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 \markboth{Behaviour of solutions to inequalities}
 {Marie-Fran\c{c}oise Bidaut-V\'eron} 

\begin{document}
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\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
USA-Chile Workshop on Nonlinear Analysis, \newline
Electron. J. Diff. Eqns., Conf. 06, 2001, pp. 29--44\newline
http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp  ejde.math.swt.edu or ejde.math.unt.edu (login: ftp)}
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%
Behaviour near zero and near infinity of solutions to elliptic 
equalities and inequalities
% 
\thanks{ {\em Mathematics Subject Classifications:}  35J55, 35J60.
 \hfil\break\indent 
{\em Key words:} A priori estimates, non-existence results, 
degenerate quasilinear inequalities. 
 \hfil\break\indent
\copyright 2001 Southwest Texas State University. 
\hfil\break\indent Published January 8, 2001.  } } 

\date{}
\author{ Marie-Fran\c{c}oise Bidaut-V\'eron }
\maketitle
\begin{abstract} 
Here we consider elliptic equations and inequalities involving 
quasilinear operators in divergence form and nonlinear lower order 
terms:
$$-\mathop{\rm div}\left(\mathcal{A}(x,u,\nabla u)\right)
\geq |x|^\sigma u^Q\quad(Q>0,\sigma \in \mathbb{R}),
$$
in dimension $N\geq 3$. We study the asymptotic behaviour of the 
solutions and give {\it a priori} estimate and non-existence results.
\end{abstract}

\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\renewcommand{\theequation}{\thesection.\arabic{equation}}
\catcode`@=11 \@addtoreset{equation}{section} \catcode`@=12 

\section{ Introduction\label{intro}}
Here we study the existence and the asymptotic behaviour
near zero and near infinity of nonnegative solutions to elliptic problems
involving quasilinear operators in divergence form. We study equalities of
the form  
\begin{equation}
-\mathop{\rm div}\left[ \mathcal{A}(x,u,\nabla u)\right] 
=|x|^{\sigma }u^{Q}, \label{equa}
\end{equation}
and more generally inequalities of the form
\begin{equation}
-\mathop{\rm div}\left[ \mathcal{A}(x,u,\nabla u)\right] \geq |x|^{\sigma }u^{Q},
\label{sal}
\end{equation}
where $Q,\sigma \in \mathbb{R}$, $Q>0$, in an open set $\Omega $ of $\mathbb{%
R}^{N}$ $(N\geq 3)$. A great part of the results extends to systems of 
the form 
\begin{equation}
\gathered
-\mathop{\rm div}\left[ \mathcal{A}(x,u,\nabla u)\right] 
=|x|^{a}u^{S}v^{R}, \\ 
-\mathop{\rm div}\left[ \mathcal{B}(x,v,\nabla v)\right] 
=|x|^{b}u^{Q}v^{T},
\endgathered\label{syl}
\end{equation}
where $Q,R,S,T\geq 0$, and to systems of inequalities; 
see for example \cite{B1,BVP}.

Let $B_{r}=\left\{ \left| x\right| <r\right\} $ with $r>0$. Let 
$\Omega $ be either $\mathbb{R}^{N}$ or $\mathbb{R}^{N}\backslash
\left\{ 0\right\} $, or an exterior or interior domain 
\begin{equation*}
\Omega _e=\left\{ x\in \mathbb{R}^{N}\left| \;\left| x\right| >1\right.
\right\} ,\quad \Omega _i=\left\{ x\in \mathbb{R}^{N}\left| \;0<\left|
x\right| <1\right. \right\} =B_{1}\backslash \left\{ 0\right\}
\end{equation*}
or the half-space $\mathbb{R}^{N\,+}=$ $\left\{ x\in \mathbb{R}^{N}\left|
\;x_{N}>0\right. \right\} $, or 
\begin{equation*}
\Omega _e^{+}=\Omega _e\cap \mathbb{R}^{N\,+},\quad \Omega
_i^{+}=\Omega _i\cap \mathbb{R}^{N\,+}.
\end{equation*}

Our aim is to point out many results on this subject and to show 
some short proofs to some results.  We cannot present a complete survey, 
because it would be too long, we rather give references that seem to 
be significant.

\section{The Laplacian case}

We begin by the model case of the Laplace operator, with the equation 
\begin{equation}
-\Delta u=|x|^{\sigma }u^{Q},  \label{eq}
\end{equation}
or the inequality 
\begin{equation}
-\Delta u\geq |x|^{\sigma }u^{Q},  \label{de}
\end{equation}
where $\sigma \in \mathbb{R}$, $Q>0,Q\neq 1$. By solution of (\ref{eq}) or (%
\ref{de}), we mean any nonnegative function $u\in C^{0}(\Omega )\cap
W_{{\rm loc}}^{1,1}(\overline{\Omega })$ with $\Delta u\in L_{{\rm loc}}^{1}(\overline{%
\Omega })$, solution in the sense of $\mathcal{D}^{\prime }($ $\Omega )$. We
set 
\begin{equation*}
Q_{\sigma }=(N+\sigma )/(N-2).
\end{equation*}
Recall that the equation admits a particular solution of the form 
\begin{equation}
u^{\ast }=C^{\ast }|x|^{-(2+\sigma )/(Q-1)},  \label{part}
\end{equation}
for some $C^{\ast }>0$ if and only if $Q>Q_{\sigma }>1$, or $Q<Q_{\sigma
}<1. $ First remark that the problem in $\Omega _i$ or $\Omega _e$ are
equivalent to solve, and in the same way in $\Omega _i^{+}$ or $\Omega
_e^{+}$, from the Kelvin transform: setting 
\begin{equation*}
u_{0}(x)=|x|^{2-N}u(y),\quad y=x/|x|^{2}
\end{equation*}
then (\ref{de}) is equivalent to 
\begin{equation}
-\Delta u_{0}\geq |y|^{\sigma _{0}}u_{0}^{Q},\quad \sigma
_{0}=(N-2)Q-(N+2+\sigma )  \label{kt}
\end{equation}
Now let us recall the Br\'{e}zis-Lions theorem in $\Omega _i$ in its
simplest form:

\begin{theorem}[\protect\cite{BrL}]
Let $w\in L_{{\rm loc}}^{1}(\Omega _i)$ be any nonnegative superharmonic
function, such that $\Delta w\in L_{{\rm loc}}^{1}(\Omega _i)$. Then $f=\Delta
w_{/\Omega _i}$ $\in L_{{\rm loc}}^{1}(B_{1})$, $w\in M_{{\rm loc}}^{N/(N-2)}(B_{1})$, 
$\left| \nabla w\right| \in M_{{\rm loc}}^{N/(N-1)}(B_{1})$ and there exists $%
\lambda \geq 0$ such that 
\begin{equation*}
-\Delta w=-\Delta w_{/\Omega _i}+\lambda \delta _{0}\quad \text{in }%
\mathcal{D}^{\prime }(B_{1}).
\end{equation*}
\end{theorem}

Then one gets a first nonexistence result concerning inequality (\ref{de}),
given in \cite{B0}. Up to some changes of variable, in the radial case of
the equation, it comes from the study of Fowler \cite{Fo1,Fo2}, of
the equation 
\begin{equation*}
-y"=r^{\theta }y^{Q},
\end{equation*}
$\theta \in \mathbb{R}$. He made a complete description of the solutions,
with the restriction $Q,\theta $ $\in \mathbb{N}$ because the phase plane
techniques for ODE's were not known; see also \cite{Be}. This result
 was extended to the inequality in the radial case with more general 
operators by Ni and Serrin \cite{NS2}. The result was also found 
again in the case $\sigma =-2$ by \cite{BrCa}.

\begin{theorem}
\label{un} Assume $Q>1$. \\
i) There exists a nontrivial solution of (\ref{de})
in $\Omega _i$ if and only if $\sigma >-2$.
\\
ii) There exists a nontrivial solution of (\ref{de}) in $\Omega _e$ if and
only if $Q>Q_{\sigma }$.
\\
iii) There exists a nontrivial solution of (\ref{de}) in $\mathbb{R}^{N}$ or 
$\mathbb{R}^{N}\backslash \left\{ 0\right\} $ if and only if 
$Q>Q_{\sigma}$ and $\sigma >-2$.
\end{theorem}

\paragraph{Proof.}
i) and ii) For the part ''if'', the particular solution (\ref{part}) is a
solution in $\mathbb{R}^{N}\backslash \left\{ 0\right\} $, hence in $\Omega
_i$ and $\Omega _e$. For the part ''only if'', \textit{the problem
reduces to the radial one}. By Kelvin transform we reduce to the case of $%
\Omega _i$. Suppose there exists a nontrivial solution $u$ of (\ref{de}%
). Let 
\begin{equation}
\overline{u}(r)=\frac{1}{\left| S^{N-1}\right| }\int_{S^{N-1}}u(r,\theta
)\,d\theta  \label{vm}
\end{equation}
be the mean value of $u$ on the sphere of center 0 and radius $r$. Then $%
\overline{u}$ also satisfies (\ref{de}), from the Jensen inequality, that is 
\begin{equation*}
-(r^{N-1}\overline{u}_{r})_{r}\geq r^{N-1+\sigma }\overline{u}^{Q},
\end{equation*}
and $\overline{u}>0$ . Then either $\lim_{r\rightarrow 0}r^{N-1}\overline{u}%
_{r}\in \left( 0,+\infty \right] ;$ then $\lim_{r\rightarrow 0}\overline{u}$ 
$=C>0$ and we reach a contradiction. Or $\overline{u}_{r}\leq 0$ near $0$.
By integration we get 
\begin{equation*}
r^{N-1}\overline{u}_{r}+\overline{u}^{Q}\int_{0}^{r}t^{N-1+\sigma }dt
\leq 0
\end{equation*}
hence $\sigma +N>0$ and 
\begin{equation*}
\overline{u}^{-Q}\overline{u}_{r}+r^{\sigma +1}/(N+\sigma )\leq 0\,.
\end{equation*}
Integrating again it implies that $\sigma >-2$, and we have the estimate
near $0$: 
\begin{equation}
\overline{u}\leq Cr^{-(2+\sigma )/(Q-1)}.  \label{aa}
\end{equation}
iii) The part ''only if'' is obvious. For the part ''if'', when $%
Q>Q_{\sigma }$ and $\sigma >-2$, the function $u(x)=c(1+|x|^{2+\sigma
})^{-1/(Q-1)}$ is a solution of (\ref{de}) in $\mathbb{R}^{N}$, hence in $%
\mathbb{R}^{N}\backslash \left\{ 0\right\} $ if $c$ is small enough. This
example can be found in \cite{MP1} when $\sigma =0$. 
\hfill$\diamondsuit$\smallskip

Now we consider the case $Q<1$. The following was proved by \cite{Ra} for
the equation, and extended in \cite{BVRa} and \cite{B1}.

\begin{theorem}
\label{ploc}Assume $Q<1$.\\
 i)There exists a nontrivial solution of (\ref{de})
in $\Omega _i$ if and only if $Q<Q_{\sigma }$.
\\
ii) There exists a nontrivial solution of (\ref{de}) in $\Omega _e$ if and
only if $\sigma <-2$.
\end{theorem}

\paragraph{Proof.}
Assume there is a nontrivial solution in $\Omega _i$. Then $u>0$, and we
can define $w=1/u$. It is subharmonic and satisfies 
\begin{equation*}
-\Delta w+|x|^{\sigma }w^{m}\leq 0
\end{equation*}
with $m=2-Q>1$. Then from Osserman's estimate (see \cite{B1}), 
\begin{equation*}
w\leq C\left\{ 
\begin{array}{c}
|x|^{-(2+\sigma )/(m-1)}\quad \text{if }\sigma \neq -2 \\ 
|\ln |x||^{-1/(m-1)}\quad \text{if }\sigma =-2
\end{array}
\right.
\end{equation*}
in $\frac{1}{2}\Omega _i$. That means 
\begin{equation}
u\geq C\left\{ 
\begin{array}{c}
|x|^{(2+\sigma )/(1-Q)}\quad \text{if }\sigma \neq -2, \\ 
|\ln |x||^{1/(1-Q)}\quad \text{if }\sigma =-2.
\end{array}
\right.  \label{bb}
\end{equation}
But $|x|^{\sigma }u^{Q}\in L_{{\rm loc}}^{1}(B(0,1))$, hence in any case $%
Q<Q_{\sigma }$. For the part ''if'', see \cite{Ra}.
\hfill$\diamondsuit$

\begin{remark} \rm
Here also \textit{the problem could be reduced to the radial one. } Indeed we
have the following property, which proves that $\overline{u}$ satisfies an
inequality of the same form as (\ref{de}).
\end{remark}

\begin{lemma}[\protect\cite{BGm}]
\label{surhar}Let $w\in C^{2}(\Omega _i)$ be any nonnegative superharmonic
function. Then there exists a constant $C(N)>0$ such that for any
 $x\in \frac{1}{2}\Omega _i$, 
\begin{equation}
w(x)\geq C(N)\overline{w}(\left| x\right| ).  \label{NA}
\end{equation}
\end{lemma}

\begin{remark} \rm
In particular, the exterior problem 
\begin{equation*}
-\Delta u\geq u^{Q}
\end{equation*}
in $\Omega _e$ has no solution except $0$ for any $0<Q<N/(N-2)$, $Q\neq 1$.
\end{remark}

\begin{remark} \rm
The non existence results are very linked to the estimates of $\overline{u}$.
In case of $\Omega _i$ we have for any solution of (\ref{de}), from (\ref
{aa}) (\ref{bb}) and the superharmonicity, 
\begin{equation*}
\overline{u}\leq C\min (r^{-(2+\sigma )/(Q-1)},r^{2-N})\quad \text{in\ }%
\frac{1}{2}\Omega _i\quad \text{if\ }Q>1,
\end{equation*}
\begin{equation*}
C_{1}r^{(2+\sigma )/(1-Q)}\leq \overline{u}\leq C_{2}r^{2-N}\quad \text{in\ 
}\frac{1}{2}\Omega _i\quad \text{if\ }Q<1.
\end{equation*}
In the case of $\Omega _e$, it follows that 
\begin{equation*}
\overline{u}\leq C\min (r^{-(2+\sigma )/(Q-1)},1)\quad \text{in\ }2\Omega
_e\quad \text{if\ }Q>1,
\end{equation*}
\begin{equation*}
C_{1}r^{(2+\sigma )/(1-Q)}\leq \overline{u}\leq C_{2}\quad \text{in\ }%
2\Omega _e\quad \text{if\ }Q<1.
\end{equation*}
\end{remark}

Now let us come to the case of the equation. In the radial case, we have a
well-known nonexistence result in whole $\mathbb{R}^{N}$.

\begin{lemma}
\label{truc} There exists a nontrivial radial solution of (\ref{eq}) in $%
\mathbb{R}^{N}$ (that means a radial ground state) if and only if 
\begin{equation}
Q\geq Q_{\sigma }^{\ast }=\frac{N+2+2\sigma }{N-2}>1  \label{tric}
\end{equation}
\end{lemma}

\paragraph{Proof.}
Assume (\ref{tric}). First one constructs a local solution near $0$ such
that $u(0)=1$ and $u_{r}(0)=0$. By concavity it extends to a solution of the
equation 
\begin{equation*}
-\Delta u=|x|^{\sigma }\left| u\right| ^{Q-1}u
\end{equation*}
in $\left[ 0,+\infty \right) $. Now suppose that $u(r_{0})=0$ for some $%
r_{0}>0$. The change of variable (first used by Fowler) 
\begin{equation*}
u(r)=r^{-\gamma }U(t)\quad \gamma =\frac{2+\sigma }{Q-1},\quad t=-\ln r,
\end{equation*}
reduces the equation to an autonomous one: 
\begin{equation*}
U_{tt}-AU_{t}-BU+\left| U\right| ^{Q-1}U=0
\end{equation*}
with $A=N-2-2\gamma >0$ and $B=\gamma ((N-2-\gamma )>0$. Then the energy
function 
\begin{equation*}
E=\frac{U_{t}^{2}}{2}-B\frac{U^{2}}{2}+\frac{\left| U\right| ^{Q+1}}{Q+1}
\end{equation*}
is nondecreasing, since $E_{t}=AU_{t}^{2}$, with $\lim_{t\rightarrow +\infty
}E(t)=0$, and $E(-\ln r_{0})\geq 0$. Then $E(t)=E_{t}(t)=0$ for $t\geq -\ln
r_{0}$, hence $U$ is constant, and we reach a contradiction. Reciprocally
suppose there exists a ground state. Then first $\sigma >-2$. Suppose $%
Q<Q_{\sigma }^{\ast }$. Then $E$ is nonincreasing, hence nonnegative, and
bounded. Then $\lim_{t\rightarrow -\infty }E(t)=$ $L>0$ and $%
\lim_{t\rightarrow -\infty }U_{t}(t)=0$, since $U_{tt}$ is bounded and $%
\int_{-\infty }^{0}U_{t}^{2}<+\infty $. Then $\lim_{t\rightarrow -\infty
}U(t)=\ell =(B(Q+1)/2)^{1/(Q-1)}$. By linearisation $U(t)\equiv \ell $,
hence a contradiction holds.
\hfill$\diamondsuit$

\begin{remark} \rm
The existence in $\mathbb{R}^{N}\backslash \left\{ 0\right\} $ is obviously
different: there exists a nontrivial radial positive solution of (\ref{eq})
in $\mathbb{R}^{N}\backslash \left\{ 0\right\} $ if and only if $Q>Q_{\sigma
}>1$. Indeed the particular solution (\ref{part}) exists in that range.
\end{remark}

Now let us come to the nonradial case. Here the results are not complete:
they require that 
\begin{equation*}
Q\leq Q_{0}^{\ast }=\frac{N+2}{N-2},
\end{equation*}
where the well-known $Q_{0}^{\ast }$ is the limit value of $Q$ for the
compacity of the Sobolev injection from $L^{Q+1}$ into $W^{1,2}$. Or they
require additional assumptions on the behaviour at infinity, see \cite{Z}.
They require difficult techniques, either linked to the Bernstein method of
a priori estimates of $\left| \nabla u\right| ^{2}$, or to the moving plane
method of Alexandroff. The pionneer works are due to Gidas, Spruck and
Caffarelli \cite{GS}, \cite{CGS}.

\begin{theorem}[\protect\cite{GS}]
i) Assume that $1<Q<Q_{0}^{\ast }$. Then any solution in $\Omega_i$
(resp. $\Omega _e)$ satisfies 
\begin{equation}
u(x)\leq C\left| x\right| ^{-(2+\sigma )/(Q-1)}\quad \text{in }\frac{1}{2}%
\Omega _i\;\text{(resp.in }2\Omega _e\text{) }  \label{res}
\end{equation}
where $C$ does not depend on $u$.
\\
ii) Assume that $1<Q_{\sigma }<Q<Q_{0}^{\ast }$. If $Q<Q_{\sigma }^{\ast }$,
then any solution $u$ in $\mathbb{R}^{N}\backslash \left\{ 0\right\} $ is
singular at $0$. In particular there is no nontrivial nonnegative
solution in $\mathbb{R}^{N}$. If $Q>Q_{\sigma }^{\ast }$, then either $%
u=u^{\ast }$ or $u$ is a solution in $\mathbb{R}^{N}$ (ground state).
\end{theorem}

\begin{remark}\rm
The result was extended to the case $Q=Q_{0}^{\ast }$ in \cite{CGS}. When $%
Q>Q_{0}^{\ast }$ the result is not known. In the case $Q=(N+1)/(N-3)$, $%
\sigma =0$, it is shown in \cite{BV} that(\ref{res}) cannot hold with a
constant independant on $u$.
\end{remark}

Now let us give a few results concerning the case of the half-space.
Concerning the inequality (\ref{de}), the usual proofs of nonexistence lie
on the use of the first eigenvalue $\lambda _{1}=N-1$ of the Beltrami
operator on the half sphere $(S^{N-1})^{+}$ with Dirichlet conditions on $%
\partial (S^{N-1})^{+}$, and the corresponding positive normalized
eigenfunction $\phi _{1}$, and extend to cones and systems. We refer for
example to \cite{Be} and \cite{BiM} . In case of the half space, we have the
following theorem.

\begin{theorem}
Assume that $N\geq 2$, and $Q>1$.
\\
i)If $Q\leq (N+1+\sigma )/(N-1)$, the problem (\ref{de}) in $\Omega
_e^{+}, $ with $u\in C^{1}(\overline{\Omega _e^{+}})$, has only the
solution $u\equiv 0$.
\\
ii) If $Q+\sigma +1\leq 0$, the problem in $\Omega _i^{+}$, with $u\in
C^{1}(\overline{\Omega _i^{+}}\backslash \left\{ 0\right\} )$ has only the
solution $u\equiv 0$.
\end{theorem}

\paragraph{Proof.}
We follow the method of \cite{CM} given in the case $u\in C^{1}(\overline{%
\mathbb{R}^{N}}\backslash \left\{ 0\right\} )$. They still show that the
problem can be reduced to a radial one, by considering the mean value
function 
\begin{equation}
u_{\sharp }(r)=\frac{1}{\left| (S^{N-1})^{+}\right| }%
\int_{(S^{N-1})^{+}}u(r,\theta )\phi _{1}\,d\theta .
\end{equation}
Namely function $u_{\sharp }$ satisfies the inequality 
\begin{equation*}
-r^{-N}((r^{N+1}(r^{-1}u_{\sharp })_{r})_{r})=-\Delta u_{\sharp }+(N-1)\frac{%
u_{\sharp }}{r^{2}}\geq r^{\sigma }u_{\sharp }^{Q}.
\end{equation*}
By Kelvin transform we are reduced to the case of $\Omega _i^{+}$. Let $%
v=r^{-1}u_{\sharp }$. Then 
\begin{equation*}
-(r^{N+1}v_{r})_{r}\geq r^{N+Q+\sigma }v^{Q},
\end{equation*}
and $\overline{u}>0$ . Then either $\lim_{r\rightarrow 0}r^{N+1}v_{r}\in
\left( 0,+\infty \right] ;$ then $\lim_{r\rightarrow 0}v$ $=C>0$ and we
reach a contradiction. Or $v_{r}\leq 0$ near $0$. By integration we get 
\begin{equation*}
r^{N+1}v_{r}+v^{Q}\int_{0}^{r}t^{N+Q+\sigma }dt\leq 0
\end{equation*}
hence $N+Q+\sigma >0$ and 
\begin{equation*}
v^{-Q}v_{r}+r^{\sigma +Q}/(N+Q+\sigma )\leq 0.
\end{equation*}
Integrating again it implies that $\sigma +Q+1>0$, and we have the estimate
near $0:$ 
\begin{equation*}
u_{\sharp }\leq Cr^{-(2+\sigma )/(Q-1)}.
\end{equation*}
\hfill$\diamondsuit$\smallskip

In the case of the equation (\ref{eq}), Gidas and Spruck have obtained a
better result:

\begin{theorem}[\protect\cite{GS1}]
Assume that $Q<(N+2)/(N-2)$. Then equation (\ref{eq}) with $\sigma =$0
has no nontrivial solution in $\mathbb{R}^{N+}$.
\end{theorem}

\section{The p-Laplacian case}

Now we consider the case of the $p$-Laplace operator ($p>1$): 
\begin{equation}
-\Delta _{p}u=-\mathop{\rm div}(\left| \nabla u\right| ^{p-2}\nabla u)=|x|^{\sigma }u^{Q}
\label{eqp}
\end{equation}
and the inequality
\begin{equation}
-\Delta _{p}u\geq |x|^{\sigma }u^{Q}.  \label{dep}
\end{equation}
In the radial case, the first estimates concerning (\ref{dep}) are due to
Guedda and V\'{e}ron \cite{GuV}, where they give the behaviour in $\Omega
_i$ and some global properties; and the first nonexistence results are
given in \cite{NS}, \cite{NS2}. Then the non-radial case was studied in \cite
{B0}, where one can also find a complete description of the radial
case.

Here one cannot use any Kelvin transform, so that the behaviour at infinity
cannot reduced to the behaviour near $0$. Also one cannot use the mean value
of $u$ since the problem is not linear. But many of the results can be
extended. The equation has a particular solution 
\begin{equation}
u^{\ast }(x)=C^{\ast }|x|^{-\Gamma },\quad \Gamma =\frac{p+\sigma }{Q-p+1}%
,\quad C^{\ast }>0,  \label{*}
\end{equation}
if and only if $Q>Q_{\sigma ,p}>p-1$, or $Q<Q_{\sigma ,p}<p-1$, where 
\begin{equation}
Q_{\sigma ,p}=(N+\sigma )(p-1)/(N-p).  \label{qsi}
\end{equation}
First theorem \ref{un} extends to the $p$-Laplacian. This was proved in \cite
{B0} for equation ( \ref{eqp}) in $\Omega _e$ without mentioning the
critical case $Q=Q_{\sigma ,p}$, but the proof extends to the general case
and we reproduce it here. The idea is the following: if (\ref{dep}) has a
solution $u$ in $\Omega _i$ or $\Omega _e$, then we can construct a 
\textit{radial} solution of (\ref{eqp}) which is less
than $u$. So that \textit{we still are reduced to the radial case}, and with
an equation.

\begin{theorem}
\label{unp} Assume $Q>p-1$. \\
i) There exists a nontrivial solution of (\ref{dep}) in $\Omega _i$ 
if and only if $\sigma >-p$.
\\
ii) There exists a nontrivial solution of (\ref{dep}) in $\Omega _e$ if
and only if $Q>Q_{\sigma ,p}$.
\end{theorem}

\paragraph{Proof.}
Let us prove for example ii). Suppose that $Q\leq Q_{\sigma ,p}$ and that (%
\ref{dep}) has a nontrivial solution $u$. Then $u>0$ from the strong maximum
principle. Let $m=\min_{\left| x\right| =2}u(x)$. Let $n\in \mathbb{N}^{\ast
}$ be fixed, such that $n>2$. By minimization we construct a sequence 
$(u_{n,k})_{k\in \mathbb{N}}$ of radial nonnegative functions with 
$u_{n,0}\equiv 0$ and 
\begin{gather*}
-\Delta _{p}u_{n,k}=|x|^{\sigma }u_{n,k-1}\text{ \quad for }2<\left|
x\right| <n, \\ 
u_{n,k}=m\quad \text{for }\left| x\right| =2, \\ 
u_{n,k}=0\quad  \text{for }\left| x\right| =n\,.
\end{gather*}
Then $0<$ $u_{n,k}\leq u_{n,k+1}$ $\leq u$ for $2<\left| x\right| <n$. And $%
(r^{N-1}\left| (u_{n,k})_{r}\right| ^{p-2}$ $(u_{n,k})_{r})_{_{k\in \mathbb{N%
}}}$ is equi-continuous on $\left[ 2,n\right] $. Thus it converges in $C^{1}(%
\left[ 2,n\right] )$ to a radial fuction $u_{n}$ such that $u_{n,k}\leq
u_{n} $ $\leq u$ and 
\begin{gather*}
-\Delta _{p}u_{n}=|x|^{\sigma }u_{n}\quad\text{for }2<\left| x\right| 
<n\,, \\ 
u_{n}=m\quad\text{for }\left| x\right| =2, \\ 
u_{n}=0\quad\text{for }\left| x\right| =n.
\end{gather*}
By extraction of a diagonal sequence, there is a subsequence of $%
(u_{n})_{n\in \mathbb{N}}$ converging in 
$C_{{\rm loc}}^{1}(\left[ 2,+\infty\right] )$ to a radial solution $w$ 
of equation (\ref{eqp}) in $2\Omega _e$, nontrivial, since $w=m$ for 
$\left| x\right| =2$. But the radial equation
has no solution when $Q\leq Q_{\sigma ,p}$, by an argument analogous to the
one of theorem \ref{un} $\medskip$. 
\hfill$\diamondsuit$\smallskip

The theorem \ref{ploc} extends immediately to the case of the $p$-Laplacian,
by the same proof, since Osserman's estimate extends.

\begin{theorem}
\label{vil} Assume $Q<p-1$. \\
i) There exists a nontrivial solution of (\ref{dep}) in 
$\Omega _i$ if and only if $Q<Q_{\sigma ,p}$.
\\
ii) There exists a nontrivial solution of (\ref{dep}) in $\Omega _e$ if
and only if $\sigma <-p$.
\end{theorem}

Now let us come to upper estimates. First the Br\'{e}zis-Lions lemma
extends in the following form, where for simplicity we supposed $p<N$.

\begin{theorem}[\protect\cite{B0}]
\label{BVlemma}Let $1<p<N$. Let $w\in C(\Omega _i)$ be any nonnegative
super-$p$-harmonic function, such that $\Delta _{p}w\in L_{{\rm loc}}^{1}(\Omega
_i)$. Then $f=-\Delta _{p}w_{/\Omega _i}\in L_{{\rm loc}}^{1}(B_{1})$, $%
w^{p-1}\in M_{{\rm loc}}^{N/(N-p)}(B_{1})$, $\left| \nabla w\right| ^{p-1}\in
M_{{\rm loc}}^{N/(N-1)}(B_{1})$ and there exists $\lambda \geq 0$ such that 
\begin{equation}
-\Delta _{p}w=-\Delta _{p}w_{/\Omega _i}+\lambda \delta _{0}\quad \text{%
in }\mathcal{D}^{\prime }(B_{1}).  \label{dis}
\end{equation}
\end{theorem}

\paragraph{Proof.}
It is divided in four steps.
\\[2pt]
i) Function $f$ is in $L^{1}(B_{1})$. In order to obtain estimates on $%
f, $ the idea is to multiply the inequality by a function $P(u)\varphi $,
with $\varphi $ with compact support in $B_{1}$, and $P$ \textit{is
decreasing in }$u$, in order to obtain some coercivity. One takes $%
P(u)=(n+1-u)^{+}$, with $n\in \mathbb{N}.$
\\[2pt]
ii) Function $w$ is in $L_{{\rm loc}}^{k}(B_{1})$ for any $1\leq k<N/(N-p$)
and it satisfies the integral estimate for $\rho \leq 1/2:$%
\begin{equation}
\int_{B\rho }w^{k}\leq C\rho ^{N-(N-p)k/(p-1)}  \label{inte}
\end{equation}
Here we use a test function introduced by Serrin \cite{S1} and capacity
methods in order to estimate $\min_{\left| x\right| =\rho }w(x)$, and then
the weak Harnack inequality.
\\[2pt]
iii) Function $\left| \nabla w\right| ^{p-1}$ is $L_{{\rm loc}}^{k}(B_{1}$)
for any $1\leq k<N/(N-1)$ and also satisfies an integral inequality.
\\[2pt]
iv) The Marcinkiewicz estimates and (\ref{dis}) hold. Here we use ideas of
P. B\'{e}nilan.
\hfill$\diamondsuit$\smallskip

This showed that we can obtain some integral estimates on $w$, even for a
nonlinear problem, replacing the estimates of the mean value for the
Laplacian. Indeed defining for any nonnegative $g$ on $\Omega $ and any $%
\omega \subset \Omega $%
\begin{equation*}
\oint_{\omega }g=\frac{1}{\left| \omega \right| }\int_{\omega }g
\end{equation*}
then (\ref{inte}) can be written 
\begin{equation*}
\Big( \oint_{B(0,\rho )}w^{k}\Big) ^{1/k}\leq C\rho ^{-(N-p)/(p-1)},
\end{equation*}
which extends the classical estimate $\overline{u}(r)\leq Cr^{2-N}$ in case
of the Laplacian. This was a motivation to extend also the estimate $%
\overline{u}(r)\leq Cr^{-(2+\sigma )/(Q-1)}$ of the problem (\ref{de}) to
the problem (\ref{dep}) and more general operators. One gets the following,
where $\mathcal{C}_{\rho _{1},\rho _{2}}=\left\{ \rho _{1}<\left| x\right|
<\rho _{2}\right\} $.

\begin{theorem}[\protect\cite{BVP}]
\label{kle}Assume that $N\geq p>1$. Let $u$ be a nonnegative solution of (%
\ref{dep}) in $\Omega _i$ $($resp. $\Omega _e)$. \\
i) If $Q>p-1$, then for
small $\rho $ (resp. for large $\rho )$ 
\begin{equation}
\Big( \oint_{\mathcal{C}_{\rho /2,\rho }\mathcal{\ }}u^{Q}\Big)
^{1/Q}\leq C\rho ^{-\Gamma }.  \label{ab}
\end{equation}
ii) If $Q<p-1$, either $u\equiv 0$, or 
\begin{equation}
u(x)\geq C\left| x\right| ^{-\Gamma }\text{\quad in }\frac{1}{2}\Omega
_i\quad (\text{resp. in }2\Omega _e).  \label{mi}
\end{equation}
\end{theorem}

\paragraph{Proof.}
We just give the proof of i). Let $u$ be a nontrivial solution of 
(\ref{dep}) , hence $u>0$. Let $1-p<\alpha <0$ . 
By computation the function $u_{\alpha }=u^{1+\alpha /(p-1)}$
 is also superharmonic and satisfies 
\begin{equation*}
-\Delta _{p}u_{\alpha }\geq C(\alpha )\left( \left| x\right| ^{\sigma
}u^{Q+\alpha }+u^{\alpha -1}\left| \nabla u\right| ^{p}\right)
\end{equation*}
for some $C(\alpha )>0$. Then we multiply by a test function $\varphi =\xi
^{\lambda }$ with $\lambda $ large enough, and $\xi \in $ $\mathcal{D}
(\Omega )$ with values in $\left[ 0,1\right] $, such that $\xi =1$
for $\rho /2\leq \left| x\right| \leq \rho $ and 
$\left| \nabla \xi \right|\leq C/\rho $. We get 
(with other constants $C=C(\alpha ,\lambda )$)
\begin{eqnarray*}
\int_{\Omega _i}\left| x\right| ^{\sigma }u^{Q+\alpha }\xi ^{\lambda
}+\int_{\Omega _i}u^{\alpha -1}\left| \nabla u\right| ^{p}\xi ^{\lambda }
&\leq & C\int_{\Omega _i}\left| \nabla u_{\alpha }\right| ^{p-1}\xi
^{\lambda -1}\left| \nabla \xi \right|\\
&\leq& C\int_{\Omega _i}u^{\alpha}\left| \nabla u\right| ^{p-1}
\xi ^{\lambda -1}\left| \nabla \xi \right|
\end{eqnarray*}
and setting $\theta =Q/(p-1+\alpha )>1$ we get from the H\"{o}lder
inequality 
\begin{eqnarray}
\lefteqn{\int_{\Omega _i}\left| x\right| ^{\sigma }u^{Q+\alpha }
\xi ^{\lambda}+\int_{\Omega _i}u^{\alpha -1}\left| \nabla u\right| ^{p}
\xi ^{\lambda}  }\nonumber\\
&\leq& C\left( \int_{\Omega _i}u^{Q}\xi ^{\lambda }\right) ^{1/\theta
}\left( \int_{\Omega _i}\xi ^{\lambda -p\theta'}\left| \nabla
\xi \right| ^{p\theta'}\right) ^{1/\theta'}  
\label{hui}
\end{eqnarray}
Now we take $\xi ^{\lambda }$ as test function directly in (\ref{dep}) and
get by using the same $\alpha $
\begin{eqnarray*}
\int_{\Omega _i}\left| x\right| ^{\sigma }u^{Q}\xi ^{\lambda } &\leq
&\lambda \int_{\Omega _i}\left| \nabla u\right| ^{p-1}\xi ^{\lambda
-1}\left| \nabla \xi \right| \\
&\leq &\lambda \int_{\Omega _i}u^{(\alpha -1)/p^{\prime }}\left| \nabla
u\right| ^{p-1}u^{(1-\alpha )/p^{\prime }}\xi ^{\lambda -1}\left| \nabla \xi
\right| \\
&\leq &C\left( \int_{\Omega _i}u^{\alpha -1}\left| \nabla u\right| ^{p}\xi
^{\lambda }\right) ^{1/p^{\prime }}\left( \int_{\Omega _i}u^{(1-\alpha
)(p-1)}\xi ^{\lambda -p}\left| \nabla \xi \right| ^{p}\right) ^{1/p}
\end{eqnarray*}
And from (\ref{hui}), choosing $\alpha $ small enough such that $\tau
=Q/(1-\alpha )(p-1)>1$ , 
\begin{eqnarray}
\int_{\Omega _i}u^{Q}\xi ^{\lambda } &\leq &C\rho ^{-\sigma }\left(
\int_{\Omega _i}u^{Q}\xi ^{\lambda }\right) ^{1/\theta p^{\prime }+1/\tau
p}\times  \notag \\
&&\left( \int_{\Omega }\xi ^{\lambda -\theta'p}\left| \nabla \xi
\right| ^{\theta'p}\right) ^{1/\theta'p^{\prime }}\left(
\int_{\Omega }\xi ^{\lambda -\tau'p}\left| \nabla \xi \right|
^{\tau'p}\right) ^{1/\tau'p}.  \label{bla}
\end{eqnarray}
And $1/\theta p^{\prime }+1/\tau p=(p-1)/Q=1-(1/\theta'p^{\prime
}+1/\tau'p)$, hence (\ref{ab}) follows.
\hfill$\diamondsuit$

\begin{remark} \rm
In the case $Q>p-1$, Theorem \ref{unp} can be found again in a longer
way by using these upper estimates. Indeed following the technique of
comparison of theorem \ref{unp}, one can prove lower estimates. Consider the
radial elementary $p$-harmonic functions in $\mathbb{R}^{N\,}\backslash
\left\{ 0\right\} $, that means functions 
\begin{equation*}
\Phi _{1,p}(r)\equiv 1,\quad \Phi _{2,p}(r)=\left\{ 
\begin{array}{ll}
r^{(p-N)/(p-1)}& \text{if }N>p, \\ 
\ln r& \text{if }N=p\,.
\end{array}
\right.
\end{equation*}
Then any super-$p$-harmonic function $u$ in $\Omega _i$ (resp. $\Omega
_e)$ satisfies 
\begin{equation*}
u\geq C\Phi _{1,p}\quad \text{in }\frac{1}{2}\Omega _i\quad (\text{%
resp.\quad }u\geq C\Phi _{2,p}\quad \text{in }2\Omega _e)\,;
\end{equation*}
see \cite{BVP}.
\end{remark}

Above all, the integral estimates can give \textit{punctual estimates} in
the case of the equation (\ref{eqp}), in the subcritical case. The following
is proved in \cite{B0} when $\sigma =0$, and in \cite{BVP} in the general
case.

\begin{theorem}
\label{souk}Assume that $N\geq p>1$. Let $u$ be a nonnegative solution 
of (\ref{eqp}) in $\Omega _i$. Assume that 
\begin{equation*}
0<Q<Q_{0,p}=N(p-1)/(N-p).
\end{equation*}
Then $u$ satisfies the Harnack inequality. Consequently, if $Q>p-1$, 
\begin{equation}
u(x)\leq C\min (\left| x\right| ^{-\Gamma },\left| x\right| ^{(p-N)/(p-1)})%
\text{\quad in }\frac{1}{2}\Omega _i;  \label{glic}
\end{equation}
if $Q<p-1$, then 
\begin{equation*}
u(x)\leq C\left| x\right| ^{(p-N)/(p-1)})\text{\quad in }\frac{1}{2}
\Omega _i.
\end{equation*}
\end{theorem}

\paragraph{Proof.}
First suppose $Q>p-1$. We write the equation under the form 
\begin{equation*}
-\Delta _{p}u=h\;u^{p-1},\quad h=\left| x\right| ^{\sigma }u^{Q-p+1}.
\end{equation*}
If $\sigma =0$, we remark that $u^{Q}\in L^{1}(B_{1/2})$ from the 
Br\'{e}zis-Lions theorem. Hence $h^{s}\in L^{1}(B_{1/2})$ for 
$s=Q/(Q-p+1)>N/p$, since $Q<Q_{0}$. Then we can apply Serrin's 
results of \cite{S1}, and conclude. 
In the general case $\sigma \in \mathbb{R}$, we use the estimate 
(\ref{ab})
\begin{equation}
\int_{\mathcal{C}_{\rho /2,\rho }}h^{s}=\int_{\mathcal{C}_{\rho /2,\rho
}}\left| x\right| ^{\sigma s}u^{Q}\leq \rho ^{\sigma s}\int_{\mathcal{C}%
_{\rho /2,\rho }}u^{Q}\leq C\rho ^{N+\sigma s-\Gamma Q}=C\rho ^{N-ps}.
\label{hi}
\end{equation}
This implies the Harnack inequality, and (\ref{glic}) follows. \medskip\ Now
suppose $Q\leq p-1$. We observe that $h(x)\leq C\left| x\right| ^{-p}$
near $0$, from (\ref{mi}) if $Q<p-1$. Then $h$ satisfies (\ref{hi}) for any $%
s>1$, and the Harnack inequality still holds.
\hfill$\diamondsuit$\smallskip

As in the case $p=2$, the question of the estimates is harder in the case $%
Q>Q_{0,p}$. Serrin and Zou have announced in January 2000 the following
beautiful result, which extends the one of \cite{GS} and of \cite{B0}:

\begin{theorem}[\protect\cite{SZ}]
Assume that $1<Q<Q_{0,p}^{\ast }=(N(p-1)+p)/(N-p)$. Then any solution of (%
\ref{eqp}) with $\sigma =0$ in $\Omega _i$ satisfies 
\begin{equation}
u(x)\leq C\left| x\right| ^{-p/(Q+1-p)}\text{\quad in }\frac{1}{2}\Omega
_i;
\end{equation}
where $C$ does not depend on $u$, and $u$ satisfies the Harnack inequality.
Moreover there is no nontrivial nonnegative solution in $\mathbb{R}^{N}$.
\end{theorem}

At last we consider the case of an halfspace. First following the ideas of
the proof of theorem \ref{kle}, we get upper estimates when $Q>p-1$:

\begin{theorem}
Assume that $N\geq p>1$, $Q>p-1$. Let $u$ be a nonnegative solution of (\ref
{dep}) in $\Omega _i^{+}($resp. $\Omega _e^{+})$ . Let $K_{a}=\left\{
x\in \mathbb{R}^{N\,+}\right| \left. x_{N}\geq a\left| x\right| \right\}$ 
for any $a>0$. Then for small $\rho $ (resp. for large $\rho )$ 
\begin{equation}
\Big( \oint_{K_{a}\cap \mathcal{C}_{\rho /2,\rho }\mathcal{\ }}u^{Q}\Big)
^{1/Q}\leq C\rho ^{-\Gamma }.
\end{equation}
\end{theorem}

Here also we can find lower estimates by comparison to the $p$-harmonic
functions which vanish on the set $x_{N}=0$. In the case $p=2$, they are
given by $x\longmapsto x_{N}$ and $x\longmapsto x_{N}/\left| x\right| ^{N}$.
In the general case, they are given by 
\begin{equation}
\Psi _{1,p}(x)=x_{N},\quad \Psi _{2,p}(x)=\frac{\varpi (x/\left| x\right| )%
}{\left| x\right| ^{\beta _{p,N}}},  \label{psi}
\end{equation}
for some unique $\beta _{p,N}>0$ and $\varpi \in C^{1}(S^{N-1})$, $\varpi
>0, $ with maximum value $1$, from \cite{KV}. The exact value of $\beta
_{p,N}$ is unknown if $p\neq 2$, except in the case $N=2$. We prove that any
super-$p $-harmonic function $u$ in $C^{1}(\overline{\Omega _i^{+}}%
\backslash \left\{ 0\right\} )$ (resp. $C^{1}(\overline{\Omega _e^{+}})$)
satisfies 
\begin{equation*}
u\geq C\Psi _{1,p}\quad \text{in }\frac{1}{2}\Omega _i\quad (\text{%
resp.\quad }u\geq C\Psi _{2,p}\quad \text{in }2\Omega _e).
\end{equation*}
So that we deduce a new nonexistence result:

\begin{theorem}[\protect\cite{BVP}]
Assume that $N\geq p>1$, and $Q>p-1.$\\
i) If $Q<q_{\sigma,p}$, where 
\begin{equation*}
q_{\sigma ,p}=p-1+(p+\sigma )/\beta _{p,N},
\end{equation*}
the problem (\ref{dep}) in $\Omega _e^{+}$, with $u\in C^{1}(\overline{%
\Omega _e^{+}})$, has only the solution $u\equiv 0$.
\\
ii) If $Q+\sigma +1<0$, the problem in $\Omega _i^{+}$, with 
$u\in C^{1}(\overline{\Omega _i^{+}}\backslash \left\{ 0\right\} )$ has
only the solution $u\equiv 0$.
\end{theorem}

\section{More general operators}

Some of the above results are still valid for problems of the form 
\begin{equation}
-\mathop{\rm div}\left[ \mathcal{A}(x,u,\nabla u)\right] =|x|^{\sigma }u^{Q},  \label{alo}
\end{equation}
or 
\begin{equation}
-\mathop{\rm div}\left[ \mathcal{A}(x,u,\nabla u)\right] \geq |x|^{\sigma }u^{Q},
\label{ali}
\end{equation}
where $\mathcal{A}:\Omega \times \mathbb{R}^{+}\times
 \mathbb{R}^{N}\rightarrow \mathbb{R}^{N}$ is a Caratheodory function, 
satisfying suitable assumptions. The radial case has been studied by
 many authors, among them we refer to \cite{PS, CM}.

We shall say that $\mathcal{A}$ is strongly $p$-coercive if 
\begin{equation}\gathered
\left| \mathcal{A}(x,u,\eta )\right|\leq K_{1}\left| \eta \right |^{p-1},\\ 
\mathcal{A}(x,u,\eta )\eta \geq K_{2}\left| \eta \right| ^{p}\,.
\endgathered\label{S}
\end{equation}
for some $K_{1},K_{2}>0$, and for all $(x,u,\eta )\in \Omega \times \mathbb{R%
}^{+}\times \mathbb{R}^{N}$. Up to some variants, the condition (\ref{S}) is
a classical frame for the study of quasilinear operators, see \cite{S1}.
It implies the weak Harnack inequality, and hence the strong
maximum principle.

We shall say that $\mathcal{A}$ is weakly $p$-coercive if 
\begin{equation}
\mathcal{A}(x,u,\eta ).\eta \geq 
K\left| \mathcal{A}(x,u,\eta )\right|^{p'}  \label{W}
\end{equation}
for some $K>0$, and for all $(x,u,\eta )\in \Omega \times \mathbb{R}%
^{+}\times \mathbb{R}^{N}$. This condition (\ref{W}) is clearly weaker than (%
\ref{S}), and does not imply the Harnack inequality. It is satisfied in
particular by the mean curvature operator 
$u\mapsto -\mathop{\rm div}(\nabla u/\sqrt{1+\left| \nabla u\right|^{2}})$
 with $p=2$.

\subsection*{Operators with a weak coercivity}

For a general weakly $p$-coercive operator, first we can extend 
Theorem \ref{kle}.

\begin{theorem}[\protect\cite{BVP}]
\label{kle2} Assume that $N\geq p>1$, and $\mathcal{A}$ is weakly 
$p$-coercive. Let $u$ be a nonnegative solution of (\ref{ali}) in 
$\Omega _i$ (resp. $\Omega _e$). If $Q>p-1$, then (\ref{ab}) holds for 
small $\rho$ (resp. for large $\rho )$. If $Q<p-1$, for any $\ell >p-1-Q$,
 then 
\begin{equation}
\Big( \oint_{\mathcal{C}_{\rho /2,\rho }\mathcal{\ }}u^{\ell }\Big)
^{1/\ell }\geq C\rho ^{-\Gamma }.
\end{equation}
\end{theorem}

\paragraph{Proof.}
It is an extension of the one of theorem \ref{kle}: we multiply the
inequality by $u^{\alpha }\varphi $, where $1-p<\alpha <0$ , and $\varphi$ 
is a test function, in order to get coercivity, then directly by $\varphi $.
\hfill$\diamondsuit$\smallskip

Then one can give nonexistence results in whole $\mathbb{R}^{N}$:

\begin{theorem}[\protect\cite{BVP}]
\label{rn} Assume that $N\geq p>1$, $Q>p-1$, and $\mathcal{A}$ is weakly $p$%
-coercive. If $Q\leq Q_{\sigma ,p}$, there exists no nontrivial solution of (%
\ref{ali}) in $\mathbb{R}^{N}$ .
\end{theorem}

\paragraph{Proof.} From the a priori estimate of theorem \ref{kle2}, one 
deduces 
\begin{equation*}
\int_{B_{\rho }}\left| x\right| ^{\sigma }u^{Q}\leq C\rho ^{\theta }
\end{equation*}
with $\theta =(N-p)(Q-Q_{\sigma })/(Q-p+1)\leq 0$. If $\theta <0$, then as $%
\rho \rightarrow +\infty $, we deduce that $\int_{\mathbb{R}^{N}}|x|^{\sigma
}u^{Q}=0$, hence $u\equiv 0$. If $\theta =0$, then $|x|^{\sigma }u^{Q}\in
L^{1}(\mathbb{R}^{N})$, hence $\lim \int_{\mathcal{C}_{2^{n},2^{n+1}}}|x|^{%
\sigma }u^{Q}=0$. And we show that 
\begin{equation*}
\int_{B_{2^{n}}}|x|^{\sigma }u^{Q}\leq C\Big( \int_{\mathcal{C}%
_{2^{n},2^{n+1}}}|x|^{\sigma }u^{Q}\Big) ^{(p-1)/Q}
\end{equation*}
hence again $u\equiv 0$.
\hfill$\diamondsuit$\smallskip

For some weakly $p$-coercive operators which only depend on the gradient of $%
u$, we can also extend the nonexistence results in $\Omega _i$ and $\Omega
_e$.

\begin{theorem}[\protect\cite{BVP}]
Assume that $\mathcal{A}(x,u,\eta )=A(\left| \eta \right| )\eta $, with 
$t\mapsto A(t)t$ non-decreasing and 
\begin{equation} \gathered 
A(t)\leq Mt^{p-2},\quad \text{for } t>0, \\ 
A(t)\geq M^{-1}t^{p-2}\quad \text{for small }t>0,
\endgathered   \label{ta}
\end{equation}
for some $M>0$. If $\sigma \leq -p$ , there exists no nontrivial solution of
(\ref{ali}) in $\Omega _i$. If $Q\leq Q_{\sigma ,p}$, there exists no
nontrivial solution of (\ref{ali}) in $\Omega _e$.
\end{theorem}

The result applies in particular to the mean curvature operator with $p=2$.

\subsection*{Operators with a strong coercivity}

For a general strongly $p$-coercive operator, one can give nonexistence
results in $\mathbb{R}^{N}\backslash \left\{ 0\right\} $. The method is a
combination of the two techniques of multiplication, either by 
$u^{\alpha}(\alpha <0)$ or by $(k-u)^{+}(k>0)$.

\begin{theorem}[\protect\cite{BVP}]
\label{rno}Assume that $N\geq p>1$, $Q>p-1$, and$\mathcal{A}$ is weakly 
$p$-coercive. If $Q<Q_{\sigma ,p}$, there exists no nontrivial solution 
of (\ref{ali}) in $\mathbb{R}^{N}\backslash \left\{ 0\right\} $.
\end{theorem}

Moreover Theorems \ref{BVlemma} and \ref{souk} extend
completely, see \cite{B00} (with more general assumptions on 
$\mathcal{A}$) and \cite{BVP}. 
This problem with  $Q\geq Q_{0,p}$ for such operators is open.

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\end{thebibliography}

\noindent{\sc Marie-Fran\c{c}oise Bidaut-V\'eron} \\ 
Laboratoire de Math\'ematiques et Physique Th\'eorique \\
CNRS UMR 6063\\
Facult\'e des Sciences et Techniques\\
Parc de Grandmont \\
F 37200 Tours France\\
e-mail: veronmf@balzac.univ-tours.fr
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