\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 $1<p<\frac{N+2}{N-2}$ there are no solutions to
$$
\Delta u +u^p=0,\ u>0\quad \mbox{in } \mathbb{R}^N.
$$
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
\begin{equation}\label{eq 0}
- \mathop{\rm div} (|\nabla u|^{p-2 }\nabla u) \geq h(x) u^q \quad
\mbox{in } \mathbb{R}^N
\end{equation}
with $h$ satisfying
\begin{equation} \label{eqh}
 h(x)=a|x|^\gamma\quad \mbox{for $|x|$ large, $a>0$ and $\gamma>-p$}.
\end{equation}
  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
\begin{equation}\label{eq 01}
-\Delta_p u =   r^\gamma u^q ,\;\;u\geq 0\quad \mbox{in } \mathbb{R}^N,
\end{equation}
Serrin and Zou have proved in \cite{SZ} that for $p-1<q< {(N+\gamma)(p-1)+p+\gamma \over N-p}$ and $\gamma\geq 0$
any non negative  solution of (\ref{eq 01}) is identically zero.

Let us recall that
Gidas and Spruck have used Liouville theorem (for $p=2$)
to obtain a priori estimates for
solutions of the following problem:
\begin{equation}\label{eq 02}
\begin{gathered}
Lu +f(x,u)=0\quad \mbox{in } \Omega\\
u=\phi  \quad \mbox{on } \partial\Omega
\end{gathered}
\end{equation}
where $L$ is a second order uniformly elliptic operator and $f$ satisfies some
growth conditions. This is done through a blow up argument (see also
\cite{BCN2}).

Analogously, Theorem \ref{th1} constitutes the first step to obtain a priori
estimates for reaction diffusion equations involving $p$-Laplacian type
operators in bounded domains. In the case of systems this was done by
C. Azizieh and Ph. Clement in \cite{AC}, it would be interesting to do
it for general non divergence form operators.

\section{The inequation}

When $N>p$  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
\begin{equation}\label{eq}
-\Delta_p u \geq h(x) u^q \quad\mbox{in } \mathbb{R}^N,
\end{equation}
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 $-\Delta_p
\varphi(x_0)<f(x_0)$. By continuity, one can choose
$r$ sufficiently small such that
$\nabla \varphi(y)\neq 0$ , as well as
$$ -\Delta_p \varphi(y)<f(y),
$$
for all $y\in B(x_0,r)$.
 Let $m = \inf_{|x-x_0|=r} \{(u(x)-\varphi(x))>0\}$, and define
$$\bar\varphi = \varphi+{m\over 2}.
$$
One has $-\Delta_p \bar\varphi<f$ in $B(x_0,r)$ and $\bar\varphi\leq u$
on $\partial B(x_0,r)$.
Using the comparison principle one gets that $\bar\varphi\leq u$ in the ball which contradicts
$\bar\varphi(x_0) = \varphi(x_0)+{m\over 2}> u(x_0)$.
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 $0<r_1<r<r_2$:
\begin{equation}\label{i2}
m(r)\geq {m(r_1)(r^\lambda-r_2^\lambda)+m(r_2)(r_1^\lambda-r^\lambda)\over r_1^\lambda-r_2^\lambda}.
\end{equation}
Let $N=p$. Then
\begin{equation}\label{i2b}
m(r)\geq {m(r_1)\log({r\over r_2})+m(r_2)\log({r_1\over r})\over\log({ r_1\over r_2})}.
\end{equation}
\end{pro}

\paragraph{Proof:} Let $N\neq p$. Let  $0<r_1<r_2$.
Let us consider $\phi(r)=C_2r^{\lambda }+ C_1$ with $C_2$ and $C_1 $ such
that
$\phi(r_1)=m(r_1)$ and $\phi(r_2)=m(r_2)$. It is easy
 to see that
$$
\phi(r)={m(r_2)(r^\lambda-r_1^\lambda)+m(r_1)(r_2^\lambda-r^\lambda)\over r_2^\lambda-r_1^\lambda}.
$$
Obviously $\phi>0$ 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 $0<r_1<R$, define
$g(r)=m(r_1)\left\{1-{[(r-r_1)^+]^{k+1}\over (R-r_1)^{k+1}}\right\}$ with $k$ such that
$$
k\geq 3 \quad\mbox{and}\quad {1\over k}< p-1.
$$
Let $\zeta(x)=g(|x|)$. Clearly for $|x|<r_1$, $u(x)> m(r_1) =\zeta(x)$
 while for
$|x|\geq R$, $\zeta(x)\leq 0 <u(x)$.
On the other hand there exists $\tilde x$ such that $|\tilde x|=r_1$ and
$u(\tilde x)=\zeta(\tilde x)$. Hence the minimum of $u(x)-\zeta(x)$ occurs for
some $\bar x$ such that $|\bar x|=\bar r$ with $r_1\leq \bar r<R$.

Let $|x|=r$, it is an easy computation to see that for
$r\geq r_1$
\begin{eqnarray}\label{i3}
\lefteqn{-\Delta_p\zeta(x)} \\
&=&\Big({(k+1)m(r_1)\over (R-r_1)^{k+1}}\Big)^{(p-1)}
\big[2(p-1)+(N-1){(r-r_1)^+\over r}\big]((r-r_1)^+)^{kp-(k+1)}.\nonumber
\end{eqnarray}
Clearly with our choice of $k$, $kp-(k+1)>0$ 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
\begin{equation}
\label{eR}m(R)\leq C m(R)^{(p-1)\over q}R^{-p-\gamma\over q}.
\end{equation}
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
\begin{equation}\label{i5}
m(R)R^{-\lambda }\leq CR^{-\lambda-{p+\gamma\over q-p+1}}.
\end{equation}
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:
 \begin{equation}\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}.
\end{equation}
 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
$N<p$. Let
$m(r)=\inf_{x\in B_r(0)}u(x)$. From Proposition \ref{p1} we know that for $0<r_1<r<r_2$
\begin{equation}\label{i7}
m(r)\geq
{m(r_1)(r_2^\lambda-r^\lambda)+m(r_2)(r^\lambda-r_1^{\lambda})\over
r_2^\lambda-r_1^\lambda,}
\end{equation}
where $\lambda={p-N\over p-1}>0$.
When we let $r_2\rightarrow
+\infty$ inequality (\ref{i7}) becomes
\begin{equation}\label{i8}
m(r)\geq m(r_1).
\end{equation}
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
\begin{equation}\label{st1}
-\Delta_p u = r^\gamma u^q,
\end{equation}
for some $\gamma \geq 0$.
If $$p-1<q< {(N+\gamma)(p-1)+p+\gamma\over N-p}$$
and  $u$ is radial then  $u\equiv 0$.
\end{thm}


\begin{rema} \rm
One can get the same result for $-\Delta_p u = Cr^\gamma u^q$ by considering $u$
 multiplied by some convenient constant.
\end{rema}

The proof given here is similar to the one given by Caffarelli, Gidas and Spruck in \cite{CGS}.

\paragraph{Proof.}
It is sufficient to consider the case
$q\geq (N+\gamma)(p-1)/(N-p)$, since the other cases are
proved in Theorem \ref{lnl}.


If $u$ is a radial solution and satisfies (\ref{st1}) in a weak sense,
then it is not difficult to
see that it satisfies in the weak sense
$$-(r^{N-1} |u'|^{p-2} u')' = r^{N-1+\gamma} u^q
$$
Integrating  between $0$ and $r$, one has
$$r^{N-1} |u'|^{p-2} u' = -\int_0^r s^{N-1+\gamma} u^q (s) ds.
$$
Since $u'<0$, $u$ is decreasing and then,
$$r^{N-1} |u'|^{p-2} u' \leq -u(r)^q {r^{N+\gamma}\over N+\gamma}.
$$
Hence
$$u' u^{-q\over p-1} \leq -cr^{1+\gamma\over p-1}
$$
and integrating one gets
$$u(r)\leq C r^{\gamma+p\over p-1-q}.$$
Coming back to the equation one obtains
$$r^{N-1} |u'|^{p-1} = \int_0^r s^{N-1+\gamma} u^q (s) ds \leq C\int_0^r
s^{N-1+\gamma } s^{(\gamma+p)q\over (p-1-q)}ds.$$
Clearly
$N+\gamma+{(\gamma+p)q\over p-1-q}\geq 0$ when
$q\geq {(N+\gamma)(p-1)\over N-p}$ and therefore
$$|u'(r)|^{p-1}\leq C r^{\gamma+  {(\gamma+p)q\over p-1-q}+1}$$
and then
$$|u' |\leq Cr^{(\gamma+q+1)\over p-1-q}.$$
In order to conclude, we need to use Pohozaiev identity:
$$(N-p)\int_B |\nabla u|^p+ p\int_{\partial B} \sigma.n (\nabla u.x)=
p\int_B\Delta_p u (\nabla u.x)+\int_{\partial B} |\nabla u|^p (x.\vec
n)$$
here $\sigma = |\nabla u|^{p-2} \nabla u$ and $B=B(0,R)$. From the equation
we know that
$$\int_B |\nabla u|^p-\int_{\partial B} (\sigma.\vec n) u = \int_B r^\gamma
u^{q+1}$$
and then
\begin{multline}
\lefteqn{(N-p)\Big(\int_B r^\gamma u^{q+1}+\int_{\partial B} \sigma.\vec n u
\Big) +p\int_{\partial B} (\sigma.\vec n ) (\nabla u.x)} \\
 =  -p\int_B r^\gamma u^q
(\nabla u.x)+\int_{\partial B} |\nabla u|^p x.\vec n.
\end{multline}
Using the fact that $u$ is radial, for $\omega_n=|B_1|$ one gets
\begin{eqnarray*}
{1\over \omega_n}\int_{B_R} r^\gamma u^q \nabla u.x dx&=& \int_0^R r^{\gamma+N} u^q(r)
u'(r) dr \\
&=&
\int_0^R r^{\gamma+N} ({u^{q+1}(r)\over q+1})' dr\\
& =& -{\gamma+N\over q+1}
\int_0^R r^{\gamma+N-1} u^{q+1}+{1\over q+1}R^{\gamma+N} u^{q+1}(R).
\end{eqnarray*}
We have
finally obtained
\begin{eqnarray*}
\lefteqn{\big(N-p-{(\gamma+N)p\over q+1}\big)
\int_0^R r^{\gamma+N-1} u^{q+1} dr }\\
&= &(N-p) |u'|(R)^{p-1}u(R) R^{N-1} + (1-p) |u'(R)|^p R^N
-{p\over q+1}R^{\gamma+N} u(R)^{q+1}.
\end{eqnarray*}
Let us note that since $q< {(N+\gamma)(p)+p-N\over N-p}$,
one has
$$ {(\gamma+N)p\over q+1}+p-N>0.
$$
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. She wishes to thank the
people of the laboratoire for the kind invitation and their welcome.
The authors wish to thank Ilkka Holopainen for providing
some interesting references.

\begin{thebibliography}{99} \frenchspacing

\bibitem{AC} C. Azizieh, Ph. Clement,
A priori estimates and continuation methods for positive solutions of $p$-Laplace equations.
{\it J. Differential Equations} 179 (2002), no. 1, 213--245.

\bibitem{BCN1} { H.Berestycki, I. Capuzzo
Dolcetta, L. Nirenberg} { Probl\'emes
Elliptiques ind\'efinis et Th\'eor\`eme de
Liouville non-lin\'eaires}, {\it C.R.A.S. S\'erie  I.}
317, 945-950, (1993).


\bibitem{BCN2} { H.Berestycki, I. Capuzzo
Dolcetta, L. Nirenberg}, Superlinear indefinite elliptic problems and nonlinear Liouville theorems. {\it Topol.
Methods Nonlinear Anal. }4 (1994), no. 1, 59--78.

\bibitem{BCDC} I. Birindelli, I. Capuzzo Dolcetta, A. Cutr\'\i,
Liouville theorems for semilinear equations on the Heisenberg group,
{\it Ann. Inst. H. Poincar\'e Anal.Nonlin\'eaire, }{\bf 14} (1997),
295-308.

\bibitem{BM} I. Birindelli, E. Mitidieri, Liouville theorems for elliptic inequalities and
applications. {\it Proc. Roy. Soc. Edinburgh }Sect. A 128 (1998), no. 6, 1217--1247.

\bibitem{CGS} L.A. Caffarelli, B. Gidas, J. Spruck, Asymptotic symmetry and local behavior of
semilinear elliptic equations with critical Sobolev growth. {\it Comm. Pure Appl. Math. }42
(1989), no. 3, 271--297.

\bibitem{ChL} { W. Chen, C.Li} { A priori estimates for
prescribing scalar curvature equations}, {\it Ann. of Math.} 48 (1997),
47-92.

\bibitem{CL}A. Cutr\'\i, F. Leoni,  On the Liouville property for fully nonlinear equations.
{\it Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire }17 (2000), no. 2, 219--245.

\bibitem{DP} L. Damascelli, F. Pacella,{  Monotonicity and symmetry results for $p$-Laplace equations and applications.} {\it Adv. Differential
Equations } 5 (2000), no. 7-9, 1179--1200.

\bibitem{Dr} P. Dr\'abek, Nonlinear eigenvalue problem for the $p$-Laplacian in $\mathbb{R}^N$, {\it
Math. Nachr. } 173 (1995), 131-139.

\bibitem{DKN}P. Dr\'abek, A. Kufner, F. Nicolosi, {\it Quasilinear elliptic Eequations with
degenerations and singularities} De Gruyter Series In Nonlinear Analysis And Applications,
Berlin, New York, 1997.

\bibitem{G}B. Gidas, Symmetry properties and isolated singularities of positive solutions of
nonlinear elliptic equations, {\it Nonlinear Partial Differential equations in engineering and
applied sciences,} Eds. R. Sternberg, A. Kalinowski and J. Papadakis, Proc. Conf. Kingston, R.I.
1979, Lect. Notes on pure appl. maths, 54, Decker, New York, 1980, 255-273.

\bibitem{GS1} B. Gidas, J. Spruck, A priori bounds for positive solutions of
nonlinear elliptic equations. {\it Comm. Partial Differential Equations} 6 (1981)

\bibitem{GS2}B. Gidas, J.  Spruck,  Global and local behavior of positive solutions of
nonlinear elliptic equations. {\it Comm. Pure Appl. Math. }34 (1981), no. 4, 525--598.

\bibitem{hkm}J. Heinonen,T.  KilpelŠinen, O.  Martio,  Nonlinear potential theory of degenerate elliptic
equations. Oxford Mathematical Monographs. Oxford Science Publications. The Clarendon Press, Oxford University Press,
New York, 1993.

\bibitem{H1} I. Holopainen, A sharp $L\sp q$-Liouville theorem for $p$-harmonic functions. {\em Israel J.
Math.} 115 (2000), 363--379.

\bibitem{H2}I. Holopainen, Volume growth, Green's functions, and parabolicity of ends. {\em Duke Math. J.} 97 (1999),
no. 2, 319--346.

\bibitem{Juu} P. Juutinen, Minimization problems for Lipschitz functions, {\it Dissertation, University of JyvŠskulŠ}, JyvŠskulŠ,
 Ann. Acad. Sci. Fenn. Math. Diss. No. 115 (1998),

\bibitem {To}  P. Tolksdorff, {Regularity for a more general class of
quasilinear elliptic equations}, {\it Journal of Differential Equations}, 51 (1984),
126-150.

\bibitem{SZ}J. Serrin, Henghui Zou,  Non-existence of positive solutions of Lane-Emden systems.
{\it Differential Integral Equations } 9 (1996), no. 4, 635--653.

\bibitem{sz} J. Serrin, Henghui Zou, Cauchy-Liouville and universal
boundedness theorems for quasilinear ellptic equations  Preprint.

\bibitem{V} { J.L. Vasquez} { A strong maximum principle for some quasilinear
elliptic equations}, {\it Appl. Math. Optim.} 12: 191-202, (1984).

\end{thebibliography}


\noindent\textsc{Isabeau Birindelli} \\
Universit\`a di Roma ``La Sapienza'' \\
Piazzale Aldo moro, 5\\
00185 Roma, Italy \\
e-mail: isabeau@mat.uniroma1.it

\medskip

\noindent\textsc{Fran\c coise Demengel}\\
Universit\'e de Cergy Pontoise,\\
Site de Saint-Martin, 2 Avenue Adolphe Chauvin\\
95302 Cergy Pontoise\\
e-mail: Francoise.Demengel@math.u-cergy.fr
\end{document}