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 $a$, $\varphi_b(a)\frac{b}{N-1}$ 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 $0**r_0$,
$$-\ln \big| \frac{u(r)}{u(r_0)} \big| \leq
a_*^{\frac{1}{p-1}} \ln \big( \frac{r}{ r_0}\big),$$
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 $p 0$ 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**

\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} \begin{thebibliography}{00} \frenchspacing \bibitem{b1} T. Boggio, Sull'equazione del moto vibratorio delle membrane elastiche, {\it Accad. Lincei, sci. fis.}, ser. 5 {\bf 16} (1907) 386--393. \bibitem{c1} P. Cl\'ement, R. Man\'asevich, and E. Mitidieri, Positive solutions for a quasilinear system via blow-up, {\it Commun. 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Soc.}, {\bf 115} (1992) 1037--1045. \end{thebibliography} \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 \end{document}