\pdfoutput=1\relax\pdfpagewidth=8.26in\pdfpageheight=11.69in\pdfcompresslevel=9 \documentclass[twoside]{article} \usepackage{amsfonts, amsmath} % used for R in Real numbers \pagestyle{myheadings} \markboth{Some Liouville theorems for the $p$-Laplacian } { Isabeau Birindelli \& Fran\c coise Demengel } \begin{document} \setcounter{page}{35} \title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent 2001-Luminy conference on Quasilinear Elliptic and Parabolic Equations and Systems,\newline Electronic Journal of Differential Equations, Conference 08, 2002, pp 35--46. \newline http://ejde.math.swt.edu or http://ejde.math.unt.edu \newline ftp ejde.math.swt.edu (login: ftp)} \vspace{\bigskipamount} \\ % Some Liouville theorems for the $p$-Laplacian % \thanks{ {\em Mathematics Subject Classifications:} 35J60, 35D05. \hfil\break\indent {\em Key words:} Liouville, $p$-Laplacian. \hfil\break\indent \copyright 2002 Southwest Texas State University. \hfil\break\indent Published Octrober 21, 2002.} } \date{} \author{Isabeau Birindelli \& Fran\c coise Demengel} \maketitle \begin{abstract} In this paper we propose a new proof for non-linear Liouville type results concerning the $p$-Laplacian. Our method differs from the one used by Mitidieri and Pohozaev because it uses a comparison principle that can be applied to nondivergence form operators. \end{abstract} \numberwithin{equation}{section} \newtheorem{thm}{Theorem}[section] \newtheorem{pro}[thm]{Proposition} \newtheorem{lem}[thm]{Lemma} \newtheorem{rema}[thm]{Remark} \section{Introduction} In 1981 Gidas and Spruck proved in their famous work \cite{GS2} that for $10\quad \mbox{in } \mathbb{R}^N. $$The proof is very difficult but a simpler proof was given by Chen and Li using the moving plane method \cite{ChL}. Similarly, non-existence results hold for the inequality$$ \Delta u + u^p \leq 0,\ u>0\quad {\rm in }\ \Sigma $$where \Sigma is a cone in \mathbb{R}^N (see Berestycki, Capuzzo Dolcetta, Nirenberg \cite{BCN2}). The values of p for which there is no positive solution depend on the cone \Sigma. For example for \Sigma=\mathbb{R}^N, p\in (0,\frac{N}{N-2}). The generalization of this result to the p-Laplacian (\Delta_p=\mathop{\rm div}(|\nabla .|^{p-2}\nabla)) is very recent. Mitidieri and Pohozaev proved among other things the following result. \begin{thm}\label{th1} 1) Suppose that N>p>1, and u\in W^{1,p}_{loc} (\mathbb{R}^N)\cap {\cal C} (\mathbb{R}^N) is a nonnegative weak solution of $$\label{eq 0} - \mathop{\rm div} (|\nabla u|^{p-2 }\nabla u) \geq h(x) u^q \quad \mbox{in } \mathbb{R}^N$$ with h satisfying $$\label{eqh} h(x)=a|x|^\gamma\quad \mbox{for |x| large, a>0 and \gamma>-p}.$$ Suppose that$$p-1< q\leq {(N+\gamma)(p-1)\over N-p}\,.$$Then u\equiv 0. \\ 2) Let N\leq p. If u\in W^{1,p}_{loc} (\mathbb{R}^N)\cap {\cal C} (\mathbb{R}^N) is a weak solution of$$-\mathop{\rm div} (|\nabla u|^{p-2 }\nabla u)\geq 0\;\;\mbox{in } \;\mathbb{R}^N $$and u is bounded below then u is constant. \end{thm} In this paper we present a new simple proof of Theorem \ref{th1}. The proof of Mitidieri and Pohozaev relies on variational methods and the use of global test function. On the other hand here we use the notion of viscosity solutions and therefore use local test functions. This kind of technique should allow us to extend Theorem \ref{th1} to a large class of non divergence operators. An example of such operators is given by:$$ Lu= |\nabla u|^{\alpha}\big({\rm Tr}(A(x)D^2u) +k D^2u: \frac{\nabla u}{|\nabla u|}\otimes\frac{\nabla u}{|\nabla u|}\big) $$where \alpha\in \mathbb{R}, and A(x) is a symmetric matrix with$$ \lambda |\xi|^2\leq A\xi\cdot\xi\leq \Lambda |\xi|^2$$and k\in\mathbb{R} satisfies \lambda+k>0. More generally this kind of proof can be used for fully nonlinear equations: Suppose that we consider F(x,\nabla u,D^2u) where for example F(x,\xi,M) satisfies for some \lambda>0 \begin{gather*} |\xi|^\alpha \lambda {\rm Tr} N \leq F(x,\xi,M+N) - F(x,\xi, M)\leq |\xi|^\alpha \Lambda {\rm Tr} N,\\ F(x,\xi, 0)=0 \end{gather*} for any symmetric and positive matrix N. Cutr\i and Leoni \cite{CL} have used similar arguments to study Liouville theorems for fully non-linear operators F(x, D^2u) which satisfy the above inequality for \alpha=0. We would like to remark that the first result of Theorem \ref{th1} is optimal in the sense that for any q>(N+\gamma)(p-1)/(N-p) we construct a nonnegative solution of (\ref{eq 0}). A similar example was given in \cite{BM} when p=2. Let us also remark that the condition on \gamma in (\ref{eqh}) is optimal. Indeed, for \gamma< -p, Dr\'abek in \cite{Dr} has proved the existence of non trivial weak solutions in \mathbb{R}^N (see e.g. Theorem 4.1 of \cite{DKN}). When treating the equation instead of the inequality, the values of q for which non existence results hold true are not the same. Precisely for the following equation $$\label{eq 01} -\Delta_p u = r^\gamma u^q ,\;\;u\geq 0\quad \mbox{in } \mathbb{R}^N,$$ Serrin and Zou have proved in \cite{SZ} that for p-1p our main non-existence result in this section is the following \begin{thm}\label{lnl} Suppose that N>p>1. Let u\in W^{1,p}_{loc} (\mathbb{R}^N)\cap {\cal C} (\mathbb{R}^N) be a nonnegative weak solution of $$\label{eq} -\Delta_p u \geq h(x) u^q \quad\mbox{in } \mathbb{R}^N,$$ with h satisfying (\ref{eqh}). If 0< q\leq {(N+\gamma)(p-1)\over N-p}, then u\equiv 0. \end{thm} The proof is inspired by the one given in \cite{CL}, where the authors treat fully nonlinear strictly elliptic equations. Let us start by one remark and two propositions. \begin{rema} \label{r1} \rm The following comparison result holds true: Let u and \phi satisfy u, \phi\in W^{1,p}(\Omega) \begin{gather*} -\Delta_pu\geq -\Delta_p\phi \quad \mbox{in } \Omega\\ u\geq \phi \quad \mbox{on } \partial\Omega\,. \end{gather*} Then u\geq \phi in \Omega. This is a standard result and it is easy to see for example by multiplying -\Delta_p u+\Delta_p \phi by (\phi-u)^+. \end{rema} \begin{pro}\label{prop10} Let \Omega be an open set in \mathbb{R}^N, and let f\in {\cal C}(\overline{\Omega}). Suppose that u\in W^{1,p}_{\rm loc} (\Omega)\cap {\cal C}(\overline{\Omega}) is a weak solution of -\Delta_p u\geq f in \Omega. If x_0\in \Omega and \varphi\in {\cal C}^2(\Omega)\cap {\cal C} (\overline{\Omega}) are such that$$ \nabla \varphi(x_0)\neq 0,\ u(x_0)- \varphi(x_0) = \inf_{y\in \Omega} u(y)-\varphi(y)\,, $$then -\Delta_p \varphi(x_0)\geq f(x_0). \end{pro} This proof is inspired by Juutinen \cite{Juu}. \paragraph{Proof.} Without loss of generality we can suppose that u(x_0)=\varphi(x_0). Let us note first that it is sufficient to prove that the property holds for every \varphi such that \varphi(y)< u(y) for all y\neq x_0 in a sufficiently small neighborhood of x_0. Indeed, suppose that the property holds for such functions then taking \varphi_\epsilon (y) = \varphi(y)-\epsilon {|y-x_0|^4} and letting \epsilon go to zero, one obtains the result for every \varphi. Suppose by contradiction that there exists some x_0\in \Omega and some {\cal C}^2 function \varphi such that \nabla \varphi(x_0)\neq 0, \varphi(x_0)= u(x_0) and \varphi(y)< u(y) on some ball B(x_0,r)\setminus \{x_0\} and -\Delta_p \varphi(x_0)0\}, and define$$\bar\varphi = \varphi+{m\over 2}. $$One has -\Delta_p \bar\varphi u(x_0). This ends the proof of Proposition \ref{prop10}. \hfill\square \smallskip Finally let us recall that if v is radial i.e. v(x)=V(|x|)\equiv V(r) for some function V in {\cal C}^2, then if x is such that V' (|x|)\neq 0,$$ \Delta_p v(x) =|V'(r)|^{p-2}\big((p-1)V^{\prime\prime}(r)+{N-1\over r}V'(r)\big). $$Hence for any constants C_1 and C_2 if N\neq p and for \lambda={p-N\over p-1} the function \phi(x)=C_2|x|^{\lambda}+ C_1 satisfies \Delta_p \phi=0 for x\neq 0. Before giving the proof of Theorem \ref{lnl} let us define m(r)=\inf_{x\in B_r}u(x) and prove the following Hadamard type inequality \begin{pro} \label{p1} Let N\neq p. Suppose that -\Delta_p u\geq 0 and u\geq 0. Let \lambda = {p-N\over p-1}. For any 00 and for i=1 and i=2, u(x)\geq m(r_i)=\phi(r_i) for x\in \partial B_{r_i}, hence u and \phi satisfy the conditions of Remark \ref{r1}. and u(x)\geq \phi(|x|) in B_{r_2}\setminus B_{r_1}. Taking the infimum we obtain that \inf_{|x|=r} u(x)\geq \phi(r) for r\in [r_1,r_2]. By the minimum principle m(r)=\inf_{|x|=r} u(x). This completes the proof of the first part of proposition \ref{p1}. For N=p consider$$ \psi(r)= {m(r_1)\log({r\over r_2})+m(r_2)\log({r_1\over r})\over\log({ r_1\over r_2})}. $$Remark that \Delta_N\psi=0 and \psi(r_1)=m(r_1) and \psi(r_2)=m(r_2). Now proceed as above. \begin{rema}\label{r3} \rm Clearly if \lambda<0 i.e. p< N, then g(r):=m(r)r^{-\lambda} is an increasing function. Just observe that r_1^\lambda-r^\lambda\geq 0 and let r_2 tend to + \infty in (\ref{i2}) and one obtains for r\geq r_1:$$ m(r)\geq { m(r_1)r^\lambda\over r_1^\lambda}. $$\end{rema} \paragraph{Proof of Theorem \ref{lnl}.} We suppose by contradiction that u\not\equiv 0 in \mathbb{R}^n, but since u\geq 0 by the strict maximum principle of Vasquez \cite{V} we get that u>0. Let 0 m(r_1) =\zeta(x) while for |x|\geq R, \zeta(x)\leq 0 0 and hence, for |x|=r_1, -\Delta_p\zeta (x)=0 while, of course, \nabla \zeta (x)=0. Now we have two cases: First case \bar r=r_1. This implies$$ u(\bar x)-m(r_1)= u(\bar x)-\zeta(\bar x)\leq u(x)-\zeta(x) $$for all x. In particular choosing x=\tilde x, one gets$$ u(\bar x)-m(r_1)\leq u(\tilde x)-\zeta(\tilde x)=0. $$Finally u(\bar x)= m(r_1) and \bar x is a minimum for u on B(0,r_1). Since -\Delta_p u\geq 0, Hopf's principle as stated in Vasquez \cite{V} implies that \nabla u(\bar x)\neq 0. On the other hand \nabla u (\bar x)= \nabla \zeta (\bar x)= 0, a contradiction. \noindent Second case: r_1< \bar r< R. Now \nabla \zeta (\bar x)\neq 0, and using Proposition \ref{prop10} one has$$ h(\bar x)u^q(\bar x)\leq -\Delta_p \zeta (\bar x). $$We choose r_1 and R sufficiently large in order that h(x) = a|x|^\gamma for |x|\geq \min (r_1, R/2). Combining this with (\ref{i3}), we obtain$$ a\bar r^\gamma m(\bar r)^q\leq a\bar r^\gamma u^q(\bar x)\leq (k+1)^{(p-1)}(N+2p-3)m(r_1)^{(p-1)}(R-r_1)^{-p}. $$Since m is decreasing we have obtained for some constant C>0$$ m(R)\leq C m(r_1)^{(p-1)\over q}\bar r^{-\gamma\over q}(R-r_1)^{-p\over q}. $$Now we choose r_1={R\over 2}, we use Remark \ref{r3} and the previous inequality becomes $$\label{eR}m(R)\leq C m(R)^{(p-1)\over q}R^{-p-\gamma\over q}.$$ First we will suppose that q\leq p-1; hence, using the monotonicity of m(R), the above inequality becomes$$ R^{p+\gamma\over q} \leq Cm(R)^{{(p-1)\over q}-1} \leq Cu(0)^{{(p-1)\over q}-1}. $$But this is absurd since the left hand side tends to infinity when R does. This conclude the proof of this case. Now suppose that q>p-1, then (\ref{eR}) becomes $$\label{i5} m(R)R^{-\lambda }\leq CR^{-\lambda-{p+\gamma\over q-p+1}}.$$ Clearly -\lambda-{p+\gamma\over q-p+1}={N-p\over p-1}-{p+\gamma \over q-p+1}\leq 0 when q\leq {(N+\gamma) (p-1)\over N-p}. If q< (N+\gamma) (p-1)/(N-p) we have reached a contradiction since the right hand side of (\ref{i5}) tends to zero for R\rightarrow +\infty while the left hand side is an increasing positive function as seen in Remark \ref{r3}. We now treat the case q=(N+\gamma)(p-1)/(N-p). Let us remark that for this choice of q we have that for some C_1>0, c>0 and r>r_1>0, with r_1 large enough: $$\label{if1} -\Delta_p u\geq ar^\gamma u^q\geq C_1r^{-N}\quad\mbox{since} \quad m(r)\geq c r^{p-N\over p-1}.$$ We choose \psi(x) = g(|x|) with$$ g(r) = \gamma_1 r^{p-N\over p-1} \log^\beta r+\gamma_2 $$where \gamma_1 and \gamma_2 are two positive constants such that for some r_1>1 and some r_2>r_1:$$ m(r_2) = g(r_2),\quad m(r_1)\geq g(r_1), $$while \beta is a positive constant to be chosen later. It is easy to see that \begin{eqnarray*} \Delta_p \psi&= &|\gamma_1|^{p-1} r^{-N} \big\vert{p-N\over p-1}\log^\beta r +\beta \log^{\beta-1} r\big\vert^{p-2} \\ &&\times \left[(p-1)\beta(\beta-1)\log^{\beta-2} r -\beta(3N-2p-2)\log^{\beta-1} r\right] \end{eqnarray*} Suppose now that p>2, and choose 0<\beta<{1\over p-1}<1, then there exists C>0 such that$$ \Delta_p \psi\geq -|\gamma_1|^{p-1}Cr^{-N}(\log r)^{\beta(p-1)-1}\geq -|\gamma_1|^{p-1}Cr^{-N}(\log r_1)^{\beta(p-1)-1}. $$On the other hand for p\leq 2 we can choose \beta=1 and a calculation similar to the one above implies that$$ \Delta_p \psi\geq -c|\gamma_1|^{p-1}r^{-N} \left(\log r_1\right)^{p-2}. $$In both cases we can choose \gamma_1 small enough to get$$ \Delta_p \psi \geq -C_1r^{-N}\geq \Delta_p u. $$Since u\geq \psi on the boundary of B_{r_2}\setminus B_{r_1}, one obtains by the comparison principle (Remark \ref{r1}) that u\geq \psi everywhere in B_{r_2}\setminus B_{r_1}. When r_2 goes to infinity it is easy to see that \gamma_2\rightarrow 0, and we obtain$$ u(x)\geq c |x|^{p-N\over p-1} \log^\beta |x|, $$for |x|\geq r_1. This implies that$$ m(r)\geq cr^{p-N\over p-1} \log r $$for r> r_1. We have reached a contradiction since$$m(r)\leq Cr^{p-N\over p-1}.$$Hence u\equiv 0. This concludes the proof of Theorem \ref{lnl}. \hfill\square We treat now the case N\leq p where the result is much stronger. \begin{thm}\label{Np} Let N\leq p. If u\in W^{1,p}_{loc}(\mathbb{R}^N)\cap {\cal C} (\mathbb{R}^N) is bounded below and is a weak solution of$$ -\Delta_p u\geq 0\;\;\mbox{in } \;\mathbb{R}^N, $$then u is constant. \end{thm} \begin{rema} \rm For N\leq p, for any q>0 and for any nonnegative h, if u\in W^{1,p}_{loc}(\mathbb{R}^N)\cap {\cal C} (\mathbb{R}^N) is a weak solution of$$ -\Delta_p u\geq h(x) u^q \;\;\mbox{in } \;\mathbb{R}^N $$then u\equiv 0. \end{rema} \paragraph{Proof of Theorem \ref{Np}.} Without loss of generality we can suppose that u\geq 0. First we will consider N0. When we let r_2\rightarrow +\infty inequality (\ref{i7}) becomes $$\label{i8} m(r)\geq m(r_1).$$ But of course m(r) is decreasing hence (\ref{i8}) implies that m(r) is constant i.e. m(r)=m(0)=u(0) for any r>0. Clearly this can be repeated with balls centered in any point of \mathbb{R}^N. Hence u is constant. For the case N=p just use inequality (\ref{i2b}) in Proposition \ref{p1} and proceed as above. This concludes the proof of Theorem \ref{Np}. \subsection*{Counterexample} We are going to show that for N>p, \gamma\geq 0 and q>(N+\gamma)(p-1)/(N-p) there exists a non-negative function u such that$$ -\Delta_p u\geq r^\gamma u^q \quad\mbox{in }\mathbb{R}^N $$hence proving that (N+\gamma)(p-1)/(N-p) is an optimal upper bound for q in Theorem \ref{lnl}. Indeed consider g(r)=C(1+r)^{-\alpha} with \alpha and C two positive constants to be determined. Clearly \Gamma(x)=g(|x|) satisfies \begin{eqnarray*} -\Delta_p\Gamma &=&C^{p-1}\alpha^{p-1}(1+r)^{-(\alpha+1)(p-2)} [-(\alpha+1)(p-1)(1+r)^{-(\alpha+2)}+\\ &&+{(N-1)\over r} (1+r)^{-(\alpha+1)}]\\ &\geq& C^{p-1}\alpha^{p-1}(1+r)^{-\alpha(p-1)-p}[N-1-(\alpha+1)(p-1)] \end{eqnarray*} with r=|x|. Now let \epsilon>0 such that q=(N+\gamma-\epsilon)(p-1)/(N-p-\epsilon) and let \alpha=(N-p-\epsilon)/(p-1). Clearly we have \alpha (p-1)+p+\gamma=N+\gamma-\epsilon=\alpha q. Furthermore N-1-(\alpha+1)(p-1)=N-p -\alpha(p-1)=\epsilon>0. Hence choosing C such that C^{p-1}\alpha^{p-1}(\epsilon)=C^q we obtain that \Gamma(x) satisfies$$ -\Delta_p \Gamma\geq C^q (1+r)^\gamma (1+r)^{-\alpha (p-1)-p-\gamma}\geq r^\gamma\Gamma^q \quad\mbox{in }\mathbb{R}^N. $$\section{The equation} In this section we are interested in studying non-existence results concerning the equation. Clearly in view of Theorem \ref{Np}, we are only interested in the case N>p. \begin{thm} Suppose that u\in W^{1,p}_{\rm loc}(\mathbb{R}^N) is nonnegative and satisfies $$\label{st1} -\Delta_p u = r^\gamma u^q,$$ for some \gamma \geq 0. If$$p-10.$$Moreover the estimates on$u$and$u'$imply that the terms$|u'|^{p-1}u(R) R^{N-1}$,$ |u'|^p(R) R^N$and$R^{\gamma+N} u^{q+1}(R)$behave respectively as$R^{N-1+{\gamma+p\over p-1-q}+{(\gamma+q+1)(p-1)\over p-1-q}}$,$R^{\gamma+N+{\gamma+p\over p-1-q}(q+1)}$and$R^{N-p\left({\gamma+q+1\over q-p+1}\right)}$. All the exponents are negative, and then$\int_0^R r^{\gamma+N-1} u^{q+1} dr\rightarrow 0$when$R\rightarrow +\infty$, hence$u\equiv 0$. This concludes the proof. \hfill$\square$\paragraph{Acknowledgments} This work was mainly done while the first author was visiting the Laboratoire d'Analyse, G\'eom\'etrie et Mod\'elisation of the University of Cergy-Pontoise. 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