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\markboth{\hfil Asymptotic behavior of solutions  \hfil EJDE--2001/77}
{EJDE--2001/77\hfil J. Fleckinger, E. M. Harrell II, \& F. de Th\'elin \hfil}
\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc  Electronic Journal of Differential Equations},
Vol. {\bf 2001}(2001), No.~77, pp. 1--14. \newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp  ejde.math.swt.edu  (login: ftp)}
 \vspace{\bigskipamount} \\
 %
  Asymptotic behavior of solutions for some nonlinear
  partial differential equations on unbounded domains 
 %
\thanks{ {\em Mathematics Subject Classifications:} 35B40, 35J60, 35J70.
\hfil\break\indent
{\em Key words:} p--Laplacian, Riccati, uncertainty principle
\hfil\break\indent
\copyright 2001 Southwest Texas State University. \hfil\break\indent
Submitted July 2, 2001. Published December 14, 2001.} }
\date{}
%
\author{Jacqueline Fleckinger, Evans M. Harrell II, \& Fran\c cois de Th\'elin}
\maketitle

\begin{abstract} 
 We study the asymptotic behavior of positive solutions $u$ of 
  $$ -\Delta_p u({\bf x})  =  V({\bf x}) u({\bf x})^{p-1}, 
  \quad p>1;\ {\bf x} \in \Omega,$$
 and related partial differential inequalities, as well
 as conditions for existence of such solutions.
 Here, $\Omega$ contains the exterior of a ball in $\mathbb{R}^N$
 $1<p<N$, $\Delta_p$ is the $p$-Laplacian, and $V$ is a 
 nonnegative function.  Our methods include generalized Riccati 
 transformations, comparison theorems, and the uncertainty
 principle.
\end{abstract}

\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}

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\section{Introduction} 

In an earlier article \cite{f2} we studied the behavior of 
solutions of equations and inequalities containing the $p$-Laplacian
near the boundary of a bounded domain.  In this article we consider unbounded 
domains, and estimate the behavior of solutions as 
$|{\bf x}| \to \infty$.   We restrict ourselves to positive
solutions of
$$ -\Delta_p u({\bf x}) = \lambda V({\bf x}) u({\bf x})^{p-1}, 
\eqno(1.1)$$
which decrease at infinity.  Here, $p>1$,  $\lambda>0$,  and 
{\bf x} runs over a domain
containing the exterior of a large ball in $\mathbb{R}^N$.  The value of
the constant $\lambda$ is important for some questions of existence, but 
it is not essential for most of our purposes and will set to 1 in the
following sections.  The
power $p-1$ on the right provides the same homogeneity in $u$ as
the $p$-Laplacian.  The weight function $V$ is always assumed nonnegative, 
and some further restrictions will be imposed.

The number of previous articles on the subject of 
asymptotics of solutions of equations like $(1.1)$ does not appear
to be large.  Many of
those of which we are aware use 
or adapt lemmas of Serrin \cite{s1} and of Ni and Serrin \cite{n1}; 
e.g., see \cite{c1}.  
In case $N>p$,  a positive radial solution $u$ of the 
partial differential inequality
$$  -\Delta_p u({\bf x}) \geq 0 $$
will satisfy bounds of the form \cite{n1},
$$\begin{gathered}
 u(r) \geq C_1 r^{-{\frac{N-p}{p-1}}} \\
 u'(r) \geq C_2 r^{-{\frac{N-1}{p-1}}}.  
\end{gathered} \eqno(1.2)
$$
Other related estimates are to be found in \cite{d2} when $u$ is
a ground--state solution of $(1.1)$.
For some conditions guaranteeing the existence of solutions to
$(1.1)$ we refer to \cite{s2}  and references therein.  
(In some circumstances our results on asymptotics will
imply nonexistence of solutions.) 
For certain equations resembling $(1.1)$, but where the degree of
homogeneity on the right differs from that of the $p$-Laplacian,
there is some work on asymptotic estimates and existence theory:
See \cite{e1,k2,y1}, and especially the books \cite{d2} and
\cite{d3} for these and background material on
equations like $(1.1)$.

A {\it ground-state solution} is understood as a positive solution
on  
$\mathbb{R}^N$ which tends to $0$ as $|{\bf x}| \to \infty$.
In case $N>p$,  
it is shown in [4], Theorem 4.1, 
that for some $\lambda =: \lambda_1$, there exists
a ground-state solution, which is in $L^{q}$ for any 
$q\in [p^*, \infty]$,
where $p* := Np/(N-p)$.
It is also remarked in \cite{d2} that the same bound applies when 
$u$ is a
positive, decaying solution on an exterior domain $\Omega$, 
given Dirichlet boundary conditions on $\partial \Omega$.

Henceforth we absorb the eigenvalue into $V$, setting $\lambda=1.$  

In this article, we assume initially that $-\Delta_p u$ is bounded from
below.  We study the radial case with a 
generalized Riccati transformation, and establish a priori bounds on the 
logarithmic derivative of $u$.  These in turn imply some 
lower bounds on $u$ or, in some circumstances, its nonexistence.

Then we turn our attention to the non-radial case.  We find it 
convenient to study averages of 
expressions containing $u$ over 
suitable sets rather than attempting pointwise estimates.
We modify the Riccati transformation for the
non-radial case and use it to
derive some bounds analogous to those of the
radial case.

In the fourth section we make the complementary assumption,
that $-\Delta_p u$ is bounded from above.  Here we 
adapt the techniques of \cite{f2}  to 
unbounded domains and establish upper bounds on $u$.


\section{The Logarithmic Rate of Decrease of Radial Solutions}

In this section we assume radial symmetry, and 
study the positive radial solutions of the inequality

$$-\Delta_p u({\bf x}) \geq  V({\bf x}) u({\bf x})^{p-1}, \; p>1. 
\eqno(2.1)$$
Since this section concerns only radial solutions,
$(2.1)$ may be written as
$$ -(r^{N-1} |u'|^{p-2}u')' \geq  V r^{N-1}u^{p-1}. \eqno(2.2)$$
In \cite{n1}, it is shown that:

\begin{proposition}[\cite{n1}] \label{prop2.1}
Assume that $V(r) > 0$ is bounded and measurable on any finite subset of
$\{ r > r_0\}$, where $r$ is the radial coordinate 
in $\mathbb{R}^N, N > p.$  Suppose that $u(r)$ is a positive radial
ground--state  solution of $(2.1)$ for $r > r_0.$ 
Then there exist two positive constants $c_1$ and $c_2$ such that
\begin{gather*}
 u(r) \geq  c_1 r ^{-\left({\frac{N-p}{p-1}}\right) },\\
 |u'(r)| \geq c_2 r^{-({\frac{N-1}{p-1}})}.
\end{gather*}
\end{proposition}

Under certain circumstances we shall improve Ni and Serrin's
bound on $u$
with a Riccati transformation adapted to the $p$-Laplacian.
An interesting aspect of this is
that our bound can be viewed as an oscillation theorem.
We shall also comment
on the consequences for the possible existence of
positive solutions. Let
$$ \rho := - 
\Big|{\frac{u'}{u}}\Big|^{p-2}\frac{u'}{u} . \eqno(2.3)$$
The sign is reasonable because we shall show that $u'<0$ for large $r$.
By inserting $(2.3)$ into $(2.2)$, we derive
$$ \rho' \geq V + (p-1)\left|{ \rho }\right|^{\frac{p}{p-1}} -(N-1)
\frac{\rho}{r}.
\eqno(2.4)$$
We note here that since $\rho$ determines the 
logarithmic derivative of $u$, bounds on 
$\rho$ as $r \to \infty$ correspond roughly to decay estimates for $u$.
Moreover, at finite $r$
a divergence of $\rho$ may simply arise from a zero of
$u;$ it may be possible to continue through the zero in 
standard ways \cite{h1}.


We consider here positive radial solutions of $(2.1)$ such that
$$ \limsup_{r \to +\infty}\rho(r) < + \infty. \eqno(2.5)$$

The following proposition states that any positive radial solution must
either satisfy an {\it a priori} bound on all $r > r_0$ or else blow up at 
some finite value of $r$. 

\begin{proposition} \label{prop2.2}
Assume $V(r) > 0$ is bounded and measurable on any finite subset of
$\{ r > r_0\}$, where $r$ is the radial coordinate 
in $\mathbb{R}^N, N \geq 2.$  Let $u(r)$ be a positive 
radial  solution of $(2.1)$ for all $r > r_0$, and define $\rho$ by $(2.3)$.  
Assume further that $\rho$ satisfies $(2.5)$. Then
 a)
$$ \rho <\rho_{cr}:= \left\{{\frac{N-1}{(p-1)r}}\right\}^{p-1}.\eqno(2.6)$$ 
 b) \
 If $u(r) > 0$, $u'(r) < 0$ for $r > r_0$, and if $(2.6)$ is violated at
$r=r_0$, then $u(r_1) = 0$ for some $r_1$ which could be
bounded above explicitly in terms of $\rho(r_0), N$, and $r_0$.
\end{proposition}

\paragraph{Proof:}
a) First we show by contradiction that if $\rho(r_0)>0$, then $\rho(r)>0$ 
for all $r>r_0$.
Suppose otherwise and let $R$ be the first zero of $\rho$ 
larger than $r_0$: $\rho(R)=0$ and $\rho'(R) \leq 0$. This contradicts  
$(2.4)$ at the point $R$, by which $\rho'(R) > 0$.  

Moreover, 
statement a) is trivial in the case where $\rho(r)<0$ for any $r$. 
We conclude that we 
may assume that $\rho(r)>0$ for any $r$.

Now consider the critical curve defined by $(2.6)$ 
in the $(r,\rho)$ plane, $\rho>0$:
$$(p-1)\rho_{cr}^{\frac{p}{p-1}} -(N-1)\frac{\rho_{cr}}{r}=0.$$ 
The function $\rho_{cr}$ decreases as $r$ increases, and by $(2.4)$, 
if $\rho \geq \rho_{cr}$, then $\rho' \, \geq\,  V \, >0$. 

Hence if $(2.6)$ were false
at $r = R$ for some finite $R$, then 
$\rho$ would be an increasing function for all $r > R$.  
Consequently, it would either
approach a finite positive limit or else diverge to $+ \infty$.  A finite 
positive limit, however, is incompatible with $(2.4)$ for large $r$,
and therefore $\rho$ would become arbitrary large as $r \to \infty$. 
For large $r$ and small positive $\varepsilon$, 
it then follows from $(2.4)$ that 
$\rho'>(p-1-\varepsilon )\rho^{\frac{p}{p-1}}$, which implies 
by comparison that
$\rho \geq \tilde \rho$ with $\tilde \rho$ a positive solution of 
$$\tilde \rho'=(p-1-\varepsilon )\tilde \rho^{\frac{p}{p-1}}.$$
It is elementary to solve the comparison equation:
We find 
$$\tilde \rho(r) = \Big[{\frac{p-1}{(p-1-\varepsilon)(r_2 - r)}}
\Big]^{p-1}, \quad 
\mbox{for some }  r_2>R.$$
Since any such solution is singular at the finite 
point $r=r_2$, any solution $\rho$ violating $(2.6)$ likewise blows up
at some finite $r_1 \leq r_2$, which contradicts $(2.5)$.  

b) The proof above shows that $\rho$ blows up at $r_1$ and therefore $u(r_1)=0$.
\hfill $\Box$ \medskip

Proposition \ref{prop2.2} makes no use of the detailed nature of $V$, and 
therefore it can be sharpened, given more information about $V:$ 

\begin{lemma} \label{lem2.3}
 For a given $b>0$ and  $x>b/(N-1)$, set 
$$\varphi_b(x):= \left({\frac{(N-1)x-b}{p-1}}\right)^{\frac{p-1}{p}} .$$
The concave function $\varphi_b$ is increasing and admits a fixed 
point if and only if
$b\leq (\frac{N-1}{p})^p$. 
By concavity there are at most two fixed points, and
we denote by $a_*(b)$, or $a_*$ for short, 
 the larger (or only) one.  An explicit bound on the fixed point is:
$$a_* \leq \left({ \frac{N-1}{p-1} } \right)^{p-1} - 
b \left({\frac {p-1}{N-1} } \right).\eqno(2.7)$$
\end{lemma}

\paragraph{Proof:} Since $\varphi_b\left(\frac{b}{N-1}\right)=0$, 
we seek $ x>\frac{b}{N-1}$ such that 
$\varphi_b( x) =  x$, which is equivalent
to $\psi( x)= b$ with $\psi(x):= (N-1)x -(p-1) x^{\frac{p}{p-1}}$.
The extremum of $\psi$ is obtained for $\tilde x= \left(
\frac{N-1}{p}\right)^{p-1}$ and $\psi(\tilde x) = (\frac{N-1}{p})^p$. 
Hence $\varphi_b$ admits  at least one fixed point  
if and only if $$b \leq  b_{\max}(N,p):= \left(\frac{N-1}{p}\right)^p.
\eqno(2.8)$$


The roots of $\psi$ are $x=0$ and $x=\hat x:=
(\frac{N-1}{p-1})^{p-1}$. Obviously $a_* \leq \hat x$.
In fact, since $\psi$ is concave, the curve $\psi$ lies below the tangent at 
point $\hat x$, which leads to the estimate $(2.7)$.
\hfill $\Box$


\begin{proposition} \label{prop2.4}
Suppose that for some $b > 0$,  $V(r) \geq b r^{-p}$ on the 
interval $[r_0; \infty)$. \begin{enumerate}

\item[a)]  If $0 <  b \leq b_{\max}(N,p) :=\frac{(N-1)^p}{p^p}$, then 
for any  solution  $u$ of $(2.1)$ on $[r_0; \infty)$,  such that
its $\rho$ satisfies $(2.5)$, we have  
$ \rho \leq a_*(b) r^{-(p-1)}$, where $a_*(b)$ is 
as defined in Lemma~\ref{lem2.3}.

\item[b)]  If $b > b_{\max}(N,p)$, then
there are no solutions $\rho$ of $(2.4)$ which satisfy $(2.5)$, 
and thus there are no positive, 
decreasing
radial solutions $u$ of $(2.1)$.  
\end{enumerate}
\end{proposition}

\paragraph{Proof:} 
a)\  We assume that $b \leq \frac{(N-1)^p} {p^p}$ and
extend  Proposition \ref{prop2.2} by a bootstrap argument.  Suppose that
it has been established that $ \rho \leq a r^{-(p-1)}$ for some $a$.
Then from $(2.4)$ it follows that $\rho' \geq 0$ provided that 
$$-((N-1)a - b) r^{-p} + (p-1) \rho^{\frac{p}{p-1}}\geq 0,$$
which corresponds to the critical curve
$$\rho_{cr}(r;a) =\varphi_b(a)r^{-(p-1)}.
$$
With the same argument as in Proposition \ref{prop2.2}, we conclude that 
$\rho$ lies below the critical curve, $\rho(r)<\varphi_b(a)r^{-(p-1)}$.
By iteration we improve the above estimate with a decreasing sequence of
$a_k$, and  
as $k \to \infty$ we obtain 
$$a_*= \left[\frac{(N-1) a_* - b}{p-1} \right]^{\frac{p-1}p}$$
as the largest fixed point of $\varphi_b$: $a_*=\varphi_b(a_*)$, which 
exists by Lemma \ref{lem2.3}.

b)\  Assume now that $b > \frac{(N-1)^p}{p^p}$ . We have shown above
that if $a > \frac{b}{N-1} $  and $\rho \leq ar^{-(p-1)}$, 
then $\rho$ also satisfies 
$\rho \leq \varphi_b(a) r^{-(p-1)}$ . Recalling Proposition~II.2, there exists 
$a_0 > \frac{b}{N-1} $ such that $\rho \leq a_0r^{-(p-1)}$ . We define 
$a_{k+1}:= \varphi_b(a_k)$.  For large $a$, $\varphi_b(a)<a$, so the
sequence $a_k$ is bounded from above. Hence either 
\par 
$(i)\;$ there exists a subsequence $a_{k_j} \to a_*>\frac{b}{N-1}$ as 
$j\to \infty$; or
\par  $(ii)\;$ there exists $k$ such that $a_{k+1} \leq \frac{b}{N-1} 
< a_k$.

\noindent
Case $(i)$ is excluded by the previous lemma. In Case $(ii)$, we may decrease 
$b$ as necessary to $b_{\epsilon} := 
(N-1)a_{k+1}- \left( \epsilon (p-1)\right)^{\frac{p}{p-1}} $ so that 
$a_{k+1} \geq \frac{b_{\epsilon}}{N-1}$. 
Then we define $\tilde a_{k+2}:=
\varphi_{b_{\epsilon}} (a_{k+1}) =\epsilon $. 
It follows that
$\rho \leq \epsilon r^{-(p-1)} $, and as $\epsilon \searrow 0$ we find
$\rho \leq 0$.  Therefore $u$ cannot be a positive decreasing solution.
\hfill $\Box$

\begin{corollary} \label{cor2.5}
If $V(r) \geq b r^{-p}$ for $0<b< b_{\max}(N,p)$, then 
any positive solution $u$ on $[r_0; \infty[$ satisfies 
$$ u(r) \geq u(r_0) \left(\frac{r}{r_0}\right)^{-a_*^{\frac{1}{p-1}}},$$
where $a_*$ is as in Lemma \ref{lem2.3}.
\end{corollary}

\paragraph{Proof:}  Because of $(2.3)$, 
$$\big(-\frac{u'}{u} \big) 
\leq a_*^{\frac{1}{p-1}} \frac1r, $$
and by integrating, for $r>r_0$, 
$$-\ln  \big| \frac{u(r)}{u(r_0)} \big| \leq  
a_*^{\frac{1}{p-1}} \ln  \big( \frac{r}{ r_0}\big),$$
which for $u>0$ implies the claim. \hfill $\Box$

\paragraph{Remark.}
Corollary \ref{cor2.5} is weaker than Proposition \ref{prop2.1} for small $b$, but
improves it for some values of $N,p$, and $b$.  More specifically,
Corollary \ref{cor2.5} is an improvement 
when  $N>p$ and
$a_*<(\frac{N-p}{p-1})^{p-1}.$  From $(2.7)$, for this
it suffices to have
$$\big({\frac{N-1}{p-1} } \big)^{p-1} - 
b \big({\frac{p-1}{N-1} } \big) <
\big({\frac{N-p}{p-1}}\big)^{p-1}.$$
For $b=b_{\max}$, this condition becomes 
$$ 1 - \big({\frac{p-1}p}\big)^p < \big(1-{\frac{p-1}
{N-1}}\big)^{p-1},$$
which is clearly true for sufficiently large $N$.

\begin{corollary} \label{cor2.6} 
Suppose that for some $C > 0$ and $m < p$,
$V(r) \geq C r^{-m}$ for all  $r > r_0$. 
Then Equation $(2.1)$ has no solutions 
which remain positive on $(r_0, \infty).$ 
\end{corollary}

\paragraph{Remark.}  The proof of this corollary is merely an application 
of Propositon \ref{prop2.4} b). This corollary is a special case of 
\cite[Theorem 3.2]{s2}.


\begin{lemma} \label{lem2.7} 
Let $u(r)$ be a positive, radial, decreasing solution of $(2.1)$ 
for $r \geq r_0,$ with $V(r) > 0$,
and define $\rho$ as before. Then 
$$\rho(r) \geq \rho(r_0) \left( \frac r{r_0} \right)^{-(N-1)}$$
for all $r > r_0$.
\end{lemma}

\paragraph{Proof:} 
Under these circumstances, it follows from $(2.4)$ that
$$ \rho' \geq - {\frac{(N-1) \rho}r},$$
so by the comparison principle, $\rho$ is bounded from below by
the solution of
$$ \tilde \rho' = - {\frac{(N-1) \tilde \rho}r},$$
which agrees with $\rho$ at $r=r_0.$  This  yields the
claimed bound.
\hfill $\Box$ \smallskip

Finally we observe that our bounds imply some simple and 
fairly standard nonexistence criteria.

\begin{corollary} \label{cor2.8} 
Suppose: \begin{enumerate}
\item[a)] that $p>N$. Then there are no ground--state solutions of $(2.1)$.

\item[b)]  that $p=N$, and that 
$V(r) \geq \left({\frac{p-1} p }\right)^p r^{-p}$ on the 
interval $[r_0; \infty).$  Then there
are no ground--state solutions of $(2.1)$.
\end{enumerate} \end{corollary}

\paragraph{Proof:} 
a)  If $p > N$, then the lower bound of Lemma \ref{lem2.7} would exceed the 
upper bound of Proposition \ref{prop2.2} for large $r$. 
Part b) is the same as Proposition \ref{prop2.4} b) whith $p=N$.  
\hfill $\Box$

\section{The Nonradial Case}

We now turn our attention to $(2.1)$ when $V$
is not necessarily radial.  We suppose throughout this
section that $u > 0$ on the exterior of some ball, and consider
$(2.1)$ on this exterior domain.  Without assumptions of symmetry
some control is lost on the decrease of the solutions.  Instead
of estimating the solutions pointwise, we shall 
estimate certain integrals over large balls and spheres.  



We frequently use the following standard notation:

\noindent $B_R$ is the  ball of radius $R$.

\noindent ${\bf n}$ is the unit radial vector.

\noindent $\omega_N$ is the surface area of $\partial B_1$,

\noindent and we adapt the definition of $\rho$ to the nonradial case:

$${\bf \rho }:=-\left|{\frac{\nabla u }u}\right|^{p-2}{\frac{\nabla u }
u}.\eqno(3.1)$$


\begin{theorem} \label{thm3.1}
Let 
$$W(R):= \int_{B_R}\left( {\frac{|\nabla u |^{p-2} \left| \nabla u \cdot {\bf n}
\right|}
{|u|^{p-1}}}\right)^{\frac{p}{p-1}}.$$
Then have the following estimates: 
\begin{gather*}
W(R) \leq (p-1)^{-p}(N-p)^{p-1} \omega_N R^{N-p}\,, \\
\int_{B_R}\left| {\frac{\partial \ln  u}{\partial r}}\right|^p 
\leq (p-1)^{-p}(N-p)^{p-1} \omega_N R^{N-p}.
\end{gather*}
\end{theorem}

\paragraph{Proof:} The second estimate is merely a simplification and 
slight weakening of the  first, 
since 
$\nabla u \cdot {\bf n} = {\frac{\partial u}{\partial r}}$.
Inequality $(2.1)$ may be rewritten as:
$$ \nabla \cdot {\bf \rho } \geq V+(p-1) \left| {\bf \rho}\right|^{p/(p-1)}. 
\eqno(3.2)$$
By integration and by Gau\ss's divergence theorem we have
$$ \int_{\partial B_r} {\bf \rho}\cdot {\bf n} \geq \int_{B_r} V + 
(p-1)\int_{B_r} \left| {\bf \rho}\right|^{p/(p-1)}.\eqno(3.3)$$ 
Set 
$$g(r)=\int_{B_r} V \eqno(3.4)$$
and
$$U(r)=\int_{\partial B_r} 
|{\bf \rho}\cdot {\bf n}| ^{p/(p-1)}. \eqno(3.5)
$$
It follows from $(3.3)$ that
$$ \int_{\partial B_r} {\bf \rho}\cdot {\bf n} \geq g(r) + 
(p-1)\int_0^r \int_{\partial B_s} [|{\bf \rho}\cdot {\bf n}|^2+{\bf
\rho}_{\tau}^2]^{p/2(p-1)}, 
$$
where ${\bf \rho}_{\tau}$ designates the tangential 
component of ${\bf \rho}$, i.e.,
${\bf \rho}_{\tau} = {\bf \rho} - ({\bf \rho}\cdot {\bf n}) {\bf n}$.
$$\int_{\partial B_r} {\bf \rho}\cdot {\bf n}
\geq g(r) + 
(p-1)\int_0^r \int_{\partial B_s} |{\bf \rho}\cdot {\bf n}|^{p/(p-1)}.
\eqno(3.6)$$ 
Set $U=W'$, so that
$$W(r)=\int_0^r U(s) ds. \eqno(3.7)$$
By H\"older's inequality and $(3.5)$,
$$  \int_{\partial B_r} {\bf \rho}\cdot {\bf n} \leq [U(r)]^{(p-1)/p}
 \omega_N^{1/p}
r^{(N-1)/p}.$$
Inequality $(3.6)$ may be rewritten as 
$$ g(r)+(p-1)W(r) \leq 
\omega_N^{1/p} \left(W'(r)\right)^{(p-1)/p} r^{(N-1)/p}.\eqno(3.8)$$
 Since $V>0$, $g>0$, and hence
$$ (p-1)W(r) \leq \omega_N^{1/p} \left(W'(r)\right)^{(p-1)/p} r^{(N-1)/p}.$$
Therefore, by integration, for any $r$ and $r_0$ satisfying $r>r_0>0$, we have:
$$ W(r)^{-(\frac 1{p-1})} \leq \left(W(r_0)\right)^{-(\frac 1{p-1})} + 
(p-1)^{(\frac p{p-1})} (N-p)^{-1} \omega_N^{-(\frac 1{p-1})}
\big[r^{{\frac{p-N}
{p-1}}} - r_0^{{\frac{p-N}
{p-1}}}\big].\eqno(3.9)$$ 
Now we claim that 
$$W(r) \leq K r^{N-p} \mbox{ for }  r\geq r_0. \eqno(3.10)$$
Assume for the purpose of contradiction that $W(r_0)>K r_0^{N-p}$.
Let $r_1$ be defined 
by $$(p-1)^{(\frac p {p-1})} (N-p)^{-1} \omega_N^{-(\frac 1{p-1})}
\big( r_1^{{\frac{p-N} {p-1}}}
-r_0^{{\frac{p-N} {p-1}}} \big)
=  (Kr_0^{N-p})^{-(\frac 1
{p-1})}.$$
We deduce from this and from $(3.9)$ that 
$$W^{(\frac 1{p-1})}(r)\geq \Big[(p-1)^{(\frac p {p-1})} (N-p)^{-1} 
\omega_N^{-(\frac 1{p-1})}
r_1^{{\frac{p-N} {p-1}}}\Big]^{-1} \times 
\Big[r^{\frac{p-N}{p-1}}-r_1^{\frac{p-N}{p-1}}\Big]^{-1}.
$$
It follows that there is some $r^* \in (r_0; r_1]$ such that  
$W(r^*)=\infty$, which is impossible since $W(r)$ is finite for
all $r$. Hence $(3.10)$ is proved with
$K= (p-1)^{-p}(N-p)^{p-1}\omega_N$.
We also conclude from $(3.8)$ that 
$$ g(r)\leq \omega_N^{1/p} \left(W'(r)\right)^{(p-1)/p} r^{(N-1)/p}.
$$ 
If $V$ grows as above, then $g(r)\geq r^{N-p}$, and 
$$ W'(r) \geq k r^{\frac{Np-p^2-N+1}{p-1}},$$
so
$$ W(r) \geq A+k r^{N-p}. \eqno(3.11)$$
\hfill $\Box$ \smallskip


As in the previous proof, let
$  {\bf \rho}_{\tau}$ designate the tangential 
component of ${\bf \rho}$. It can be controlled as follows:

\begin{corollary} \label{cor3.2}
Under the same conditions as in 
Theorem \ref{thm3.1},
$$\int_0^R \int_{B_r} |{\bf \rho}_{\tau}|^{\frac p {p-1}} 
\leq K R^{N-p+1}.$$
\end{corollary}

\paragraph{Proof:} From $(3.3)$, we have
$$ \int_{\partial B_r} {\bf \rho}\cdot {\bf n} \geq
(p-1)\int_{B_r} |{\bf \rho}_{\tau}|^{(p/(p-1))}.$$ 
The desired estimate follows by integrating this from $0$ to $R$ and applying
H\"older's inequality.\hfill $\Box$

\paragraph{Remark.} If $u$ is radially symmetric, and if $\rho$ satisfies 
the estimate $ \rho(r) \leq a_*r^{-(p-1)}$ with
$a_* \leq \left({\frac{N-p}{p-1}}\right)^{p-1} $, then by integration on
$B_r$, we obtain 
$$W(r) \leq \left(a_*\right)^{\frac p {p-1}} \int_0^r s^{-p+N-1} \omega_N ds
\leq K r^{N-p},$$
where $K=\left(a_*\right)^{\frac p {p-1}} \frac{\omega_N}{N-p}
\leq \left({\frac{N-p} {p-1}}\right)^p \frac{\omega_N}{N-p}$,
which implies the result of Theorem~\ref{thm3.1} for $u$ radial.

\section{Decay Estimates Using the Uncertainty Principle}

Our purpose in this section is to find decay estimates for Equation 
$(1.1)$ to complement those of the previous section, which essentially apply
to the partial differential inequality $(2.1).$  In this section we
pose the opposite inequality.
Specifically, we shall assume, in contrast to $(2.1)$, that
$$ -\Delta_pu(x) \leq  V(x) u^{p-1}(x), \quad  x \in \mathbb{R}^N.
\eqno(4.1)$$
An important tool will be an $L^p$ uncertainty principle from \cite{f2}, 
which is 
analogous to Hardy's inequality as used in \cite{e2,d1,f2}.   
Indeed, like some other Hardy--type inequalities, it can be derived
from an inequality of Boggio \cite{b1},
as generalized and
discussed in \cite{f2}.  For further information about Hardy--type 
inequalities,
see \cite{m1,o1} and references therein.

In what follows, $k$ will denote positive constants with various 
values that we do not compute precisely.  Henceforth let
$$ c_p \, := 
\, \left(\frac p{N-p} \right)^{\left({\frac{p-1} p} \right)}.$$
In \cite{f2} we generalized the uncertainty--principle lemma,
classical when $p=2$ (cf.~\cite{f1,k1,r1}), to arbitrary $p<N$.
Here we recall \cite[Corollary II.4]{f2}, and extend it by closure to the
 Sobolev space
${\cal D}^{1,p}(\mathbb{R}^N)$:

\begin{lemma} \label{lem4.1}
Assume that $p<N$. For any 
$u\in {\cal D}^{1,p}(\mathbb{R}^N)$, we have 
$$  \int_{\mathbb{R}^N} \left|{\frac u r} \right|^p \leq   
c_p^p \int_{\mathbb{R}^N} | \nabla u |^p  .  \eqno(4.2)$$ 
\end{lemma}

\paragraph{Proof:}:
In \cite[Corollary II.4]{f2}, $(4.2)$ was  established for functions in
${\cal C}^{\infty}_0 (\mathbb{R}^N)$.
For any $u \in {\cal D}^{1,p}(\mathbb{R}^N)$,
there exists $u_n \in 
{\cal C}^{\infty}_0 (\mathbb{R}^N)$,
tending to $u$ with respect to the ${\cal D}^{1,p}$ norm.  We infer that
$$\int_{\mathbb{R}^N} | \nabla u_n |^p  \to \int_{\mathbb{R}^N} |
\nabla u |^p,$$
and  $u_n$ converges to $u$ strongly in $L^{p^*}$, and therefore $a.e.$ in
$\mathbb{R}^N$ (for a subsequence still denoted by $u_n$).
Then from Fatou's Lemma, 
\begin{align*}  \int_{\mathbb{R}^N}  {\lim}\left|{\frac u  r}\right|^p \; &\leq \;
{\limsup}\int_{\mathbb{R}^N} \left|{ \frac{u_n} r}\right|^p \\
&\leq  c_p^p \; \;
{\limsup}\int_{\mathbb{R}^N} | \nabla u_n |^p  \; = \; 
c_p^p \; \int_{\mathbb{R}^N} 
| \nabla u |^p.
\tag*{$\Box$}\end{align*}
Consider $u > 0$ in the Sobolev space $ {\cal D}^{1,p}(\mathbb{R}^N)$, and 
satisfying $(4.1)$.  We assume further:

\paragraph{Hypothesis (H):} \begin{enumerate}
\item[(1)] $V=V(|x|)=V(r) >0$ and  $V\in L^{N/p}(\mathbb{R}^N)$,
with $N>p$.
\item[(2)]
There exists $\mu > p$ such that
$$ V(r) \leq { \frac k {(1+r^2)^{{\frac{\mu} 2}+{\frac p {2c}}}}},$$
where $ c \, := \, \hat m c_p, $ and the positive constant
$\hat m = \hat m(p,N)$ is defined in \cite{f2}.
\end{enumerate}

\begin{theorem} \label{thm4.2} 
Assume Hypothesis {\bf (H)}.  Then 
there exists $k>0$ such that for any $\varepsilon >0$,
any positive solution $u$ 
of $(4.1)$ 
satisfies the estimate:
$$ \int_{ \{r> \left( {\frac1{\varepsilon}} \right)^c \}} \left(  {\frac{ u } 
 r} \right)^p  \leq  
 k \varepsilon^p \| u \|_{{\cal D}^{1,p}}^p .  $$ 
\end{theorem}

\paragraph{Proof:}
Let $\varphi$ be piecewise ${\cal C}^1$; by Lemma \ref{thm3.1} 
of \cite{f2}, we have 
$$\int_{\mathbb{R}^N}| \nabla (\varphi u ) |^p \leq  \hat m^p  
\int_{\mathbb{R}^N} | u \nabla \varphi |^p + \hat k \int_{\mathbb{R}^N} u | 
\varphi |^p (-\Delta_p u ) . $$ 
Here 
$\hat k = 2^{{\frac{p-2}2}} p^{2-p}(p-1)^{p-1}$. 
By Lemma \ref{lem4.1} combined with the definition of $c$, 
we get 
$$\int_{\mathbb{R}^N} \left|{\frac{\varphi u } r} \right|^p \leq  c^p 
\int_{\mathbb{R}^N} | u\nabla \varphi |^p + \hat k c_p^p \int_{\mathbb{R}^N}  
V | u \varphi| ^p, $$ 
which is equivalent to
$$ \int_{\mathbb{R}^N} \left( \left| {\frac{\varphi}  r} \right|^p - c^p | 
\nabla \varphi 
|^p \right) u^p
 \leq   \hat k  c_p^p 
\int_{\mathbb{R}^N}  V | \varphi |^p  u^p .  \eqno(4.3)$$ 
We define 
$$ \varphi ({\bf x}) \, := \, \varphi (r)  := 
 \min \big( r^{{\frac 1 c }}, {\frac 1
{\varepsilon}} \big). \eqno(4.4)$$
By a computation, if $r < \left( {\frac1 {\varepsilon}} \right)^c$, then
$$ | \nabla \varphi | = \varphi'(r)= {\frac 1 c} r^{{\frac 1 c }-1} 
= 
{\frac 1 c} {\frac{\varphi} r}.$$  From $(4.4)$ it follows that
$$ \int_{ \{r> \left( {\frac 1{\varepsilon}} \right)^c \}} \left|{\frac{\varphi} 
r} \right|^p u^p \leq  
\hat k  c_p^p 
\int_{\mathbb{R}^N}  V | \varphi |^p  u^p .  
\eqno(4.5)$$ 
Since $\mu >p$, Hypothesis {\bf (H)} implies that 
\begin{align*}
\int_{\mathbb{R}^N}  V | \varphi |^p  u^p \, &\leq \, k \int_{\mathbb{R}^N} 
 \frac {  u^p}{ (1+r^2)^{\frac {\mu} 2}} \\ 
&\leq 
 k \Big( \int_{\mathbb{R}^N} u^{p^*} \Big)^{p/p^*} \Big( \int_{\mathbb{R}^N} 
 {\frac{  1 }{ (1+r^2)^{\frac{N \mu } {2 p} }}} \Big)^{p/N}
\\
& \leq  k \| u \|_{L^{p^*}}^p \leq   k \| u 
\|_{{\cal D}^{1,p}}^p . \tag*{(4.6)}\end{align*}
  From $(4.4)$ and $(4.6)$, we derive 
$$ \int_{ \{r> \left( {\frac1 {\varepsilon}} \right)^c \}} \left(  {\frac{ u } 
 r} \right)^p  \leq  
 k \varepsilon^p \| u \|_{{\cal D}^{1,p}}^p .  \eqno(4.7)$$ 
\hfill $\Box$

\paragraph{Remark.}  To illustrate these bounds, 
assume that asymptotically 
$u(r) \simeq r^{-\alpha}$ as $r\to \infty.$  Then by 
Theorem \ref{thm4.2}, 
$$ \int_{\varepsilon^{-c}}^{+\infty} r^{N-1-p(\alpha + 1)} dr \leq  k 
\varepsilon^p. \eqno(4.8)$$
This implies that $N<p(\alpha +1)$ and $\varepsilon^{cp(\alpha +1) -cN }  
\leq  k \varepsilon^p $ and therefore 
$\alpha \geq {\frac1  c} + {\frac{N-p} p}$. 

\paragraph{Examples}  \begin{enumerate}
\item[(i)]
Suppose that $p=2$ and $N=3$; in this case $c_p=\sqrt{2}$, $\hat m=1$;
 $c=\sqrt{2}$, $\hat k=1$. 
Hence, from $(4.8)$, $\alpha \geq {\frac{1+ \sqrt{2}} 2}$. 
 
\item[(ii)]
Suppose that $2<p<N<2p$;  in this case $c_p= \left( \frac p{N-p} 
\right)^{\left({\frac{p-1} p} \right)}$;  
$\hat m= (p-1) 2^{{\frac{p-2}{2p}}}$; 
$c=\hat m c_p$. 
 Hence from 
$(4.8)$, $\alpha \geq \frac1 c + {\frac{N-p} p}$. 
\end{enumerate}

Finally suppose that
$u\in {\cal D}^{1,p}(\mathbb{R}^N)$ is a positive solution of (4.1)
with $V=V_1-V_2$, where $V_1$ satisfies {\bf (H)} and $V_2(r) \geq 0$ for
any $r \geq 0$.
Since $k \geq 1$, Equation (4.5) implies that
\begin{align*} \int_{ \{r> \left( {\frac1 {\varepsilon}} \right)^c \}} 
 V_2 \left(  {\frac{u} {\varepsilon}} \right)^p  \, 
 &\leq  k \int_{\mathbb{R}^N}  V_2 | \varphi |^p  u^p +
\int_{ \{r> \left({\frac1 {\varepsilon}} \right)^c \}} 
\left|{\frac\varphi r} \right|^p u^p \cr
&\leq \, k \int_{\mathbb{R}^N}  V_1 | \varphi |^p  u^p  \leq k.\end{align*}
Hence we obtain the following:

\begin{corollary} \label{cor4.2}
Assume that $V=V_1-V_2$, where $V_1$ satisfies 
{\bf (H)}
 and $V_2(r) \geq 0$ for any $r \geq 0$.
Then there exists $k>0$ such that for any  $\varepsilon >0$, 
any positive solution $u$ 
of $(4.1)$ 
satisfies:
$$ \int_{ \{r> ( 1/\varepsilon)^c \}} V_2 u^p  
\leq  k \varepsilon^p .  $$ 
\end{corollary}

 
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\subsection*{ERRATUM: Submitted on April 28, 2003}

In Section 4, before Lemma 4.1, the constant $c_p$ defined as
$$ 
c_p  :=  \Big( \frac{p}{N-p} \Big)^{(\frac{p-1}{p})}.
$$
should be replaced by
$$ 
c_p := \Big(\frac{p}{N-p}\Big) 
$$
(no exponent).  This error propagated to the examples at the end of 
the article. Example (i) should read:
\par
\noindent
(i)  Suppose that $p=2$ and $N=3$; in this case $c_p=2$, $\hat m=1$;
 $c=2$, $\hat k=1$.
Hence, from $(4.8)$, $\alpha \geq 1$.
\medskip

\noindent
In Example (ii), the expression
$$ 
c_p := \Big(\frac{p}{N-p}\Big)^{(\frac{p-1}{p})}.
$$
should be replaced by 
$$ 
c_p := \Big( \frac{p}{N-p}\Big) .
$$
(no exponent)
\medskip

\noindent\textsc{Jacqueline Fleckinger }\\
\textsc{CEREMATH \& UMR MIP}, Universit\'e Toulouse-1\\
Universit\'e Toulouse-1 \\
21 all\'ees de Brienne, 31000 Toulouse, France\\
e-mail: jfleck@univ-tlse1.fr  \smallskip 

\noindent\textsc{Evans M. Harrell II }\\
School of Mathematics\\
Georgia Institute of Technology\\
Atlanta, GA 30332-0160, USA \\
e-mail: harrell@math.gatech.edu \smallskip

\noindent\textsc{Fran\c cois de Th\'elin }\\
\textsc{UMR MIP}, Universit\'e Paul Sabatier\\ 
31062 Toulouse, France\\
e-mail: dethelin@mip.ups-tlse.fr


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