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\markboth{\hfil nonlinear elliptic variational inequalities \hfil EJDE--2002/14}
{EJDE--2002/14\hfil Luka Korkut, Mervan Pa\v si\'c, \& Darko \v Zubrini\'c
\hfil}
\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc  Electronic Journal of Differential Equations},
Vol. {\bf 2002}(2002), No. 14, 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} \\
 %
A class of nonlinear \\
elliptic variational inequalities:\\
qualitative properties and existence of solutions
 %
\thanks{ {\em Mathematics Subject Classifications:} 35J65, 35J85, 35B05.
\hfil\break\indent
{\em Key words:} variational inequalities, double
obstacle, qualitative properties, \hfil\break\indent Schwarz symmetrization,
 generating singularities.
\hfil\break\indent
\copyright 2002 Southwest Texas State University. \hfil\break\indent
Submitted December 15, 2001. Published February 9, 2002.} }
\date{}
%
\author{Luka Korkut, Mervan Pa\v si\'c, \& Darko \v Zubrini\'c}
\maketitle

\begin{abstract}
 We study a class of nonlinear elliptic variational inequalities 
 in divergence  form. In the recent paper \cite{KPZ1}, we obtained
 results on the local control of essential infimum and supremum of
 solutions of quasilinear elliptic equations, and here we extend this
 point of view to the case of variational inequalities.
 It implies a new qualitative property of solutions in $W^{1,p}(\Omega )$
 which we call ``jumping over the control obstacle.''
 Using the Schwarz symmetrization technique, we give an existence
 and symmetrization theorems in
 $W_0^{1,p}(\Omega )\cap L^{\infty }(\Omega )$
 which agree completely with previous qualitative results. Also we consider
 generating singularities of weak solutions in $W^{1,p}(\Omega )$
 of variational inequalities.
\end{abstract}


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\section{Problem setting and main results}

Let $\Omega$ be a bounded open subset of $\mathbb{R}^N$, $N\geq 1$, and $p\in
(1,\infty)$. We are concerned with the following
nonlinear elliptic double obstacle
problem: Find $u\in W^{1,p}(\Omega)$ such that $\omega_1\leq u\leq
\omega _2$ in $\Omega$  and
\begin{equation}\label{osn}
\int_{\Omega }a(x,u,\nabla u)\cdot\nabla (v-u)\,dx\geq \int_{\Omega }
\bigl[f(x,u)+g(x,u)|\nabla u|^{p}\bigl](v-u)\,dx,
\end{equation}
for all $v\in W^{1,p}(\Omega )$ such that $v-u\in L^{\infty
}(\Omega )$ and $\omega_1\leq v\leq \omega _2$
in $\Omega$.

The obstacles $\omega_1$ and $\omega _2$ are two measurable functions
without any global regularity, such that $\omega_1\leq \omega _2$ in
$\Omega$. For any measurable set $A$ in $\mathbb{R}^N$ we say
that a property holds ``in $A$''
if it holds in the a.e.\ sense. The leading term on the left-hand side is the
Carath\'{e}odory vector function $a(x,\eta ,\xi)$
satisfying general
structure conditions of Leray-Lions type, see (\ref{ll1})--(\ref{ll2}).
The leading terms on the right-hand
side are Carath\'{e}odory real functions $f(x,\eta )$ and $g(x,\eta )$
which will essentially influence the main results.

In order to describe our main goals of this paper, we introduce an
additional obstacle $\omega _{c}$, that we call control obstacle,
which is taken to be a
measurable real function
defined on $\Omega$, satisfying the following natural condition
relative to $\omega_1$ and $\omega _2$:
\begin{equation}\label{oc}
\omega_1\leq \omega _{c}\leq \omega _2\quad \mbox{in }\Omega\,.
\end{equation}
Furthermore, we assume that there exist two balls $B_{r}$ and $B_{\rho }$ in
$\Omega$ such that $B_{2r}\subseteq
\Omega $, $B_{2\rho }\subseteq \Omega $, $B_{2r}\cap B_{2\rho
}=\emptyset$,
\begin{equation}\label{mm}
\begin{gathered}
m_1={\mathop{\rm ess\,inf}}_{B_{2r}}\omega_1,
\quad M_{c}={\mathop{\rm ess\,sup}}_{B_{2r}}\omega _{c},
\quad m_2={\mathop{\rm ess\,inf}}_{B_{2r}}\omega _2,  \\
-\infty <m_1<M_{c}<m_2<\infty ,
\end{gathered}
\end{equation}
and
\begin{equation}\label{MM}
\begin{gathered}
M_1={\mathop{\rm ess\,sup}}_{B_{2\rho }}\omega_1,
\quad m_{c}={\mathop{\rm ess\,inf}}_{B_{2\rho}}\omega _{c},
\quad M_2={\mathop{\rm ess\,sup}}_{B_{2\rho }}\omega _2,  \\
\quad -\infty <M_1<m_{c}<M_2<\infty .
\end{gathered}
\end{equation}
Relation (\ref{mm}) (respectively (\ref{MM})) has the following meaning:
the obstacles $\omega_1$ and $\omega _2$ are bounded from below
(respectively from above) on the corresponding ball,
and the obstacles $\omega _{c}$ and $\omega _2$
(respectively $\omega _{c}$ and $\omega_1)$ are strictly
separated on the corresponding balls.

In this paper we find sufficient conditions on functions $f(x,\eta )$
and $g(x,\eta )$ which will give us the
following three types of results: (i)~jumping over a prescribed control
obstacle on prescribed balls,
(ii)~existence of at least one essentially bounded weak solution,
(iii)~generating of singularities, bumping on the upper obstacle
and pushing to the upper obstacle.

\paragraph{(i)}
We say that a solution $u$ of (\ref{osn}) {\em jumps over the
prescribed control obstacle\/} $\omega_c$ if it satisfies
\begin{equation}\label{Oc}
\big|\{x\in \Omega :u(x)>\omega _{c}(x)\}\big|\neq 0,
\qquad  \big|\{x\in \Omega :u(x)<\omega _{c}(x)\}\big|\neq 0\,,
\end{equation}
where $|A|$ denotes the Lebesgue measure of a subset $A$ of $\mathbb{R}^N$.
Taking $B_{r}$, $B_{\rho }$, $m_1$, $M_1$, $m_{c}$, $M_{c}$, $m_2$ and $%
M_2$ as in (\ref{oc})--(\ref{MM}), and $\alpha_0 $, $a_0(x)$,
$a_1$, $a_2$ as in structure conditions
(\ref{ll1})--(\ref{ll2}), we now
impose two crucial sets of hypotheses. First those corresponding
to ball $B_{2r}$:
\begin{enumerate}
\item[(H1)] $ g(x,\eta )\geq 0$ in $B_{2r}$, for all
$\eta \in I_1=(m_1,M_{c})$

\item[(H2)] There exists $f_1\in L^{1}(B_{2r})$, such that
$f(x,\eta )\geq f_1(x)$ in $B_{2r}$, for all $\eta \in I_1$,
$f_1(x)\geq 0$ in $\ B_{2r}\setminus B_{r}$, and
$$
\int_{B_{r}}f_1(x)\,dx>D_1\frac{m_2-m_1}{m_2-M_{c}},
$$
where $D_1=\overline{d}\int_{B_{2r}}[a_0(x)+a_1\widehat{m}^{p-1}
]^{p^{\prime}}\,dx+\bigl(\frac{p}{d}\bigl)^{p-1}
\frac{(2^N-1)|B_{r}|}{r^{p}}$, \\
$\widehat{m}=\max \{ |m_1|,|M_{c}|\}$,
$\overline{d}=\frac{\alpha_0 }{a_2^{p'}(m_2-m_1)}$,
$d=\frac{p'}{2^{p'-1}}\overline{d}$.
\end{enumerate}
Now we impose the dual hypotheses corresponding to ball $B_{2\rho}$.
\begin{enumerate}
\item[(H3)] $g(x,\eta )\leq 0$ in $B_{2\rho }$, for all
$\eta \in I_2=(m_{c},M_2)$

\item[(H4)]
There exists $f_2\in L^{1}(B_{2\rho })$, such that $f(x,\eta )\leq f_2(x)$
 in $B_{2\rho }$ for all $\eta \in I_2$,
$f_2(x)\leq 0$ in $B_{2\rho }\setminus B_{\rho }$,
$$\int_{B_{\rho }}f_2(x)\,dx<-D_2\frac{M_2-M_1}{m_{c}-M_1},
$$
where $D_2=\overline{d}\int_{B_{2\rho }}[a_0(x)+a_1\widehat{m}
^{p-1}]^{p'}\,dx+\bigl(\frac{p}{d}\bigl)^{p-1}
\frac{(2^N-1)|B_{\rho}|}{\rho ^{p}}$, \\
$\widehat{m}=\max \{\ |m_{c}|,|M_2|\ \},$
$\overline{d}=\frac{\alpha_0 }{a_2^{p'}(M_2-M_1)}$,
$d=\frac{p'}{2^{p'-1}}\overline{d}$,
with $p'$ satisfying $1/p+1/p'=1$.
\end{enumerate}
Two complementary situations occur: the hypotheses (H1)--(H2) require
that the function $f(x,\eta )$ be sufficiently large and positive in
the strip $B_{2r}\times I_1$, and that $g(x,\eta )$ be
non-negative in the same strip
(respectively, the hypotheses (H3)--(H4) require that $f(x,\eta )$
be sufficiently large and negative in the strip $B_{2\rho}\times
I_2$, and that $g(x,\eta )$ be non-positive in the strip). These
conditions will imply, see Theorem~\ref{main}, that each solution $u$
of (\ref{osn}) satisfy
\begin{equation}\label{ne}
\big|\{x\in B_{2r} :u(x)>\omega _{c}(x)\}\big|\neq 0,\qquad
 \big|\{x\in B_{2\rho}:u(x)<\omega _{c}(x)\}\big| \neq 0\,,
\end{equation}
that is to say, there are two measurable sets $E_{r}\subseteq B_{2r}$ and $%
E_{\rho }\subseteq B_{2\rho }$, $|E_{r}|\neq 0$, $|E_{\rho }|\neq
0$, satisfying
$u(x)>\omega _{c}(x)$ for each $x\in E_{r}$ and
$u(x)<\omega _{c}(x)$ for each $x\in E_{\rho }$.

Since $B_{2r}\cap B_{2\rho }=\emptyset$, both pairs of hypotheses
(H1)--(H2) and (H3)--(H4) are independent of each other, which allows us to
combine them and derive the main result of this paper:

\begin{theorem}[Jumping over the Control obstacle in $W^{1,p}(\Omega )$]
\label{main}
 Under assumptions (\ref{oc})--(\ref{MM}), let the Carath\'eodory vector
function $a(x,\eta ,\xi )$ satisfy:
\begin{equation} \label{ll1}
\exists \alpha_0 >0,\ a(x,\eta ,\xi )\cdot \xi \geq
\alpha_0 |\xi |^{p}\quad \mbox{in }\Omega ,\ \eta \in \mathbb{R},\ \xi
\in \mathbb{R}^N,
\end{equation}
\begin{equation}\label{ll2}
\begin{gathered}
\exists a_0=a_0(x)\geq 0,\quad a_0\in L^{p'}(\Omega ),\quad
\exists a_1\geq 0,\quad \exists a_2>0,\\
|a(x,\eta ,\xi )|\leq a_0(x)+a_1|\eta |^{p-1}+a_2|\xi |^{p-1}\quad
\mbox{in }\Omega ,\; \eta \in \mathbb{R},\; \xi \in\mathbb{R}^N.
\end{gathered}
\end{equation}
If the Carath\'eodory functions  $f(x,\eta )$  and $g(x,\eta)$
satisfy the hypotheses (H1)--(H4), then for each solution
$u\in W^{1,p}(\Omega )$ of (\ref{osn}) satisfies (\ref{ne}).
\end{theorem}

To prove (\ref{ne}) we argue by contradiction. First we
choose appropriate test functions in order to localize the balls in
$\mathbb{R}^N$,
then integrate over level sets of the form $\{u>t\}$ and
$\{u<t\}$, and then use
several elementary inequalities in $\mathbb{R}$ in order to obtain contradiction.
We want to indicate that the proof of property (\ref{ne}) is
not difficult. As pointed out in the abstract, the same method
has been exploited in \cite{KPZ1} in order
to obtain the local control of essential infimum and
supremum of solutions of elliptic equations, and in \cite{KPZ2}
to obtain some
new qualitative properties of solutions.
In (H2) and (H4) we have to impose
slightly different conditions on nonlinear term
$f(x,\eta)$ than we did in \cite[Theorem 5]{KPZ1}, in order to
have the desired control effect.

\paragraph{(ii)} The second type of result is the existence of at least one solution
of (\ref{osn}) in $W_0^{1,p}(\Omega )\cap L^{\infty }(\Omega )$ that satisfy
jumping condition (\ref{Oc}). This condition is fulfilled if
$|f(x,\eta )|$ is uniformly
bounded in $\Omega \times \mathbb{R}$ and $|g(x,\eta )|$ is uniformly small enough
in $\Omega \times \mathbb{R}$, or conversely, $|g(x,\eta)|$ is uniformly
bounded and $|f(x,\eta)|$ uniformly small enough.
Precisely, we will impose one more hypotheses on these functions,
which does not contradict (H1)--(H4):
\begin{enumerate}
\item[(H5)]
There exist $f_0>0$ and $g_0\geq 0$  such that
$|f(x,\eta )|\leq f_0$, $|g(x,\eta )|\leq g_0$,
in $\Omega \times \mathbb{R}$, and
\[ f_0^{p'-1}g_0<\Big(\frac{\alpha_0 NC_{N}^{1/N}}{2\mid \Omega
\mid ^{1/N}}\Big)^{p'}\frac{p'}{N(p'+1)}.
\]
\end{enumerate}
Then we have the following result on existence and symmetrization of
solutions in  $W_0^{1,p}(\Omega )\cap L^{\infty }(\Omega )$).

\begin{theorem} \label{thm2}
 Under the structure
assumptions (\ref{oc})--(\ref{MM}) where $\omega_1\leq 0\leq \omega _2$
in $\Omega $ and $\omega_1,\omega _2\in L^{p}(\Omega )$, let the
Carath\'eodory vector function $a(x,\eta ,\xi )$ satisfy the hypotheses
(\ref{ll1}), (\ref{monoton}), and
\begin{equation}\label{monoton}
(a(x,\eta ,\xi )-a(x,\eta ,\xi ^{*}))\cdot (\xi -\xi ^{*})>0\quad
\mbox{in }\Omega ,\; \eta \in \mathbb{R},\; \xi ,\xi^{*}\in \mathbb{R}^N,
\;\xi\ne \xi ^{*}.
\end{equation}
Assume that
Carath\'eodory functions $f(x,\eta )$ and $g(x,\eta )$ satisfy
conditions
(H1)--(H5). Then there exists a solution $u\in W_0^{1,p}(\Omega )\cap
L^{\infty }(\Omega )$ of (\ref{osn}) satisfying (\ref{Oc}). Moreover,
\begin{equation}\label{sharp<}
\begin{gathered}
u^{\#}(x)\leq v^{\#}(x)=v(x)\quad \mbox{in } \Omega ^{\#},  \\
\|u\|_{L^{\infty }(\Omega )}\leq \|v\|_{L^{\infty }(\Omega ^{\#})}
\quad \mbox{and}\quad \|\nabla u\|_{L^{p}(\Omega )}\leq
\|\nabla v\|_{L^{p}(\Omega ^{\#})},
\end{gathered}
\end{equation}
where $u^{\#}$ is the Schwarz symmetrization of $u$, and $v$ is
the unique solution of the symmetrized problem
\begin{equation}\label{sharp=}
\begin{gathered}
-\mathop{\rm dvi}(\alpha_0 |\nabla v|^{p-2}\nabla v)
=f_0+g_0|\nabla v|^{p} \quad \mbox{in } \Omega ^{\#}, \\
v\in W_0^{1,p}(\Omega ^{\#})\cap L^{\infty }(\Omega ^{\#}),
\mbox{ $v$ is positive and spherically symetric.}
\end{gathered}
\end{equation}
Here $\Omega ^{\#}$ is a ball in $\mathbb{R}^N$ centered at the origin,
with the same volume as $\Omega$.
\end{theorem}

Applications of the Schwartz symmetrization to
partial differential equations can be seen for
instance in \cite{ALT,FP,Tal}, while
applications to variational inequalities are treated in
\cite{BMP,BM, FPR}. Let us mention that the additional condition
(H5), that is to
say, the ``smallness condition'' on the data $f(x,\eta )$ and $g(x,\eta )$
is used only sufficient for existence of a solution $v$ of the
symmetrized equation (\ref{sharp=}). This problem is treated in
detail in \cite{P3,KPZ3,Tu}.

In contrast to the proof of qualitative property
(\ref{ne}), the proof of existence result requires a
more complicated procedure. Here we exploit the method of penalty
functions as approximation step, and the method of Schwartz symmetrization of
penalty equation in order to derive a priori estimates which are independent
on the
approximative process. This construction has already been announced in
the recent paper \cite{P1}, but without proof and details.

\paragraph{(iii)} The third type of results concerns the possibility
to generate singularities of solutions in a given point. In
particular, this enables to obtain nonexistence result for
essentially bounded weak solutions.

It will be convenient to define
essential supremum $u^*$ and essential infimum $u_*$
of a measurable function $u\:\Omega\to\overline{\mathbb{R}}$ in the
point $x_0\in\Omega$:
\begin{equation}
u^*(x_0)=\lim_{r\to0}{\mathop{\rm ess\,sup}}_{B_r(x_0)}u(x),\quad
u_*(x_0)=\lim_{r\to0}{\mathop{\rm ess\,inf}}_{B_r(x_0)}u(x).
\end{equation}
We say that $u$ has singularity at $x_0$ if $u^*(x_0)=+\infty$.
The following theorem shows that it is possible to generate a singularity of
all solutions of variational problem (\ref{osn})
in a given point $x_0\in\Omega$. Of course, the upper obstacle $w_2$
also has to be singular in this point.

\begin{theorem}\label{sing}
{\rm(Generating singularities of solutions)}
Assume that $a(x,\eta,\xi)$ satisfies conditions (\ref{ll1}) and
(\ref{ll2}).
Let there exist $x_0\in \Omega$ and $\beta\in\mathbb{R}$ such that:
\begin{equation}\label{intBr}
\int_{B_{r}}[a_0(x)+a_1 \widehat m^{p-1}]^{p'}dx=O(r^\beta),\quad as\,\, r\to0,
\end{equation}
where $B_r=B_r(x_0)$.
Assume that there exist positive constants $\alpha$, $\gamma$, $R$,
$C_1$, $C_2$,
such that for a.e.\ $x\in B_R$ and $\eta\ge{\mathop{\rm ess\,inf}}_{B_R}
\omega_1$,
\begin{equation}\label{f}
g(x,\eta)\ge0,\quad f(x,\eta)\ge\frac {C_1}{|x-x_0|^\gamma},\quad
w_2(x)\ge\frac{C_2}{|x-x_0|^\alpha}.
\end{equation}
If
\begin{equation}\label{ac}
\alpha<\frac{\gamma-p}{p-1},
\end{equation}
and
\begin{equation}\label{beta}
\alpha+\beta+\gamma> N,
\end{equation}
then any solution $u$ of problem (\ref{osn}) is singular at $x_0$,
that is, $u^*(x_0)=\infty$. In particular, problem (\ref{osn}) has no
solutions in $W^{1,p}(\Omega)\cap L^{\infty}(\Omega)$.
\end{theorem}


\paragraph{Remark} Note that conditions (\ref{f}), (\ref{ac}) and
$\alpha>0$ imply that
$p<\gamma<N$.
This shows that Theorem~\ref{sing} is in accordance with the Sobolev
imbedding theorem, since we have $p<N$ and singular solutions.


\paragraph{Remark} If $a_0(x)\equiv0$ and $a_1=0$, then condition
(\ref{beta}) can be dropped, since then we can take $\beta$
arbitrarily large, in order to ensure (\ref{beta}).
Also, if $a_0(x)=O(r^{\overline \beta})$ for some
$\overline\beta\ge0$, or $a_1>0$,
then condition (\ref{beta}) is fulfilled with $\beta=N$ or
$\beta<\frac{\alpha+\gamma}p$ respectively. It is also possible to consider
the case of $\alpha=(\gamma-1)/(p-1)$ in Theorem~\ref{sing}.

Under simple additional conditions we can ensure
that the solution will bump on the upper obstacle infinitely many
times near its singular point $x_0$, that is, along an infinite
sequence converging to $x_0$.

\begin{theorem} [Bumping on the Upper Obstacle
near the Singularity] \label{Bump}
Assume that $a(x,\eta,\xi)=|\xi|^{p-2}\xi$,
and let there exist positive constants $\alpha$, $\gamma$, $R$,
$C_1$, $C_2$, such that for a.e.\ $x\in B_R$ and $\eta\ge
{\mathop{\rm ess\,inf}}_{B_R(x_0)}\omega_1$,
\begin{gather}
g(x,\eta)\ge0,\quad f(x,\eta)\ge\frac {C_1}{|x-x_0|^\gamma},\\
\omega_2(x)\le C_2|x-x_0|^{-\alpha} \mbox{ a.e.  on $B_R(x_0)$.}\label{o2}
\end{gather}
Let
$\alpha<\frac{\gamma-p}{p-1}$
and let a solution $u$ of (\ref{osn}) and lower obstacle $w_1$
satisfy the condition
\begin{gather}
 u_*(x)>w_1^*(x) \quad \mbox{for all } x\in B_R \mbox{ for some $R>0$,}\\
u(x)\ge 0\quad \mbox{a.e. on $B_R$,}
\end{gather}
Then for any $r>0$ there
exists $x_r\in B_r(x_0)\setminus \{x_0\}$, such that
\begin{equation}
u^*(x_r)=w_{2*}(x_r).
\end{equation}
\end{theorem}


\begin{theorem}[Pushing to the upper obstacle] \label{Pushing}
Assume that $a(x,\eta,\xi)=|\xi|^{p-2}\xi$.
Let $x_0$ be a given point in $\Omega$ such that $w_{2*}(x_0)<\infty$.
Assume that
$$
f(x,\eta)\ge C\cdot|x-x_0|^{-\gamma}\quad\mbox{for a.e. } x\in
B_R=B_R(x_0),\; \eta\in (\overline m_1,\overline m_2),
$$
where $\overline m_1={\mathop{\rm ess\,inf}}_{B_R} w_1$ and
$\overline m_2={\mathop{\rm ess\,sup}}_{B_R}w_2$.
If $p<c<N$, then for any solution $u$ of variational problem
(\ref{osn}), we have
$u^*(x_0)=w_{2*}(x_0)$.
\end{theorem}

\section{Proof of results}

\paragraph{Proof of Theorem \ref{main}}
As we have described in Introduction, we will
restrict our attention to the first case in (\ref{ne}) only,
because the dual case can be
obtain in analogous way. We start by an
easy localization fact from harmonic analysis (see for instance
\cite{KPZ2}) that we
state in the following form. For any $c_0>1$ and $r>0$ there
exists a function $\Phi \in C_0^{\infty }(\mathbb{R}^N)$,
$0\leq \Phi \leq 1$ in $\mathbb{R}^N$ such that 
\begin{equation}\label{local}
\begin{gathered}
\Phi (x)=1\quad\mbox{for }x\in B_{r}\quad\mbox{and}
\quad \Phi (x)=0\quad\mbox{for }x\in \mathbb{R}^N\setminus B_{2r}  \\
\Phi (x)>0\quad\mbox{for }x\in B_{2r}\quad\mbox{and}\quad
|\nabla \Phi |\leq c_0/r\quad\mbox{in }\mathbb{R}^N\,. 
\end{gathered}
\end{equation}
Taking $B_{r}$, $m_1$, $M_{c}$ and $m_2$ such that
conditions (\ref{oc})--(\ref{MM}) are satisfied,
let us choose for any $c_0>1$ an appropriate test function $\varphi $
defined by
\begin{equation}\label{test}
\varphi =(u-t)^{-}\Phi ^{p}+u\quad\mbox{with }t\in (M_{c},m_2]\,.
\end{equation}
Here and in the sequel $u$ is a solution of (\ref{osn}) and $\eta ^{-}=\max
\{0,-\eta \}$.\ The basic step it is to check that $\varphi$ has the
properties
\begin{equation}\label{uvjeti}
\varphi \in W^{1,p}(\Omega ),\quad\varphi -u\in L^{\infty
}(\Omega ),\quad\mbox{and}\quad \omega_1\leq \varphi \leq \omega _2\quad
\mbox{in}\Omega \,.
\end{equation}

Arguing by contradiction, we assume that there holds
the opposite of (\ref{ne}), say $|\{x\in
B_{2r} :u>\omega _{c}\}|=0$. In other words, $u\leq \omega _{c}$
in $B_{2r}$. Remark that we already have $\omega_1\leq u$ in
$\Omega$.
Taking one-sided supremum and infimum over $B_{2r}$ in the previous two
inequalities, we deduce:
\begin{equation}\label{muM}
m_1\leq u\leq M_{c}\quad\mbox{ \ \ in }B_{2r}\,.
\end{equation}
Since
\[
\nabla (u-t)^{-}=\left\{
\begin{array}{ll}
-\nabla u & \mbox{in }\{u<t\}, \\
0 & \mbox{in }\{u\geq t\},
\end{array}
\right.
\]
substituting
test function $\varphi $ from (\ref{test})
into (\ref{osn}),
and using the structure condition (\ref{ll1}), we obtain:
\begin{eqnarray}\label{level}
\lefteqn{\alpha_0 \int_{\{u<t\}_{2r}}|\nabla u|^{p}\Phi ^{p}\,dx } \nonumber\\
&\leq&
p\int_{\{u<t\}_{2r}}|a(x,u,\nabla u)|\Phi ^{p-1}(t-u)|\nabla \Phi
|\,dx \\
&&-\int_{\{u<t\}_{2r}}f(x,u)(t-u)\Phi ^{p}\,dx-
\int_{\{u<t\}_{2r}}g(x,u)|\nabla u|^{p}(t-u)\Phi ^{p}\,dx. \nonumber
\end{eqnarray}
where $\{u<t\}_{2r}=\{u<t\}\cap B_{2r}$. It is interesting to mention that
in the light of (\ref{muM}) the level set $\{u<t\}_{2r}$ satisfies:
\[
\{u<t\}_{2r}=\{u\leq M_{c}\}_{2r}\cup \{M_{c}<u<t\}_{2r}=\{u\leq
M_{c}\}_{2r}=B_{2r}.
\]
Thus, we are able to reduce all integrations over $\{u<t\}_{2r}$,
appearing in (\ref{level}),
to $B_{2r}$.

In the sequel, due to (H1) we can drop the last term
on the
right-hand side of (\ref{level}). This together with the structure condition (\ref{ll2}) and $%
t-u=(t-u)^{\frac{1}{p'}}(t-u)^{\frac{1}{p}}$
leads us to the inequality
\begin{eqnarray} \label{B2r}
\lefteqn{\alpha_0 \int_{B_{2r}}|\nabla u|^{p}\Phi ^{p}\,dx}\nonumber \\
&\leq&
\int_{B_{2r}}\bigl[\bigl(a_0(x)+a_1|u|^{p-1}+a_2|\nabla u|^{p-1}\bigl)
\Phi ^{p-1}(t-u)^{\frac{1}{p'}}\bigl]\cdot\bigl[p|\nabla \Phi |(t-u)^{
\frac{1}{p}}\bigl]\,dx \nonumber\\
&&-\int_{B_{2r}}f(x,u)(t-u)\Phi ^{p}\,dx\,.
\end{eqnarray}
Now we consider the product under the
first integral on the right-hand side of (\ref{B2r}).
Applying the following
two elementary inequalities, $a(pb)\leq \frac{d}{p'}a^{p'}+(%
\frac{p}{d})^{p-1}b^{p}$ and $(a+b)^{p'}\leq 2^{p'-1}(a^{p'}+b^{p'})$ for all $a\geq 0$, $b\geq 0$, and $%
d>0$, we obtain
\begin{eqnarray} \label{0}
0&=&\big[\alpha_0 -a_2^{p'}\overline{d}(m_2-m_1)\big]
\int_{B_{2r}}|\nabla u|^{p}\Phi ^{p}\,dx\nonumber\\
&\leq&
\overline{d}\int_{B_{2r}}\bigl(a_0(x)+a_1|u|^{p-1}\bigl)^{p'}\Phi ^{p}(t-u)\,dx
+\bigl(\frac{p}{d}\bigl)^{p-1}\int_{B_{2r}}|\nabla \Phi
|^{p}(t-u)\,dx \nonumber\\
&&-\int_{B_{2r}}f(x,u)(t-u)\Phi ^{p}\,dx,
\end{eqnarray}
where the numbers $d$ and $\overline{d}$ are defined in (H2).

Now we are in the position to exploit properties of the
localization function $\Phi $ in (\ref{0}).
First, since $f(x,\eta )\geq f_1(x)$ in $B_{2r}$ for
all $\eta \in I_1$ (see $(H2)$), then using (\ref{local}) and (\ref{muM})
we derive
\begin{eqnarray*}
0&\leq& (t-m_1)\overline{d}\int_{B_{2r}}[a_0(x)+a_1\widehat{m}%
^{p-1}]^{p'}\,dx\\
&&+(t-m_1)\bigl(\frac{p}{d}\bigl)^{p-1}|B_{2r}\setminus B_{r}|\bigl
(\frac{c_0}{r}\bigl)^{p}-(t-M_{c})\int_{B_{r}}f_1(x)\,dx.
\end{eqnarray*}
Setting $s=t-M_{c}$, using  $|B_{2r}\setminus B_{r}|=(2^N-1)|B_{r}|$, and
passing to the limit as $c_0\to1$, we obtain
\[
\int_{B_{r}}f_1(x)\,dx\leq D_1\frac{\bigl[s+(M_{c}-m_1)\bigl]}{s},\quad\mbox{
for all }s\in (0,m_2-M_{c}].
\]
Since in the previous inequality the function appearing on the right-hand side
is decreasing with respect to $s$,
we can set $s=m_2-M_{c}$ to obtain
\[
\int_{B_{r}}f_1(x)\,dx\leq D_1\frac{m_2-m_1}{m_2-M_{c}}.
\]
However, this contradicts (H2), and the theorem is proved.
\hfill$\diamondsuit$


\paragraph{Sketch of the proof of Theorem \ref{thm2}} 
The penalty method is carried out in the following three steps 
(see for instance \cite{L} for the case of 
$Au=f\in V'$).

Firstly, we associate to (\ref{osn}) an $\varepsilon$-problem,
the so called penalty equation
\begin{equation}\label{penal}
\begin{gathered}
-\mathop{\rm dvi}a(x,u_{\varepsilon },\nabla u_{\varepsilon })
+\frac{1}{\varepsilon }\beta (x,u_{\varepsilon })
=f(x,u_{\varepsilon})+g(x,u_{\varepsilon })|\nabla u_{\varepsilon }|^{p}  
\quad\mbox{in } D'(\Omega ), \\
u_{\varepsilon }\in W_0^{1,p}(\Omega )\cap L^{\infty }(\Omega ),
\end{gathered}
\end{equation}
where the penalty function $\beta (x,\eta )$ is a Carath\'eodory function
defined by
\begin{equation}
\beta (x,\eta )=((\eta -\omega _2(x))^{+})^{p-1}-((\eta -\omega
_1(x))^{-})^{p-1},\quad\mbox{ \ in }\Omega \times \mathbb{R}.
\end{equation}
Since $\omega_1(x)\leq 0\leq \omega _2(x)$ in $\Omega$,
the penalty function $\beta$ has
the following three important
properties:
\begin{gather}\label{beta0}
\beta (x,v)=0\quad\mbox{in }\Omega \quad\mbox{if and only if}\quad
\omega_1\leq v\leq \omega _2\quad\mbox{in }\Omega ,\\
\label{beta-}
(\beta (x,\eta_1)-\beta (x,\eta _2))(\eta_1-\eta _2)>0\quad
\mbox{in }\Omega ,\; \eta_1\neq \eta _2\in \mathbb{R}, \\
\label{betasgn}
\beta (x,\eta )\mathop{\rm sgn}(\eta )\geq 0\quad\mbox{in }\Omega ,
\; \eta \in \mathbb{R}.
\end{gather}

In the first step, by means of the Schwartz symmetrization we derive
some basic a priori estimates for $u_{\varepsilon }$ in $W_0^{1,p}(\Omega
)\cap L^{\infty }(\Omega )$.

\begin{prop}  \label{prop1}
 Under the assumptions of Theorem \ref{thm2},
for each $\varepsilon >0$  there exist a solution $u_{\varepsilon }$ of
(\ref{penal}), and two constants $C_1$ and $C_2$  
which are independent on $\varepsilon $,  such that
\begin{equation}\label{ue}
\begin{gathered}
u_{\varepsilon }^{\#}(x)\leq v^{\#}(x)=v(x) \quad\mbox{in }\Omega^{\#},  \\
\|u_{\varepsilon }\| _{L^{\infty }(\Omega )}\leq
C_1=\|v\|_{L^{\infty }(\Omega ^{\#})}, \\
\|\nabla u_{\varepsilon }\|_{L^{P}(\Omega )}\leq C_2=\|\nabla
v\|_{L^{p}(\Omega ^{\#})}, 
\end{gathered}
\end{equation}
where $u_{\varepsilon }^{\#}$ is the Schwarz symmetrization of $u$, and
$v$ is the unique solution of~(\ref{sharp=}).
\end{prop}

Having in mind the sign condition (\ref{betasgn}) for the penalty function $\beta
(x,\eta )$, the proof of Proposition 1 is very similar to the proofs of
\cite[Theorems 2, 3, 4]{P2}.

Next, we consider the relative compactness of the sequence $u_{\varepsilon}$.
According to previous estimates and the relative compactness results
from \cite{BMP}, one can show the following proposition.

\begin{prop}  \label{prop2}
Under the assumptions of Theorem \ref{thm2}, let
$u_{\varepsilon }$ be a solution of (\ref{penal}).  Then there
exist a subsequence of $u_{\varepsilon}$, still denoted by
$u_{\varepsilon}$,
 and a function $u\in W_0^{1,p}(\Omega )\cap L^{\infty }(\Omega
 )$,  such that as $\varepsilon \rightarrow 0$, 
 \begin{equation}\label{ueto}
\begin{gathered}
u_{\varepsilon }\rightarrow u \quad\mbox{strongly in }W_0^{1,p}(\Omega ), \\
a(x,u_{\varepsilon },\nabla u_{\varepsilon })\rightarrow a(x,u,\nabla u) 
\quad \mbox{weakly in }L^{p'}(\Omega ), \\
f(x,u_{\varepsilon })\rightarrow f(x,u),\quad
g(x,u_{\varepsilon })|\nabla u_{\varepsilon }|^{p}\rightarrow g(x,u)|\nabla
u|^{p} \quad \mbox{weakly in }L^{1}(\Omega ).
\end{gathered}
\end{equation}
\end{prop}

\paragraph{Proof} 
According to (\ref{ue}), and using the reflexivity of 
$W_0^{1,p}(\Omega )$ and compactness of imbedding of $W_0^{1,p}(\Omega )$ 
into $L^{p}(\Omega )$, we immediately conclude that there exist a 
subsequence of $u_{\varepsilon}$, still denoted by $u_{\varepsilon}$, 
and a function $u\in W_0^{1,p}(\Omega )\cap L^{\infty }(\Omega )$, such that
\begin{equation}\label{uetow}
\begin{gathered} u_{\varepsilon }\rightarrow u \quad
\mbox{weakly in $W_0^{1,p}(\Omega )$ and strongly in $L^{p}(\Omega )$},  \\
\mbox{a.e.\ in $\Omega $ and weak$^{*}$ in $L^{\infty }(\Omega )$.} 
\end{gathered}
\end{equation}
By means of the monotonicity assumption (\ref{monoton}) we are able to repeat all steps
from the proof of  \cite[Lemma 4,  p.\ 189]{BMP}, and to derive:
\begin{equation}\label{ue-}
\int_{\Omega }[a(x,u_{\varepsilon },\nabla u_{\varepsilon
})-a(x,u_{\varepsilon },\nabla u)]\cdot\nabla (u_{\varepsilon }-u)\,dx\rightarrow 0.
\end{equation}
Now, with the help of the compactness result
from \cite[Lemma 5, p.\ 190]{BMP},
and using the convergence result  from \cite[Lemma 3.2]{L}, together
with (\ref{uetow}) and (\ref{ue-}),
we derive all claims in (\ref{ueto}).\hfill$\diamondsuit$\smallskip

Finally, as a consequence of two previous propositions, we
obtain the following statement.

\begin{prop}  Under the assumptions of Theorem \ref{thm2}, let
$u_{\varepsilon }$ be a solution of $(\ref{penal})$ satisfying
(\ref{ue}), and let $u\in
W_0^{1,p}(\Omega )\cap L^{\infty }(\Omega )$ be a function
satisfying
(\ref{ueto}). Then we have:

\begin{enumerate}
\item[{\rm(i)}] $\omega_1\leq u\leq \omega _2$ \ {in} $\Omega $

\item[{\rm(ii)}] $u$ is a solution of (\ref{osn}).
\end{enumerate}
\end{prop}

\paragraph{Proof} Using (\ref{beta0})
we see that in order to prove (i) it suffices to check that
$\beta (x,u)=0$ in $\Omega $.
Let us remark that with the help of (\ref{ll2}), $(H5)$ and (\ref{ue}) we obtain the
existence of three positive constants $c_1$, $c_2$ and $c_{3}$
such that
\begin{gather*}
\int_{\Omega }a(x,u_{\varepsilon },\nabla u_{\varepsilon })\cdot\nabla
u_{\varepsilon }\,dx<c_1,\quad
\big|\int_{\Omega }f(x,u_{\varepsilon})u_{\varepsilon }\,dx\big|<c_2,\\
\big|\int_{\Omega }g(x,u_{\varepsilon})|\nabla u_{\varepsilon }|^{p}
u_{\varepsilon}\,dx\big|<c_{3}.
\end{gather*}
Furhtermore, testing (\ref{penal}) with the function $\varphi =\varepsilon
u_{\varepsilon }$, we obtain
\begin{eqnarray*}
\lefteqn{\varepsilon \int_{\Omega }a(x,u_{\varepsilon },\nabla u_{\varepsilon
})\cdot\nabla u_{\varepsilon }\,dx+\int_{\Omega }\beta (x,u_{\varepsilon
})u_{\varepsilon }\,dx}\\
&=&\varepsilon \int_{\Omega }f(x,u_{\varepsilon})u_{\varepsilon }\,dx+
\varepsilon \int_{\Omega }g(x,u_{\varepsilon })|\nabla
u_{\varepsilon }|^{p}u_{\varepsilon }\,dx.
\end{eqnarray*}
Passing to the limit as $\varepsilon \rightarrow 0$, from the previous
estimates we obtain
\[
\int_{\Omega }\beta (x,u)u\,dx=0,
\]
which together with the ``sign'' condition (\ref{betasgn})
implies that $\beta (x,u(x))u(x)=0$ in $\Omega $.
Since $\beta (x,0)=0$, we obtain that also $\beta (x,u(x))=0$
in $\Omega$.\smallskip

Now we proceed with the proof of the second claim of the proposition. 
Testing the penalty equation (\ref{penal}) with the function $\varphi
=v-u_{\varepsilon }$, where $v\in W^{1,p}(\Omega)\cap L^\infty(\Omega)$ 
such that $\omega_1\leq v\leq \omega _2$ \ in $\Omega $, we
have
\begin{eqnarray}\label{1e}
\lefteqn{\int_{\Omega }a(x,u_{\varepsilon },\nabla u_{\varepsilon })\cdot\nabla
(v-u_{\varepsilon })\,dx+\frac{1}{\varepsilon }\int_{\Omega }\beta
(x,u_{\varepsilon })(v-u_{\varepsilon })\,dx }\nonumber\\
&=&\int_{\Omega }f(x,u_{\varepsilon })(v-u_{\varepsilon })\,dx+\int_{\Omega
}g(x,u_{\varepsilon })|\nabla u_{\varepsilon
}|^{p}(v-u_{\varepsilon })\,dx.
\end{eqnarray}
Furthermore, by means of (\ref{beta0}) and (\ref{beta-}) we also have
\begin{equation}\label{betaue}
\frac{1}{\varepsilon }\int_{\Omega }\beta (x,u_{\varepsilon
})(v-u_{\varepsilon })\,dx=-\frac{1}{\varepsilon }\int_{\Omega }(\beta
(x,v)-\beta (x,u_{\varepsilon }))(v-u_{\varepsilon })\,dx\leq 0.
\end{equation}
 From  (\ref{1e}) and using (\ref{betaue}) we derive
\begin{eqnarray}\label{uev}
\lefteqn{\int_{\Omega }a(x,u_{\varepsilon },\nabla u_{\varepsilon })\nabla
(v-u_{\varepsilon })\,dx}\\
&\geq& 
\int_{\Omega }f(x,u_{\varepsilon})(v-u_{\varepsilon })\,dx
+\int_{\Omega }g(x,u_{\varepsilon })|\nabla u_{\varepsilon }|^{p}
(v-u_{\varepsilon })\,dx. \nonumber
\end{eqnarray}
Finally, according to (\ref{ueto})--(\ref{uetow}), and passing to
the limits in (\ref{uev}), we
deduce that the function $u$ is a solution of the equation
(\ref{osn}).\hfill$\diamondsuit$\smallskip



\paragraph{Proof of Theorem~\ref{sing}}
 We assume without loss of generality that $x_0=0$, and
denote $r=|x|$. Let us fix a ball $B_{2r}$, $r<R/2$, and let us define
\begin{eqnarray*}
f_1(x)=C_1(2r)^{-\gamma},\quad\overline m_1={\mathop{\rm ess\,inf}}_{B_R}
\omega_1, \\
 m_2(r)=C_1(2r)^{-\alpha},\quad M_c(r)=L\cdot m_2(r),
\end{eqnarray*}
where $L\in(0,1)$ is a given fixed number and $C_1>0$.
If we show that for any $r>0$ sufficiently small condition
\begin{equation}\label{h2}
\int_{B_r}f_1(x)\,dx>D_1(r)\cdot\frac{m_2(r)-\overline
m_1}{m_2(r)-M_c(r)},
\end{equation}
is satisfied, see (H2), than the claim will follow from
Theorem~\ref{main},  since $M_c(r)\to\infty$ as $r\to0$, and
$\cap_{r>0}B_{2r}=\{0\}$. Denoting the left-hand side of
(\ref{h2}) by $F(r)$, we have (note that $\gamma<N$ since $f_1(x)\le
f(x,u)\in L^1(B_{2r})$):
\begin{equation}
F(r)=\frac{C_1\omega_N N}{2^{\gamma}(N-\gamma)}r^{N-\gamma},
\end{equation}
where $\omega_N=|B_1(0)|$.
To estimate the right-hand side of (\ref{h2}), that we denote by
$G(r)$, note that for a given and fixed $k>1$ we have that
 $m_2(r)-\overline m_1\le k^{1/p}\cdot m_2(r)$ for all $r$ small enough.
 Also, the left-hand side of (\ref{intBr}) can be estimated by $C\cdot
 r^\beta$, where $C$ is a positive constant.
 Hence,
 \begin{eqnarray}
 \lefteqn{G(r)} \nonumber\\
 &\le& [\alpha_0 a_2^{-p'}C\cdot r^{\beta}
 +2k\left(\frac{p-1}{\alpha_0}\right)^{p-1}
 a_2^p(2^N-1)\omega_N  m_2(r)^p\cdot r^{N-p}] 
 \frac{2^\alpha r^\alpha}{(1-L)C_1} \nonumber \\
&=& D_1 r^{\alpha+\beta}+D_2r^{-\alpha(p-1)+N-p},
\end{eqnarray}
where $D_1\ge0$ and $D_2>0$ are explicit positive constants
independent of $r$.
In order to ensure $F(r)>G(r)$ for all $r>0$ small enough, see
(\ref{h2}), it
suffices have:
$$
\frac{C_1\omega_N}{2^{\gamma}(N-\gamma)}r^{N-\gamma}>D_1
r^{\alpha+\beta}+D_2r^{-\alpha(p-1)+N-p},
$$
that is,
$$
\frac{C_1\omega_N}{2^{\gamma}(N-\gamma)}r^{\alpha(p-1)+p-\gamma}>D_1
r^{\alpha+\beta+\alpha(p-1)-N+p}+D_2.
$$
This inequality holds for all $r>0$ small enough due to
$\alpha(p-1)+p-\gamma<0$ and
$\alpha(p-1)+p-\gamma<\alpha+\beta+\alpha(p-1)-N+p$.
\hfill$\diamondsuit$\smallskip


\paragraph{Proof of Theorem~\ref{Bump}}
 Assume, contrary to claim of the theorem, that
there exists a ball $B_r=B_r(x_0)$ such that
$u^*(x)<w_{2*}(x)$ for all $x\in B_r$. Since we have strict
inequalities
$$
w^*_1(x)<u_*(x),\quad
u^*(x)<w_{2*}(x),\quad\forall x\in B_r,
$$
it is easy to see that solution $u$ of variational inequality
(\ref{osn}) is in fact a solution of
quasilinear elliptic equation on $B_r$ in the sense of
distributions:
\begin{equation}
-\Delta_pu=f(x,u)+g(x,u)\cdot|\nabla u|^p,\quad \mbox{in ${\cal D}'(B_r)$},
\end{equation}
since for any test function $\varphi\in {\cal D}(B_r)$ the function
 $v=u+t\varphi$ is admissible in (\ref{osn}) for $t$ small enough.
Therefore,
$$
-\Delta_pu\ge\frac{C_1}{|x-x_0|^{\gamma}},\quad\mbox{in ${\cal D_+'}(B_r)$.}
$$
Since $u\ge0$ on $\partial B_r$, then using
 \cite[Theorem~3]{dz-sing} we
conclude that solution $u$ has singularity at least of order
$\frac{\gamma-p}{p-1}$ at $x_0$. However, this is a impossible, since
$u(x)$ is dominated from above by obstacle $w_2(x)$,
 which has singularity at $x_0$ at most of order
equal precisely to $\alpha$, see (\ref{o2}), such that
$\alpha<\frac{\gamma-1}{p-1}$.
\hfill$\diamondsuit$\smallskip


\paragraph{Remark}
 Using the same method it is possible to show that
Theorem~\ref{Bump} holds also for $p$-Laplace like operators,
%in the sense of Kilpel\"ainen and Maly,
that is
for $a(x,\eta,\xi)=A(x,\xi)$ satisfying Leary--Lions conditions
with $|A(x,\xi)|\le a_2|\xi|^{p-1}$.


\paragraph{Proof of Theorem~\ref{Pushing}}
 Let us assume that $x_0=0$ and denote $r=|x|$.
Here we define $f_1(x)=C\cdot|x|^{-\gamma}$,
$m_2(r)={\mathop{\rm ess\,inf}}_{B_{2r}} w_2$,
$M_c(r)=m_2(r)-r^\varepsilon$, where we take
$\varepsilon>0$ small enough.  Similarly as in the proof of 
Theorem~\ref{sing} we
obtain that condition (\ref{h2}) is satisfied for all $r>0$
small enough if we have
$r^{N-\gamma}>a\cdot r^{N-p-\varepsilon}$,
where $a$ is a positive constant independent of $r$. Thus we have
to secure that
$r^{\gamma-p-\varepsilon}<1/a$
for $r>0$ small, and this is possible by taking
$\varepsilon\in(0,\gamma-p)$.
The claim follows from Theorem~\ref{main}, since $M_c(r)\to w_{2*}(x_0)$ as
$r\to0$.
\hfill$\diamondsuit$\smallskip

As a final remark, we note that our main Theorem~\ref{main} can
be formulated in a much more general context.

\begin{theorem}
Assume that $a(x,\eta,\xi)$ satisfies conditions (\ref{ll1}) and
(\ref{ll2}).
Let $A$ be a measurable subset of $\Omega$, such that
$A_r\subseteq\Omega$ and $|A_r\setminus A|<\infty$. Let $M_c$ be a given
number such that
\begin{equation}
M_c\in(m_1,m_2),\quad m_i={\mathop{\rm
ess\,inf}}_{A_r}\omega_i(x),\quad i=1,2.
\end{equation}
Assume that
\begin{gather}
g(x,\eta)\ge0 \quad\mbox{for a.e. }x\in A_r \quad\mbox{and}\quad
\eta\in I_1=(m_1,M_c),  \\
\exists f_1\in L^1(A_r),\quad f(x,\eta)\ge f_1(x)\quad\mbox{for a.e. }
x\in A_r,\; \eta\in I_1,\\
f_1(x)\ge0 \quad\mbox{on } A_r\setminus A.
\end{gather}
Furthermore, assume that
\begin{equation}
\int_A f_1(x)\,dx> D_1\cdot\frac{m_2-m_1}{m_2-M_c},
\end{equation}
where
\begin{equation}
D_1=\overline d\int_{A_r}[a_0(x)+a_1 \widehat m^{p-1}]^{p'}dx+\left(\frac
pd\right)^{p-1}\frac{|A_r\setminus A|}{r^p},
\end{equation}
with $\widehat m$, $\overline d$ and $d$ defined in the same way as in
Theorem~\ref{main}. Then for any solution $u$ of (\ref{osn}) we have
\begin{equation}
|\{x\in A_r\: u(x)>M_c\}|\ne0.
\end{equation}
\end{theorem}

{\bf Proof.} The proof is the same as the proof of
Theorem~\ref{main}. We only
have to change $B_r$ to $A$, $B_{2r}$ to $A_r$, and to use our
general result about localization of measurable sets stated in
\cite[Lemma 5]{KPZ2}.

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\noindent{\sc Luka Korkut} (e-mail: luka.korkut@fer.hr) \\
{\sc  Mervan Pa\v si\'c} (e-mail: mervan.pasic@fer.hr) \\
{\sc Darko \v Zubrini\'c} (e-mail: darko.zubrinic@fer.hr) \\[2pt]
Department of mathematics \\
Faculty of Electrical Engineering and Computing \\
University of Zagreb\\
Unska 3, 10000 Zagreb, Croatia



\end{document}