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\markboth{\hfil
Discontinuous functional equations \hfil EJDE--2002/61}
{EJDE--2002/61\hfil S. Carl \&  S. Heikkil\"a \hfil}
\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc  Electronic Journal of Differential Equations},
Vol. {\bf 2002}(2002), No. 61, pp. 1--10. \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} \\
 %
  Existence of solutions for discontinuous functional equations and
  elliptic boundary-value problems
 %
\thanks{ {\em Mathematics Subject Classifications:} 35J65, 35R05, 35R10.
\hfil\break\indent
{\em Key words:} functional equations, elliptic boundary-value problems,
\hfil\break\indent discontinuous  nonlinearities.
\hfil\break\indent
\copyright 2002 Southwest Texas State University. \hfil\break\indent
Submitted: January 15, 2002. Published June 25, 2002.} }
\date{}
%
\author{Siegfried Carl \&  Seppo Heikkil\"a}
\maketitle

\begin{abstract}
 We prove existence results for discontinuous functional equations
 in general $L^p$-spaces and apply these results to the solvability of
 implicit and explicit elliptic boundary-value problems involving
 discontinuous nonlinearities.  The main tool in the proof is a
 fixed point result in lattice-ordered Banach spaces proved by the
 second author.
\end{abstract}

\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{example}[theorem]{Example}
\newtheorem{remark}[theorem]{Remark}

\section{Introduction}

In this paper we shall first prove existence results for the
functional equations
$$
h(x)=f(x,\phi(h(x)),h(x))\quad \text {and} \quad h(x)=g(x,\phi(h(x)))
$$
in the space $L^p(\Omega)$, $1 \le p < \infty$,
where  $\Omega$ is a measure space,
$f:\Omega\times \mathbb{R}\times \mathbb{R}\to \mathbb{R}$, $g:\Omega\times \mathbb{R}\to \mathbb{R}$
and $\phi:L^p(\Omega)\to
L^p(\Omega)$.
The proofs  are based on a fixed point result in \cite{h1}, which is
derived by applying a recursion principle introduced in \cite{h2}.

Then the existence results are applied to study the existence of
weak solutions to boundary-value problems of elliptic differential
equations of the form
$$
\Lambda u(x)= f(x,u(x),\Lambda u(x)) \quad \text{and}\quad
 \Lambda u(x)= g(x,u(x)),
$$
where
$$
\Lambda u(x):=
-\sum_{i,j=1}^N \frac{\partial}{\partial x_i}
\Big(a_{ij}(x) \frac{\partial u(x)}{\partial x_j}\Big) +q(x,u(x)).
$$
The functions $f$, $g$, and  the mapping $\phi$ may be discontinuous in
their arguments. Concrete and worked examples are provided to demonstrate
the applicability of the results obtained.

\section{Existence results for functional equations}

In this section we assume that  $\Omega=(\Omega,\mathcal{A},\mu)$ is a measure space, and that
the space $L^p(\Omega)$, $1\le p < \infty$, is ordered a.e. pointwise.

In the proof of our existence theorem for the functional equation
\begin{equation}
h(x)=f(x,\phi(h(x)),h(x))\ \text { a.e. in } \Omega,\label{e2.1}
\end{equation}
we make use of the following fixed point result.

\begin{lemma} \label{lm2.1}
Assume that a mapping
$G:L^p(\Omega)\to L^p(\Omega)$ is increasing, and that
$\|Gh\|_p\le M + \psi(\|h\|_p)$, where
$\psi:\mathbb{R}_+\to\mathbb{R}_+$ is increasing, and
$M+\psi(R)\le R$ for some $R > 0$. Then $G$ has a fixed point.
\end{lemma}

\paragraph{Proof}
Choose an  $R > 0$ such that
$M+\psi(R)\le R$. Because $\psi$ is increasing, then
$G$ maps the set $P= \{h\in L^p(\Omega)\mid \|h\|_p\le R\}$ into itself.
Thus $G$ has by \cite[Corollary 5]{h1} a fixed point in $P$.
\hfill$\diamondsuit$\smallskip

For the functions $\phi:L^p(\Omega)\to L^p(\Omega)$ and
$f:\Omega\times \mathbb{R}\times \mathbb{R}\to \mathbb{R}$
we have the following hypotheses:
\begin{enumerate}
\item[($\phi$)] $\phi$ is increasing, and
$\|\phi\circ h\|_p\le m+\kappa\|h\|_p$ for some $m\ge 0$ and $\kappa > 0$.
\item[(f1)] $f$ is sup-measurable, i.e.,
$x\mapsto f(x, u(x),v(x))$ is measurable in $\Omega$ whenever
$u,\,v:  \Omega\to \mathbb{R}$
are measurable.
\item[(f2)] $|f(x,y,z)|\le k(x)+c_1(x)
|y|^\alpha+c_2(x)|z|^\beta$  for a.e. $x\in\Omega$ and for all
$y,\,z\in \mathbb{R}$, where $k\in L^p(\Omega)$, and
 either \begin{enumerate}
\item[(i)] $0 < \alpha,\,\beta < 1$, $c_1\in L^{\frac p{1-\alpha}}(\Omega)$, $c_2\in L^{\frac p{1-\beta}}(\Omega)$, and $f(x,\cdot,\cdot)$ is increasing for a.e. $x\in \Omega$, or
\item[(ii)]  $\alpha=\beta=1$, $\kappa\|c_1\|_\infty + \|c_2\|_\infty < 1$, where $\kappa$ is the constant in ($\phi$), and the function $(y,z)\mapsto f(x,y,z)+\lambda z$ is increasing for a.e. $x\in \Omega$ and for some $\lambda\ge 0$.
\end{enumerate}
\end{enumerate}
Our existence result for the functional equation \eqref{e2.1} reads as follows.

\begin{theorem} \label{thm2.1}
Under the assumptions ($\phi$), (f1), and (f2),
Equation \eqref{e2.1} has a solution $h$ in $L^p(\Omega)$.
\end{theorem}

\paragraph{Proof} The hypotheses ($\phi$) and (f1) imply that for each
$h\in L^p(\Omega)$ the relation
\begin{equation}
 Gh:=f(\cdot,\phi(h(\cdot)),h(\cdot)) \label{e2.2}
\end{equation}
defines a measurable function $Gh:\Omega\to \mathbb{R}$. To show that
\eqref{e2.2} defines a mapping $G:L^p(\Omega)\to L^p(\Omega)$ we have to prove that $Gh\in L^p(\Omega)$.
Applying the growth condition of (f2), the hypothesis ($\phi$) and the H\"older inequality we obtain
$$\begin{aligned}
\|Gh\|_p&=\|f(\cdot,\phi(h(\cdot)),h(\cdot))\|_p
\le \|k\|_p + \|c_1(\cdot)|\phi(h(\cdot))|^\alpha\|_p
+ \|c_2(\cdot)|h(\cdot)|^\beta\|_p\\
&\le \|k\|_p+ (\|c_1(\cdot)^p\|_{\frac 1{1-\alpha}}\|\,
|\phi(h(\cdot)|^{p\alpha}\|_{\frac 1\alpha})^{1/p}
+ (\|c_2(\cdot)^p\|_{\frac 1{1-\beta}}\|\,
|h(\cdot)|^{p\beta}\|_{\frac 1\beta})^{1/p}\\
&= \|k\|_p+ \|c_1\|_{\frac p{1-\alpha}}\|\phi\circ h\|^{\alpha}_p
+ \|c_2\|_{\frac p{1-\beta}}\|h\|^{\beta}_p\\
&\le \|k\|_p+ \|c_1\|_{\frac p{1-\alpha}}(m+\kappa\|h\|_p)^\alpha
+ \|c_2\|_{\frac p{1-\beta}}\|h\|^{\beta}_p\,.\end{aligned}
$$
Thus $Gh\in L^p(\Omega)$, and
\begin{equation}
 \|Gh\|_p\le M + \psi(\|h\|_p),\label{e2.3}
\end{equation}
where   $M =\|k\|_p$ and
$\psi(r):=\|c_1\|_{\frac p{1-\alpha}}(m+\kappa r)^\alpha
+ \|c_2\|_{\frac p{1-\beta}}r^\beta$.

\noindent\textbf{a)} Assume first that the hypotheses (f2) (i) hold.
If $h_1,h_2\in L^p(\Omega)$, $h_1\le h_2$, then $\phi(h_1)\le \phi(h_2)$
by ($\phi$). Since $f(x,\cdot,\cdot)$ is increasing, then
for  a.e. $x\in\Omega$,
$$
Gh_1(x)=f(x,\phi(h_1(x)),h_1(x))\le f(x,\phi(h_2(x)),h_2(x))=Gh_2(x)\,.
$$
This proves that $G$ is increasing.
Since $0 < \alpha,\,\beta< 1$, then the mapping $\psi:\mathbb{R}_+\to\mathbb{R}_+$ defined in \eqref{e2.3} is increasing, and
$r-\psi(r)\to \infty$ as $r\to \infty$. Thus $M+\psi(R)\le R$ when $R$ is large enough,
whence $G$ has a fixed point by Lemma \ref{lm2.1}.

\noindent\textbf{b)} Assume next that the hypothesis (f2) (ii) holds
with $\lambda = 0$. Then $f(x,\cdot,\cdot)$ is increasing, whence $G$
is increasing by the above proof. Since $\alpha=\beta=1$, then $\psi$
given by \eqref{e2.3} is of the form
$\psi(r)=\|c_1\|_\infty(m+\kappa r)+\|c_2\|_\infty r$. If $\kappa\|c_1\|_\infty + \|c_2\|_\infty < 1$, then $M+\psi(R)\le R$
when $R$ is sufficiently large. Thus $G$ has a fixed point by Lemma \ref{lm2.1}

The above proof shows that in the cases a) and b) $G$ has a fixed point
$h\in L^p(\Omega)$.
This implies by \eqref{e2.2} that
$h(x)=Gh(x)=f(x,\phi(h(x)),h(x))$ a.e. in $\Omega$.

\noindent\textbf{c)} Assume finally that the hypotheses (f2) (ii) hold with
$\lambda > 0$.
Then a function $\tilde f:\Omega\times \mathbb{R}\times \mathbb{R}$, defined by
\begin{equation}
\tilde f(x,y,z)=\frac{f(x,y,z)+\lambda z}{1+\lambda}, \quad x\in\Omega, \ y,\,z\in \mathbb{R},
\label{e2.4}
\end{equation}
is sup-measurable, $\tilde f(x,\cdot,\cdot)$ is increasing, and
$$|\tilde f(x,y,z)|\le \|\tilde k\|_2 + \tilde c_1(x)|y| + \tilde c_2(x)|z|,
$$
 where
$\tilde k_2= \frac{k_2}{1+\lambda}$,  $\tilde c_1=\frac{c_1}{1+\lambda}$,
$\tilde c_2=\frac{c_2 + \lambda}{1+\lambda}$.
Since $\kappa\|c_1\|_\infty + \|c_2\|_\infty < 1$, then
$$ \kappa\|\tilde c_1\|_\infty + \|\tilde c_2\|_\infty =
\frac{\kappa\|c_1\|_\infty + \|c_2\|_\infty +\lambda}{1+\lambda}
< \frac{1+\lambda}{1+\lambda} = 1
$$
Thus $\tilde f$ satisfies the hypotheses (f1) and (f2) (ii) with $\lambda=0$.
The proof of the case b) above implies an existence of a
$h\in L^p(\Omega)$ such that
$h(x)=\tilde f(x,\phi(h(x)),h(x))$, or equivalently, by \eqref{e2.4},
 $h(x)=f(x,\phi(h(x)),h(x))$ a.e. in $\Omega$.
This concludes the proof. \hfill$\diamondsuit$\smallskip

As a consequence of Theorem \ref{thm2.1} we obtain an existence result for the
 equation
\begin{equation}
h(x)=g(x,\phi(h(x)))\quad \text { a.e. in } \ \Omega.\label{e2.5}
\end{equation}
For the next proposition we assume the following hypotheses:
\begin{enumerate}
\item[(g1)] $g$ is sup-measurable, and $g(x,\cdot)$ is increasing for a.e.
$x\in \Omega$.
\item[(g2)] $|g(x,y)|\le k(x)+c_1(x)|y|^\alpha$  for a.e. $x\in\Omega$
and for all $y\in \mathbb{R}$, where $k\in L^p(\Omega)$,
and either $0 < \alpha < 1$ and   $c_1\in L^{\frac p{1-\alpha}}(\Omega)$,
or $\alpha = 1$ and \ $\kappa\|c_1\|_\infty < 1$.
\end{enumerate}


\begin{proposition} \label{prop2.1}
Assume that $\phi:L^p(\Omega)\to L^p(\Omega)$ satisfies the hypothesis
($\phi$) and that $g:\Omega\times \mathbb{R}\to \mathbb{R}$
satisfies (g1) and (g2). Then \eqref{e2.5} has a solution $h$ in $L^p(\Omega)$.
\end{proposition}


\begin{remark} \label{rmk2.1}\rm
 The hypotheses of Theorem \ref{thm2.1} and
Proposition \ref{thm2.1} allow the functions $f$ and $g$ to be
discontinuous in all their arguments. Even the mapping $\phi$ may
be discontinuous.
\end{remark}

\section{Applications to elliptic boundary-value problems}

Let $\Omega\subset \mathbb{R}^N$, $N\ge 3$, be a bounded domain with a Lipschitz boundary $\partial\Omega$.
In this section we study the existence of weak solutions of
the implicit elliptic BVP
\begin{equation}
\begin{gathered} \Lambda u(x)= f(x,u(x),\Lambda u(x)) \quad\text{in } \Omega,\\
u=0\quad \text{on } \partial\Omega,
\end{gathered} \label{e3.1}
\end{equation}
where
$$
\Lambda u(x):=-\sum_{i,j=1}^N \frac{\partial}{\partial x_i}
\Big(a_{ij}(x) \frac{\partial u(x)}{\partial x_j}\Big)+q(x,u(x)).
$$
Theorem \ref{thm2.1} will be the main tool in our investigations.
We assume that the coefficients $a_{ij}\in L^{\infty}(\Omega)$
satisfy the ellipticity condition
\begin{equation}
\sum_{i,j=1}^N a_{ij}(x) \xi_i\xi_j\ge \gamma\,\sum_{i=1}^N \xi_i^2 \label{e3.2}
\end{equation}
for a.e. $x\in \Omega$, all $\xi_1,\dots, \xi_N\in \mathbb{R}$, and
some $\gamma>0$.

Let $W_0^{1,2}(\Omega)$ denote the usual Sobolev space of square
integrable functions having generalized homogeneous boundary values, and denote its dual space by $W^{-1,2}(\Omega)$.
We are going to introduce conditions which ensure that
\eqref{e3.1} has a weak solution in the following sense.

\paragraph{Definition}
A function $u\in W_0^{1,2}(\Omega)$ is called
a {\it weak solution} of the BVP \eqref{e3.1} if there exists a function
$h\in L^2(\Omega)$ such that
\begin{equation}
h(x) = f(x,u(x),h(x)) \quad\text{ for a.e. } x\in \Omega,\label{e3.3}
\end{equation}
and $u$ is a weak solution of the semilinear BVP
\begin{equation} \begin{gathered}
\Lambda u(x)=h(x) \quad \text{in }  \Omega, \\
u = 0 \quad\text{on } \partial\Omega. \end{gathered}\label{e3.4}
\end{equation}

We shall first prove an existence, uniqueness and comparison result for the BVP
\eqref{e3.4} assuming that $W^{1,2}_0(\Omega)$ and $L^2(\Omega)$ are equipped with the natural partial ordering of functions defined by the order cone
$L^2_+(\Omega)$ of all nonnegative functions of $L^2(\Omega)$.
On the function $q$ we assume the following hypotheses:
\begin{enumerate}
\item[(q1)] $q$ is a
Carath\'eodory function and  $q(x,\cdot)$ is increasing for a.e. $x\in\Omega$.
\item[(q2)] $|q(x,y)|\le k_0(x)+ c_0(x)\,| s|^{p_0-1}$  for
a.e. $x\in\Omega$ and for all $x\in\mathbb{R}$,
where $k_0\in L^\frac{p_0}{p_0-1}(\Omega)$, $c_0\in L^\infty_+(\Omega)$
and $1 < p_0\le 2^*:= \frac{2N}{N-2}$ (critical exponent).
\end{enumerate}

\begin{lemma} \label{lm3.1}
Assume that $q:\Omega\times\mathbb{R}\to\mathbb{R}$ satisfies (q1) and (q2).
Then \eqref{e3.4} has a unique weak solution $u$ for each $h\in L^2(\Omega)$.
Moreover, $u$ is increasing with respect to $h$ and there exist constants
$m\ge 0$ and $\kappa > 0$ such that
\begin{equation}
\|u\|_{1,2}\le m + \kappa \|h\|_2.\label{e3.5}
\end{equation}
\end{lemma}

\paragraph{Proof} It is well-known that
\begin{equation}
 a(u,v):=\int_\Omega \sum_{i,j=1}^Na_{ij}(x)
\frac{\partial u(x)}{\partial x_i}\frac{\partial v(x)}{\partial x_j},
\quad u,\,v\in W^{1,2}_0(\Omega)\label{e3.6}
\end{equation}
defines a bounded  bilinear form $a:W^{1,2}_0(\Omega)\times
W^{1,2}_0(\Omega)\to\mathbb{R}$.
The assumptions (q1) and  (q2) imply that
the mapping $A:W^{1,2}_0(\Omega)\to W^{-1,2}(\Omega)$ given by
\begin{equation}\langle
Au,v\rangle:= a(u,v)+
\int_{\Omega}q(x,u(x))\,v(x)\,dx,
\quad u,\,v\in W^{1,2}_0(\Omega),\label{e3.7}
\end{equation}
is well-defined, continuous and strongly monotone, and hence bijective by
\cite[Theorem 26.A]{z1}.

To each $h\in L^2(\Omega)$ there corresponds a unique functional $\tilde h\in
W^{-1,2}(\Omega)$ given by
\begin{equation}
\langle\tilde h,v\rangle =\int_\Omega h(x)v(x)\,dx, \quad  v\in
W^{1,2}_0(\Omega).\label{e3.8}
\end{equation}
Denoting $u=A^{-1}\tilde h$, we then have $Au=\tilde h$,
which by  \eqref{e3.7} and \eqref{e3.8} is equivalent to
\begin{equation}
a(u,v)+\int_\Omega q(x,u(x))\,v(x)\,dx=\int_\Omega h(x)v(x)\,dx,\
v\in W^{1,2}_0(\Omega).
\label{e3.9}
\end{equation}
Thus $u$ is, by definition, a weak solution of  \eqref{e3.4}.
To prove that $u$ is  increasing with respect to $h$,
let $h_1,\ h_2\in L^2(\Omega)$ satisfy $h_1\le h_2$.
Denoting by $u_i$ the weak solutions of \eqref{e3.4} with $h=h_i$,  $i=1,2$, it
follows from \eqref{e3.9} that
\begin{eqnarray}
\lefteqn{a(u_1-u_2,v)+\int_{\Omega} (q(x,u_1(x))-q(x,u_2(x)))v(x)\,dx}\nonumber\\
&=& \int_\Omega(h_1(x)-h_2(x))v(x)\,dz\le 0\label{e3.10}
\end{eqnarray}
for all $v\in (W^{1,2}_0(\Omega))_+:= W^{1,2}_0(\Omega)\cap L^2_+(\Omega)$.
Choosing in \eqref{e3.10}
$v=(u_1-u_2)^+$,
and noticing that due to the
monotonicity of $q(x,\cdot)$ the inequality
$$\int_{\Omega}
(q(x,u_1(x))-q(x,u_2(x))) (u_1-u_2)^+(x)\,dx\ge 0
$$
holds, and that $a$ is coercive and $a((u_1-u_2)^-,(u_1-u_2)^+)=0$,
we obtain from  \eqref{e3.10}
$$ c \|(u_1-u_2)^+\|_{1,2}^2\le a((u_1-u_2)^+,(u_1-u_2)^+)
=a(u_1-u_2,(u_1-u_2)^+)\le 0.
$$
This result implies that $(u_1-u_2)^+=0$,
i.e. $u_1\le u_2$, and hence proves that the weak solution $u$ of \eqref{e3.4}
is increasing with respect to $h$.

To prove estimate \eqref{e3.5}, let  $h\in L^2(\Omega)$ be given, and let
$u\in W^{1,2}_0(\Omega)$ be the weak solution of \eqref{e3.4}.
The monotonicity of $q(x,\cdot)$ along with the continuous embedding  $W^{1,2}_0(\Omega)\subset L^{p_0}(\Omega)$ and the
coercivity of $a$ yield the following estimate:
$$
\begin{aligned} c\|u\|_{1,2}^2 &\le a(u,u)\le
a(u,u)+\int_{\Omega}(q(x,u(x))-q(x,0))u(x)\,dx\\
&=\int_\Omega h(x)u(x)\,dx-\int_\Omega q(x,0)u(x)\,dx\\
&\le \|h\|_2\|u\|_2+\|k_0\|_\frac{p_0}{p_0-1}\|u\|_{p_0}\le
(b\,\|k_0\|_\frac{p_0}{p_0-1}+\|h\|_2)\|u\|_{1,2},
\end{aligned}
$$
for some positive constant $b$.
Thus \eqref{e3.5} holds with $m=\frac bc\|k_0\|_\frac{p_0}{p_0-1}$ and
$\kappa = 1/c$. \hfill$\diamondsuit$ \smallskip

As an application of Theorem \ref{thm2.1} and Lemma \ref{lm3.1},
 we shall prove the following existence result for \eqref{e3.1}.

\begin{theorem} \label{thm3.1}
Assume that $q:\Omega\times\mathbb{R}\to\mathbb{R}$ satisfies the
hypotheses (q1) and (q2), and that
$f:\Omega\times \mathbb{R}\times \mathbb{R}\to \mathbb{R}$
satisfies the hypotheses (f1) and (f2) with $p=2$.
Then the BVP \eqref{e3.1} possesses a weak solution.
\end{theorem}

\paragraph{Proof}
If follows from Lemma \ref{lm3.1} that the mapping
$\phi:L^2(\Omega)\to  L^2(\Omega)$, which assigns to each
$h\in L^2(\Omega)$ the weak solution
$u:=\phi(h)\in  W^{1,2}_0(\Omega)\subset L^2(\Omega)$ of the BVP \eqref{e3.4},
is increasing. Moreover, the  inequality \eqref{e3.5}
holds, whence
$$ \|\phi\circ h\|_2= \|u\|_2\le \|u\|_{1,2}\le m + \kappa \|h\|_2, \quad
h\in L^2(\Omega).$$
This proves that $\phi$ satisfies the hypothesis ($\phi$). Thus  the
hypotheses of Theorem \ref{thm2.1} hold when $p=2$, whence
there exists a function $h\in L^2(\Omega)$ such that
$$ h(x)=f(x,\phi(h(x)),h(x))=f(x,u(x),h(x)) \ \text{ a.e. in } \ \Omega,
$$
and $u$ is the weak solution of \eqref{e3.4}.  This implies by Definition 3.1
that $u$ is a weak solution of \eqref{e3.1}.  \hfill$\diamondsuit$\smallskip

As a consequence of Theorem \ref{thm3.1}, we obtain an existence result
for the (explicit) BVP
\begin{equation}
\begin{gathered} \Lambda u(x)= g(x,u(x)) \quad \text{ a.e. in }  \Omega,\\
u=0\quad \text{on } \partial\Omega, \end{gathered}\label{e3.11}
\end{equation}
where
$$\Lambda u(x):=-\sum_{i,j=1}^N \frac{\partial}{\partial x_i}
\Big(a_{ij}(x) \frac{\partial u(x)}{\partial x_j}\Big)+q(x,u(x)).
$$

\begin{proposition} \label{prop3.1}
 Assume that $q:\Omega\times\mathbb{R}\to\mathbb{R}$ satisfies the
 hypotheses (q1) and (q2),
and that $g:\Omega\times \mathbb{R}\to \mathbb{R}$ satisfies the hypotheses
(g2) and (g2) with $p=2$. Then the BVP
\eqref{e3.11} has a weak solution.
\end{proposition}

\begin{remark} \label{rmk3.1} \rm
(i) The hypotheses of Theorem \ref{thm3.1} and
Proposition \ref{prop3.1} allow both functions $f$ and $g$ to be
discontinuous in all their arguments.
\\
(ii) Theorem \ref{thm3.1} and Proposition \ref{prop3.1}
 also apply to problems in domains $\Omega$ of dimensions $N=1$ and $N=2$,
since in these cases the critical exponent $2^*=\infty$ and Lemma \ref{lm3.1}
 is valid with an exponent $p_0$ satisfying $1< p_0 < \infty$.
\\
(iii) If the coefficients $a_{ij}$ are uniformly Lipschitz continuous,
it follows by the regularity result \cite[Theorem 8.8]{c3} that the weak
solutions of problems \eqref{e3.1} and \eqref{e3.11} satisfy their differential equation
a.e. pointwise. This holds, in particular, when $a_{ij}=\delta_{ij}$
which is the case in the following examples, where $[z]$ denotes the
greatest integer $\le z\in\mathbb{R}$.
\end{remark}

\begin{example} \label{exm3.1} \rm
 Assume that ${\mathbb{R}}^4$ is
equipped with the Euclidean norm $|\cdot|$. Choose
$\Omega=\{x\in{\mathbb{R}}^4:\frac 12 < |x|< 1\}$, and consider
the BVP
\begin{equation}
\begin{gathered}
\Lambda u(x)=5+[6|x|]+7[10^9u(x)]^\frac 13 +8[10^{10}\Lambda u(x)]^\frac 15,
\quad\text{a.e. in } \Omega, \\
u=0\quad \text{on }\partial\Omega,
\end{gathered} \label{e3.12}
\end{equation}
where
$\Lambda u(x):=-\Delta u(x)+u(x)^3$, for $x\in\Omega$.
The BVP \eqref{e3.12} is of the form \eqref{e3.1}, where
$$a_{ij}(x)\equiv\delta_{ij}, \ q(x,y)=y^3\
  \text{ and } \
f(x,y,z)=5 + [6|x|]+7[10^9y]^{1/3}+8[10^{10}z]^{1/5}.$$
The critical exponent here is $2^*=4$ and
it is easy to see that the hypotheses (q), (f1) and (f2) with $p=2$ hold,
whence the BVP \eqref{e3.12} has by Theorem \ref{thm3.1} a weak solution.
\end{example}

\begin{example} \label{exm3.2} \rm
 For $\Omega = (0,1)$, consider the
boundary-value problem
\begin{equation}
\begin{gathered}
 -u''(x) = 2+2[2-2x]+2\,[(2u(x)-2x)^\frac 13]+[(-u''(x)-1)^\frac 13] \
\text{ a.e. in } (0,1)\\
u(0) = u(1) = 0\,.\end{gathered}\label{e3.13}
\end{equation}
Problem \eqref{e3.13} is of the form \eqref{e3.1}, with
\begin{equation}
 q(x,y)\equiv -1, \ \text{ and } \ f(x,y,z) =1+2[2-2x]+
 2\,[(2y-2x)^\frac 13]+[(z-1)^\frac 13].\label{e3.14}
\end{equation}
By elementary calculations one can  show that for each
$h\in L^2(\Omega)$ the function
\begin{equation}
\begin{aligned} u(x)&= \phi(h(x))=(1-x)\int_0^xt(1+h(t))dt +
x\int_x^1(1-t)(1+h(t))dt\\
&= \frac{x-x^2}2+(1-x)\int_0^xth(t)dt + x\int_x^1(1-t)h(t)dt, \quad x\in
[0,1]\end{aligned}\label{e3.15}
\end{equation}
is a unique solution of the BVP
\begin{gather*}
\Lambda u(x):= -u''(x)-1 = h(x) \ \text{  in } \ (0,1), \\
u(0)=u(1)=0
\end{gather*}
 in $W^{1,2}_0(0,1)$, and that
$$ \|u\|=\|\phi\circ h\|_2\le  \frac 1{2\sqrt{30}}(1+2\|h\|_2).
$$
Thus the hypothesis ($\phi$) holds.
Obviously, $f$ is sup-measurable, i.e., (f1) is fulfilled. Since
$$|f(x,y,z)|\le 15+4|y|^{1/3}+|z|^{1/3},
$$
then the hypothesis (f2) (i) is satisfied.
It then follows from Theorem 3.1 that the BVP \eqref{e3.13} has a solution.
\end{example}

\begin{example} \label{exm3.3} \rm
 For $\Omega = (0,1)$, consider the boundary-value problem
\begin{equation} \begin{gathered}
 -u''(x) = 2+2\,[u(x)-2x+1]+\frac {[-u''(x)-2]}2 \ \text{ a.e. in } (0,1),\\
u(0) = u(1) = 0.\end{gathered}\label{e3.16}
\end{equation}
Problem \eqref{e3.16} is of the form \eqref{e3.1}, where
\begin{equation}
 q(x,y)\equiv -2, \ \text{ and } \ f(x,y,z) =
 2\,[y-2x+1]+\frac {[z-2]}2.\label{e3.17}
\end{equation}
For each $h\in L^2(\Omega)$ the function
\begin{equation}
\begin{aligned} u(x)&= \phi(h(x))=(1-x)\int_0^xt(2+h(t))dt + x\int_x^1(1-t)(2+h(t))dt\\
&= x-x^2+(1-x)\int_0^xth(t)dt + x\int_x^1(1-t)h(t)dt, \quad x\in [0,1]
\end{aligned} \label{e3.18}
\end{equation}
is a unique solution of the BVP
\begin{gather*}
\Lambda u(x):= -u''(x)-2 = h(x) \quad \text{in }  (0,1)\\
u(0)=u(1)=0
\end{gather*}
in $W^{1.2}_0(0,1)$. Moreover,
$$ \|\phi\circ h\|_2\le  \frac 1{\sqrt{30}}(1+\|h\|_2).
$$
Thus the hypothesis ($\phi$) holds with $m=\kappa = 1/\sqrt{30}$.
Obviously, $f$ is sup-measurable, i.e. the hypothesis (f1)
is satisfied. Since
$$ |f(x,y,z)|\le 6+4|x|+2|y|+\frac 12|z|,
$$
and since $(2/\sqrt{30}) + (1/2) < 1$, then
also the hypothesis (f2) (ii) is fulfilled. Thus
it follows from Theorem \ref{thm3.1} that the BVP \eqref{e3.16}
possesses a solution.
\end{example}

\begin{remark} \label{rmk3.2} \rm
 (i) Based on the method of proof of
the abstract fixed point result obtained in \cite[Corollary 5]{h1}
an algorithm has been developed to calculate approximations for
Examples \ref{exm3.2} and \ref{exm3.3}, which can be used to infer
the exact solutions. Computational results will be given in a
forthcoming paper.
\\
(ii) We have restricted to homogeneous
boundary value problems only for the sake of simplicity.
Nonhomogeneous Dirichlet boundary conditions as well as
Neumann or Robin type boundary conditions involving even discontinuous
nonlinearities can be treated.
\\
(iii) As for existence results for discontinuous
explicit and implicit elliptic BVP's different from those presented in this
paper, see e.g., \cite{c1, c2, h2, m1} and the
references therein.  The existence of extremal solutions is considered in
\cite{c1, c2, h2}.
\end{remark}

\begin{thebibliography}{00} \frenchspacing

\bibitem{c1} S. Carl and  S. Heikkil\"a,
\textit{Discontinuous implicit elliptic boundary value problems},
Differential Integral Equations, Vol. 11(1998), No. 6, 823--834.

\bibitem{c2} S. Carl and  S. Heikkil\"a, \textit{Nonlinear Differential
Equations in Ordered Spaces}, Chapman \& Hall/CRC, London, 2000.

\bibitem{c3} D. Gilbarg and N. S. Trudinger,  \textit{Elliptic Partial
Differential Equations of Second Order},
Springer-Verlag, Berlin, 1983.

\bibitem{h1} S. Heikkil\"a, \textit{A method to solve discontinuous boundary
value problems}, Proc. 3rd World Congress of Nonlinear Analysts,
Nonlinear Anal., Vol. 47(2001), No. 4, 2387--2394.

\bibitem{h2} S. Heikkil\"a V. and Lakshmikantham, \textit{Monotone Iterative
Techniques for Discontinuous Nonlinear Differential Equations},
Marcel Deccer, Inc., New York, Basel, Hong Kong, 1994.

\bibitem{m1} S. A. Marano, \textit{Implicit elliptic differential equations},
Set-Valued Anal. Vol. 2(1994), 545--558.

\bibitem{z1}  E. Zeidler, \textit{Nonlinear Functional Analysis and its
Applications}, Vol. II A/B, Springer-Verlag, Berlin, 1990.

\end{thebibliography}

\noindent\textsc{Siegfried Carl }\\
Current address:\\
Florida Institute of Technology,
Department of Mathematical Sciences,\\
150 West University Boulevard,
Melbourne, FL 32901, USA,\\
e-mail: scarl@fit.edu\\
permanent address:\\
Martin-Luther-Universit\"at Halle-Wittenberg,\\
Fachbereich Mathematik und Informatik,\\
Institut f\"ur Analysis,
06099 Halle, Germany \\
email: carl@mathematik.uni-halle.de \medskip

\noindent\textsc{Seppo Heikkil\"a }\\
Department of Mathematical Sciences, University of Oulu, \\
Box 3000, FIN-90014,
University of Oulu, Finland\\
%. Fax: 358 8 5531730.
e-mail: sheikki@cc.oulu.fi

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