\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2015 (2015), No. 182, pp. 1--7.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2015 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2015/182\hfil A Liouville type theorem]
{A Liouville type theorem for $p$-Laplace equations}

\author[C. Enache \hfil EJDE-2015/182\hfilneg]
{Cristian Enache}

\address{Cristian Enache \newline
Simion Stoilow Institute of Mathematics of the Romanian Academy,
010702,\newline  Bucharest, Romania}
\email{cenache23@yahoo.com}

\thanks{Submitted October 18, 2014. Published July 1, 2015.}
\subjclass[2010]{70H25}
\keywords{p-Laplace equation; Liouville theorem; entire solutions}

\begin{abstract}
 In this note we study solutions defined on the whole space $\mathbb{R}^N$
 for the  $p$-Laplace equation
 $$
 \operatorname{div}(| \nabla u| ^{p-2}\nabla u)+f(u)=0.
 $$
 Under an appropriate condition on the growth of $f$, which is weaker
 than conditions previously considered in McCoy \cite{McC07} and
 Cuccu-Mhammed-Porru \cite{CMP10}, we prove the non-existence of non-trivial
 positive solutions.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\allowdisplaybreaks

\section{Introduction}

In this note we  improve some Liouville type results previously
obtained in  McCoy \cite{McC07} and  Cuccu-Mohammed-Porru \cite{CMP10} 
for solutions to the $p$-Laplace equation
\begin{equation}
\operatorname{div}( | \nabla u| ^{p-2}\nabla u) +f(u)=0\quad
\text{in }\mathbb{R}^{N},\; p>1,   \label{e1.1}
\end{equation}
where the nonlinearity $f$ is a real differentiable function.

The classical Liouville Theorem states that 
\emph{any harmonic function on
the whole Euclidian space} $\mathbb{R}^{N}$, $N\geq 2$, \emph{which is bounded
from one side, must be identically constant}.
Nowadays it is already known
that this property is not anymore a prerogative of harmonic functions, since
it is also shared by bounded (from below and/or above) entire solutions to
many other differential equations (we refer the reader to the survey paper
of  Farina \cite{Fa07} for an overview on Liouville type theorems in
PDEs). For instance, when $p=2$ in \eqref{e1.1},  McCoy \cite{McC07} has
proved that \emph{if }$f$\emph{ is differentiable and satisfies}
\begin{equation}
f'(t) \leq \frac{N+1}{N-1}\frac{f(t) }{t}\quad
\text{for all }t>0,    \label{e1.2}
\end{equation}
\emph{then any positive solution of \eqref{e1.1} must be a constant}.
 Later, this result was extended to the more general case $p>1$ by 
 Cuccu,  Mohammed and  Porru \cite{CMP10}, as follows:
 \emph{if} $f$ \emph{is differentiable and satisfies}
\begin{equation}
f'(t) \leq (p-1) \frac{N+1}{N-1}\frac{f(t) }{t}\quad \text{for all }t>0, \label{e1.3}
\end{equation}
\emph{then any positive solution of \eqref{e1.1} must be a constant.}

Adapting the main idea from the above mentioned works, we are going to
show that, under a weaker condition on the growth of $f$, the above
Liouville type results still hold. More precisely, we have:


\begin{theorem} \label{thm1.1}
Assume that $u(\mathbf{x})>0$ satisfies \eqref{e1.1}. 
If $f$ is differentiable and satisfies
\begin{equation}
f'(t) \leq \beta (p-1) \frac{N+1}{N-1}\frac{f(t) }{t}\quad \text{for all }t>0, 
\label{e1.4}
\end{equation}
where
\begin{equation}
\beta \in \begin{cases}
[ 1,\frac{N-1}{N-p}), &\text{when }1<p<N,  \\
[ 1,\infty ), &\text{when }p\geq N,
\end{cases}  \label{e1.5}
\end{equation}
then $u(\mathbf{x}) $ must be a constant. As a
consequence, if $f(t)=0$ has no positive roots, then \eqref{e1.1} has no
positive weak solutions.
\end{theorem}

\section{Proof of Theorem \ref{thm1.1}}

We first state a lemma which plays some role in the proof of Theorem \ref{thm1.1}.

\begin{lemma}[Cuccu-Mohammed-Porru \cite{CMP10}] \label{lem2.1}
Let $p>1$ be a real number and $N\geq 2$. If $u( x) $ is a $C^2$
function and $u_{i}$ denotes partial differentiation with respect to $x_{i}$,
then
\begin{equation}
(p-1) u_{11}^2+\sum_{i=2}^N
u_{ii}^2\geq \frac{(p-1) ( N-1) +1}{N-1}u_{11}^2-
\frac{2}{N-1}\Delta uu_{11}+\frac{1}{N-1}( \Delta u) ^2.
 \label{e2.1}
\end{equation}
\end{lemma}

Let us now introduce the  auxiliary function
\begin{equation}
P(u;\mathbf{x}):=\frac{| \nabla u(\mathbf{x})| ^2}{
u^{2\beta }(\mathbf{x})}.    \label{e2.2}
\end{equation}
Let us also consider a point $\mathbf{x}^{\ast }$ where 
$| \nabla u| >0$. From a seminal work of  Tolksdorf \cite{To84} we know
that $u(\mathbf{x}) $ is smooth in $\omega :=\{ \mathbf{x}
\in \Omega :| \nabla u| (\mathbf{x})>0\} $. Therefore, we may compute 
in $\omega $, successively, the following derivatives:
\begin{gather}
P_{k}=\frac{2}{u^{2\beta }}u_{ik}u_{i}-\frac{2\beta }{u^{2\beta +1}}
| \nabla u| ^2u_{k}  \label{e2.3}
\\
\begin{aligned}
P_{kl}&=  \frac{2}{u^{2\beta }}u_{ik}u_{il}+\frac{2}{u^{2\beta }}
u_{ikl}u_{i}-\frac{4\beta }{u^{2\beta +1}}u_{ik}u_{i}u_{l}-\frac{4\beta }{
u^{2\beta +1}}u_{il}u_{i}u_{k}  \\
&\quad +\frac{2\beta ( 2\beta +1) }{u^{2\beta +2}}| \nabla
u| ^2u_{k}u_{l}-\frac{2\beta }{u^{2\beta +1}}| \nabla
u| ^2u_{kl}.
\end{aligned}  \label{e2.4}
\end{gather}
Now, performing eventually a translation and/or rotation if necessary, we
choose the coordinate axes such that at $\mathbf{x^{\ast }}$ we have
\begin{equation}
| \nabla u| =u_{1}\,\quad u_{i}=0\quad \text{for }i=2,\dots ,N. \label{e2.5}
\end{equation}
Using \eqref{e2.5} in \eqref{e2.4} we find that
\begin{equation}
\begin{aligned}
P_{11}&=  \frac{2}{u^{2\beta }}u_{1i}u_{1i}+\frac{2}{u^{2\beta }}
u_{111}u_{1}-\frac{8\beta }{u^{2\beta +1}}u_{11}u_{1}^2  \\
&\quad +\frac{2\beta ( 2\beta +1) }{u^{2\beta +2}}u_{1}^{4}
-\frac{2\beta }{u^{2\beta +1}}u_{11}u_{1}^2,
\end{aligned} \label{e2.6}
\end{equation}
respectively
\begin{equation}
\begin{aligned}
\Delta P&=  \frac{2}{u^{2\beta }}u_{ik}u_{ik}+\frac{2}{u^{2\beta }}
(\Delta u) _{1}u_{1}-\frac{8\beta }{u^{2\beta +1}}u_{11}u_{1}^2
  \\
&\quad +\frac{2\beta ( 2\beta +1) }{u^{2\beta +2}}u_{1}^{4}-\frac{
2\beta }{u^{2\beta +1}}u_{1}^2\Delta u.
\end{aligned}  \label{e2.7}
\end{equation}
It then it follows that
\begin{equation}
\begin{aligned}
&\Delta P+( p-2) P_{11} \\
& =\frac{2}{u^{2\beta }}[ (
\Delta u) _{1}+( p-2) u_{111}] u_{1}+\frac{2}{
u^{2\beta }}[ u_{ik}u_{ik}+( p-2) u_{1j}u_{1j}] \\
& \quad -\frac{2\beta }{u^{2\beta +1}}[ \Delta u+( 5p-6)
u_{11}] u_{1}^2+\frac{2\beta ( 2\beta +1) }{u^{2\beta +2}
}(p-1) u_{1}^{4}.
\end{aligned} \label{e2.8}
\end{equation}
On the other hand, differentiating \eqref{e1.1} with respect to $x_{1}$ and
evaluating the result making use of \eqref{e2.5}, we obtain that at 
$\mathbf{x}^{\ast }$ we have
\begin{equation}
\begin{aligned}
&u_{1}^{p-1}[ ( \Delta u) _{1}+( p-2)
u_{111}+2( p-2) \sum_{j=2}^N
u_{1j}^2u_{1}^{-1}]   \\
&\quad +( p-2) [ \Delta u+( p-2) u_{11}
] u_{11}u_{1}^{p-3}+f'u_{1}=0.
\end{aligned}  \label{e2.9}
\end{equation}
Also, evaluating equation \eqref{e1.1} at $\mathbf{x^{\ast }}$ we have
\begin{equation}
\Delta u+( p-2) u_{11}=-f( u) u_{1}^{2-p}.  \label{e2.10}
\end{equation}
Inserting \eqref{e2.10} in \eqref{e2.9} we obtain
\begin{equation}
( \Delta u) _{1}+( p-2) u_{111}=2( 2-p)
\sum_{j=2}^N u_{1j}^2u_{1}^{-1}+( p-2)
u_{11}fu_{1}^{1-p}-f'u_{1}^{3-p}.   \label{e2.11}
\end{equation}
Making now use of Lemma \ref{lem2.1}, we also have
\begin{equation}
\begin{aligned}
&u_{ij}u_{ij}+( p-2) u_{1j}u_{1j}\\
& \geq (p-1)
u_{11}^2+\sum_{i=2}^N u_{ii}^2
+p\sum_{j=2}^N u_{1j}^2 \\
&\geq \frac{(p-1) ( N-1) +1}{N-1}u_{11}^2
-\frac{2}{N-1}\Delta uu_{11}  
+\frac{1}{N-1}( \Delta u) ^2+p\sum_{j=2}^N  u_{1j}^2.
\end{aligned} \label{e2.12}
\end{equation}
Therefore, inserting \eqref{e2.11} and \eqref{e2.12} in \eqref{e2.8} 
leads to
\begin{equation}
\begin{aligned}
&\Delta P+( p-2) P_{11} \\
& \geq \frac{2}{u^{2\beta }}\Big\{( 4-p) \sum_{j=2}^N u_{1j}^2
+(p-2) u_{11}fu^{2-p}-f'u_{1}^{4-p}\\
&\quad +\frac{(p-1) ( N-1) +1}{N-1}
u_{11}^2-\frac{2}{N-1}\Delta uu_{11}
+\frac{1}{N-1}( \Delta u)^2   \\
&\quad-\beta [ \Delta u+( 5p-6) u_{11}]
\frac{u_{1}^2}{u}+\beta ( 2\beta +1) (p-1) \frac{
u_{1}^{4}}{u^2}\Big\} .
\end{aligned}  \label{e2.13}
\end{equation}
From \eqref{e2.3} and \eqref{e2.5} we  have
\begin{equation}
P_{i}=\frac{2}{u^{2\beta }}u_{i1}u_{1}\quad \text{for }i=2,\dots ,N.  \label{e2.14}
\end{equation}
Therefore
\begin{equation}
\frac{4}{u^{4\beta }}u_{1}^2\sum_{i=2}^N u_{i1}u_{i1}
=\sum_{i=2}^N P_{i}^2
\leq | \nabla P| ^2,    \label{e2.15}
\end{equation}
so that 
\begin{equation}
\frac{2}{u^{2\beta }}( 4-p) \sum_{j=2}^N u_{1j}^2
\geq -| 4-p| \frac{| \nabla P| ^2}{2P}.   \label{e2.16}
\end{equation}
Finally, since
\begin{equation}
\begin{gathered}
\frac{(p-1) ( N-1) +1+2( p-2) +(p-2) ^2}{N-1}
=\frac{(p-1) ( p+N-2) }{N-1},  \\
\frac{2}{N-1}+\frac{2( p-2) }{N-1}=\frac{2(p-1) }{N-1},
\end{gathered}  \label{e2.17}
\end{equation}
using \eqref{e2.10} and \eqref{e2.16} in \eqref{e2.13} we are lead to
\begin{equation}
\begin{aligned}
&\Delta P+  ( p-2) P_{11}\\
&\geq -| 4-p| \frac{| \nabla P| ^2}{2P}
+\frac{2}{u^{2\beta }}\Big\{-f'u_{1}^{4-p}+\frac{(p-1) ( p+N-2) }{N-1}
u_{11}^2   \\
&\quad +\big( p-2+\frac{2(p-1) }{N-1}\big)
fu_{1}^{2-p}u_{11}+\frac{1}{N-1}( fu_{1}^{2-p}) ^2  \\
&\quad -\beta [ 4(p-1) u_{11}-fu_{1}^{2-p}
] \frac{u_{1}^2}{u}+\beta ( 2\beta +1) (p-1)
\frac{u_{1}^{4}}{u^2}\Big\} .
\end{aligned}  \label{e2.18}
\end{equation}
Now, evaluating \eqref{e2.3} at $\mathbf{x^{\ast }}$, by using  \eqref{e2.5}, we
have
\begin{equation}
P_{1}=\frac{2}{u^{2\beta }}u_{11}u_{1}-\frac{2\beta }{u^{2\beta +1}}
u_{1}^{3},   \label{e2.19}
\end{equation}
so that
\begin{equation}
u_{11}=\frac{P_{1}u^{2\beta }}{2u_{1}}+\beta \frac{u_{1}^2}{u}.
 \label{e2.20}
\end{equation}
Inserting \eqref{e2.20} into \eqref{e2.18} we obtain
\begin{equation}
\begin{aligned}
&\Delta P+  ( p-2) P_{11}\\
&\geq -| 4-p| \frac{| \nabla P| ^2}{2P}
+\frac{2}{u^{2\beta }}\Big\{-f'u_{1}^{4-p}  \\
&\quad +\frac{(p-1) ( p+N-2) }{N-1}\Big( \frac{
P_{1}u^{2\beta }}{2u_{1}}+\beta \frac{u_{1}^2}{u}\Big) ^2
  \\
&\quad +\Big( p-2+\frac{2(p-1) }{N-1}\Big)
fu_{1}^{2-p}\Big( \frac{P_{1}u^{2\beta }}{2u_{1}}+\beta \frac{u_{1}^2}{u
}\Big)    \\
&\quad +\frac{1}{N-1}( fu_{1}^{2-p}) ^2
-\beta \Big[ 4(p-1) \Big( \frac{P_{1}u^{2\beta }}{2u_{1}}
+\beta \frac{u_{1}^2}{u} \Big) -fu_{1}^{2-p}\Big]
\frac{u_{1}^2}{u}   \\
&\quad +\beta ( 2\beta +1) (p-1) \frac{u_{1}^{4}}{u^2}\Big\} .
\end{aligned}
  \label{e2.21}
\end{equation}
Next, using the restriction \eqref{e1.4} on $f$ we note that 
\begin{equation}
\begin{aligned}
&-f'u^{4-p}+\Big( p-2+\frac{2(p-1) }{N-1}\Big) \beta
\frac{f}{u}u_{1}^{4-p}+\beta \frac{f}{u}u_{1}^{4-p}  \\
& =u^{4-p}\big[ -f'+\beta (p-1) \frac{N+1}{N-1}\frac{f}{u}\big] \geq 0.
\end{aligned} \label{e2.22}
\end{equation}
We also note that
\begin{equation}
\frac{(p-1) ( p+N-2) }{N-1}\Big( \frac{
P_{1}u^{2\beta }}{2u_{1}}\Big) ^2\geq 0.  \label{e2.23}
\end{equation}
Therefore
\begin{equation}
\begin{aligned}
\Delta P+( p-2) P_{11}
&\geq -| 4-p| \frac{| \nabla P| ^2}{2P}  \\
&\quad  +\frac{(p-1) ( p+N-2) }{N-1}\Big(
2\beta \frac{P_{1}u_{1}}{u}+2\beta ^2\frac{u_{1}^{4}}{u^{2\beta +2}}
\Big)   \\
&\quad  +\Big( p-2+\frac{2(p-1) }{N-1}\Big)
P_{1}fu_{1}^{1-p}+\frac{2}{N-1}\frac{( fu_{1}^{2-p}) ^2}{
u^{2\beta }}  \\
&\quad  -4\beta (p-1) \frac{P_{1}u_{1}}{u}+2(p-1) 
( \beta -2\beta ^2) \frac{u_{1}^{4}}{u^{2\beta +2}}\,.
\end{aligned}  \label{e2.24}
\end{equation}
Moreover, using the following two identities
\begin{gather}
\frac{2(p-1) ( p+N-2) }{N-1}-4(p-1) =
\frac{2(p-1) ( p-N) }{N-1},  \label{e2.25} \\
\frac{2(p-1) ( p+N-2) }{N-1}\beta ^2+2(p-1) ( \beta -2\beta ^2) 
=\frac{p-1}{N-1}[ 2\beta ^2( p-N) +2\beta ( N-1) ] ,  \label{e2.26}
\end{gather}
one may easily see that \eqref{e2.24} becomes
\begin{equation}
\begin{aligned}
\frac{\Delta P+( p-2) P_{11}}{P} 
& \geq -| 4-p| \frac{| \nabla P| ^2}{2P^2}+2\beta
\frac{(p-1) ( p-N) }{N-1}\frac{P_{1}u_{1}}{Pu}  \\
&\quad +\Big( p-2+\frac{2(p-1) }{N-1}\Big) 
\frac{P_{1}fu_{1}^{1-p}}{P}+\frac{2}{N-1}( fu_{1}^{1-p}) ^2
\\
&\quad +2\beta [ \beta ( p-N) +N-1] \frac{p-1}{N-1}\frac{u_{1}^2}{u^2}.
\end{aligned}\label{e2.27}
\end{equation}
Next, let us consider a point $\mathbf{x_{0}}\in \mathbb{R}^{N}$ and define
\begin{equation}
J(\mathbf{x}) =( a^2-r^2) ^2P,  \label{e2.28}
\end{equation}
where $a>0$ is a constant and $r:=| \mathbf{x}-\mathbf{x_{0}}
| $. Let us denote by $B$ the ball centered at $\mathbf{x_{0}}$
and of radius $a$. Then we immediately notice that
\begin{equation}
J(\mathbf{x}) \geq 0\text{ in }B, \quad 
J( \mathbf{x }) =0 \text{ on }\partial B.    \label{e2.29}
\end{equation}
Consequently, $J(\mathbf{x}) $ must attain its maximum at some
(interior) point $\mathbf{x^{\ast }}$.

Now, if $| \nabla u| ( \mathbf{x^{\ast }}) =0$, then $P\equiv 0$ in $B$. 
Since the ball was chosen arbitrarily,  $
P\equiv 0$ in every ball, so that $\nabla u\equiv 0$ in $\mathbb{R}^{N}$ and
our theorem follows. It thus remain to analyze the case 
$| \nabla u| ( \mathbf{x^{\ast }}) >0$. In such a case, we have
the following complementary inequality at $x^{\ast }$
 (see  Cuccu-Mohammed-Porru \cite[p. 227]{CMP10}  for the proof; they  used a
different auxiliary function $P$, but the proof is identical, since the
form of $P$ does not really play a role in the proof):
\begin{equation}
\frac{\Delta P+( p-2) P_{11}}{P}
\leq \frac{Ca^2}{(a^2-r^2) ^2},\quad 
C:=24+4N+28| p-2| .  \label{e2.30}
\end{equation}
Combining \eqref{e2.27} and \eqref{e2.30} we obtain
\begin{equation}
\begin{aligned}
\frac{Ca^2}{( a^2-r^2) ^2} 
& \geq -| 4-p| \frac{| \nabla P| ^2}{2P^2}+2\beta
\frac{(p-1) ( p-N) }{N-1}\frac{P_{1}u_{1}}{Pu} \\
&\quad +\Big( p-2+\frac{2(p-1) }{N-1}\Big) \frac{
P_{1}fu_{1}^{1-p}}{P}+\frac{2}{N-1}( fu_{1}^{1-p}) ^2 \\
&\quad +2\beta [ \beta ( p-N) +N-1] \frac{p-1}{N-1}\frac{u_{1}^2}{u^2}.
\end{aligned}  \label{e2.31}
\end{equation}
On the other hand, differentiating \eqref{e2.28} we obtain that 
at $\mathbf{x^{\ast }}$
(the point of maximum for $J$ in $B$) we have
\begin{equation}
J_{i}=-2( a^2-r^2) ( r^2) _{i}P+(a^2-r^2) ^2P_{i}=0,    \label{e2.32}
\end{equation}
so that
\begin{equation}
P_{1}=2\frac{( r^2) _{1}P}{a^2-r^2},\quad 
\nabla P=2\frac{\nabla r^2P}{a^2-r^2}.   \label{e2.33}
\end{equation}
From \eqref{e2.5} and \eqref{e2.33} we then conclude that
\begin{gather}
\frac{P_{1}u_{1}}{P}=\frac{\nabla P\nabla u}{P}=2\frac{\nabla r^2\nabla
u}{a^2-r^2},    \label{e2.34}
\\
\frac{| \nabla P| }{P}=\frac{2| \nabla (
r^2) | }{a^2-r^2}=\frac{4r}{a^2-r^2}. \label{e2.35}
\end{gather}
Now using \eqref{e2.34} and \eqref{e2.35} in \eqref{e2.31} we obtain
\begin{equation}
\begin{aligned}
\frac{Ca^2}{( a^2-r^2) ^2} 
& \geq -| 4-p| \frac{8r^2}{( a^2-r^2) ^2}+4\beta \frac{
(p-1) ( p-N) }{N-1}\frac{\nabla r^2\nabla u}{
( a^2-r^2) u}  \\
&\quad +2\Big( p-2+\frac{2(p-1) }{N-1}\Big) fu_{1}^{1-p}
\frac{\nabla r^2\nabla u}{a^2-r^2}+\frac{2}{N-1}(
fu_{1}^{1-p}) ^2 \\
&\quad +2\beta [ \beta ( p-N) +N-1] \frac{p-1}{
N-1}\frac{| \nabla u| ^2}{u^2}.
\end{aligned}  \label{e2.36}
\end{equation}
Moreover, by  classical inequalities we have
\begin{equation}
\begin{aligned}
&4\beta (p-1) ( p-N) \frac{\nabla r^2\nabla u}{( a^2-r^2) u}\\
&\geq -\beta ^2\gamma (p-1) ^2 \frac{| \nabla u| ^2}{u^2}-4( p-N) ^2
\frac{4r^2}{\gamma ( a^2-r^2) ^2},  
\end{aligned} \label{e2.37}
\end{equation}
and
\begin{equation}
\begin{aligned}
&2( p-2) +\frac{2(p-1) }{N-1}fu_{1}^{-p}\frac{\nabla
r^2\nabla u}{a^2-r^2} \\
& \geq -4( | p-2| + \frac{2(p-1) }{N-1}) | f| u_{1}^{p-1}
\frac{r}{a^2-r^2}  \\
& \geq -\frac{2}{N-1}( fu_{1}^{1-p}) ^2-\widetilde{C}\frac{
r^2}{( a^2-r^2) ^2},
\end{aligned}  \label{e2.38}
\end{equation}
with $\gamma >0$ to be chosen and 
$\widetilde{C}:=2[ ( N-1)( p-2) +2(p-1) ] ^2/( N-1) $.
Inserting now estimates \eqref{e2.37} and \eqref{e2.38} into \eqref{e2.36} we find
\begin{equation}
\begin{aligned}
\frac{Ca^2}{( a^2-r^2) ^2} 
& \geq -| 4-p| \frac{8r^2}{( a^2-r^2) ^2}
 -4(p-N) ^2\frac{4r^2}{\gamma ( a^2-r^2) ^2}
 - \widetilde{C}\frac{r^2}{( a^2-r^2) ^2}  \\
& \quad+[ 2\beta ^2( p-N) +2\beta ( N-1)
 -\beta ^2\gamma (p-1) ] \frac{p-1}{N-1}\frac{| \nabla u| ^2}{u^2}.
\end{aligned}  \label{e2.39}
\end{equation}
Now let us analyze separately the following two cases:
\smallskip

\noindent\textbf{Case 1.} when $1<p<N$ and $\beta \in [ 1,\frac{N-1}{N-p}) $,
 we have
\begin{equation}
2\beta ^2( p-N) +2\beta ( N-1) >0.  \label{e2.40}
\end{equation}
Therefore, we can choose $\gamma $ small enough so that we have
\begin{equation}
[ 2\beta ^2( p-N) +2\beta ( N-1) -\beta ^2\gamma (p-1) ] >0.   \label{e2.41}
\end{equation}
smallskip

\noindent\textbf{Case 2.} when $p\geq N$ and $\beta \in [ 1,+\infty ) $, 
as above, we can again choose a small enough $\gamma $ so that \eqref{e2.41} holds.

In conclusion, for a well chosen $\gamma$, there exists a
constant $K=K( N,p,\beta ,\gamma ) $ such that
\begin{equation}
\frac{| \nabla u| ^2}{u^2}
\leq \frac{Ka^2}{( a^2-r^2) ^2}.   \label{e2.42}
\end{equation}
Moreover, since $u(\mathbf{x}) $ is positive, there exists a
constant $L>0$ such that $u^{2-2\beta }\leq L$. Therefore, at some point 
$\mathbf{x}^{\ast }$ we have
\begin{equation}
J( \mathbf{x}^{\ast }) =\frac{| \nabla u| ^2
}{u^2}\frac{1}{u^{2\beta -2}}( a^2-r^2) ^2\leq KLa^2.
  \label{e2.43}
\end{equation}
But $\mathbf{x}^{\ast }$ is a point of maximum for 
$J( \mathbf{x}) $ in $B$, so that we have
\begin{equation}
J( \mathbf{x}_{0}) =\frac{| \nabla u| ^2}{u^{2\beta }}a^{4}\leq KLa^2. \label{e2.44}
\end{equation}
It follows that at $\mathbf{x=x}_{0}$ we have
\begin{equation}
\frac{| \nabla u| ^2}{u^{2\beta }}\leq \frac{KL}{a^2}.   \label{e2.45}
\end{equation}
Letting $a\to \infty $ we find that $\nabla u=0$ at $x_{0}$. 
Since $x_{0}$ is arbitrary, we must have $\nabla u=0$ in $\mathbb{R}^{N}$. 
The proof is thus achieved. 


\subsection*{Acknowledgments} 
This work was supported by a CNCSIS (Romania) research grant
(PN-II-ID-PCE-2012-4-0021 : Variable Exponent Analysis: Partial
Differential Equations and Calculus of Variations).

\begin{thebibliography}{9}

\bibitem{CMP10} F. Cuccu, A. Mohammed, G. Porru;
\emph{Extensions of a theorem of Cauchy-Liouville}, 
J. Math. Anal. Appl., \textbf{369} (2011),
222--231.

\bibitem{Fa07} A. Farina;
\emph{Liouville-type theorems for elliptic problems},
 Handbook of differential equations: stationary partial differential
equations, \textbf{4} (2007), 61--116.

\bibitem{McC07} J. A. McCoy;
\emph{Bernstein properties of solutions to some
higher order equations}, Diff. Int. Equ., \textbf{20} (2007), 1153--1166.

\bibitem{To84} P. Tolksdorf;
\emph{Regularity for a more general class of
quasilinear elliptic equations}, J. Differential Equations, \textbf{51}
(1984), 126--150.

\end{thebibliography}

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