minimizer of the energy functional, then $u$ is a stable solution of \eqref{problem:main}. \begin{remark} \label{rmk1.2} \rm Let $u$ be a stable solution of \eqref{problem:main}. Then \begin{equation}\label{definition:stable} \int_{\mathbb{R}^N} e^u\varphi^2 dx \le (p-1) \int_{\mathbb{R}^N} |\nabla u|^{p-2} |\nabla\varphi|^2 dx \end{equation} for every $\varphi \in C_c^1(\mathbb{R}^N)$. \end{remark} The nonexistence and stability of solutions to nonlinear elliptic partial differential equations have drawn much attention in the last decades. Readers can find recent developments on stable solutions in the monograph \cite{LD11} by Dupaigne, and on related problems in \cite{HBTCYMAR96,DCMS15,LDAF10,PMVR96}. We should mention here the results in \cite{AF05,AF07_2} for Lane-Emden-Fowler equation $-\Delta u = |u|^{m-1}u$ where it is proved that there is no nontrivial stable solution if $1 0$ and a constant $C = C(p,\alpha) > 0$ such that for any function $\eta\in C_c^1(\mathbb{R}^N)$ with $0\le\eta\le1$ we have \begin{equation}\label{s3} \int_{\mathbb{R}^N} e^{(p\alpha + 1) u} \eta^{pm} \,dx \le C\int_{\mathbb{R}^N} |\nabla\eta|^{p(p\alpha + 1)} \,dx. \end{equation} \end{theorem} The method of proof is inspired by the techniques developed in \cite{LDAFBSEV09,NDAF07,AF07}. Our main result is the following theorem, which is a generalization of the Theorem \ref{thm:AF07}. \begin{theorem}\label{thm:instability} For $N < \frac{p(p+3)}{p-1}$, there is no stable $C^1$ solution of \eqref{problem:main}. \end{theorem} Theorem \ref{thm:instability} is sharp when $p=2$ as already pointed out in \cite{AF07}. However, the optimality of the dimension $N$ in terms of $p$ is still an interesting open question for $p>2$. \subsection*{Open problem} For $N \ge \frac{p(p+3)}{p-1}$, does equation \eqref{problem:main} admit a stable $C^1$ solution? As far as we know, there is no result on nonexistence of stable solutions for \eqref{problem:main} on the case $p<2$. Hence, this case should be also an interesting topic for future research. \section{Proofs} In the sequel, we denote by $C$ a generic constant whose concrete values may change from line to line or even in the same line. If this constant depends on an arbitrary small number $\varepsilon$, then we will denote it by $C_\varepsilon$. We also use Young inequality in the form $ab \le \varepsilon a^p + C_\varepsilon b^q$ for $p,q>0$ satisfying $\frac{1}{p} + \frac{1}{q} = 1$. \begin{proof}[Proof of Theorem \ref{thm:estimate}] We split the proof into two steps. \smallskip \noindent\textbf{Step 1.} For any $\varepsilon \in (0, p\alpha)$ and for any nonnegative function $\psi\in C_c^1(\mathbb{R}^N)$, there exists a constant $C_\varepsilon = C(p,\varepsilon) > 0$ such that \begin{equation}\label{s1} (p\alpha-\varepsilon)\int_{\mathbb{R}^N} |\nabla u|^p e^{p\alpha u} \psi^p \,dx \le C_\varepsilon\int_{\mathbb{R}^N} e^{p\alpha u} |\nabla\psi|^p \,dx + \int_{\mathbb{R}^N} e^{(p\alpha + 1) u} \psi^p \,dx. \end{equation} To prove this, using $\varphi=e^{p\alpha u}\psi^p$ as a test function in \ref{definition:solution}, since $$ \nabla\varphi = p\alpha e^{p\alpha u}\psi^p\nabla u + pe^{p\alpha u}\psi^{p-1}\nabla\psi, $$ we obtain $$ p\alpha\int_{\mathbb{R}^N} |\nabla u|^p e^{p\alpha u} \psi^p \,dx + p\int_{\mathbb{R}^N} |\nabla u|^{p-2} e^{p\alpha u} \psi^{p-1} (\nabla u, \nabla\psi) \,dx = \int_{\mathbb{R}^N} e^{(p\alpha + 1) u} \psi^p \,dx. $$ Therefore, \begin{align*} &p\alpha\int_{\mathbb{R}^N} |\nabla u|^p e^{p\alpha u} \psi^p \,dx\\ &\le p\int_{\mathbb{R}^N} |\nabla u|^{p-1} e^{p\alpha u} \psi^{p-1} |\nabla\psi| \,dx + \int_{\mathbb{R}^N} e^{(p\alpha + 1) u} \psi^p \,dx\\ &\le \int_{\mathbb{R}^N} \varepsilon \Big(|\nabla u|^{p-1} e^{(p-1)\alpha u} \psi^{p-1} \Big)^{\frac{p}{p-1}} + C_\varepsilon\big(e^{\alpha u} |\nabla\psi| \big)^p \,dx + \int_{\mathbb{R}^N} e^{(p\alpha + 1) u} \psi^p \,dx\\ &= \varepsilon\int_{\mathbb{R}^N} |\nabla u|^p e^{p\alpha u} \psi^p \,dx + C_\varepsilon\int_{\mathbb{R}^N} e^{p\alpha u} |\nabla\psi|^p \,dx + \int_{\mathbb{R}^N} e^{(p\alpha + 1) u} \psi^p \,dx, \end{align*} which implies \eqref{s1}. \smallskip \noindent\textbf{Step 2.} For any $\varepsilon \in (0, p\alpha)$, we set $$ \beta_\varepsilon = 1 - \Big(\frac{(p-1)p^2\alpha^2}{4}+\varepsilon\Big) \frac{1}{p\alpha-\varepsilon} $$ and we claim that there exists a constant $C_\varepsilon = C(p,\varepsilon) > 0$ such that \begin{equation}\label{s2} \beta_\varepsilon\int_{\mathbb{R}^N} e^{(p\alpha + 1) u} \psi^p \,dx \le C_\varepsilon\int_{\mathbb{R}^N} e^{p\alpha u} |\nabla\psi|^p \,dx. \end{equation} To prove this, we use the stability assumption with $\varphi = e^{\frac{p\alpha u}{2}} \psi^{\frac{p}{2}}$. Since $$ \nabla\varphi = \frac{p\alpha}{2} e^{\frac{p\alpha u}{2}} \psi^{\frac{p}{2}} \nabla u + \frac{p}{2}e^{\frac{p\alpha u}{2}} \psi^{\frac{p-2}{2}} \nabla\psi, $$ using \eqref{definition:stable} we obtain \begin{align*}% \label{s2.1} \int_{\mathbb{R}^N} e^{(p\alpha + 1) u} \psi^p \,dx &\le (p-1)\int_{\mathbb{R}^N} |\nabla u|^p \left(\frac{p\alpha}{2}\right)^2 e^{p\alpha u} \psi^p \,dx\\ &\quad + (p-1)\int_{\mathbb{R}^N} |\nabla u|^{p-1} \frac{p^2\alpha}{2} e^{p\alpha u} \psi^{p-1} |\nabla\psi| \,dx\\ &\quad + (p-1)\int_{\mathbb{R}^N} |\nabla u|^{p-2} \left(\frac{p}{2}\right)^2 e^{p\alpha u} \psi^{p-2} |\nabla\psi|^2 \,dx. \end{align*} Now we use Young inequality to estimate the last two terms \begin{align*} &(p-1)\int_{\mathbb{R}^N} |\nabla u|^{p-1} \frac{p^2\alpha}{2} e^{p\alpha u} \psi^{p-1} |\nabla\psi| \,dx\\ &\le \int_{\mathbb{R}^N} \frac{\varepsilon}{2} \left(|\nabla u|^{p-1} e^{(p-1)\alpha u} \psi^{p-1}\right)^{\frac{p}{p-1}} + C_\varepsilon\left(e^{\alpha u}|\nabla\psi|\right)^p \,dx\\ &= \frac{\varepsilon}{2}\int_{\mathbb{R}^N} |\nabla u|^p e^{p\alpha u} \psi^p \,dx + C_\varepsilon\int_{\mathbb{R}^N} e^{p\alpha u} |\nabla\psi|^p \,dx, \end{align*} and \begin{align*} &(p-1)\int_{\mathbb{R}^N} |\nabla u|^{p-2} \left(\frac{p}{2}\right)^2 e^{p\alpha u} \psi^{p-2} |\nabla\psi|^2 \,dx\\ &\le \int_{\mathbb{R}^N} \frac{\varepsilon}{2}\left(|\nabla u|^{p-2} e^{(p-2)\alpha u} \psi^{p-2}\right)^{\frac{p}{p-2}} + C_\varepsilon\left(e^{2\alpha u}|\nabla\psi|^2\right)^{\frac{p}{2}} \,dx\\ &= \frac{\varepsilon}{2}\int_{\mathbb{R}^N} |\nabla u|^p e^{p\alpha u} \psi^p \,dx + C_\varepsilon\int_{\mathbb{R}^N} e^{p\alpha u} |\nabla\psi|^p \,dx. \end{align*} Plugging these two estimates into the previous one, we obtain \begin{align*} &\int_{\mathbb{R}^N} e^{(p\alpha + 1) u} \psi^p \,dx \\ &\le \Big(\frac{(p-1)p^2\alpha^2}{4}+\varepsilon\Big) \int_{\mathbb{R}^N} |\nabla u|^p e^{p\alpha u} \psi^p \,dx + C_\varepsilon\int_{\mathbb{R}^N} e^{p\alpha u} |\nabla\psi|^p \,dx\\ &\le \Big(\frac{(p-1)p^2\alpha^2}{4}+\varepsilon\Big) \frac{1}{p\alpha-\varepsilon}\int_{\mathbb{R}^N} e^{(p\alpha + 1) u} \psi^p \,dx + C_\varepsilon\int_{\mathbb{R}^N} e^{p\alpha u} |\nabla\psi|^p \,dx. \end{align*} We have used \eqref{s1} in the last inequality. The claim \eqref{s2} is now proved. We are now in a position to prove the Theorem \ref{thm:estimate}. Since $\lim_{\varepsilon\to 0}\beta_\varepsilon = 1 - \frac{\alpha p(p-1)}{4} > 0$, we can find some $\varepsilon\in(0, 1)$ depending on $p$ and $\alpha$ such that $\beta_\varepsilon > 0$. Next we choose some $m$ large enough satisfying $(m-1)\frac{p\alpha + 1}{\alpha} \ge pm$ and apply \eqref{s2} for $\psi=\eta^m$ to obtain \begin{align*} \int_{\mathbb{R}^N} e^{(p\alpha + 1) u} \eta^{pm} \,dx &\le C\int_{\mathbb{R}^N} e^{p\alpha u} \eta^{p(m-1)} |\nabla\eta|^p \,dx\\ &\le \int_{\mathbb{R}^N} \varepsilon \Big(e^{p\alpha u} \eta^{p(m-1)}\Big)^{\frac{p\alpha+1}{p\alpha}} + C_\varepsilon (|\nabla\eta|^p)^{p\alpha + 1} \,dx\\ &\le \varepsilon \int_{\mathbb{R}^N} e^{(p\alpha + 1) u} \eta^{pm} \,dx + C_\varepsilon \int_{\mathbb{R}^N} |\nabla\eta|^{p(p\alpha + 1)} \,dx. \end{align*} Hence, \eqref{s3} follows. \end{proof} \begin{proof}[Proof of Theorem \ref{thm:instability}] By contradiction, we suppose that \eqref{problem:main} admits a stable solution for $N < \frac{p(p+3)}{p-1}$. Since $$ \lim_{\alpha\to\frac{4}{p(p-1)}} N-p(p\alpha + 1) = N - \frac{p(p+3)}{p-1} < 0, $$ we may find some $\alpha \in \left(0,\frac{4}{p(p-1)}\right)$ such that $N-p(p\alpha + 1) < 0$. We then apply Theorem \ref{thm:estimate} for a test function $\eta_R\in C_c^1(\mathbb{R}^N)$ satisfying $0\le\eta_R\le1$ in $\mathbb{R}^N$, $\eta_R=1$ in $B(0,R)$ and $\eta_R=0$ in $\mathbb{R}^N \setminus B(0,2R)$ to obtain $$ \int_{B(0,R)} e^{(p\alpha + 1) u} \,dx \le C R^{N-p(p\alpha + 1)}. $$ Letting $R\to\infty$ in the last inequality we obtain $\int_{\mathbb{R}^N} e^{(p\alpha + 1) u} \,dx = 0$, a contradiction. This completes the proof. \end{proof} \subsection*{Acknowledgments} The author would like to thank the anonymous referees for their comments that improved this article. \begin{thebibliography}{10} \bibitem {HBTCYMAR96} H. Brezis, T. Cazenave, Y. Martel, A. Ramiandrisoa; \emph{Blow up for $u_t - \Delta u=g(u)$ revisited}, Adv. Differential Equations, \textbf{1} (1996), 73--90. \bibitem {DCPEBS09} D. Castorina, P. Esposito, B. Sciunzi; \emph{Low dimensional instability for semilinear and quasilinear problems in $\mathbb{R}^N$}, Commun. Pure Appl. Anal., \textbf{8} (2009), 1779--1793. \bibitem {DCMS15} D. Castorina, M. Sanch\'{o}n; \emph{Regularity of stable solutions to semilinear elliptic equations on Riemannian models}, Adv. Nonlinear Anal., \textbf{4} (2015), 295--309. \bibitem {LDAFBSEV09} L. Damascelli, A. Farina, B. Sciunzi, Enrico Valdinoci; \emph{Liouville results for $m$-Laplace equations of Lane-Emden-Fowler type}, Ann. Inst. H. Poincar\'{e} Anal. Non Lineaire, \textbf{26} (2009), 1099--1119. \bibitem {NDAF07} N. Dancer, A. Farina; \emph{On the classification of solutions of $-{\Delta} u = e^u$ on $\mathbb{R}^N$: {S}tability outside a compact set and applications}, Proc. Amer. Math. Soc., \textbf{137} (2009), 1333--1338. \bibitem {LD11} L. Dupaigne; \emph{Stable solutions of elliptic partial differential equations}, Chapman and Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, \textbf{143}, Boca Raton, FL, 2011. \bibitem {LDAF10} L. Dupaigne, A. Farina; \emph{Stable solutions of $-\Delta u=f(u)$ in $\mathbb{R}^N$}, J. Eur. Math. Soc., \textbf{12} (2010), 855--882. \bibitem {AF05} A. Farina; \emph{Liouville-type results for solutions of $-{\Delta} u=|u|^{p-1}u$ on unbounded domains of $\mathbb{R}^N$}, C. R. Math. Acad. Sci. Paris, Ser. I, \textbf{341} (2005), 415--418. \bibitem {AF07_2} A. Farina; \emph{On the classification of solutions of the {L}ane-{E}mden equation on unbounded domains of $\mathbb{R}^N$}, J. Math. Pures Appl., \textbf{87} (2007), 537--561. \bibitem {AF07} A. Farina; \emph{Stable solutions of $ −\Delta u = e^u $ on $\mathbb{R}^N$}, Comptes Rendus Mathmatique, \textbf{345} (2007), 63--66. \bibitem {AFAF16} A. Farina, A. Ferrero; \emph{Existence and stability properties of entire solutions to the polyharmonic equation $(-\Delta)^mu=e^u$ for any $m\ge1$}, Ann. Inst. H. Poincar\'e Anal. Non Lineaire, \textbf{33} (2016), 495--528. \bibitem {XHDY16} X. Huang, D. Ye; \emph{Existence of stable solutions to $(-\Delta)^mu=e^u$ in $\mathbb{R}^N$ with $m\ge3$ and $N>2m$}, J. Differential Equations, \textbf{260} (2016), 6493--6503. \bibitem {PMVR96} P. Mironescu, V. Radulescu; \emph{The study of a bifurcation problem associated to an asymptotically linear function}, Nonlinear Anal., \textbf{26} (1996), 857--875. \end{thebibliography} \end{document}