\documentclass[twoside]{article} \usepackage{amsfonts, amsmath} % used for R in Real numbers \pagestyle{myheadings} \markboth{Semi-classical analysis and vanishing properties } { Yves Belaud } \begin{document} \setcounter{page}{9} \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 9--22. \newline http://ejde.math.swt.edu or http://ejde.math.unt.edu \newline ftp ejde.math.swt.edu (login: ftp)} \vspace{\bigskipamount} \\ % Semi-classical analysis and vanishing properties of solutions to quasilinear equations % \thanks{ {\em Mathematics Subject Classifications:} 35K55, 35P15. \hfil\break\indent {\em Key words:} evolution equations, $p$-Laplacian, porous-medium, strong absorption, \hfil\break\indent regularizing effects, semi-classical limits. \hfil\break\indent \copyright 2002 Southwest Texas State University. \hfil\break\indent Published October 21, 2002.} } \date{} \author{Yves Belaud} \maketitle \begin{abstract} Let $\Omega$ be an open bounded subset of $\mathbb{R}^N$ and $b$ a measurable nonnegative function in $\Omega$. We deal with the time compact support property for $$ u_t - \Delta u + b(x)|u|^{q-1} u = 0 $$ for $p \geq 2$ and $$ u_t - \mathop{\rm div} ( |\nabla u|^{p-2} \nabla u ) + b(x)|u|^{q-1} u = 0 $$ with $m \geq 1$ where $0 \leq q <1$. We give criteria associated to the first eigenvalue of some quasilinear Schr\"odinger operators in semi-classical limits. We also provide a lower bound for this eigenvalue. \end{abstract} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}{Lemma}[section] \newtheorem{remark}{Remark}[section] \newtheorem{corollary}{Corollary}[section] \numberwithin{equation}{section} \section{Introduction} Let $\Omega$ be a regular bounded domain of $\mathbb{R}^N$ ($N \geq 1$) and $q \in [0,1)$. We consider the weak solution of the degenerate parabolic equations subject to the Neumann boundary condition: \begin{equation} \label{eq} \begin{gathered} u_t - \Delta u + b(x)|u|^{q-1} u = 0 \quad\mbox{in } \Omega \times (0 , \infty ), \\ \partial_\nu u = 0 \quad\mbox{on } \partial \Omega, \\ u(x,0) = u_0(x) \quad\mbox{in } \Omega, \end{gathered} \end{equation} and more generally, \begin{equation} \label{eqp} \begin{gathered} u_t - \mathop{\rm div} ( |\nabla u|^{p-2} \nabla u ) + b(x)|u|^{q-1} u = 0 \quad\mbox{in } \Omega \times (0 , \infty ), \\ \partial_\nu u = 0 \quad\mbox{on } \partial \Omega, \\ u(x,0) = u_0(x) \quad\mbox{in } \Omega, \end{gathered} \end{equation} with $p \geq 2$, or \begin{equation} \label{eqm} \begin{gathered} u_t - \Delta ( u^m ) + b(x)|u|^{q-1} u = 0 \quad\mbox{in } \Omega \times (0 , \infty ), \\ \partial_\nu u = 0 \quad\mbox{on } \partial \Omega, \\ u(x,0) = u_0(x) \quad\mbox{in } \Omega, \end{gathered} \end{equation} with $m \geq 1$. Many authors have already dealt with such equations giving a wide range of applications in physical mathematics. Now, our task is to describe a compact compact support property, in time. \paragraph{Definition.} A solution $u$ satisfies the Time Compact Support property (for short \textbf{TCS} property) if there exists a time $T$ such that for all $t \geq T$ and all $x \in \Omega$, $u(x,t)=0$. \smallskip First, we study some simple cases for (\ref{eq}): \noindent 1) Suppose that there exists a real $\gamma$ such as $b(x) \geq \gamma > 0 $ a.e. in $\Omega$. From the maximum principle, $u(x,t) \leq ( 1 - \gamma (1-q) t )^\frac{1}{1-q}$ in $\Omega \times (0 , \infty ) $. The nonlinear absorption is stronger than the diffusion and the \textbf{TCS} property holds. \noindent 2) We have a different feature if we assume that there exists a connected open set $\omega$ such as $b(x) = 0$ a.e. in $\omega$ (no absorption in $\omega$). Then usually, $u$ has not the compact support property. Indeed, if we denote by $\lambda(\omega)$ the first eigenvalue of $-\Delta$ in $W^{1,2}_0 (\omega)$ and $\zeta$ the first eigenfunction with $\|\zeta\|_{L^{\infty}(\omega)} = 1$ and $\zeta \geq 0$, then from the maximum principle, $u(x,t) \geq \zeta (x) \; e^{-\lambda (\omega) t} $ for all $x$ in $\omega$ and for all $t \geq 0$. Up to some minor changes, the previous examples are also valid if $u$ satisfies (\ref{eqp}) and (\ref{eqm}). The compact support property is related to $ \{x: b(x) = 0 \} $ and the behaviour of the function $b$ in a neighbourhood of this set. \section{The time compact support property} The starting idea was in the article of Kondratiev and V\'eron \cite{KV}. They established this property for (\ref{eq}) with the help of the following quantities \[ \mu_n = \inf \Big\{\int_{\Omega} ( |\nabla v|^2 + 2^n b(x) |v|^2 ) dx : v \in W^{1,2} (\Omega) , \int_{\Omega} |v|^2 \, dx = 1 \Big\} , \] with $n$ positive integer number. More precisely, up to a small change, they proved the following theorem. \begin{theorem} Suppose that $u$ is a solution of (\ref{eq}) and \[ \sum_{n=0}^{+\infty} \frac{\ln \mu_n}{\mu_n} < + \infty , \] then there exists some $T>0$ such that $u(x,t)=0$ for $(x,t) \in \Omega \times [T,+\infty)$. \end{theorem} We see that $\mu_n$ are linked to well-known questions in the semi-classical limit of Schr\"odinger operator of the type $-\Delta + 2^n b(.)$. In \cite{BHV}, the authors give a first extension of this theorem by replacing the sequence $2^n$ by any decreasing sequence going to zero. For the sake of simplicity, we denote by $\mu(\alpha)$ the lowest eigenvalue of the Neumann realization of the Schr\"odinger operator $-\Delta + \alpha^{q-1} b(.)$ in $W^{1,2}(\Omega)$, that is, \begin{equation} \mu(\alpha) = \inf \Big\{ \int_{\Omega} ( |\nabla v|^2 + \alpha^{q-1} b(x) |v|^2 ) dx : v \in W^{1,2} (\Omega) , \int_{\Omega} |v|^2 \, dx = 1 \Big\} . \end{equation} They proved the following theorem \cite[page 50]{BHV}. \begin{theorem} \label{theorembhv} Assume that $(\alpha_n)$ is a decreasing sequence of positive numbers such that \begin{equation} \sum_{n=1}^{+ \infty} \frac{1}{\mu(\alpha_n)} \big( \ln ( \mu(\alpha_n )) + \ln ( \frac{\alpha_n}{\alpha_{n+1}} ) + 1 \big) < + \infty , \end{equation} then any solution of (\ref{eq}) satisfies the \textbf{TCS} property. \end{theorem} The proof is based on an iterative method using the following lemma. \begin{lemma} \label{lemmabhv} Suppose that $b \geq 0$ a.e. in $\Omega$, $0 \leq q <1$ and $u$ is a bounded weak solution of (\ref{eq}) such that $\|u_0\|_{L^\infty (\Omega)} \leq \alpha$ for some $\alpha >0$. Then \begin{equation} \|u(.,t)\|_{L^\infty (\Omega)} \leq \min \big(1,C(\mu(\alpha))^{N/4} e^{-t \mu(\alpha)}\big) \|u_0\|_{L^\infty(\Omega)} , \end{equation} where $C=C(\Omega)$ is a positive real number. \end{lemma} \paragraph{Outline of the proof.} We use $u$ as test-function and since $u^{1-q} \geq \alpha^{1-q}$, we have \[ \frac{1}{2} \frac{d}{dt} \int_\Omega u^2 \, dx + \int_\Omega ( |\nabla u|^2 + b\alpha^{q-1}u^2 ) \, dx \leq 0. \] The definition of $\mu(\alpha)$ and H\" older's inequality gives \[ \|u(.,s)\|_{L^2 (\Omega)} \leq e^{-s \mu (\alpha)} |\Omega|^{1/2} \|u_0\|_{L^\infty (\Omega)} , \] for all positive real number $s$. The regularizing effects associated to this type of equation can be write under the following form \cite{VE79,VE76}: \[ \|u(.,t)\|_{L^\infty (\Omega)} \leq C ( 1+\frac{1}{t-s} )^{N/4} \|u(.,s)\|_{L^2 (\Omega)} , \] for all $t>s$. Taking $t - s = 1 / \mu(\alpha)$ completes the proof of the lemma. \hfill $\square$ \paragraph{Sketch of the proof of the theorem \ref{theorembhv}.} $(\alpha_n)$ is a decreasing sequence which tends to zero. We shall construct an increasing sequence $(t_n)$ such that for all $n$, \[ \forall t \geq t_n, \; \|u(.,t)\|_{L^\infty (\Omega)} \leq \alpha_n . \] If $\lim_{n \to + \infty} t_n = T < + \infty$ then $u$ satisfies the \textbf{TCS} property. To do this, we use an iterative method to find an upper bound for $\sum_n t_{n+1}-t_n$ under the form of a convergent series. We set $t_0=0$ and $\alpha=\alpha_0=\|u_0\|_{L^\infty (\Omega)}$. Applying Lemma \ref{lemmabhv} gives an upper bound for $\|u(.,t)\|_{L^\infty (\Omega)}$. $t_1$ is defined by \[ C(\mu(\alpha_0))^{N/4} e^{-(t_1-t_0) \mu(\alpha_0)} \alpha_0 = \alpha_1. \] A this point, we apply Lemma \ref{lemmabhv} but for time $t \geq t_1$ with $\alpha=\alpha_1$. Iterating this process provide us the formula \[ C(\mu(\alpha_n))^{N/4} e^{-(t_{n+1}-t_n) \mu(\alpha_n)} \alpha_n = \alpha_{n+1}. \] So we obtain an upper bound for the series $\sum_n t_{n+1}-t_n$. \hfill $\square$ An analoguous result can be proved for (\ref{eqp}). But before, we recall the regularizing effects for this type of equation \cite{VE79,VE76}. \begin{theorem} \label{thmA} Let $p>1$. Suppose that $u$ is a weak solution of \begin{gather*} u_t - \mathop{\rm div} ( |\nabla u|^{p-2} \nabla u ) + B(x,t,u) = 0 \quad\mbox{in } \Omega \times (0 , \infty ), \\ \partial_\nu u = 0 \quad\mbox{on } \partial \Omega, \\ u(x,0) = u_0(x) \in L^r(\Omega), \end{gather*} where $B$ is a Caratheodory functions which satisfies $B(x,t,\rho)\rho \geq 0$ a.e. in $\Omega \times (0, \infty)$. If $r \geq 1$, $r>N(2/p -1)$ then \[ \|u(.,t) \|_{L^\infty(\Omega)} \leq C \big( 1 + \frac{1}{t} \big)^{\delta(r)} \|u(.,0)\|_{L^r(\Omega)}^{\sigma(r)}, \] with $C = C(\Omega,p)$, $ \delta(r) = \frac{N}{rp+N(p-2)}$ and $ \sigma(r) = \frac{rp}{rp+N(p-2)}$. \end{theorem} In a similar way, we introduce \[ \mu(\alpha,p) = \inf \Big\{ \int_{\Omega} ( |\nabla v|^p + \alpha^{q-(p-1)} \; b(x) \; |v|^p ) dx : v \in W^{1,p} (\Omega) , \int_{\Omega} |v|^p \,dx = 1 \Big\} . \] Here $\mu(\alpha,p)$ is the first eigenvalue in $W^{1,p}(\Omega)$ for the Neumann boundary condition of \[ u \mapsto - \Delta_p u + \alpha^{q-(p-1)} b(.) u^{p-1} . \] The theorem states as follows \cite{BE}: \begin{theorem} \label{theoremp} Let $0 \leq q < 1$, $p > 2$ and assume that there exist two sequences of positive real numbers $(\alpha_n)$ and $(r_n)$ such that $(\alpha_n)$ is decreasing and \begin{eqnarray} \sum_{n=0}^\infty \frac{r_n^{p-1}}{\alpha_{n+1}^{p-2} \mu(\alpha_n,p)^{\sigma(r_n)}} < + \infty. \end{eqnarray} Then any solution of (\ref{eqp}) with initial bounded data satisfies the \textbf{TCS} property. \end{theorem} Consequently, if $r_n = \ln \mu (\alpha_n,p)$, we have the following statement. \begin{corollary} \label{corollaryp} Under the same assumptions on $q$ and $p$, if there exists a decreasing sequence of positive real numbers $(\alpha_n)$ such that \begin{eqnarray} \sum_{n=0}^\infty \frac{( \ln \mu(\alpha_n,p) )^{p-1}}{\alpha_{n+1}^{p-2} \mu(\alpha_n,p)} < + \infty, \end{eqnarray} then any solution of (\ref{eqp}) satisfies the \textbf{TCS} property. \end{corollary} Theorem \ref{theoremp} comes from the following lemma. \begin{lemma} \label{lemmap} Suppose there exists a measurable function $u$ in $\Omega \times \mathbb{R}^+$ which satisfies weakly (\ref{eqp}) with $\|u_0\|_{L^\infty(\Omega)} \leq \alpha$ for some $\alpha>0$. Then \begin{eqnarray} \|u(.,t)\|_{L^r(\Omega)} \leq \Big( \frac{1}{\|u(.,0)\|_{L^r(\Omega)}^{2-p}+C_1 \mu(\alpha,p) t } \Big)^{\frac{1}{p-2}} , \end{eqnarray} where $C_1 = C_1(\Omega,r,p)$ is a positive real constant and there exist two positive real numbers $C=C(\Omega,p)$ and $C_2=C_2(r,p)$ such that \[ \|u(.,t)\|_{L^\infty(\Omega)} \leq \min \Big( C ( 1 + \frac{2}{t} )^{\delta(r)} \Big( \frac{1}{\|u(.,0)\|_{L^\infty(\Omega)}^{2-p} + C_2 \mu(\alpha,p) t } \Big)^{\frac{\sigma(r)}{p-2}} , 1 \Big) , \] with $ \delta(r) = \frac{N}{rp+N(p-2)}$ and $ \sigma(r) = \frac{rp}{rp+N(p-2)}$. \end{lemma} \paragraph{Idea in the proofs.} The principle to prove them remains true. It is a bit more complicated because the term $u_t$ is not homogenuous with $u^{p-1}$ but it follows exactly the Kondratiev-Véron method as shown in the proof of Theorem \ref{theorembhv}. The main differences are technical. Instead of using $u$ as test-function, we use $u|u|^{r_n-1}$ at each step of the iteration. An estimate of the asymptotic behaviour when $r \to + \infty$ for the constant $C_2 = C_2(r,p)$ is needed. The proof of the theorem ends with sharp upper bounds for the series $\sum_n t_{n+1}-t_n$. \hfill $\square$ Now, let us talk about equation \ref{eqm}. Formally, replacing $p-1$ by $m$ give the same results \cite{VE79,VE76}: \begin{theorem} \label{thmB} Let $m>0$ and $u$ be a weak solution of \begin{gather*} u_t - \Delta(u^m) + B(x,t,u) = 0 \quad\mbox{in }\Omega \times (0 ,\infty ),\\ \partial_\nu u = 0 \quad\mbox{on } \partial \Omega, \\ u(x,0) = u_0(x) \in L^r(\Omega), \end{gather*} where $B$ is a Caratheodory function satisfying $B(x,t,\rho)\rho \geq 0$ a.e. in $\Omega \times (0, \infty)$. If $r \geq 1$ and $r>N(1-m)/2$, then \[ \|u(.,t) \|_{L^\infty(\Omega)} \leq C ( 1 + \frac{1}{t} )^{\delta(r)} \|u(.,0)\|_{L^r(\Omega)}^{\sigma(r)}\,, \] with $C = C(\Omega,m)$, $ \delta(r) = \frac{N}{2r+N(m-1)}$ and $ \sigma(r) = \frac{2r}{2r+N(m-1)}$. \end{theorem} We set quantities adapted to the problem \[ \mu'(\alpha,m) = \inf \Big\{ \int_{\Omega} ( |\nabla v|^2 + \alpha^{q-m} b(x) |v|^2 ) dx : v \in W^{1,2} (\Omega) , \int_{\Omega} |v|^2 \, dx = 1 \Big\} . \] Thus, \begin{theorem}[\cite{BE}] \label{theoremm} Let $0 \leq q < 1$, $m > 1$ and assume that there exists two sequences of positive real numbers $(\alpha_n)$ and $(r_n)$ such that $(\alpha_n)$ is decreasing and \begin{eqnarray} \sum_{n=0}^\infty \frac{r_n^m}{\alpha_{n+1}^{m-1} \mu'(\alpha_n,m)^{\sigma(r_n)}} < + \infty. \end{eqnarray} Then any solution of (\ref{eqm}) with initial bounded data satisfies the \textbf{TCS} property. \end{theorem} With $r_n = \ln \mu'(\alpha_n,m)$, we deduce the following statement. \begin{corollary} \label{corollarym} Under the above assumptions on $q$ and $m$, if there exists a decreasing sequence of positive real numbers $(\alpha_n)$ such that \[ \sum_{n=0}^\infty \frac{( \ln \mu'(\alpha_n,m) )^m}{\alpha_{n+1}^{m-1} \mu'(\alpha_n,m)} < + \infty, \] then any solution of (\ref{eqm}) satisfies the \textbf{TCS} property. \end{corollary} The proof of Theorem \ref{theoremm} also comes from the following lemma. \begin{lemma} We suppose there exists a measurable function $u$ in $\Omega \times \mathbb{R}^+$ which satisfies weakly (\ref{eqm}) with $\|u_0\|_{L^\infty(\Omega)} \leq \alpha$ for some $\alpha >0$. Then \begin{eqnarray} \|u(.,t)\|_{L^r(\Omega)} \leq \Big( \frac{1}{ \|u(.,0)\|_{L^r(\Omega)}^{1-m} + C_1 \mu'(\alpha,m) \; t }\Big)^{1/(m-1)}, \end{eqnarray} with $C_1=C_1(\Omega,r,m)$ and there exist two positive real numbers $C=C(\Omega,m)$ and $C_2=C_2(r,m)$ such that \[ \|u(.,t)\|_{L^\infty(\Omega)} \leq \min \Big( C \big( 1 + \frac{2}{t} \big)^{\delta(r)} \big( \frac{1}{\|u(.,0)\|_{L^\infty(\Omega)}^{1-m} + C_2 \mu'(\alpha,m) t }\big)^{\frac{\sigma(r)}{m-1}} , 1 \Big), \] where $\delta(r)$ and $\sigma(r)$ are defined in Theorem \ref{thmB} \end{lemma} The assumptions in Theorem \ref{theorembhv} and Corollaries \ref{corollaryp}, \ref{corollarym} admit a simpler form. A comparaison between series and integral gives the following theorem. \begin{theorem}[Integral criterion \cite{BHV,BE}] Let $0 \leq q <1$. 1) If $p \geq 2$ and \[ \int_0^1 \frac{( \ln \mu(t,p) )^{p-1}}{t^{p-1} \mu(t,p)} dt < + \infty , \] then all solutions of (\ref{eqp}) satisfy the \textbf{TCS} property.\\ 2) If $m \geq 1$ and \[ \int_0^1 \frac{( \ln \mu'(t,m) )^m}{t^m \mu'(t,m)} dt < + \infty , \] then all solutions of (\ref{eqm}) satisfy the \textbf{TCS} property. \end{theorem} We remark that $\mu(t)=\mu(t,2)$ and that (\ref{eq}) is a particular case of (\ref{eqp}) for $p=2$ and (\ref{eqm}) for $m=1$. The proof is first establish for $p=2$ \cite[page 51]{BHV} and then for $p>2$ and $m>1$ \cite{BE}. What is remarkable is that this criterion has a same simple form in all cases. For applications, $\mu(t,p)$ and $\mu'(t,m)$ have to be linked directly to the function $b$. We recall that $\mu(\alpha,p)$ is the first eigenvalue in $W^{1,p}(\Omega)$ for the Neumann boundary condition of $u \mapsto -\Delta_p u + \alpha^{q-(p-1)} b(.) u^{p-1}$. The aim of semi-classical analysis is to describe the behavior of the spectrum of the operator $u \mapsto -\Delta_p u + h^{-p} V(.) u^{p-1}$ in particular $\lambda_1(h)$ the lowest eigenvalue. $V$ is a function which holds in our case \begin{equation} \label{V} V \in L^{\infty} (\Omega) , \quad\mathop{\rm ess\,inf}_\Omega V = 0 \quad \mbox{and} \quad \int_\Omega V(x) \,dx > 0 . \end{equation} We denote by $\gamma$ a positive number which satisfies: \begin{equation} \label{gamma} \gamma \begin{cases} = \frac{N}{p} & \mbox{ for } 1 < p < N,\\ \in (1 , + \infty) & \mbox{ for } p = N,\\ = 1 & \mbox{ for } p > N, \end{cases} \end{equation} \begin{corollary} If (\ref{V}) holds then for $h$ small enough, \begin{equation} \label{usualinequality} \lambda_1(h) ( \mathop{\rm meas} \{ x : V(x) \leq h^p \lambda_1(h) \} )^{1/\gamma} \geq C, \end{equation} where $C=C(p,N,\gamma,\Omega,V)$ is a positive constant. \end{corollary} $\mu(t,p)$ can be written as $\mu(t,p)=\lambda_1 ( t^\frac{(p-1)-q}{p} )$ which after a change of variables gives \[ \int_0^1 \frac{( \ln \mu(t,p) )^{p-1}}{t^{p-1} \mu(t,p)} dt = \int_0^1 \frac{( \ln \lambda_1(h) )^{p-1}}{h^\frac{p(p-1)-(1+q)}{p-(1+q)} \; \lambda_1(h) } dh . \] If we have an estimate of the type \[ \lambda_1(h) \geq C \frac{1}{h^\theta}, \] where $C$ and $\theta$ are two positive real numbers, then the integral criterion holds for $p > 2$ provided \begin{equation} \label{theta} \theta > \frac{p(p-2)}{p-(1+q)}. \end{equation} Similar expressions can be found for $p=2$ and $m>1$. Finally, we obtain next theorem. \begin{theorem}[$1/b$ criterion \cite{BHV,BE}] Let $0 \leq q <1$ and $b$ be a bounded measurable function such that \[ \mathop{\rm ess\,inf}_\Omega b =0 \quad \mbox{and} \quad \int_\Omega b(x)\,dx > 0 . \] 1) If $p=2$ and $\ln (1/b) \in L^s(\Omega)$ for some $s>N/2$ then equation (\ref{eq}) satisfies the \textbf{TCS} property. \\ 2) If $p>2$ and $(1/b)^s \in L^1(\Omega)$ for some $s$ with \[ s > \begin{cases} \frac{p-2}{1-q} (\frac{N}{p})& \mbox{for } p \leq N,\\ \frac{p-2}{1-q} & \mbox{for } p > N, \end{cases} \] then equation (\ref{eqp}) satisfies the \textbf{TCS} property. \\ 3) If $m>1$ and $(1/b)^s \in L^1(\Omega)$ for some $s$ with \[ s > \begin{cases} \frac{m-1}{1-q} ( \frac{N}{2})& \mbox{for } N \geq 2,\\ \frac{m-1}{1-q} & \mbox{for } N = 1, \end{cases} \] then equation (\ref{eqm}) satisfies the \textbf{TCS} property. \end{theorem} \paragraph{Outline of the proof.} the three cases are based on Marcinkiewicz type inequalities. For 1) \[ \mathop{\rm meas} \left\{ x \in \Omega \; : \; \ln \frac{1}{b(x)} \geq \ln \frac{1}{h^2 \lambda_1(h)} \right\} \leq \frac{1}{\big( \ln \frac{1}{h^2\lambda_1(h)}\big)^s} \int_\Omega \big( \ln \frac{1}{b(x)}\big)^s dx , \] and for 2) \[ {\rm meas} \left\{ x : \frac{1}{b(x)} \geq \frac{1}{h^p \lambda_1(h)} \right\} \leq ( h^p \lambda_1(h) )^s \int_\Omega \big( \frac{1}{b(x)} \big)^s dx. \] The proof ends with estimates such as (\ref{theta}) and some technical arguments. \hfill $\Box$ \begin{remark}\rm In the case where $p=2$ and $N \leq 2$, estimate (\ref{usualinequality}) is not enough sharp so we use the formula of Lieb and Thirring. See \cite{BHV} for details. \end{remark} Now we apply the previous theorem to the radial functions. \begin{corollary} Suppose that $0 \in \Omega$. 1) If $b(x) = \exp ( -\frac{1}{\|x\|^\beta})$ with $\beta < 2$ then any solution of (\ref{eq}) satisfies the \textbf{TCS} property.\\ 2) If $b(x) = \|x\|^\beta$ with $p \leq N$ and $\beta < p(1-q)/(p-2)$ then any solution of (\ref{eqp}) satisfies the \textbf{TCS} property.\\ One has the same conclusion if $p > N$ and $\beta < N(1-q)/(p-2)$.\\ 3) If $b(x) = \|x\|^\beta$ with $N \geq 2$ and $\beta < 2(1-q)/(m-1)$ then any solution of (\ref{eqm}) satisfies the \textbf{TCS} property.\\ One has the same conclusion if $N=1$ and $\beta < (1-q)/(m-1)$. \end{corollary} \section{A lower bound for the first eigenvalue} This section is dedicated to estimating the first eigenvalue, in $W^{1,p}(\Omega)$, of the operator $u \mapsto -\Delta_p u + h^{-p} V(.) u^{p-1}$. We have seen that a lower bound is fundamental for applications. First,we introduce a sequence of definitions. We consider a non-empty connected open subset $\Omega \subset \mathbb{R}^N$ and a mesurable function $V$ defined in $\Omega$. We set \[ W^{1,p,V}(\Omega) = \{ \psi \in W^{1,p} (\Omega) \; : \; V(x) |\psi^p| \in L^1 (\Omega)\} . \] If $W^{1,p,V}(\Omega) \ne \{ 0 \}$ and $\psi \in W^{1,p,V}(\Omega)$, we set \begin{equation} \label{Fpsi} F_V(\psi) = \int_{\Omega} |\nabla \psi |^p + V(x) |\psi|^p \,dx, \end{equation} and define \begin{equation} \label{lambda} \lambda_1 = \inf \left\{ F_V(\psi) : \psi \in W^{1,p,V}(\Omega) , \int_{\Omega} |\psi|^p \,dx = 1 \right\} , \end{equation} and for $h>0$, \begin{equation} \label{lambdaone} \lambda_1(h) = \inf \left\{ F_{h^{-p}V}(\psi) : \psi \in W^{1,p,V}(\Omega) , \int_{\Omega} |\psi|^p \,dx = 1 \right\}, \end{equation} Thus $\lambda_1(h)$ is the first eigenvalue of the operator \begin{equation} u \mapsto -\Delta_p u + h^{-p} V(.) |u|^{p-2} u . \end{equation} in $W^{1,p,V}(\Omega)$ with Neumann boundary condition if the infimum is achieved by a regular enough element of $W^{1,p,V}(\Omega)$ and $\partial \Omega$ ${\cal C}^1$.\\ We start with a simple result which enlights our arguments. On the contrary to the linear case ($p=2$), our proof is not based on the theory of pseudodifferential operators but on the continuous injections of $W^{1,p}(\Omega)$ into the $L^s$ spaces for suitable $s$. \begin{theorem} \label{pRN} Suppose $N>p>1$. Then either $\lambda_1=-\infty$ or \begin{eqnarray} \Big( \int_{V(x) \leq \lambda_1} ( \lambda_1 - V(x))^{N/p} \, dx \Big)^{p/N} \geq C(p,N), \end{eqnarray} where $C=C(p,N)>0$ is the positive constant of the Sobolev inequality.\\ In addition, if there exists a minimizer in $W^{1,p,V}(\mathbb{R}^N)$, \begin{eqnarray} \Big( \int_{V(x) < \lambda_1} ( \lambda_1 - V(x))^{N/p} \, dx \Big)^{p/N} \geq C(p,N). \end{eqnarray} \end{theorem} \paragraph{Proof.} Let $\psi$ be in $W^{1,p,V}(\mathbb{R}^N)$ with $\|\psi\|_{L^p(\mathbb{R}^N)}=1$ then \[ \int_{\mathbb{R}^N} |\nabla \psi |^p \,dx + \int_{\mathbb{R}^N} V(x) |\psi|^p \,dx = F_V(\psi) = F_V(\psi) \int_{\mathbb{R}^N} |\psi|^p \,dx. \] The integral with $V$ is split in two parts, that is,\\ $\mathbb{R}^N = \{ x : V(x) < F_V(\psi) \} \cup \{ x : V(x) \geq F_V(\psi) \}$. Therefore, \begin{eqnarray} \label{VIF} \int_{\mathbb{R}^N} |\nabla \psi |^p \,dx \leq \int_{V(x) < F_V(\psi)} (F_V(\psi) - V(x)) |\psi|^p \, dx. \end{eqnarray} H\"older's inequality leads to \begin{multline} \label{depart} \int_{\mathbb{R}^N} |\nabla \psi |^p \, dx\\ \leq \Big( \int_{V(x) < F_V(\psi)} ( F_V(\psi) - V(x) )^{N/p} \, dx \Big)^{p/N} \Big( \int_{\mathbb{R}^N}|\psi|^{p^*} \, dx \Big)^{1-\frac{p}{N}}. \end{multline} since $\{ x : V(x) < F_V(\psi) \} \subset \mathbb{R}^N$. Non zero constants do not belong to $W^{1,p,V}(\mathbb{R}^N)$ and so all functions $\psi$ satisfy $\int_{\mathbb{R}^N} |\nabla \psi |^p \,dx > 0$. We can apply Sobolev inequality. The Beppo-Levi theorem completes the proof. \hfill $\square$ \begin{remark} \rm If $\Omega$ is any open domain of $\mathbb{R}^N$, we define \[ W^{1,p,V}_0(\Omega) = \{ \psi \in W^{1,p}_0 (\Omega) : V(x) |\psi^p| \in L^1 (\Omega)\}, \] and if $W^{1,p,V}_0(\Omega) \ne \{ 0 \}$, \[ \tilde{\lambda_1} = \inf \big\{ F_V(\psi) : \psi \in W^{1,p,V}_0(\Omega) , \int_{\Omega} |\psi|^p \,dx = 1 \big\} , \] then the estimates in Theorem \ref{pRN} hold for $\tilde{\lambda_1}$. \end{remark} When $\Omega$ is a ${\cal C}^1$ bounded domain of $\mathbb{R}^N$ and $V$ is a measurable function such that \begin{equation} \label{VO} V \in L^{\infty} (\Omega) ,\quad \mathop{\rm ess \, inf}_\Omega V = 0 \quad \mbox{and} \quad \int_\Omega V(x) \, dx > 0, \end{equation} we set $u_h$ the first eigenfunction related to the first eigenvalue $\lambda_1(h)$. Recall that $\gamma$ is a positive number which satisfies \begin{equation} \label{gammaO} \gamma \begin{cases} = \frac{N}{p} & \mbox{for } 1 < p < N,\\ \in (1 , + \infty) & \mbox{for } p = N,\\ = 1 & \mbox{for } p > N, \end{cases} \end{equation} with $\frac{\gamma}{\gamma -1} = + \infty$ if $\gamma =1$. This $\gamma$ is such that $W^{1,p}$ imbeds $L^q(\Omega)$ continuously with $q=p\frac{\gamma}{\gamma-1}$. \begin{theorem} \label{usual} Assume that (\ref{VO}) holds. Then for $h$ small enough, \[ \Big( \int_{V(x) < h^p \lambda_1(h)} \big( \lambda_1(h) - \frac{V(x)}{h^p} \big)^\gamma \,dx \Big)^{1/\gamma} \geq C , \] where $C = C(p,N,\gamma,\Omega,V)$ is a positive real constant. \end{theorem} \paragraph{Proof.} We start with (\ref{depart}) because the beginning is similar. Replacing $\mathbb{R}^N$, $\psi$ and $V$ by $\Omega$, $u_h$ and $\frac{V}{h^p}$ the H\"older's inequality gives \[ \int_{\Omega} |\nabla u_h |^p \,dx \leq \Big( \int_{V(x) 0 . $$ Suppose that there exists a sequence $(h_n)$ of positive real numbers which goes to zero such that $$ \lim_{n \to +\infty} \|\nabla u_{h_n}\|_{L^p(\Omega)} = 0 . $$ Hence $(u_{h_n})$ is bounded in $W^{1,p}(\Omega)$, so there exists a function $u_0$ in $W^{1,p}(\Omega)$ such that, up to a subsequence, $u_{h_n} \rightharpoonup u_0$ weakly in $W^{1,p}(\Omega)$. Obviously, $\|\nabla u_0\|_{L^p(\Omega)}=0$. Therefore, $u_0=C$ where $C$ is a real. Thanks to the Rellich-Kondrachov theorem, up to a subsequence, $u_{h_n} \to C$ strongly in $L^p(\Omega)$ so $ C = (\frac{1}{{\rm meas}(\Omega)} )^\frac{1}{p}$. We deduce that $ \lim_{n \to +\infty} h^p_n \lambda_1(h_n) = \frac{\int_\Omega V(x) \,dx}{{\rm meas}(\Omega)}$. But from lemma 3.2 in \cite{BHV}, $ \lim_{h \to 0} h^p \lambda_1(h) =0$ which leads to a contradiction. \hfill $\square$ A simpler form is provided in the following corollary. \begin{corollary} If (\ref{VO}) holds then for $h$ small enough, \[ \lambda_1(h) ( \mbox{meas} \{ x : V(x) < h^p \lambda_1(h) \} )^\gamma \geq C, \] where $C=C(p,N,\gamma,\Omega,V)$. \end{corollary} We end this section by quoting a theorem. For $\Omega$ a domain of $\mathbb{R}^N$ bounded or not, regular or not and $V$ a mesurable function defined on $\Omega$ such that $W^{1,p,V}(\Omega) \ne \{ 0 \}$, we define a well for a mesurable function $V$ \cite{BE}. \paragraph{Definition.} We say that $V$ has a well in $U$ if $U$ is a ${\cal C}^1$ bounded, connected, non-empty open set of $\Omega$ and if there exists $\psi_0 \in W^{1,p,V}(\Omega)$ with $\|\psi_0\|_{L^p(\Omega)}=1$ such that $\displaystyle \int_\Omega V(x) |\psi_0|^p \; dx < a = \mathop{\rm essinf}_{\Omega \backslash U} V$ with $\mathop{\rm meas}(\Omega \backslash U) > 0$. The term of well generalizes the definition in \cite{HE}. \begin{theorem}[\cite{BHV}] If $V$ has a well in $U$, for $h$ small enough, \[ \Big( \int_{V(x) \leq h^p \lambda_1(h)} ( \lambda_1(h) - h^{-p} V(x) )^\gamma \,dx \Big)^{1/\gamma} \geq C, \] where $C$ is a positive constant which does not depend on $h$. In addition, if there exists a minimizer in $W^{1,p,V}(\Omega)$, \[ \Big( \int_{V(x) < h^p \lambda_1(h)} ( \lambda_1(h) - h^{-p} V(x) )^\gamma \,dx \Big)^{1/\gamma} \geq C. \] \end{theorem} The proof is technical but some arguments have already been used for Theorem \ref{usual}. \section{Summary and open questions} For the sake of completeness, we quote another theorem of. \begin{theorem}[\cite{BHV}] Suppose that $b$ is a continuous and nonnegative function defined in $\overline{\Omega}$ which satisfies for some $x_0 \in \Omega$ \[ \lim_{r \to 0} r^2 \ln (1/\|b\|_{L^\infty(B_r(x_0))}) = \infty . \] If $u$ is a weak solution of (\ref{eq}) then $u$ does not satisfies the \textbf{TCS} property. \end{theorem} Up to now, we have the following: \medskip \noindent\begin{tabular}{|c|c|c|c|} \hline & $p = 2$ & $p > 2$ & $m > 1$\\ \hline & & &\\ \parbox{2cm}{Integral \\ criterion} &$\int_0^1 \frac{\ln \mu(t)}{t \mu(t)} dt < \infty$ &$\int_0^1 \frac{( \ln \mu(t,p) )^{p-1}}{t^{p-1} \mu(t,p)} dt < \infty$ &$\int_0^1 \frac{( \ln \mu'(t,m) )^m}{t^m \mu'(t,m)} dt < \infty$ \\ & & & \\ \hline \parbox{20mm}{$1/b$ criterion\\ with} & \parbox{25mm}{\begin{center}$\ln(1/b) \in L^s$\\ $s > \frac{N}{2}$\end{center}} & \parbox{30mm}{\begin{center}$1/b \in L^s$ \\ $s > \frac{p-2}{1-q} \frac{N}{p},\, N \geq p$ \\ $s>\frac{p-2}{1-p},\, N < p$\end{center}} &\parbox{30mm}{\begin{center}$1/b \in L^s$\\ $s > \frac{m-1}{1-q} \frac{N}{2}, N \geq 2$\\ $s > \frac{m-1}{1-q} ,\, N = 1$ \end{center}} \\ \hline \parbox{20mm}{Radial case\\ for $\beta \geq 0$ \\ and} & \parbox{25mm}{\begin{center}$\exp ( -1/\|x\|^\beta )$ \\ $\beta < 2$\end{center}} & \parbox{30mm}{\begin{center}$\|x\|^\beta $\\ $\frac{p(1-q)}{p-2} ,\, N \geq p$\\ $\beta < \frac{N(1-q)}{p-2} ,\, N < p$\end{center}} & \parbox{30mm}{\begin{center}$\|x\|^\beta$ \\ $\beta < \frac{2(1-q)}{m-1} ,\, N \geq 2$\\ $\beta < \frac{(1-q)}{m-1} ,\, N = 1$\end{center}}\\ \hline Converse & yes & no & no\\ \hline \parbox{20mm}{Non \textbf{TCS}\\ property for} & \parbox{25mm}{\begin{center}$\exp ( -1/\|x\|^\beta)$ \\ $\beta > 2$ \end{center}} & $\vdots$ & $\vdots$ \\ \hline \end{tabular} \subsection*{Open questions} \begin{enumerate} \item What happens for $p = 2$ and $\beta = 2$ ? It does not seem within sight. \item We have no genuine converse for $p>2$ and $m>1$. A converse has been found for $p=2$ because $L^2(\Omega)$ has an inner product. More precisely, for $p>2$, $ \int_\Omega u^{p-1} v \,dx \ne \int_\Omega v^{p-1} u \,dx$ in general. We search for another test-functions (see \cite{BHV} for details). \item When $p > 2$, we have a good generalization of the Cwikel, Lieb and Rosenblyum formula, that is, for large dimension ($N > p$). The estimate for $N \leq p$ is far from the optimum. When $p=2$, the Lieb and Thirring formula works well. We hope that we will find an equivalent. \item In \cite{KV}, they also deal with second order elliptic equations with a strong absortion, i.e., $u_{tt} + \Delta u - a(x) u^q = 0$. Heuristically speaking, changing $\mu(\alpha)$ into $\sqrt{\mu(\alpha)}$ gives a sufficient condition for the \textbf{TCS} property. We are working on this type of equation when $a$ depends also on $t$. \item More generally, the following problem $\Delta_p u - a(x) u^{p-1}=0$ in an outside domain is difficult to handle. On $\mathbb{R}^N$ minus a ball, a similar technique may be possible. \end{enumerate} \begin{thebibliography}{00} \frenchspacing \bibitem{BE} Y. Belaud, {\it Time vanishing properties for solutions of some degenerate parabolic equations with strong absorption}, Advanced Nonlinear Studies {\bf 1}, 2 (2001), 117-152. \bibitem{BR} H. Brezis, Analyse fonctionnelle. Théorie et applications, Collection Mathématiques appliquées pour la maîtrise, Masson, 1986. \bibitem{BHV} Y. Belaud, B. Helffer, L. Véron, {\it Long-time vanishing properties of solutions of sublinear parabolic equations and semi-classical limit of Schrödinger operator}, Ann. Inst. Henri Poincarré Anal. nonlinear {\bf 18}, 1 (2001), 43-68 \bibitem{BS} F.A. Berezin, M.A. Shubin, The Schrödinger Equation, Kluwer Academic Publishers, 1991. \bibitem{CW} M. Cwikel, {\it Weak type estimates for singular value and the number of bound states of Schrödinger operator}, Ann. Math. {\bf 106} (1977), 93-100. \bibitem{GT} D. Gilbarg, N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer Verlag, 1977. \bibitem{KV} V.A. Kondratiev and L. Véron, {\it Asymptotic behaviour of solutions of some nonlinear parabolic or elliptic equations}, Asymptotic Analysis {\bf 14} (1997), 117-156. \bibitem{HE} B. Helffer, Semi-classical analysis for the Schrödinger operator and applications, Lecture Notes in Math. 1336, Springer-Verlag, 1989. \bibitem{LT} E. H. Lieb, W. Thirring, {\it Inequalities for the moments of the eigenvalues of the Schrödinger Hamiltonian and their relations to Sobolev Inequalities}, In Studies in Math. Phys., essay in honour of V. Bargmann, Princeton Univ. Press, 1976. \bibitem{RO} G. V. Rosenblyum, {\it Distribution of the discrete spectrum of singular differential operators}, Doklady Akad. Nauk USSR {\bf 202} (1972), 1012-1015. \bibitem{VE79} L. V\'eron, {\it Effets r\'egularisants de semi-groupes non lin\'eaires dans des espaces de Banach}, Annales facult\'e des Sciences Toulouse {\bf 1} (1979), 171-200. \bibitem{VE76} L. V\'eron, {\it Coercivit\'e et propri\'et\'es r\'egularisantes des semi-groupes non lin\'eaires dans les espaces de Banach}, Publication de l'Universit\'e Fran\c cois Rabelais - Tours (1976). \end{thebibliography} \noindent\textsc{Yves Belaud}\\ Laboratoire de Math\' ematiques et Physique Th\' eorique,\\ CNRS ESA 6083, Facult\'e des Sciences et Techniques,\\ Universit\'e Fran\c cois Rabelais, 37200 Tours\\ e-mail: belaud@univ-tours.fr \end{document}