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\controldates{21-SEP-2004,21-SEP-2004,21-SEP-2004,21-SEP-2004}
 
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\begin{document}
\title[An upper bound for positive solutions]{An upper bound for positive 
solutions\linebreak[1] of the equation $\Delta u=u^\alpha$}

\author{S. E. Kuznetsov}
\address{Department of Mathematics,  University of Colorado, Boulder, CO
80309-0395}
\email{Sergei.Kuznetsov@Colorado.edu}
\thanks{Partially supported by the National Science Foundation Grant
DMS-9971009}

\subjclass[2000]{Primary 35J15; Secondary 35J25}

\commby{Mark Freidlin}

\date{April 5, 2004}

\begin{abstract}
In 2002 Mselati proved that every positive solution of the
equation $\Delta u=u^2$ in a bounded domain of class $C^4$ is the
limit of an increasing sequence of moderate solutions. (A solution
is called moderate if it is dominated by a harmonic function.)  As
a part of his proof, he established an  upper bound (in terms of the
capacity of $K$) for solutions vanishing off a compact subset $K$
of $\partial E$. We use a different kind of capacity  (we call it the
Poisson capacity) and we establish in terms of this capacity an
upper bound for solutions of $\Delta u=u^\alpha$ with $1<\alpha\le
2$. This is a part of the program: to classify all positive
solutions of this equation.
\end{abstract}

\maketitle

\section{Introduction}
\subsection{Main result}
Let $E\subset \rd$ be a bounded smooth domain of class $C^4$ in
$\rd$. For $x\in E$, we denote by $\rho(x)$ the distance to the
boundary $\pE$ and by $k(x,y)$  the Poisson kernel in $E$ for the
Laplacian $\Delta$.

Let $\M(S)$ stand for the set of all finite measures on a
measurable space $S$.  For every  $\nu\in\M(\pE)$, we denote by
$h_\nu$  the harmonic function $h_\nu(x)= \int_{\partial E} k(x,y)
\nu(dy)$.

For every $\myaa>1$ and every Radon measure $m$ on $E$, there exists
a Choquet capacity given on  compact subsets of $\pE$ by the
formula
\begin{equation}
\label{1.2} \cp(K)=\sup\limits_{\nu\in\myP(K)}  \E(\nu)^{-1}
\end{equation}
where $\myP(K)$ is the  set of all probability measures on $K$ and
\begin{equation}
\label{1.1} \E(\nu)=\int_Eh_\nu(x)^\myaa m(dx).
\end{equation}
We call $\cp$ the Poisson capacity.

Our goal is to establish the following theorem.

\begin{thm}
\label{main thm}
  Suppose $\cp$ is the Poisson capacity corresponding
to $1<\myaa\le 2$ and the measure $m(dx)=\rho(x)dx$. Let $K$ be a
compact subset of $\partial E$. There exists a constant $C(E)$
depending only on $E$, such that, for every compact $K\subset \pE$
and every solution $u$ of the boundary value problem
\begin{equation}
\label{main}
\begin{cases}\Delta u = u^\alpha \qt{ in } E,\\
u = 0 \qt{ on } \partial E\setminus K,
\end{cases}
\end{equation}
we have
\begin{equation}
\label{1.3} u(x)\le C(E) \rho(x) \dist(x,K)^{-d}
\cp(K)^{1/(\myaa-1)}.
\end{equation}
\end{thm}

\subsection{Equivalent definitions of the Poisson capacity}
Put
\begin{equation}
\label{1.4} \hat Kf(y)=\int_Ef(x)m(dx)k(x,y).
\end{equation}

The following definitions of the Poisson capacity are equivalent
to \eqref{1.2}:
\begin{equation}
\label{1.4a} \cp(K)^{1/\myaa}=\sup\{\nu(K): \nu\in\M(K), \E(\nu)\le
1\}
\end{equation}
and
\begin{equation}
\label{1.5} \cp(K)^{1/\myaa} =\inf\{\|f\|_{\myaa'}:\hat Kf\ge 1 \text{ on } K\}
\end{equation}
where $\myaa'=\myaa/(\myaa-1)$ and $\|f\|_{\alpha'}$ stands for the norm
in $L_{\myaa'}(m)$.

The equivalence of  \eqref{1.4a} and \eqref{1.5} is proved, for
instance, in \cite{Dy02} (see Theorem 5.1 in Chapter 13). To prove
the equivalence of \eqref{1.2} and \eqref{1.4}, we note that
$\nu\in\M(K)$ is equal to $t\mu$ where $t=\nu(K)$ and
$\mu=\nu/t\in\myP(K)$ and
\begin{align*}
\sup\limits_{\nu\in\M(K)}\{\nu(K):\E(\nu)\le1\}&=
\sup\limits_{\mu\in\myP(K)}\sup\limits_{t\ge 0}\{t:t^\myaa\E(\mu)\le 1\}\\
&=\sup\limits_{\mu\in\myP(K)}\E(\mu)^{-1/\myaa}=(\cp(K))^{1/\myaa}.
\end{align*}

\subsection{Notation} We denote by $B_r(x)$ a ball of radius $r$
centered at $x$. Let $H$ be a compact subset of $\partial E
\cap B_r(x)$ and let $\phi$ be a $C^\infty$ function on $E$ such
that $0\le \phi \le 1$. We call $\phi$ an \textit{$(R,x)$-truncating
function for $H$} if $\phi = 0$ in a neighborhood of $H$ and
$\phi(y) = 1$ if $\dist(x, y)\ge R$. We call $\phi$ an
\textit{$R$-localizing function} if $\phi= 0$ in a neighborhood of
$H$ and $\phi(y) = 1$ if $\dist(y,H )\ge R$.


\section{Bounds in a halfspace}
\subsection{}
First, we establish some  bounds in the case when $E=(0,\infty)\times
\R^{d-1}$. A generic element of $E$ is denoted  by $z$ or
$(s,x),s\in (0,\infty), x\in \R^{d-1}$. We use the notation $f(s,x)$
and $f^s(x)$ for functions on $E$.

Denote by $\EP$ an infinite strip $\{(s,x): 0\le s<1\}$. The measure
$m(dz)=1_{[0,1]}(s) s\,ds\,dx$ is concentrated on $\EP$. We denote
by $\|f\|_{\alpha}$ the norm in $L_\alpha(m)$. The Poisson kernel
$k$ can be represented by the formula
\[
k((s,x),y)=C q^s(x-y)
\]
where $C$ is a constant depending only on the dimension, and
\begin{equation}
\label{2.1} q^s(x)=\frac {s} {(|x|^2+s^2)^{d/2}}.
\end{equation}
\begin{lem}
\label{Lemma 3.1.1} Suppose that $z_0\in \partial E$ and $H$ is a
compact subset of $\partial E\cap B_1(z_0)$. If $\cp(H)>0$, then
there exists a $(3/2,z_0)$-truncating function $\beta$ for $H$
such that $\beta^s(x)=1$ for $s\ge 1$ and
\begin{equation}
\label{3.1.1-0} \|\nabla^2 \beta\|^{\alpha'} _{\alpha'} + \|
|\nabla \beta|^2\|^{\alpha'}_{\alpha'}+
\|\nabla\beta\|^{\alpha'}_{\alpha'}+\left\|\frac 1s
\frac{\partial\beta}{\partial s}\right\|^{\alpha'}_{\alpha'} \le
C(d) \cp (H)^{1/(\myaa-1)}
\end{equation}
where the constant $C(d)$ depends only on $d$. If $\cp(H)=0$,
then, for every $\epsilon > 0$, there exists a
$(3/2,z_0)$-truncating function $\beta$ for $H$ such that
$\beta^r(x)=1$ for $r\ge 1$ and
\begin{equation}
\label{3.1.1-0a} \|\nabla^2 \beta\|^{\alpha'} _{\alpha'} + \|
|\nabla \beta|^2\|^{\alpha'}_{\alpha'}+
\|\nabla\beta\|^{\alpha'}_{\alpha'}+\left\|\frac 1s
\frac{\partial\beta}{\partial s}\right\|^{\alpha'}_{\alpha'} <
\epsilon.
\end{equation}
\end{lem}
\begin{proof}  Let $\cp(H)>0$. By \eqref{1.5}, there exists a
function $f$ on $E$ such that $\|f\|_{\alpha'}^{\alpha'} \le 2
\cp(H)^{\alpha'/\alpha}$ and $\hat K f\ge 1$ on $H$. We may assume
that $f\ge 0$ (otherwise we just replace $f$ with $f^+$).

Let $A(t), 0\le t<\infty$, be an increasing $C^2$ function such
that $A(t)=0 $ for $t\le 1$ and $A(t) = 1$ for $t\ge \sqrt{2}$.
We set
\begin{equation}
\label{2.2} (\T f)^t(y) = \int_0^1 s\,ds
A(\sqrt{s/t})\int_{\R^{d-1}} f^s(x) q^s(y-x)\,dx \qt{for } t>0
\end{equation}
and
\begin{equation}
\label{2.3} (\T f)^0(y)= \lim\limits_{t\down 0}(\T f)^t(y)=\hat
Kf(y)
\end{equation}
(cf. \cite[formula (6.3), p.~175]{Dy02}).


By \cite[Theorem 13.6.1]{Dy02}, we have
\[
\|\T f\|_{\alpha'}+\|\nabla\T f\|_{\alpha'}+\|\nabla^2 \T
f\|_{\alpha'}+ \left\|\frac 1s \frac{\partial\T f}{\partial
s}\right\|_{\alpha'}\le  C \|f\|_{\alpha'}.
\]
Let $g(s,y)=a(s)b(y)$ be such that $0\le a,b\le 1$, $a,b$ are
$C^2$ functions, $a=1$ in a neighborhood of $0$, $b=1$ in a
neighborhood of $\partial E\cap B_1(z_0)$ and $g=0$ outside
$B_{3/2}(z_0)$. Let $h(t)$ be an increasing $C^2$ function on
$[0,\infty)$ such that $h(t) = 0$ if $t\le 1/4$ and $h(t) = 1$ if
$t\ge 3/4$. As in the proof of Lemma 13.6.5 from \cite{Dy02}, we
put
\[
u=\T f,\quad v = gu, \quad \phi = h(v)
\]
and, finally,
\[
\beta = 1- \phi = 1- h(g \T f).
\]
By \eqref{2.3}, $\T f=\hat K f \ge 1$ on $H$, and $\beta = 0$ in a
neighborhood of $H$ by the choice of $g$ and $h$. By direct
computation,\footnote{See \cite[pp.~181--182]{Dy02}, or
\cite[Section 3]{Ku98}.} we get
\begin{gather}
\label{2.6}
|v|+ |\nabla v|+ |\nabla^2 v| \le C(|u|+ |\nabla u|+ |\nabla^2 u|),\\
\label{2.7} \left|\frac 1s \frac{\partial v}{\partial s}\right| =
\left(\left|\frac as \frac{\partial u}{\partial s} + \frac {a'}s
u\right|\right)b\le C \left(|u|+ \frac 1s \left|\frac{\partial
u}{\partial s}\right|\right).
\end{gather}
More computation yields
\begin{gather}
\label{2.8}
|\nabla \phi|\le C |\nabla v|,\\
\label{2.9}
|\nabla^2 \phi| \le C \left(|\nabla^2 v| + \frac{|\nabla v|^2}v\right),\\
\label{2.10} |\nabla \phi|^2 \le C \left(\frac{|\nabla
v|^2}v\right).
\end{gather}
Therefore
\[
\|\nabla^2 \beta\| _{\alpha'} + \| |\nabla \beta|^2\|_{\alpha'}+
\|\nabla\beta\|_{\alpha'}+\left\|\frac 1s
\frac{\partial\beta}{\partial s}\right\|_{\alpha'} \le C
\|f\|_{\alpha'} \le C \cp(H)^{1/(\myaa-1)}.
\]


If $\cp(H) = 0$, then $\|f\|_{\alpha'}$ can be made arbitrary
small, and the same construction yields \eqref{3.1.1-0a}.
\end{proof}

\subsection{}
For a set $H\subset R^{d-1}$, we put $\lambda H = \{\lambda x:x\in
H\}$.


\begin{lem}
\label{Lemma A} For every compact set $H\subset R^{d-1}$ and every
$0<\lambda<1$,
\begin{equation}
\label{cap-scale} \cp(\lambda H)^{1/(\myaa-1)} \le
\lambda^{d-2\alpha'+1} \cp(H)^{1/(\myaa-1)}.
\end{equation}
\end{lem}
\begin{proof} Let $\myll>0$ and $\nu \in \myP(H)$.
Then $\nu_\lambda(A) = \nu(A/\lambda)$ is concentrated on $\lambda
H$. Note that $q^{\myll s}(\myll x)=\myll^{d-1}q^s(x)$ and  therefore
\begin{equation}
\label{A-1} h_{\nu}(s,x) = \myll^{d-1} h_{\nu_\myll}(\myll s, \myll x).
\end{equation}
Formula \eqref{A-1} and change of variables $t=\myll s, y=\myll x$
yield
\begin{equation*}
\begin{split}
\E(\nu) &= \int_0^1 \int_{\R^{d-1}} h_\nu^\myaa(s,x) sdsdx \\&=
\int_0^1 \int_{\R^{d-1}} \myll^{(d-1)\myaa} h_{\nu_\myll}^\myaa (\myll s,
\myll x) sdsdx \\&= \int_0^\myll \int_{\R^{d-1}} \myll^{(d-1)\myaa}
\myll^{-(d+1)} h_{\nu_\myll}^\myaa (t, y) tdtdy \\&\le \myll^{(d-1)\myaa}
\myll^{-(d+1)} \E(\nu_\myll)
\end{split}
\end{equation*}
and  \eqref{1.2} implies
\begin{equation}
\label{A-2} \cp(H)\ge \myll^{d+1 - (d-1)\myaa} \cp(\myll H).
\end{equation}
Formula \eqref{cap-scale} follows from \eqref{1.2} because
$d-2\myaa'+1=-[d+1-(d-1)\myaa]/(\myaa-1)$.
\end{proof}
\begin{lem}
\label{Lemma B} Let $z_0\in\partial E, 0<\delta<1$ and let
$\Gamma$ be a compact subset of $\partial E\cap B_{\delta}(z_0)$.
Suppose $\cp(\Gamma)>0$. There exists a
$(3\delta/2,z_0)$-truncating function $\mygg=\mygg_{\Gamma,\delta}$
for $\Gamma$ such that
\begin{equation}
\label{B-0} \|\nabla^2 \mygg\|^{\alpha'} _{\alpha'} + \| |\nabla
\mygg|^2\|^{\alpha'}_{\alpha'}+ \left\|\frac
1\delta\nabla\mygg\right\|^{\alpha'}_{\alpha'}+\left\|\frac 1s
\frac{\partial\mygg}{\partial s}\right\|^{\alpha'}_{\alpha'} \le C(d) \cp
(\Gamma)^{1/{\myaa-1}}
\end{equation}
where the constant $C(d)$ depends only on $d$. If $\cp(\Gamma)=0$,
then the left side of \eqref{B-0} can be made smaller than any
$\ee>0$.
\end{lem}
\begin{proof} Let $H = \Gamma/\dd$ and let $\bb(s,x)$ be the function constructed in Lemma \ref{Lemma 3.1.1} applied to $H$ and $z_0/\dd$.
Put $\mygg(s,x) = \beta(r/\delta,x/\delta)$. Since $\beta(s,x) = 1$
if $s\ge 1$, $\mygg(s,x)= 1$ if $s\ge\dd$. Also,
\[
\nabla\mygg(s,x) = (1/\dd) \nabla \beta(s/\dd,x/\dd),\qquad \nabla^2
\mygg(s,x) = (1/\dd^2) \nabla^2 \beta(s/\dd,x/\dd)
\]
and therefore
\[
\begin{split}
\|\nabla \mygg\|_{\myaa'}^{\myaa'} &= \int_0^\dd \int_{\R^{d-1}} |\nabla
\mygg(s,x)|^{\myaa'}\, s\,ds\,dx\\
&= \int_0^\dd \int_{\R^{d-1}} \dd^{-2\myaa'} |\nabla \bb(s/\dd,x/\dd)|^{\myaa'}\, s\,ds\,dx\\
&=\int_0^1 \int_{\R^{d-1}} \dd^{d+1-2\myaa'} |\nabla
\bb(s,x)|^{\myaa'}\, s\,ds\,dx = \dd^{d+1-2\myaa'} \|\bb\|_{\myaa'}^{\myaa'}.
\end{split}
\]
In a similar way,
\[
\| |\nabla \mygg|^2\|^{\alpha'}_{\alpha'} = \dd^{d+1-2\myaa'}\|
|\nabla \bb|^2\|^{\alpha'}_{\alpha'},\qquad \left\|\frac
1\delta\nabla\mygg\right\|^{\alpha'}_{\alpha'} = \dd^{d+1-2\myaa'}
\|\nabla\bb\|^{\alpha'}_{\alpha'}
\]
and
\[
\left\|\frac 1s \frac{\partial\mygg}{\partial
s}\right\|^{\alpha'}_{\alpha'} = \left\|\frac 1s
\frac{\partial\bb}{\partial s}\right\|^{\alpha'}_{\alpha'}.
\]
Therefore \eqref{B-0} follows from \eqref{3.1.1-0} and Lemma
\ref{Lemma A}.
\end{proof}

\section{Bounds in a unit ball}
\subsection{}
Now let $E$ be a ball of radius 1 in
$\rd$ centered at a point $z_0$ with coordinates $s=1,
x=0$. As before, let $\EP =\{(s,x): 0\le s < 1\}$. For a point
$z=(s,x)\in\EP$, we denote by
\[
\phi(z) = x/(1-s)
\]
a projection of $z$ to $\R^{d-1}$ with center at $z_0$. For a
point $z\in E\cap\EP$, we put
\[
\psi(z) = (1-|z-z_0|, \phi(z))
\]
(cf. \cite{Ms02}, Section 3.1.1). The mapping $\psi$ defines a 1-1
correspondence between $E\cap \EP$ and $\EP$.

For a set $H\subset \pE$, denote by $\cp_E(H)$ the Poisson
capacity of $H$ with respect to the domain $E$ and the measure
$m(dz) = \dist(z, \pE) \,dz$. For a set $K\subset \R^{d-1}$, we
denote by $\cp_{\EP}(K)$ the Poisson capacity with respect to the
halfspace and the measure $m(ds,dx) = 1_{[0,1)}(s)\,s\,ds\,dx$.

\begin{lem}
\label{Lemma C} Let $H$ be a compact subset of $\pE$ that is contained in
a ball of radius $1/4$ centered at zero, and let $K =
\psi(H)$. There exists a constant $C$ depending only on the
dimension, such that
\[
C^{-1} \cp_{\EP}(K) \le \cp_E(H) \le C \cp_{\EP}(K).
\]
\end{lem}

\begin{proof} Let $\mu$ be a probability measure on $H$ and let $\nu$ be a measure on $K$ defined by the formula $\nu(\Gamma) = \mu(\psi^{-1}(\Gamma))$. It is enough to show that
\begin{equation}
\label{C-0} C^{-1} \E_E(\mu) < \E_{\EP}(\nu)0$. There exists a $(3\delta/8,0)$-truncating function
$\gamma$ such that
\begin{equation}
\label{s3.1.2-0} \|\nabla^2 \gamma\|^{\alpha'} _{\alpha'} + \|
|\nabla \gamma|^2\|^{\alpha'}_{\alpha'}+ \left\|\frac
1\delta\nabla\gamma\right\|^{\alpha'}_{\alpha'}+\left\|\frac 1\rho
\frac{\partial\gamma}{\partial \rho}\right\|^{\alpha'}_{\alpha'}
\le C(d) \cp_E (H)^{1/(\myaa-1)},
\end{equation}
where the constant $C(d)$ depends only on $d$. If $\cp_E(H)=0$,
then the left side of \eqref{s3.1.2-0} can be made smaller than
any $\ee>0$.
\end{lem}

\begin{proof} We apply Lemma \ref{Lemma B} to the set $K=\psi(H)$. 
Let $\gamma_K$ be the function constructed in Lemma  \ref{Lemma B}. We put
$\gamma(s,x) = \gamma_K(\psi(s,x))$ if $s<1$, and $\gamma(s,x)=0$
otherwise. Similarly to the proof of \cite[Sublemma 3.1.2]{Ms02},
we show that
\begin{equation}
\label{s3.1.2-2}
\begin{split}
\|\nabla^2 \gamma\|^{\alpha'} _{E,\alpha'} &+ \| |\nabla
\gamma|^2\|^{\alpha'}_{E,\alpha'}+ \left\|\frac
1\delta\nabla\gamma\right\|^{\alpha'}_{E,\alpha'}+\left\|\frac
1\rho \frac{\partial\gamma}{\partial
\rho}\right\|^{\alpha'}_{E,\alpha'} \\&\le \|\nabla^2
\gamma_K\|^{\alpha'} _{\EP, \alpha'} + \| |\nabla
\gamma_K|^2\|^{\alpha'}_{\EP, \alpha'}+ \left\|\frac
1\delta\nabla\gamma_K\right\|^{\alpha'}_{\EP,
\alpha'}+\left\|\frac 1s \frac{\partial\gamma_K}{\partial
s}\right\|^{\alpha'}_{\EP, \alpha'}
\end{split}
\end{equation}
where $\|\cdot\|_E$ and $\|\cdot\|_{\EP}$ stand for
$L_{\myaa'}$-norms in $E$ and $\EP$. Finally, we apply Lemma
\ref{Lemma C}.
\end{proof}

\subsection{Localizing functions} Let $H$ be a subset of $\pE$ and let $\gamma$ be a $C^2$-function on $E$.
We call $\gamma$ an $\ee$-localizing function for $H$ if $0\le
\gamma\le 1$, $\gamma = 1$ in a neighborhood of $H$ and $\gamma(z)
= 0$ if $\dist(z,\myH)>\ee$.
\begin{lem}
\label{Lemma 3.1.2} There exists a constant $C(d)$ such that, for
every compact subset $K$ of $\partial E$ with $\cp(K)>0$ and
$\diam (K) \le 4\delta$, there exists a $\delta/2$-localizing
function $\gamma=\gamma_{\delta, K}$ for $K$ such that
\begin{equation}
\label{3.1.2-0} \|\nabla^2 \gamma\|^{\alpha'} _{\alpha'} + \|
|\nabla \gamma|^2\|^{\alpha'}_{\alpha'}+ \left\|\frac
1\delta\nabla\gamma\right\|^{\alpha'}_{\alpha'}+\left\|\frac 1\rho
\frac{\partial\gamma}{\partial \rho}\right\|^{\alpha'}_{\alpha'}
\le C(d) \cp(K)^{1/(\alpha-1)}.
\end{equation}
\end{lem}

\begin{proof} As in \cite[Lemma 3.1.2]{Ms02}, we cover the set $K$ by
finitely many balls $B_{\delta/8}(y_k)$ (the number $n$ of
the balls depends only on the dimension $d$). We apply Lemma
\ref{sublemma 3.1.2} to each of the sets $H_k = K\cap
B_{\delta/8}(y_k)$. Denote by $\gamma_k$ the corresponding
truncating function constructed in Lemma \ref{sublemma 3.1.2} (we
choose $\epsilon = \cp(\Gamma)^{1/(\alpha-1)}$ if $\cp(H_k)=0$ for
some $k$). We set
\[
\gamma = \gamma_1 \cdots \gamma_n.
\]
Note that
\[
|\nabla\gamma| \le \sum_k |\nabla \gamma_k|
\]
and
\[
\begin{split}
|\nabla^2\gamma| &\le \sum_k |\nabla^2 \gamma_k|+ \sum_{k\ne l}
|\nabla \gamma_k|\, |\nabla \gamma_l| \\
&\le \sum_k |\nabla^2 \gamma_k|+ \frac {(n-1)}2 \sum_{k} |\nabla
\gamma_k|^2.
\end{split}
\]
By applying Minkowski inequality, we get
\begin{equation}
\label{3.1.2-1}
\begin{split}
\|\nabla^2 \gamma\|^{\alpha'} _{\alpha'} &+ \| |\nabla
\gamma|^2\|^{\alpha'}_{\alpha'}+ \left\|\frac
1\delta\nabla\gamma\right\|^{\alpha'}_{\alpha'}+\left\|\frac 1\rho
\frac{\partial\gamma}{\partial \rho}\right\|^{\alpha'}_{\alpha'}\\
&\le C(d,n) \sum_k \cp (H_k)^{1/(\alpha-1)} + n C(n)\epsilon
\\&\le n (C(d,n)+ C(n)) \cp (\Gamma)^{1/(\alpha-1)}.
\end{split}
\end{equation}
\end{proof}

\begin{lem}
\label{Sublemma 3.1.4} Let $K$ be a compact subset of $\partial E$
such that $\diam(K)\le 4\delta$ and $\cp(K)>0$. Let $\gamma$ be
the $\delta/2$-localizing function constructed in Lemma \ref{Lemma
3.1.2}. Then
\begin{equation}
\label{3.8}
\begin{split}
\int_E u \gamma^{2\alpha'-1} |\Delta \gamma| \rho &\le C 
\left(\int_E u^\alpha \gamma^{2\alpha'}\rho\right)^{1/\alpha} 
\cp(K)^{1/\alpha},\\
\int_E u \gamma^{2\alpha'-2} |\nabla \gamma|^2 \rho &\le C \left(\int_E 
u^\alpha \gamma^{2\alpha'}\rho\right)^{1/\alpha} \cp(K)^{1/\alpha},\\
\int_E u \gamma^{2\alpha'-1} |\nabla \gamma| \rho &\le C \left(\int_E 
u^\alpha \gamma^{2\alpha'}\rho\right)^{1/\alpha}\delta \cp(K)^{1/\alpha},\\
\int_E u \gamma^{2\alpha'-1} \left|\frac{\partial \gamma}{\partial
\rho}\right|  &\le C \left(\int_E u^\alpha
\gamma^{2\alpha'}\rho\right)^{1/\alpha} \cp(K)^{1/\alpha}
\end{split}
\end{equation}
whenever $u$ satisfies \eqref{main}.
\end{lem}

\begin{proof} The assertion
follows from the H\"older inequality, Lemma \ref{Lemma 3.1.2},
the identity $(\alpha-1)\alpha' = \alpha$  and the inequality
$\gamma^{(2\alpha'-1)\alpha} \le
\gamma^{(2\alpha'-2)\alpha}=\gamma^{2\alpha'}$. For instance, for
the last line in \eqref{3.8}, we have
\[
\begin{split}
\int_E u \gamma^{2\alpha'-1} \left|\frac{\partial \gamma}{\partial
\rho}\right| &= \int_E u \gamma^{2\alpha'-1} \left|\frac 1\rho
\frac{\partial \gamma}{\partial \rho}\right|\rho \\&\le \left(
\int_E u^\alpha
\gamma^{(2\alpha'-1)\alpha}\rho\right)^{1/\alpha}\left( \int_E
\left|\frac 1\rho \frac{\partial \gamma}{\partial
\rho}\right|^{\alpha'}\rho \right)^{1/\alpha'} \\&\le \left(\int_E
u^\alpha \gamma^{2\alpha'}\rho\right)^{1/\alpha} C(d)
\cp(K)^{1/[\alpha'(\alpha-1)]}.
\end{split}
\]
\end{proof}


\begin{lem}
\label{Lemma 3.1.3} Let $K, \gamma, u$ be as in Lemma
\ref{Sublemma 3.1.4}. There exists  a constant $C(d)$ such that
\begin{equation}
\label{3.1.3-0} \int_E u^\alpha \gamma^{2\alpha'} \rho \le C(d)
\cp(K)^{1/(\alpha-1)}
\end{equation}
whenever $u$ satisfies \eqref{main}.
\end{lem}
\begin{proof} This is an adaptation of Lemma 3.1.3 in \cite{Ms02}.
Let $E_s = B(s, z_0)$ and $r=|z-z_0|$. By replacing $u^2$ and
$\gamma^4$ with $u^\alpha$ and $\gamma^{2\alpha'}$ in the arguments
of \cite{Ms02}, we get a bound
\begin{equation}
\label{Ms-3.3.7}
\begin{split}
\int_E u^\alpha \gamma^{2\alpha'} \rho \le &\int_E u \Delta
(\gamma^{2\alpha'}(1-r^2))- 4 \int_E u \frac{\partial
(\gamma^{2\alpha'})}{\partial r} r \\&+ \liminf_{s\to
1-}\int_{\partial E_s} \frac{\partial}{\partial r}
(u\gamma^{2\alpha'}(1-r^2)).
\end{split}
\end{equation}
As in \cite[Sublemma 3.1.3]{Ms02}, one can show that the last term
in \eqref{Ms-3.3.7} is negative and can be dropped.

Finally, we note that $1-r^2\le 2\rho$ and therefore
\begin{equation}
\label{Ms-3.3.9}
\begin{split}
&\left|\int_E u \Delta (\gamma^{2\alpha'}(1-r^2))\right| \\
&\qquad\le 2\alpha' \int_E u \gamma^{2\alpha'-1} |\Delta \gamma|
(1-r^2)+
2\alpha'(2\alpha'-1) \int_E u \gamma^{2\alpha'-2} |\nabla \gamma|^2 \rho \\
&\qquad \le 4\alpha' \int_E u \gamma^{2\alpha'-1} |\Delta \gamma|
(1-r^2)+
4\alpha'(2\alpha'-1) \int_E u \gamma^{2\alpha'-2} |\nabla \gamma|^2 \rho \\
&\qquad \le C \left(\int_E u^\alpha
\gamma^{2\alpha'}\rho\right)^{1/\alpha} \cp(K)^{1/\alpha}.
\end{split}
\end{equation}
In a similar way,
\begin{equation}
\label{Ms-3.3.10}
\begin{split}
\left|\int_E u \frac{\partial (\gamma^{2\alpha'})}{\partial r} r\right| &= 2\alpha'\left|\int_E u \gamma^{2\alpha'-1}\frac{\partial \gamma}{\partial r} r\right| \le 2\alpha'\left|\int_E u \gamma^{2\alpha'-1}\frac{\partial \gamma}{\partial r} \right| \\
&\le C \left(\int_E u^\alpha
\gamma^{2\alpha'}\rho\right)^{1/\alpha} \cp(K)^{1/\alpha}.
\end{split}
\end{equation}
From \eqref{Ms-3.3.7}, \eqref{Ms-3.3.9} and \eqref{Ms-3.3.10}, we
get
\[
\int_E u^\alpha \gamma^{2\alpha'} \rho \le C \left(\int_E u^\alpha
\gamma^{2\alpha'}\rho\right)^{1/\alpha} \cp(K)^{1/\alpha},
\]
which implies \eqref{3.1.3-0}.
\end{proof}

Combining \eqref{3.8} with Lemma \ref{Lemma 3.1.3}, we get:
\begin{lem}
\label{Lemma 3.1.4} Let $K,\gamma,u$ be as in Lemma \ref{Sublemma
3.1.4}. Then
\begin{equation}
\label{3.8bis}
\begin{split}
\int_E u \gamma^{2\alpha'-1} |\Delta \gamma| \rho &\le C  
\cp(K)^{1/(\alpha-1)},\\
\int_E u \gamma^{2\alpha'-2} |\nabla \gamma|^2 \rho &\le C  
\cp(K)^{1/(\alpha-1)},\\
\int_E u \gamma^{2\alpha'-1} |\nabla \gamma| \rho &\le C \delta 
\cp(K)^{1/(\alpha-1)},\\
\int_E u \gamma^{2\alpha'-1} \left|\frac{\partial \gamma}{\partial
\rho}\right|  &\le C  \cp(K)^{1/(\alpha-1)}.
\end{split}
\end{equation}
\end{lem}

\subsection{}
The rest of the proof of Theorem \ref{main thm} is very close to
the corresponding part of the proof of
\cite[Theorem 3.1.1]{Ms02}. We begin with
\begin{lem}
\label{Lemma 3.1.6} Let $K,\gamma,u$ be as in Lemma \ref{Sublemma
3.1.4}, and let $G_E$ be the Green operator of $E$. For every
$y \in E, \beta >0$,
\[
\gamma^\beta(y)u(y) \le \frac12 G_E\left(\gamma^\beta \Delta u -
\Delta (\gamma^\beta u)\right)(y).
\]
\end{lem}

This is a version of \cite[Lemma 3.1.6]{Ms02} (in \cite{Ms02},
$\beta = 4$). Bounds from \cite[Sublemma 3.1.5]{Ms02} must be
replaced with those of \cite[Theorem 7.1]{Dy02}. Other
modifications are obvious.

As a first step, we establish
\begin{lem}
\label{Lemma 3.1.5} There exists a constant $C(d)$ such that the
inequality \eqref{1.3} holds whenever
\[
\dist(x,K) \ge \frac{\diam(K) }4.
\]
\end{lem}

\begin{proof} The proof is an appropriate modification of the proof of
\cite[Lemma 3.1.5]{Ms02}.
Let $\delta = \dist(x,K)$ and let $\gamma$ be the
$\delta/2$-localizing function constructed in Lemma \ref{Lemma
3.1.2}. Clearly, $\gamma=1$ in a neighborhood of $x$ and therefore
\begin{equation}
\label{3.11} u(x) = \gamma^{2\alpha'}(x)u(x) \le \frac12
G_E(\gamma^{2\alpha'} \Delta u - \Delta(\gamma^{2\alpha'}u))(x)
\end{equation}
by Lemma \ref{Lemma 3.1.6}.  As in \cite{Ms02}, the right side can
be evaluated by means of Green's formula applied to the domain
$B(z_0,r)\setminus B(x,\ee)$ and passage to the limit as $\ee \to
0, r\to 1$. Namely, we get
\[
\begin{split}
G_E(\gamma^{2\alpha'} \Delta u &- \Delta(\gamma^{2\alpha'}u))(x)
\\ &= \int_E (2 \nabla_y g_E(x,y)\nabla (\gamma^{2\alpha'}(y)
- g_E(x,y) \Delta (\gamma^{2\alpha'})(y))u(y)\,dy ).
\end{split}
\]
Together with \eqref{3.11}, this implies
\begin{equation}
\label{3.13}
\begin{split}
u(x)&\le  2\alpha'\int_E u(y) \gamma^{2\alpha'-1}(y) \nabla \gamma(y) \nabla_y g_E(x,y)\,dy \\
& \quad +\alpha' \int_E u(y) \gamma^{2\alpha'-1}(y) \Delta \gamma(y) g_E(x,y) \,dy \\
&\quad +\alpha'(2\alpha'-1) \int_E u(y) \gamma^{2\alpha'-2} (y) |
\nabla \gamma(y)|^2 g_E(x,y)\,dy
\end{split}
\end{equation}
(cf. \cite[(3.13)]{Ms02}). Now, $\gamma(y)=1$ for all $y$ such
that $\dist(y,K)>\delta/2$, in particular for all $y$ such that
$|x-y|<\delta/2$, and therefore the integrands are equal to $0$ for
such $y$. Following \cite{Ms02}, from the bounds for the Green's
function and its gradient, we get bounds for the integrals on the
right side of \eqref{3.13} in terms of integrals
\eqref{3.8bis}. For instance,
\[
g_E(x,y) \le C \rho(x)\rho(y) |x-y|^{-d},
\]
and therefore
\[
\begin{split}
{}&\int_E u(y) \gamma^{2\alpha'-2} (y) | \nabla \gamma(y)|^2
g_E(x,y)\,dy \\&\qquad\le C \rho(x) \int_E u(y) \gamma^{2\alpha'-2} (y) |
\nabla \gamma(y)|^2 \rho(y) |x-y|^{-d}\,dy \\ &\qquad=\int_{E\setminus
B(x,\delta/2)} u(y) \gamma^{2\alpha'-2} (y) | \nabla \gamma(y)|^2
\rho(y) |x-y|^{-d}\,dy
\\ &\qquad\le C \rho(x) \delta^{-d}\int_E u \gamma^{2\alpha'-2}(y) |\nabla \gamma|^2(y) \rho
(y)\,dy.
\end{split}
\]
In a similar way, we get
\begin{equation} 
\label{3.17}
\begin{split}
{}&\int_E u(y) \gamma^{2\alpha'-1}(y) \nabla \gamma(y) \nabla_y g_E(x,y)\,dy\\
&\qquad\le C \rho(x) \delta^{-d} \int_E u \gamma^{2\alpha'-1} 
\left|\frac{\partial \gamma}{\partial \rho}\right|  + C\rho(x) 
\delta^{-d-1}\int_E u \gamma^{2\alpha'-1} |\nabla \gamma| \rho, \\
{}&\int_E u(y) \gamma^{2\alpha'-1}(y) \Delta \gamma(y) g_E(x,y) \,dy
\le C \rho(x) \delta^{-d}\int_E u \gamma^{2\alpha'-1} |\Delta
\gamma| \rho.
\end{split}
\end{equation}
It remains to use the bounds of Lemma \ref{Lemma 3.1.4}.
\end{proof}

Theorem \ref{main thm} can be derived from this by using the same
construction as in \cite{Ms02}. Let $x\in E$ and $K,u$ be as in
Theorem \ref{main thm}. Let $\delta = \dist(x, K)$. We set
\[
K_1 = K\cap \overline{B(x,2\delta)},
\]
and
\[
K_n = K\cap\left( \overline{B(x, 2^n\delta)}\setminus B(x,
2^{n-1}\delta)\right),\quad n\ge 2.
\]
Since $K = \bigcap K_n$, we have
\[
u\le u_K\le \sum u_{K_n}
\]
where $u_K$ stands for the maximal solution of problem
\eqref{main}. By construction, we have $\dist (x, K_n )\ge
2^{n-1}\delta$ and $\diam (K_n)\le 2^{n+1}\delta$. Therefore Lemma
\ref{Lemma 3.1.5} is applicable to every $K_n$ and we get
\[
u_{K_n} \le C \rho(x) 2^{-nd} \delta ^{-d}
\cp(K_n)^{1/(\alpha-1)}\le C \rho(x) 2^{-nd} \delta ^{-d}
\cp(K)^{1/(\alpha-1)},
\]
which implies
\[
u(x) \le \sum u_{K_n} \le C \rho(x) \delta ^{-d}
\cp(K)^{1/(\alpha-1)} \sum 2^{-nd}.
\]

The extension of the theorem to arbitrary $C^4$ domains is a
simple modification of the arguments by Mselati \cite{Ms02}.



\bibliographystyle{amsplain}
\begin{thebibliography}{3}

\bibitem{Dy02} E. B. Dynkin, \emph{Diffusions, superdiffusions and partial
differential
  equations},
American Mathematical Society, Providence, RI, 2002. 
\MR{1883198 (2003c:60001)}

\bibitem{Ku98} S. E. Kuznetsov,  \emph{Polar boundary sets for superdiffusions
and removable lateral singularities for nonlinear parabolic PDEs},
Comm. Pure Appl. Math. \textbf{51} (1998), 303--340. 
\MR{1488517 (99c:35111)}

\bibitem{Ms02} B. Mselati, \emph{Classification et repr\'{e}sentation
probabiliste des solutions positives de $\Delta u=u^2$ dans un
domaine}, Th\'{e}se de Doctorat de l'Universit\'{e} Paris 6,
2002. 

\end{thebibliography}

\end{document}

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