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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\centerline{\Large\bf Existence of bounded solutions}
\medskip
\centerline{\Large\bf for nonlinear elliptic equations}
\medskip
\centerline{\Large\bf in unbounded domains}
\bigskip\bigskip

\centerline{\large A. Dall'Aglio\footnote{Dipartimento di Metodi e
Modelli Matematici per le Scienze
Applicate, Universit\`a di Roma I ``La Sapienza'', Via Antonio Scarpa
16, 00161 Roma, Italy}, V. De
Cicco$^1$, D. Giachetti$^1$, J.-P. Puel\footnote{Laboratoire De
Math\'ematiques Appliqu\'ees,
Universit\'e De Versailles St. Quentin, 45 Avenue Des Etats Unis,
78035, Versailles Cedex, France} }


\bigskip\bigskip

\salta{
\centerline{\large DRAFT, \today}

\bigskip\bigskip
}
\centerline{\bf Abstract}
\medskip
\noindent{\small
In this paper we study the existence of bounded weak solutions in unbounded
domains for some nonlinear Dirichlet
problems. The principal part of the operator behaves like the
$p$-laplacian operator, and the lower order
terms, which depend on the solution $u$ and its gradient $\D u$, have
a power growth of order
$p-1$ with respect to these variables, while they are bounded in the
$x$ variable. The source term belongs to
a Lebesgue space with a prescribed asymptotic behaviour at infinity. }

\bigskip

\bigskip

Key words and phrases: existence, nonlinear elliptic equations,
p-Laplacian, unbounded domains, $L^{\infty}$-estimate, homogeneous lower order
terms.

\bigskip

A.M.S. classification: 35J15, 35J25, 35J60.

\bigskip

\section{Introduction}
In this paper we deal with existence of bounded weak solutions of 
some nonlinear
elliptic problems
in a possibly unbounded open subset $\Om$ of $\R^N$. More precisely,
we refer to the problem
\be
      \left\{ \arrstre\begin{array}{ll}
	    -\De_{p}(u) + \nu_0|u|^{p-2}u + \gamma(x)\cdot \D u |\D u|^{p-2} =
      g(x) &
\mbox{in $\Omega$,} \\
		 u \in W^{1,p}_0(\Omega)\,,
          &  \end{array}\right.
\label{eq.00}
\ee
as a model case. Here $p\geq 2$, $\nu_0>0$, $\gamma(x) \in
(L^{\infty}(\Om))^N$,
$-\De_{p}(u)=-\dive(|\D u|^{p-2}\D u)$ is the $p$-laplacian operator
and $g\in L^q_{\rm loc}(\Om)$,
$q>N/p$, with a prescribed asymptotic behavior at infinity.
The term $\ga(x)\cdot \D u |\D u|^{p-2}$ may be replaced by any
function $b(x,\D u)$ such that
$$
      |b(x,\D u)| \leq d\, |\D u|^{p-1}
$$
for some positive constant $d$.
Let us point out that, even if $\Om$ is a bounded open set, the
operator associated to the problem
is, in general, not coercive.\par
Many authors have considered problem \rife{eq.00}
    in bounded domains since the earliest
paper \cite{St} by Stampacchia, where $|\gamma(x)|$ is supposed to
belong to $L^N(\Om)$, with a
sufficiently small $L^N$-norm. We refer for example to \cite{A-T},
\cite{B-M}, \cite{G-T},
\cite{L}
for the linear case ($p=2$) and to \cite{DV-P} for the nonlinear
case. In particular in \cite{B-M} the
authors deal also with  unbounded domains and with a function
$\gamma(x)$ not necessarily small in the
$L^N$-norm. Similar hypotheses on $\gamma(x)$ and $\Om$ ({\em i.e.\/},
$|\gamma(x)|\in L^s(\Om)\cap L^\infty(\Om)$ for some $1\leq s\leq 
+\infty$, $\Om$
unbounded) are made
in \cite{D-G}, where the principal part is a quasi-linear operator
and more general growth conditions
on the lower order term in the gradient are allowed.\par
It is well known
that $L^p$ spaces in domains having infinite measure are not comparable
and therefore the hypothesis $|\gamma(x)| \in L^{\infty}(\Om)$ does
not imply any
summability property.\par
The main existence result for the case of bounded
coefficients in domains having
infinite measure is due to P.L. Lions (see also \cite{C-V} for further
results
).  This result is strictly confined to the framework of linear
operators ($p=2$)  because of the techniques used there (duality
argument, linear interpolation).\par

We point out that we are interested in solutions having finite energy, that is,
solutions in $W^{1,p}_0(\Om)$. If we just look for distributional
solutions in $W^{1,p}_{\rm loc}(\Om)$,
we can drop several assumptions on the operator (in particular the
assumption $p\geq 2$)
and on the datum. This is the subject of the last section of this paper.
\par

The strategy we use consists in approximating the domain $\Om$ by bounded sets
$\Om_n$, and solving a more regular problem in $\Om_n$.
The main difficulty that arises is to get a uniform (with respect to
$n$) $L^p$-estimate
on the solution $u_n$. Such an estimate is obtained in a two-step
process: first we give an a priori
$L^\infty$-estimate for the solutions. To obtain an $L^p$-estimate
outside of compact subsets of $\Om$, we
use a method of sub and supersolutions, together with a comparison
result. Since we need to find explicit
supersolutions, we are forced to require some differentiability
assumptions on the operator, which would not
be necessary otherwise. However we point out that several
intermediate results do not use such strong assumptions.

Once an $L^p$-estimate on the solution is obtained, a simple use of
Young's inequality will give a
uniform estimate
of the $W^{1,p}_0$-norms and therefore the weak convergence, up to a
subsequence,
to a function $u\in W^{1,p}_0(\Om)$, which is shown to be a solution
of problem \rife{eq.00}. The
limit process does not present  any real difficulty since the
principal part is a strongly monotone
operator, which gives a strong local convergence of the gradients in
the $L^p$-norm.

The plan of the paper is as follows. In Section 2 we give the precise
assumptions, the notations, and
state the main result. In order to get uniform
$L^p$-estimates for the approximate solutions
$u_n$ we first need  a uniform
$L^\infty$-estimate, which will be proved in Section 3. In the next
Section 4 we
    prove a comparison principle (which also holds for domains of
infinite measure) for solutions
    to equation \rife{eq.00}.
We remark that this is the only step where the assumption $p\geq 2$
is required.
Since our proof relies heavily on such a comparison result, we are currently
unable to treat the case where the principal part depends directly on $u$.
In Section 5 we construct a fixed supersolution of
    the approximate problems
in the exterior domain
$\Om_n\setminus B$, where $B$ is a fixed ball centered at the origin.
This will give
the required $L^p$-regularity in $\Om\setminus B$ and consequently in
$\Om$. Section 6 is devoted to
the passage to the limit, which concludes the proof of the existence
theorem. Section 7
deals with the case $p=2$, with much more general hypotheses on the
datum $g$. The method used here relies on
the reduction to the linear case and on the estimates proved by P.-L.
Lions in \cite{L1} for solutions of
linear equations in unbounded domains.  Finally, in Section 8, we
show that, if we limit ourselves to look
for distributional solutions in
$W^{1,p}_{\rm loc}(\Om)$, we can obtain an existence theorem for a
much more general operator (in
particular, we may take $p>1$),  under the only hypothesis $g\in
L^q(\Om)$, with $q$ large enough (but
finite). The boundary condition is assumed in the sense that there
exists $\de>1$ such that $|u|^\de$
belongs to
$W^{1,p}_{0}(\Om)$. However this does not allow to conclude that
$u\in W^{1,p}_{0}(\Om)$.



The case where the
first-order term of \rife{eq.00} has a
``natural'' growth of order $p$ (instead of $p-1$) with respect to
the gradient will be the subject of a forthcoming paper. While it is clear
that in the case of bounded domains terms with growth $p$ are more
difficult to handle than terms
with growth $p-1$, in the case of domains having infinite measure the
respective difficulties are not
comparable. Indeed the case of order $p-1$ is not simpler than the
case of order $p$, since $|\D u|$
may be less than $1$ on a set of infinite measure.

The problem when the principal part is a general Leray-Lions type
operator (just monotone in the gradient variable) and the datum $g$ is in
$L^{\frac{p}{p-1}}(\Om)$, or in $L^{\frac{Np}{Np-N+p}}(\Om)$ ($p<N$)
(which are more natural spaces)
is, as far as we know, still open.




\section{Notations, assumptions and main result}
Let $\Omega$ be an open subset of $\R^{N}$, possibly unbounded.
We consider the following nonlinear problem
$$
	 \left\{ \arrstre\begin{array}{ll}
	    A(u) := -\dive(a(x,\D u)) + b(x,\D u) + c(x,u) = g(x) &
\mbox{in $\Omega$,} \\
		 u \in W^{1,p}_0(\Omega).
          &  \end{array}\right.
\eqno{\rm (P)}
$$
All the functions
$a(x,\xi)=(a_1(x,\xi),\dots,a_N(x,\xi)):\Omega\times \R^{N}\to
\R^{N}$, $b(x,\xi):\Om\times \R^{N}\to \R$, $c(x,s):\Om \times \R \to \R $
are always assumed to be Carath\'eodory
functions, {\em i.e.\/}, they are continuous with respect to the
second variable, for almost every $x\in
\Om$, and measurable with respect to $x$ for every fixed $\xi$, or
$s$, respectively.


Moreover, in order to prove the main result we will require  the
following hypotheses:
$$
p\geq 2\,;
$$
\begin{itemize}
	 \item[(A1)]
$a(x,\xi)$ is  differentiable with
respect to $\xi$ for almost
every $x\in \Om$, and weakly differentiable with respect to $x$ for
every $\xi \in \R^N$; moreover
there exist positive constants $M_1$ and $M_2$ such that, for almost
every $x\in \Om$ and every $\xi \in
\R^N$,
\be
     |a_{\xi}(x,\xi)|\leq M_1|\xi|^{p-2}
\label{ex.A4}
\ee
and
\be
     |a_x(x,\xi)|\leq M_2 |\xi|^{p-1}\,;
\label{ex.A5}
\ee

\item[(A2)]  there exists a constant $\beta> 0$ such that
$$
|a(x,\xi)| \leq \beta |\xi|^{p-1}
$$
for almost every $x\in \Omega$ and for every $\xi\in\R^N$;
\end{itemize}

\begin{itemize}
	 \item[(A3)]  there exists a constant $\nu> 0$ such that
$$
[a(x,\xi)-a(x,\eta)]\cdot (\xi-\eta) \geq \nu |\xi-\eta|^p
$$
for almost every $x\in\Omega$ and for every $\xi,\eta\in\R^N$.

\end{itemize}


Let us observe that from (A2) and (A3) it follows that
\begin{itemize}
	 \item[(A4)] $a(x,\xi)\cdot\xi\geq\nu|\xi|^p$ for almost every
$x\in\Om$ and every $\xi\in\R^N$.
\end{itemize}

We remark that the differentiability assumptions (A1)
are only needed for the results of Section 5, while the strong monotonicity
(A3) is used for the comparison principle (Section 4).\par
\bigskip

Moreover we assume the further conditions:

\begin{itemize}
\item[(B1)]  there exists a constant $d>0$ such that for almost every $x\in
\Om$ and for every $\xi \in \R^{N}$
$$
|b(x,\xi)| \leq d|\xi|^{p-1}\,;
$$
\end{itemize}

\begin{itemize}
\item[(B2)]  $b$ is a locally Lipschitz function with respect to
$\xi$ uniformly with respect to $x$,
{\em i.e.\/}, for every $k>0$ there exists $L(k)>0$ such that for
almost every $x\in \Om$ and for every
$\xi,\eta \in \R^{N}$, with $|\xi| \leq k$, $|\eta| \leq k$,
$$
	 	 |b(x,\xi)-b(x,\eta)| \leq L(k) |\xi-\eta|\,;
$$
\end{itemize}


\begin{itemize}
	 \item[(C1)]  there exists a constant $\La>0$
such that for almost every $x\in\Om$ and for every $s \in \R$
$$
      |c(x,s)| \leq \La |s|^{p-1};
$$
\end{itemize}

\begin{itemize}
\item[(C2)]
there exists a constant $\nu_0>0$ such that for almost every
$x\in\Om$ and for every $s, t \in \R$
$$
[c(x,s)-c(x,t)]\,(s-t) \geq \nu_0 |s-t|^p.
$$
\end{itemize}

\noindent
Let us observe that from (C1) and (C2)  it follows that
\begin{itemize}
\item[(C3)] for  almost every $x\in
\Om$ and for every $s \in \R$
$$
c(x,s)s \geq \nu_0 |s|^p.
$$
\end{itemize}

We point out that assumptions (B2) and (C2) are only
required for the comparison principle
(Section 4). In the rest of the paper only assumptions (B1), (C1),
(C3) are needed on the lower order terms.

\begin{rem}\rm
Typical examples of operators satisfying the previous assumptions are
$$
      A(u) = -\De_p(u) + \nu_0 |u|^{p-2} u + \ga(x)\cdot \D u |\D u|^{p-2}\,,
$$
where $\ga(x)\in \left(L^\infty(\Om)\right)^N$, $\De_p(u)=-\dive(|\D
u|^{p-2} \D u)$ is the $p$-laplacian
operator, and
$$
      A(u) =-\De_p(u) + \nu_0|u|^{p-2} u + \ga(x)|\D u|^{p-1}\,,
$$
with $\ga(x)\in L^\infty (\Om)$.
\end{rem}



\noindent{\bf Assumptions on $g(x)$:}
\begin{itemize}
	 \item[(G1)]  There exists $R_0>0$ such that
$g(x)\biggl|_{\Om\cap B_{R_0}} \!\!\!\in
     L^q(\Om\cap
B_{R_0})$ for some
     $q>\max\left\{{N\over p}, 1\right\}$ (here and in the sequel
$B_{R_0}$ denotes the ball with center
0 and radius
     $R_0$);

	 \item[(G2)]  there exist $K>0$, $r>N/p'$ (with $1/p
+1/p'=1$) such that, for
almost every $x\in
	 \Om$, with $|x|\geq R_{0}$ (same $R_0$ as in (G1)),
$$
|g(x)| \leq \frac{K}{|x|^{r}}\,.
$$
\end{itemize}
Let us remark that condition (G2) implies that $g\in L^{p'}(\Om
\setminus B_{R_0})$.
\bigskip

We are interested in finding weak solutions of problem (P), i.e. solutions of
(P) in the sense of distributions.

The main result in this paper is the following.

\begin{theo}\label{main theorem}
Assume that $p\geq 2$, and that the assumptions {\rm (A1)--(A3),
(B1), (B2), (C1), (C2), (G1), (G2)}
hold. Then there exists a bounded weak solution $u$ for problem {\rm (P)}.
\end{theo}

The proof of the main theorem will be carried out in sections 5 and
6, by approximating our problem
by more regular ones on bounded domains. In Sections 3 and 4 we will
prove some preliminary
results ($L^{\infty}$ estimates and a comparison theorem) which will
be applied to the
approximate problems.

We now introduce some notation and results which will be useful in the sequel.
For $k>0$, we will denote by $G_k(s)$ the function
$$
      G_k(s) = \cases{s-k & if $s > k$, \cr
                      0 & if $|s| \leq k$, \cr
                      k+s & if $s < -k$.}
$$
If $v$ is a measurable function on $\Om$, and $k>0$, we denote by $A_k$ the set
$$
      A_k = A_k(v) = \{x\in \Om: |v(x)|>k\}\,.
$$
We state some  classical results which will be used in the next sections.

\begin{lemma} \label{lemma:stampacchia} {\rm (Stampacchia, see \cite{St}).}
Let $\vp$ be a nonnegative, nonincreasing function defined on the
half line $[k_0,\infty)$. Suppose
that there exist positive constants $A$, $\ga$, $\beta$, with
$\beta>1$, such that
$$
      \vp(h) \leq \frac{A}{(h-k)^\ga}\vp(k)^\beta
$$
for every $h>k\geq k_0$. Then
$\vp(k)=0$ for every $k\geq k_1$, where
$$
      k_1 = k_0 + A^{1/\ga} 2^{\beta/(\beta-1)} \vp(k_0)^{(\beta-1)/\ga}\,.
$$
\end{lemma}

\begin{lemma}\label{GN}{\rm (Gagliardo and
Nirenberg, see \cite{Ni}). }
Let $v$ be a function in $W^{1,r}_0(\Om) \cap L^m(\Om)$, with $r \geq
1$, $m \geq 1$.
Then there exists a positive constant $C$, depending only on $N$, $r$
and
$m$, such that
\be
\norma{v}{L^\si(\Om)} \leq C\,\norma{\D
v}{(L^r(\Om))^N}^\theta\,\norma{v}{L^m(\Om)}^{1-\theta}\,,
\label{gani}
\ee
for every $\theta$ and $\si$ satisfying
\be
0 \leq \theta \leq 1\,,
\quad
1 \leq \si < +\infty\,,
\qquad
\frac1\si = \theta\,\left( \frac1r - \frac 1N \right) + \frac{1-\theta}{m}\,.
\label{teta}
\ee
\end{lemma}

\begin{lemma}\label{poincare}{\rm (Poincar\'e inequality, see \cite{G-T}). }
Let $\Om$ be an open subset of $\R^N$ with finite Lebesgue measure. Then there
exists a positive constant $C=C(N)$ such that
\be
\norma u{L^p(\Om)}\leq C|\Om|^{1/N}\norma {\nabla u}{L^p(\Om)}
\ee
for every $u\in W^{1,p}_0(\Om)$.
\end{lemma}


\section{$L^{\infty}$-estimate}

In this section we show that every solution of problem (P) is
bounded, and we obtain an
{\it a priori\/} $L^{\infty}$-estimate for the solutions. We point out
that the estimate also holds
for domains having infinite measure, that the result of this
section is valid for every
$p>1$, and that only very weak hypotheses on the operator are
assumed. In the case of unbounded
domains, this result is new  also if the first order term
$b(x,\D u)$ is absent.

\begin{prop}\label{prop:Linfty}
	 Let $\Om$ be an open subset of $\R^{N}$.
We assume the hypotheses  {\rm (A4),  (B1), (C3)} and
\be
    g=g_1 + g_2\,, \quad\mbox{with $g_1\in L^\infty(\Om)$ and
    $g_2\in L^q(\Om)$, $\dys q>\max \left\{\frac{N}{p}\,,\,
1\right\}\,, \, p>1$.}
\label{eq:g}
\ee
Let $u$ be a weak solution of
\begin{equation}
	 	 	 \left\{ \arrstre\begin{array}{ll}
	    A(u) = g(x) & \mbox{in $\Om$,} \\
		 u \in W^{1,p}_{0}(\Om)\,.
          &  \end{array}\right.
\label{eq:Linfty1}
\end{equation}
Then $u$ is essentially bounded in $\Om$ and
\be
      \norma{u}{L^{\infty}(\Om)}\leq D\,,
\label{eq:Linfty2}
\ee
where $D$  depends on
$N, p, q, d,
\nu,
\nu_0,
\norma{g_1}{L^{\infty}(\Om)}$ and $\norma{g_2}{L^q(\Om)}$
but not on $\norma{u}{W^{1,p}_{0}(\Om)}$.
\end{prop}

\begin{rem} \rm
We point out that hypotheses (G1) and (G2) imply \rife{eq:g}.
\end{rem}


\dimo
For simplicity we use the notations
$\norma{v}{r}$ and $\norma{\nabla v}{r}$ instead of
$\norma{v}{{L^r(\Om)}}$ and $\norma{\nabla v}{(L^r(\Om))^n}$ respectively.
\par

\noindent{\em Step 1.\/}
We first show that every solution $u$ of \rife{eq:Linfty1} is
bounded, and moreover that, if $k_0$
satisfies
\be
      \nu_0 k_0^{p-1} \geq \norma{g_1}{\infty}
\label{eq.h1}
\ee
and
\be
      d \left(\frac{\left|A_{k_0}(u)\right|}{\om_N}\right)^{1/N} \leq
\frac{\nu}{2}\,,
\label{eq.h2}
\ee
where $\om_N$ is the measure of the unit ball in $\R^N$, then
\be
      \norma{u}{\infty} \leq c=c\left(k_0, \left|A_{k_0}(u)\right|\right)\,.
\label{eq.h3}
\ee
Actually we will prove in Step 2 that is possible to choose $k_0$
independent of the norm of $u$ in $W^{1,p}_{0}(\Om)$ such that
\rife{eq.h1} and \rife{eq.h2} hold. We take $v=G_k(u)$, with $k\geq k_0$,
as test function in
\rife{eq:Linfty1}. From (A4), (C3) and (B1) we
obtain
\bean
\lefteqn{\nu\,\int |\D G_k(u)|^{p} + \nu_0\int
|u|^{p-1}|G_k(u)| }\label{eq.h4}\\
& \leq &
d \int |\D G_k(u)|^{p-1}|G_k(u)| + \int |g_1|\, |G_k(u)|
+\int |g_2|\,|G_k(u)|\,.\nonumber
\eean
If $k_0$ satisfies \rife{eq.h1}, since $|u| > k_0$ on the set where
$G_k(u)\not=0$, the second
integral of the right-hand side of \rife{eq.h4} is less than or equal
to the last integral of the
left-hand side. Moreover, by H\"older's and Poincar\'e's inequalities
(see, for instance, \cite{G-T},
Section 7.8),
\bea
      d\int|\D G_k(u)|^{p-1} |G_k(u)|
      &\leq&
      d \left[\int |\D G_k(u)|^p\right]^{\frac{1}{p'}}
\left[\int |{G_k(u)}|^{p}\right]^{\frac{1}{p}} \\
      &\leq&
      d \left(\frac{|A_{k_0}|}{\om_N}\right)^{\frac{1}{N}} \int|\D
G_k(u)|^{p}\,.
\eea
Therefore, if
\rife{eq.h2} holds, from
\rife{eq.h4} we get
\be
      \frac{\nu}{2} \int |\D G_k(u)|^p
      \leq
      \int |g_2|\, |G_k(u)|
      \leq
      \norma{g_2}{q}\, \norma{G_k(u)}{q'}\,.
\label{eq.h5}
\ee
We now deal separately with the three cases $p<N$, $p=N$, $p>N$.
\medskip

\noindent {\it Case} $p<N$.
As usual, we denote by $p^*=\frac{Np}{N-p}$ Sobolev's conjugate
exponent of $p$. Using Sobolev's and
H\"older's inequalities, since
$q' < p^*$, from
\rife{eq.h5} we obtain
$$
      c_1 \norma{G_k(u)}{p^*}^p
      \leq
      \norma{\D G_k(u)}{p}^p
      \leq
      \frac{2}{\nu}\, \norma{g_2}{q} \norma{G_k(u)}{q'}
      \leq
      \frac{2}{\nu}\, \norma{g_2}{q} \norma{G_k(u)}{p^*}
|A_k(u)|^{\frac{1}{q'}-\frac{1}{p^*}}\,,
$$
where $c_1= c_1(N,p)$ is the reciprocal of the Sobolev constant.
Therefore, if $h>k$ and $G_k(u)\not= 0$ (if $G_k(u)=0$ the result is
trivial)
$$
      c_1 (h-k)^{p-1} |A_h(u)|^{\frac{p-1}{p^*}}
      \leq
      \frac{2}{\nu}\, \norma{g_2}{q} |A_k(u)|^{\frac{1}{q'}-\frac{1}{p^*}}\,,
$$
from which it follows
$$
      |A_h(u)|
      \leq
c_2\left(N,p,\nu,\norma{g_2}{q}\right)\frac{|A_k(u)|^{\frac{p^*-q'}{q'(p-1)}}}{(
h-k)^{p^*}}\,.
$$
Since
$$
      \frac{p^*-q'}{q'(p-1)} >1
$$
for $q>N/p$, we can apply Lemma \ref{lemma:stampacchia} with
$\varphi(k)=|A_k(u)|$ to obtain \rife{eq.h3}.
\medskip

\noindent {\it Case} $p=N$. Let $r<N$ be such that $r^*=rN/(N-r)>q'$.
   From \rife{eq.h5} we obtain
\bea
      \norma{\D G_k(u)}{N}^N
      &\leq&
      \frac{2}{\nu}\, \norma{g_2}{q} \norma{G_k(u)}{r^*}
|A_k(u)|^{\frac{1}{q'}-\frac{1}{r^*}} \\
      &\leq&
      \frac{2c_3(N,r)}{\nu }\, \norma{g_2}{q} \norma{\D G_k(u)}{r}
|A_k(u)|^{\frac{1}{q'}-\frac{1}{r^*}} \\
      &\leq &
      \frac{2c_3(N,r)}{\nu}\, \norma{g_2}{q} \norma{\D G_k(u)}{N}
|A_k(u)|^{\frac{1}{q'}}\,.
\eea
Therefore, using an approximation procedure of $A_k(u)$ by open sets and
Poincar\'e's inequality (see Lemma \rife{poincare}),
\bea
      |A_h(u)|^{\frac{N-1}{N}}|A_k(u)|^{\frac{1-N}{N}} (h-k)^{N-1}
      &\leq&
      |A_k(u)|^{\frac{1-N}{N}} \norma{G_k(u)}{N}^{N-1}
      \leq
      c_4(N) \norma{\D G_k(u)}{N}^{N-1} \\
      &\leq &
      c_5 \left(N,r,\nu, \norma{g_2}{q}\right) |A_k(u)|^{\frac{1}{q'}}\,,
\eea
from which we conclude again using Lemma \ref{lemma:stampacchia}.
\medskip

\noindent {\it Case} $p>N$.
This is the easiest case, since it suffices to take $k=k_0$, so that,
by \rife{eq.h5} and Sobolev's
inequality,
\bea
      \norma{\D G_{k_0}(u)}{p}^p
      &\leq&
      \frac{2}{\nu}\, \norma{g_2}{q} \norma{G_{k_0}(u)}{\infty}
|A_{k_0}(u)|^{\frac{1}{q'}} \\
      &\leq&
      \frac{2}{\nu c_1(N,p)}\, \norma{g_2}{q} \norma{\D G_{k_0}(u)}{p}
|A_{k_0}(u)|^{\frac{1}{N}- \frac{1}{p} +\frac{1}{q'}}\,.
\eea
Therefore an estimate for $\norma{\D G_{k_0}(u)}{p}$ follows
immediately in terms of $|A_{k_0}(u)|$.
By Sobolev's imbedding, \rife{eq.h3} is proved, and Step 1 is completed.
\bigskip

\noindent{\em Step 2.\/}
In view of Step 1, it suffices to show that there exists some $k_0>0$
(depending only on the data of
the problem) such that \rife{eq.h2} holds,
and to do this we only need to prove that there exists $\de =
\de(p,q,d,\nu,\nu_0)>1$,
$k_1=k_1(p,\nu_0,\norma{g_1}{\infty})$ and
$C_1=C_1(N,p,q,d,\nu,\nu_0,\norma{g_2}{q})$
such that
\be
\norma{G_{k_1}(u)}{\delta} \leq C_1
\label{eq.gkm1}.
\ee
To show this, we use $|G_{k_1}(u)|^{\alpha}G_{k_1}(u)$ as test function in (1),
where $\alpha>0$ and
$k_1>0$ will be chosen later. For the sake of brevity we set
$v=G_{k_1}(u)$. We obtain, using Young's and H\"older's inequalities,
$$
\nu(\alpha+1)\int |\D v|^{p}|v|^{\alpha} + \nu_0\int
|u|^{p-1}|v|^{\alpha+1} \leq
d \int |\D v|^{p-1}|v|^{\alpha+1} + \int (g_1+g_2)|v|^{\alpha+1}\,\leq
$$
$$
\leq {\nu(\alpha+1)\over p'} \int |\D v|^{p}|v|^{\alpha} +
   {d^p\over {p(\nu(\alpha+1)})^{p-1}}\int |v|^{p+\alpha} +
\norma{g_1}{\infty}\int |v|^{\alpha+1}+
\norma{g_2}{{q}}\bigg(\int |v|^{(\alpha+1)q'}\bigg)^{1\over q'}.
$$
Choosing $\alpha$ such that
\begin{equation}
{d^p\over {p(\nu(\alpha+1)})^{p-1}}<{\nu_0\over 4}
	 \label{1 condiz.}
\end{equation}
and $k_1$ such that
\be
{\nu_0\over 4}k_1^{p-1}\geq\norma{g_1}{\infty}\,,
\label{eq:cond-k_0}
\ee
and defining
$$
      w=|v|^{\frac{\al+p}{p}}\,,
$$
we obtain
\be
      c_1(\al,\nu,p) \int |\D w|^p
      +
      \frac{\nu_0}{2} \int w^p
      \leq
      \norma{g_2}{q}
      \left[ \int w^{\frac{pq'(\al+1)}{\al+p}} \right]^{\frac{1}{q'}}\,.
\label{eq.h6}
\ee
We wish to apply Lemma \ref{GN} with $\si=\frac{pq'(\al+1)}{\al+p}$,
$r=m=p$. To show that the
hypotheses of Lemma \ref{GN} are satisfied, we have to prove that
\be
      p
      \leq
      \frac{pq' (\al+1)}{\al + p}
      \leq
      p^*
      \qquad
      \mbox{if $p<N$}
\label{eq.h7}
\ee
and
\be
      p
      \leq
      \frac{pq' (\al+1)}{\al + p}
      \qquad
      \mbox{if $p\geq N$.}
\label{eq.h8}
\ee
The first inequality of \rife{eq.h7}, and of course \rife{eq.h8}, are
true if we assume the further
condition
\be
      \al \geq \frac{p-q'}{q'-1}\,.
\label{eq.h8'}
\ee
The second inequality of \rife{eq.h7} is always true for $q>N/p$.
Therefore, from \rife{eq.h6} and
Lemma  \ref{GN} we obtain
\bean
      c_1 \int |\D w|^p
      +
      \frac{\nu_0}{2} \int w^p
      &\leq&
      c_2(N,p,\al)
      \norma{g_2}{q}
      \left[\int |\D w|^p \right]^{\frac{\theta(\al+1)}{\al+p}}
      \left[\int w^p \right]^{\frac{(1-\theta)(\al+1)}{\al+p}} \nonumber \\
      &\leq&
      c_2(N,p,\al) \norma{g_2}{q}
      \left(\eps \int |\D w|^p + \eps \int w^p + c_3(\eps)\right)\,,
\label{eq.h9}
\eean
where $\eps$ is any positive number. By choosing it small enough, we
get an estimate for the
left-hand side of \rife{eq.h9}:
$$
      \int |\D w|^p + \int w^p \leq
      c_4\left(N,p,q, \norma{g_2}{q}, \nu, \nu_0,\al\right)
$$
and this implies \rife{eq.gkm1} with $\delta=\alpha+p$.

\qed









\section{Comparison principle}
The aim of this section is to obtain a comparison result for solutions
of equation (P).

In order to prove Theorem 2.2, we will apply this result to each of the
approximate problems defined in next section (see $({\rm P}\!_n)$,
$\rife{eq:app1}$
and $\rife{eq:app2})$,
for which the following global Lipschitz continuity property holds.


\begin{itemize}
\item[(B2$'$)] There exists a constant $L$ such that for almost every $x\in
\Om$ and for every
$\xi,\eta \in \R^{N}$
$$
	 	 |b(x,\xi)-b(x,\eta)| \leq L |\xi-\eta|.
$$
\end{itemize}




\begin{theo}\label{theo:confronto}
Let $\Om$ an open subset of $\R^N$. Let us assume that $p\geq 2$,
   {\rm (A2), (A3),  (B1), (B2\,$'$), (C1), (C2)}.
Let $u,v \in W^{1,p}(\Om)$ such
that
\begin{equation}
	 	 	 \left\{ \arrstre\begin{array}{ll}
	    A(u) \leq A(v) & \mbox{in ${\cal D'}(\Om)$,} \\
		 u \leq v \,,
          &  \mbox{on $\partial\Om$.}
	\end{array}\right.
\label{eq:confronto1}
\end{equation}
Then $u\leq v$ almost everywhere in $\Om$.
\end{theo}

\salta{
\begin{rem}\label{remark:lipschitz0}\rm
In order to prove Theorem \ref{main theorem} we will applt the result
of Theorem
\ref{theo:confronto} to each of the approximate problems defined in
next section, for which global
Lipschitz continuity (B3$'$) holds.
\end{rem}
}

\begin{rem}\label{remark:unic}\rm
Under the hypotheses of Theorem \rife{theo:confronto} the solution of problem
(P) is unique.
\end{rem}


\begin{rem}\label{remark:lipschitz}\rm  Let us observe that the same
result holds if (B2) is assumed instead of (B2$'$), provided $u$ and $v$ belong to
$W^{1,\infty}(\Om)$.
This is true, for instance, in the case where the domain $\Om$ and the function
$a(x,\xi)$ which defines the principal part of the operator are
smooth enough (see
for instance \cite{G} for $W^{1,\infty}$-regularity result).
\end{rem}


\dimo
Let
$\la =
\sup(u-v)$, and we suppose by contradiction that
$\la>0$.  We confine ourselves to the case $\la < \infty$, since
otherwise the proof can be easily
adapted. Let us take as a test function in \rife{eq:confronto1} the
positive function $w=(u-v-h)_+=\max\{u-v-h,0\}$
with
$h$ such that
$\la/2<h<\la$. By (A3), (B2$'$), (C2),
if we consider the set of finite measure $E_h=\{x\in\Om:u-v\geq
h,\nabla u\not=\nabla v\}$, then
\bea
\nu\int_{E_h}|\D w|^{p} + \nu_0 \int_{E_h} (u-v)^{p-1}w
& \leq & \int_{E_h}\left|b(x,\D u)-b(x,\D v)\right| w \leq
L\int_{E_h} |\D w| w\leq\\
& \leq &  \frac{\nu}{2}\int_{E_h}|\D w|^{p} + c_1 \int_{E_h} w^{p'}\,,
\eea
where
$c_1=c_1(p, L,\nu)$.
Then we get
\be
\frac{\nu}{2}\int_{E_h}|\D w|^{p} + \nu_0 \int_{E_h} (u-v)^{p-1}w
\leq  c_1 \int_{E_h} w^{p'}\,.
\label{eq:confronto2}
\ee
Let us first consider the case $p>2$.
Then, since $p'<p$, for every $\e>0$ we have
\be
      w^{p'} \leq \eps w + c_2(\eps) w^p\,.
\label{eq:confronto3}
\ee
On the other hand, by the choice of $h$, on the set
$E_h$ we have
$$
      (u-v)^{p-1} \geq \left(\frac{\la}{2}\right)^{p-1} \,,
$$
and choosing $\dys \eps = \frac{\nu_0}{c_1}
\left(\frac{\la}{2}\right)^{p-1} $ in
\rife{eq:confronto3}, from \rife{eq:confronto2} we get
\be
      \int_{E_h}|\D w|^{p} \leq  c_3 \int_{E_h} w^{p}\,,
\label{eq:confronto4}
\ee
where $c_3=c_3(p, \nu, \nu_0, \lambda, L)$. In the simpler case
$p=2$, \rife{eq:confronto4}
follows directly from \rife{eq:confronto2}. We now consider the
following two cases.
\medskip

\noindent
{\it Case I: $p<N$.\/} By Sobolev's embedding theorem and H\"older's
inequality,
we obtain
$$
      \left[\int_{E_h} w^{p^{*}}\right]^{\frac{p}{p^{*}}} \leq
	c_4 \left[\int_{E_h} w^{p^{*}}\right]^{\frac{p}{p^{*}}}
	\left|E_{h}\right|^{1-\frac{p}{p^{*}}}\,,
$$
where
$c_4$ depends on $c_3$, $N$ and $p$.
This implies that
$|E_{h}|\geq c_5>0$, with $c_5$ independent on $h$. Then we have a
contradiction since  as $h\to \sup
(u-v)$ we have $|E_{h}| \to | \{ x: u(x)-v(x) = \sup (u-v), \D u(x)
\not = \D v(x)\}| = 0$.
\medskip

\noindent
{\it Case II: $p\geq N$.\/} Fix $r<N$ such that $r^*=\frac{Nr}{N-r} > p$. Then
\bea
      \left[\int_{E_h} w^{r^*}\right]^\frac{p}{r^*}
      &\leq &
      c_6(N,r)	\left[ \int_{E_h} |\D w|^r\right]^{\frac{p}{r}}
      \leq
      c_6 |E_h|^{\frac{p-r}{r}}
      \int_{E_h} |\D w|^p \\
	 &\leq& c_3 c_6 |E_h|^{\frac{p-r}{r}}
      \int_{E_h} w^p
      \leq
      c_3 c_6
      \left[\int_{E_h} w^{r^*}\right]^\frac{p}{r^*} |E_h|^{\frac{p}{N}}\,.
\eea
Since $c_3$ and $c_6$ do not depend on $h$, we can conclude as in Case I.
\qed





\section{Approximate problems and $W^{1,p}_0$-estimates}
We consider the following approximate problems
$$
	 \left\{ \arrstre\begin{array}{ll}
	    A_n(u_{n}) = g(x)& \mbox{in $\Om_{n}$,} \\
		 u_n \in W^{1,p}_{0}(\Om_{n}),
          &  \end{array}\right.
\eqno{({\rm P}\!_n)}
$$
where $\{\Om_n\}$ is an increasing sequence of bounded open sets
invading $\Om$ (for instance
$\Om_n
= \Om\cap B_n$) and the operator $A_n$ is defined by:
\be
A_n(v) := -\dive(a(x,\D v)) + b_n(x,\D v) + c(x,v),
\label{eq:app1}
\ee
with
\be
	b_n(x,\xi)= \left\{ \arrstre\begin{array}{ll}
	 b(x,\xi)& \mbox{if }|\xi|< n,   \\
b\bigg(x,\dys{\xi\over{|\xi|}}n\bigg)& \mbox{if }|\xi|\geq n.
           \end{array}\right.
\label{eq:app2}
\ee
Let us remark that, by (B2), for every fixed $n\in \N$ the function
$b_n(x,\cdot)$ satisfies
condition (B2$'$). The existence of a solution of $({\rm P}\!_n)$ is well known
(see for instance \cite{LL}).
\par
We are interested now in $W^{1,p}_0$-estimates for $u_n$ independent of $n$.

\begin{prop}\label{prop:stima W1,p}
Let $u_n$ be a solution of $({\rm P}\!_n)$.  Under the hypotheses of Theorem
{\ref{main theorem}}
there exists a positive constant $M$,
independent of $n$ such that
$$
\norma{u_n}{W^{1,p}_0(\Om_n)}\leq M.
$$
\end{prop}


\dimo
We note that by taking $u_n$ as test function in $({\rm P}\!_n)$ and using
Young's inequality, it will be
sufficient  to prove a uniform $L^p$-estimate on $u_n$. The result is
an immediate consequence of
Proposition  \ref{prop:Linfty} if $\Om$ has finite measure. The only
difficult case is when $\Om$ has
infinite measure. By Proposition
\ref{prop:Linfty} we just need an $L^p$-estimate in $\Om_n\setminus
B_{R_1}$, where $R_1\geq R_0$
(see hypothesis (G2)). In order to do this, we look for a
supersolution of the equation
$$
A_n(z)=g \quad {\rm in}\quad \Om_n\setminus B_{R_1}
$$
for $n$ sufficiently large ({\it i.e.\/}, $n$ such that
$\Om_n\setminus \overline B_{R_1}\not=\emptyset)$.
We consider
$$
z(x)={H\over{|x|^q}}\,,
$$
where $H$ and $q$ will be chosen later.
It is easy to check that by \rife{ex.A4} and \rife{ex.A5} we have
$$
-\dive{(a(x,\nabla z))}\geq -\tilde c\,
{{H^{p-1}}\over{\,|x|^{(q+1)(p-1)}}}\,\,\,,
$$
where $\tilde c=\tilde c(N,p,q,M_1,M_2)$ is a positive constant.
Therefore, by (B1) and (C3),
\be
      A_n(z) \geq \frac{H^{p-1}}{|x|^{q(p-1)}}
      \left\{\nu_0
            - \frac{\tilde c + d\, q^{p-1}}{|x|^{p-1}}
            \right\}\,.
\label{eq:stimalp1}
\ee
If we choose $R_1$ large enough (depending on $N$, $\nu_0$, $q$, $p$, $d$,
$M_1$,
$M_2)$, then, for every $|x| > R_1$ , we have
$$
      A_n(z) \geq \frac{\nu_0}{2}\frac{H^{p-1}}{|x|^{q(p-1)}}\,.
$$
By (G2), if we choose $q$, $H$ such that
$$
      q(p-1) = r\,, \qquad \frac{\nu_0}{2} H^{p-1} > K\,,
$$
then the function $z(x)$ satisfies $A_n(z) \geq g$ in  $\R^N\setminus B_{R_1}$.
Moreover, by Proposition \ref{prop:Linfty} there exists a positive
constant $D$ independent of
$n$ such that
$$
\norma{u_n}{\infty}\leq D.
$$
If we assume the further condition on $H$
$$
      \frac{H}{R_1^q} \geq D,
$$
then we get $z\geq u_n$ on $\partial B_{R_1}$. Since  $z(x)\geq u_n(x)$ on
$\partial\Om_n$, we can apply Theorem \ref{theo:confronto} in $\Om_n
\setminus \overline B_{R_1}$
to the operator $A_n$
and we obtain
$ u_n(x) \leq z(x) = H/|x|^q$ in this set. In a similar way one can
prove that $u_n(x) \geq -z(x)$.
By
the choice of
$q = r/(p-1)$ we have $q > N/p$ and therefore
$z\in L^p(\R^N \setminus
\overline B_{R_1})$. This implies a uniform estimate for $u_n$ in
$L^p(\Om_n \setminus
\overline B_{R_1})$ and therefore in $L^p(\Om_n)$.
\qed

\salta{
\comment{IMPORTANTE per JPP:
Ci sono due remark importanti da fare da qualche parte nell'articolo.
Essi potrebbero costringerci a
cambiare drasticamente alcune parti. Quindi prima di procedere
vorremmo sapere come la pensi:

1) Sotto le ipotesi della proposizione 3.1 si ottiene una stima
$L^\infty$ e quindi
$L^p_{\rm loc}$. Questo basta per avere soluzioni locali (nel senso
delle distribuzioni, ma SENZA
condizioni al bordo). In realt\`a la stima $L^\infty$ \`e fin troppo,
bisognerebbe vedere quali
sono le ipotesi minime su $g$ per avere la stima $L^p_{\rm loc}$.

2) Se $g_1\equiv 0$ nella proposizione 3.1, allora si ottiene una
stima in $W^{1,p}_0$ di $u^\de$,
per $\de$ molto grande. Quindi in realt\`a si pu\`o trovare una
soluzione nel senso delle distribuzioni
(di energia globale infinita), ma con ``dato al bordo'' nullo. Anche
in questo caso probabilmente la
stima
$L^\infty$ \`e esagerata, dal momento che serve solo poter utilizzare
$u^\al$ come funzione test.
}

}


\section{Proof of the main theorem}
By using the results obtained in the previous sections
we will now give the proof of Theorem \ref{main theorem}. We will
continue to denote by $u_n$
the zero-extensions of $u_n$ outside of $\Om_n$.
\par\bigskip\noindent
{\bf Proof of Theorem \ref{main theorem}.}
By Proposition {\ref{prop:stima W1,p}} it follows, passing to a
subsequence, that there exists $u\in
W^{1,p}_0(\Om)$ such that
\be
     u_n\weak u \quad{\rm weakly  \quad in} \quad W^{1,p}_0(\Om)
\label{eq:conv-debole}
\ee
and
\be
      u_n\to u  \quad{\rm strongly  \quad in} \quad L^{p}_{\rm loc}(\Om).
\label{eq:conv-forte}
\ee
Let us take $A\subset\subset\Om$ and $\psi\in C^{\infty}_0(\Om)$,
$0\leq \psi\leq 1$, $\psi\equiv 1$
on
$A$. If we take $n$ large enough, we have $\supp \psi \subset\subset
\Om_n$. We use $(u_n-u)\psi$ as
test function in $({\rm P}\!_n)$ and we obtain
$$
\int a(x,\nabla u_n) \cdot\nabla (u_n-u)\psi+\int
a(x,\nabla u_n)\cdot \nabla \psi(u_n-u)+
\int c(x,u_n) (u_n-u)\psi\leq
$$
$$
\leq d \int |\nabla u_n|^{p-1}|u_n-u|\psi+\int |g| |u_n-u|\psi.
$$
Then
\bean
\lefteqn{\int [a(x,\nabla u_n)-a(x,\nabla u)] \cdot\nabla
(u_n-u)\psi\leq}\nonumber \\
&\leq& -\int a(x,\nabla u) \cdot\nabla (u_n-u)\psi-\int
a(x,\nabla u_n)\cdot \nabla
\psi (u_n-u)\label{star} \\
& & \null -\int c(x,u_n) (u_n-u)\psi +d \int |\nabla
u_n|^{p-1}(u_n-u)\psi+\int |g|
|u_n-u|\psi\,. \nonumber
\eean
Each term in the right-hand side of  \rife{star} tends to 0 as $n$
goes to $\infty$, by (A2), (C1),
\rife{eq:conv-debole} and
\rife{eq:conv-forte}. Therefore, using (A3), one obtains that $\nabla
u_n\to\nabla u$ strongly in $L^p_{\rm
loc}(\Om)$. Thus
$$
\int a(x,\nabla u) \cdot\nabla \Phi +  \int c(x,u) \Phi
+\int b(x,\D u)\Phi=\int g\Phi
$$
for every
$\Phi\in C^{\infty}_0(\Om)$ and, by density, for every $\Phi\in
W^{1,p}_0(\Om)$.
\qed













\section{Further results in the case $p=2$}
In this section we show that, in the special case $p=2$, using the
estimates proved by P.-L. Lions in
\cite{L1} for linear problems, it is possible to obtain an existence
result under weaker hypotheses on the
datum $g$. More precisely, we consider the following problem
$$
	 \left\{ \arrstre\begin{array}{ll}
    -\dive(a(x,u)\D u) + b(x,\D u) + c(x,u) = g(x) &
\mbox{in $\Om$,} \\
		 u \in H^{1}_0(\Om),
          &  \end{array}\right.
\eqno{({\rm P}')}
$$
where: $a(x,s)=\left\{a_{ij}(x,s)\right\}_{i,j}:\Om \times \R \to
\R^{N\times N}$ is a Carath\'eodory matrix,
such that
\be
     |a(x,s)| \leq \beta\,, \qquad
     \sum_{i,j} a_{ij}(x,s)\xi_i\xi_j \geq \nu |\xi|^2\,,
\label{r1}
\ee
for almost every $x\in \Om$, every $s\in \R$ and every
$\xi=(\xi_1,\dots,\xi_N)\in \R^N$, and $\beta$,
$\nu$ are positive constants;

$b(x,\xi)$ satisfies (B1), $c(x,s)$ satisfies (C1), (C3) (for $p=2$), while
$g(x)$ satisfies
\be
     g(x)\in L^2(\Om)+L^q(\Om)\,,
\label{r2}
\ee
with $q=1$ if $N=1$, $1<q<2$ if $N=2$, $q=2N/(N+2)$ if $N\geq 3$.

\begin{theo} \label{th.9}
Under the assumptions $p=2$, {\rm (B1)}, {\rm (C1)}, {\rm (C3)}, {\rm
\rife{r1}}, {\rm \rife{r2}}, there
exists a weak solution of problem $({\rm P}')$.
\end{theo}

\dimo
We consider the approximate problems $({\rm P}_n)$ defined as in
Section 5. Let $\{u_n\}$ be a sequence of
solutions of $({\rm P}_n)$. Then $u_n$ is also solution of the {\em
linear\/} equation
$$
	 \left\{ \arrstre\begin{array}{ll}
    -\dive(\tilde a_n(x)\D u) + \tilde b_n(x)\cdot \D u_n + \tilde
c_n(x)u_n = g_n(x) &
\mbox{in $\Om_n$,} \\
		 u \in H^{1}_0(\Om_n),
          &  \end{array}\right.
\eqno{({\rm P}_n)}
$$
where
$$
     \tilde a_n(x)= a(x,u_n(x))\,,
$$

$$
     \tilde b_n(x)= \left\{ \arrstre\begin{array}{ll}
     \dys \frac{b_n(x,\D u_n(x)) \D u_n(x)}{|\D u_n(x)|^2} &
\mbox{if $\D u_n(x)\not=0$,} \\
		 0
          &  \mbox{if $\D u_n(x)=0$,}\end{array}\right.
$$
$$
     \tilde c_n(x)= \left\{ \arrstre\begin{array}{ll}
     \dys \frac{c_n(x,u_n(x)) u_n(x)}{(u_n(x))^2} &
\mbox{if $u_n(x)\not=0$,} \\
		 \nu_0
          &  \mbox{if $u_n(x)=0$.}\end{array}\right.
$$
It is easy to check that the matrix $\tilde a_n$, the vector $\tilde
b_n$ and the function $\tilde c_n$ are
uniformly bounded, that $\tilde a_n$ is uniformly elliptic, and that
$\tilde c_n(x) \geq \nu_0 >0$.

Let us assume for the moment that $g(x)$ belongs to $L^2(\Om)$. Then
it is possible to apply the
linear estimate by P.-L. Lions (see (5) in  \cite{L1}), and obtain
that the sequence $\{u_n\}$ is bounded in
$L^2(\Om)$. As observed in Section 5, this implies a uniform estimate
in $H^1_0(\Om)$.
On the other hand, if $g(x)\in L^q(\Om)$
($q=1$ if $N=1$, $1<q<2$ if $N=2$, $q=2N/(N+2)$ if $N\geq 3$),
then, again by P.L. Lions' theorem one obtains that the sequence
$\{u_n\}$ is bounded in
$L^q(\Om)$. Taking $u_n$ as test function in $({\rm P}_n)$, and using
Sobolev's inequality, one again
obtains an estimate in $H^1_0(\Om)$.
In the general case $g(x)\in L^2(\Om)+L^q(\Om)$, using the linearity
of problem $({\rm P}_n)$, we can write
$\{u_n\}$ as the sum of two bounded sequences in $H^1_0(\Om)$.
Once the estimate in $H^1_0(\Om)$ is proved, the existence follows in
a standard way.
\qed











\salta{
\section{Further results in the case $p=2$}
Questo paragrafo e' stato sostituito perche' i risultati con le stime
di Lions sono migliori - 5.10.2001)
\par

In this section we deal with the following nonlinear problem with growth $p=2$
$$
	 \left\{ \arrstre\begin{array}{ll}
    -\Delta u + b(x,\D u) + c(x,u) = g(x) &
\mbox{in $\Om$,} \\
		 u \in H^{1}(\Om),
          &  \end{array}\right.
\eqno{({\rm P}')}
$$
where the functions $b(x,\xi)$ and $c(x,s)$ satisfy the hypotheses
(B1), (B2), (C1), (C2) with $p=2$, while $g$ satisfies (G1) and
\begin{itemize}
	 \item[(G2$'$)]  there exists a radial function $h(|x|)\in
L^2(\R^N)$ such that
$$
      |g(x)| \leq h(|x|) \quad \mbox{for every $x$ such that $|x|>R_0$,}
$$
where $R_0$ is the same as in (G1).
\end{itemize}
We may always assume that $h(|x|)$ is bounded near the origin.
Condition (G2$'$) is less restrictive than (G2).  Since (G2) has only
been used to construct a
supersolution
$z$  in $\R^N \setminus \overline B_{R_0}$,  we try to do the same
under the weaker assumption (G2$'$).  We obtain the following result:

\begin{theo} \label{th.3}
If $p=2$, under the assumptions {\rm (B1), (B2), (C1), (C2), (G1) and
(G2$'$)}, there exists at least a
solution of problem $({\rm P}')$.
\end{theo}



We devote the remaining part of this Section to the proof of Theorem
\ref{th.3}.
Let us consider the function of
one real variable
\be
      v(t) = v_\la(t) = (G*h)(t)\,,
\label{eq:v}
\ee
where
$$
G(t)={1\over{2\lambda}}e^{-\lambda |t|}\,,
$$
with $\la>0$ to be chosen below, and $h(t)$ is extended to negative
$t$ by reflection. Then it is
easy to see that $v$ satisfies
\be
-v''(t)+\lambda^2 v(t)=h(t)
\label{eq:eqv}
\ee
in the sense of distributions. Under our assumption on $h$, we know
that $h(t) \in L^2_\eta(\R)$,
where
$\eta(t) = 1+|t|^{N-1}$ and
$$
L^2_\eta(\R) = \biggl\{w: \norma{w}{L^2_\eta(\R)}^2=\int_\R w^2(t)
\eta(t)\,dt <+\infty  \biggr\}\,.
$$

\begin{lemma} \label{lemma:radial1}
There exists a constant $c_1=c_1(\la, N)$ such that
\be
      (G*\eta)(t) \leq c_1 \eta(t)\,.
\label{eq:radial1}
\ee
\end{lemma}

\dimo
Since all the functions involved are even, we may restrict ourselves
to the case $t>0$. Then
\bea
      (G*\eta)(t) &=& \int_{\R} G(t-y) \eta(y)\,dy \\
      &=&
      \frac{e^{-\la t}}{2 \la} \int_{-\infty}^0 e^{\la y} \eta(y)\,dy
      + \frac{e^{-\la t}}{2 \la} \int_{0}^t e^{\la y} \eta(y)\,dy
      + \frac{e^{\la t}}{2 \la} \int_t^{+\infty} e^{-\la y} \eta(y)\,dy \\
      &=& I_1(t) + I_2(t) + I_3(t)\,.
\eea
It is easy to see that
$$
      I_1(t) \leq c(\la) e^{-\la t}\,,
      \qquad
      \lim_{t\to +\infty} \frac{I_2(t)}{\eta(t)} = \lim_{t\to +\infty}
\frac{I_3(t)}{\eta(t)}
      = \frac{1}{2\la^2}\,.
$$
Therefore \rife{eq:radial1} follows easily. \qed


\begin{lemma} \label{lemma:radial2}
The function $v$ satisfies
\be
      \norma{v}{L^2_\eta(\R)}^2 \leq c_1 \norma{G}{L^1(\R)}
\norma{h}{L^2_\eta(\R)}^2\,,
\label{eq:radial2}
\ee
where $c_1$ is the constant which appears in Lemma \ref{lemma:radial1}.
Moreover $v$ satisfies
\be
      |v'(t)| \leq \la v(t)\qquad
      \mbox{for every $t\in \R$.}
\label{eq:radial3}
\ee
\end{lemma}

\dimo
By H\"older's inequality one has
$$
      v(t) = \int_\R G(t-y) h(y)\, dy
      \leq \norma{G}{L^1(\R)}^{1/2} \left(\int_\R G(t-y) h^2(y)\,
dy\right)^{1/2}\,.
$$
Therefore, by Tonelli's theorem,
\bea
      \norma{v}{L^2_\eta(\R)}^2 &=& \int_\R v^2(t) \eta(t)\, dt
      \leq \norma{G}{L^1(\R)} \int_\R \left(\int_\R G(t-y) h^2(y)\,
dy\right)\eta(t)\,dt \\
      &=& \norma{G}{L^1(\R)} \int_\R (G*\eta)(y) h^2(y)\,dy\,.
\eea
By Lemma \ref{lemma:radial1}, inequality \rife{eq:radial2} follows
easily. Inequality
\rife{eq:radial3} follows from the similar property of
$G$.
\qed


Let us now define $w(x)=w_{\la,k}(x)=k v_\la(|x|):\R^N\to\R$. By
\rife{eq:radial2} and
\rife{eq:radial3},
$w\in H^1(\R^N)$. We want to show that, with an appropriate choice of
$k$ and $\la$, this
function satisfies
$$
      -\Delta w + b(x,\D w) + c(x,w) \geq g(x)
$$
in $\Om\setminus B_{R_0}$. Indeed this is true if
\be
      -v''(t)-{{N-1}\over t}v'(t)+\nu_0 v(t)- d|v'(t)| \geq \frac{h(t)}{k}\qquad
      \mbox{for every $t\geq R_0$.}
\label{eq:radial4}
\ee
Using \rife{eq:eqv} and \rife{eq:radial3}, it is easy to check that
\rife{eq:radial4} holds if
$k\geq 1$,  and $\lambda>0$ is such that
$$
      \la^2 +\left(\frac{N-1}{R_0}+ d\right)\la-\nu_0\leq 0\,.
$$
If we moreover take $k$ such that $k v(R_0) \geq D$ ($D$ is
the $L^\infty$-bound for $u_n$, see Proposition \ref{prop:Linfty}),
we can conclude the proof of the
existence as in Sections 5 and 6.
}




\section{Solutions in $W^{1,p}_{\rm loc}(\Om)$}
In this Section we will drop the request for a
solution to belong to
$W^{1,p}_0(\Om)$ and we will show that it is possible, under much weaker
assumptions, to prove the existence of a
distributional solution of following problem
\be
	 \left\{ \arrstre\begin{array}{ll}
	    A(u) := -\dive(a(x,\D u)) + b(x,\D u) + c(x,u) = g(x) &
\mbox{in $\Omega$,} \\
		 u=0\quad {\rm on}\quad \partial\Om.
          &  \end{array}\right.
\label{ploc}
\ee


\noindent In fact we will prove that there
exists $\de>1$ such that
\be
      |u|^\de \in W^{1,p}_{0}(\Om)\,,
\label{eq.y0}
\ee
which gives a meaning to the boundary condition.

Since in this case we do not need to use a comparison principle, or
to find supersolutions, we may
assume $p>1$, and drop several assumptions on the operator (in
particular (A1), (B2),
(C2),
(G2), while the strong monotonicity assumption (A3) can be relaxed).
Similarly, we do not need
to obtain $L^\infty$ estimates, and this leads to still weaker
hypotheses on the datum $g$.

More precisely, we will assume that the operator $A$ defined in \rife{ploc}
satisfies $p>1$, (A2), (A4), (B1), (C1), (C3),

\begin{itemize}
\item[(A3$'$)]  for almost every $x\in\Omega$ and for every
$\xi,\eta\in\R^N$, with $\xi \not =\eta$,
$$
[a(x,\xi)-a(x,\eta)]\cdot(\xi-\eta) >0\,.
$$
\end{itemize}


\begin{theo} \label{th.4}
Let $p>1$, and assume that $A$ satisfies {\rm (A2), (A3$'$),
(A4),  (B1), (C1),
(C3)}, and that
\be
      g(x) \in L^q(\Om)\,, \qquad\mbox{with } q>  \dys
      \cases{\dys \frac{N(\bar\al+p)}{N(p-1) + p(\bar\al+1)} &
             if $p<N$, \cr
              & \cr
             1 &
             if $p\geq N$,}
\label{eq.y0'}
\ee
where $\bar\al$ satisfies
\be
      \left(\frac{d}{p}\right)^p \left(\frac{p-1}{\nu (\bar\al +
1)}\right)^{p-1} = \nu_0\,.
\label{eq.y1}
\ee
Then there exists a solution $u$ of
{\rm\rife{ploc}} in the sense of distributions, 
i.e. $u\in W^{1,p}_{\rm loc}(\Om)$ and
{\rm\rife{eq.y0}} holds for a suitable $\de =
\de(N,p,\nu,\nu_0,d,q)>1$.
\end{theo}

\begin{rem}\label{rem.y1}\rm
Let us point out that, with minor modifications to the proof, one can
also allow the function $a(x,\D
u(x))$ which appears in the principal part of the operator to depend
also on the value $u(x)$, with the usual
growth conditions with respect to $u$.
\end{rem}

\noindent{\bf Proof of Theorem \ref{th.4}. }
Again we approach problem \rm\rife{ploc} by approximate problems $({\rm
P}\!_n)$ as in Section 5. In order to
obtain an {\it a priori\/} estimate for the solutions, we follow
similar calculations to those used in
Step 2 of the Proof of Proposition \ref{prop:Linfty}. More precisely,
if we take $|u_n|^\al u_n$ as
test function in $({\rm P}\!_n)$, with $\al$ such that
\be
      \al > \bar \al\,,
\label{eq.y2}
\ee
and using \rife{eq.y2} to cancel the term involving $b(x,\D u_n)$, we
obtain easily (for
brevity of notation we omit the index $n$ everywhere)
\be
      \int{} \Bigl| \D ( |u|^\frac{\al+p}{p} )\Bigr|^p
      +
      \int |u|^{\al + p}
      \leq
      c_1
      \norma{g}{q}
      \left[\int |u|^{q'(\al+1)}\right]^{\frac{1}{q'}}\,,
\label{eq.y2'}
\ee
where $c_1=c_1(\al,\nu,\nu_0,p,d)$. In order to use
Gagliardo-Nirenberg's embedding, we will show
that, under our assumptions on $q$, there exists $\al >\bar\al$ such that
\be
      \al +p
      \leq
      q' (\al+1)
      \leq
      \frac{p^* (\al + p)}{p}
      \qquad
      \mbox{if $p<N$}
\label{eq.y3}
\ee
or
\be
      \al + p
      \leq
      q' (\al+1) \qquad
      \mbox{if $p\geq N$.}
\label{eq.y4}
\ee
Assume for a moment that \rife{eq.y3}, \rife{eq.y4} hold. Then, using
Lemma \ref{GN}, and proceeding
as in the proof of Proposition \ref{prop:Linfty}, one easily obtains
a bound for the left-hand side of
\rife{eq.y2'}. Therefore we have obtained the {\em a priori\/} estimate
\be
      \norma{ |u|^\de}{W^{1,p}_0(\Om)} \leq
c_2\left(N,p,\al,\nu,\nu_0,d,\norma{g}{q}\right)\,,
      \qquad\mbox{with $\de = \dys\frac{\al+p}{p} >1$,}
\label{eq.y6}
\ee
where $\al$ is any number such that \rife{eq.y2}, \rife{eq.y3} and
\rife{eq.y4} hold.
   From \rife{eq.y6} it follows immediately that $u$ is bounded in
$L^p_{\rm loc} (\Om)$. This implies
that $u$ is also bounded in $W^{1,p}_{\rm loc} (\Om)$. Indeed for
$\Om_0\subset\subset \Om$ we can
take $u\vp$ as test function in the equation satisfied by $u$ (where
$\vp$ is a cut-off function such
that $\vp \equiv 1$ on $\Om_0$), to obtain easily an estimate for
$\int_{\Om_0} |\D u|^p$.

Let's go back to the approximating problems $({\rm P}\!_n)$. By the estimates
just proved, one can extract a
subsequence, still denoted by $\{u_n\}$, such that
\be
      \cases{ u_n \weak u
      & weakly in $W^{1,p}_{\rm loc} (\Om)$, \cr\cr
      u_n \to u &
      almost everywhere in $\Om$, \cr\cr
      |u_n|^\de \weak |u|^\de &
      weakly in $W^{1,p}_{0} (\Om)$, \cr\cr
      u_n \weak u &
      weakly in $L^{\de p^*}(\Om)$,
      }
\label{eq.y7}
\ee
where $\de=(\al +p)/p$. In order to pass to the limit in the
distributional formulation of
problem $({\rm P}\!_n)$, the only
difficulty is to prove the strong convergence of the gradients $\D
u_n$ in $L^p_{\rm loc}(\Om)$. By
J.L. Lions' Lemma and (A3$'$) (see \cite{JLLions}), it is enough to 
show that, for
every $\vp \in C^\infty_0(\Om)$,
$$
      \lim_{n\to +\infty} \int \left[a(x,\D u_n)- a(x,\D
u)\right]\cdot(\D u_n - \D u) \vp = 0\,,
$$
and this can be shown as in the proof of Theorem \ref{main theorem},
taking into account \rife{eq.y0'} and
\rife{eq.y7}.

We only have to show that it is possible to choose $\al>\bar\al$ such
that \rife{eq.y3} and
\rife{eq.y4} are satisfied. If $p<N$, condition \rife{eq.y3} is the same as
\be
      y_1(\al) = \frac{\al+p}{\al+1} < q' < \frac{p^*(\al+p)}{p(\al+1)}
= y_2(\al)\,.
\label{eq.y5}
\ee
By studying the behaviour of the functions $y_1(\al)$ and $y_2(\al)$, for
$\al>\bar\al$, it is easy to
see that condition \rife{eq.y5} can be fulfilled for every $q$
satisfying \rife{eq.y0'}.
If $p\geq N$, only the function $y_1(\al)$ is involved, and since
$\lim_{\al\to+\infty}y_1(\al)=1$, we
only need $q'>1$, which is equivalent to $q <+\infty$. \qed



\begin{thebibliography}{99}
\bibitem{A-T}  A. Alvino, G. Trombetti:
Equazioni ellittiche con termini di ordine inferiore e riordinamenti,
{\it Rend. Acc. Naz. Lincei,} {\bf 66} (1979), 1--5.

\bibitem{B-M}  G. Bottaro, M.E. Marina:
Problema di Dirichlet per equazioni ellittiche di tipo variazionale
su insiemi non limitati,
{\it Boll. U.M.I.} {\bf (4)} (1973), no. 8, 46--56.

\bibitem{C-V}  M. Chicco, M. Venturino:
Dirichlet problem for a divergence form elliptic equation with unbounded
coefficients in an unbounded domain,
{\it Ann. Mat. Pura Appl. {\rm(4)}} {\bf (178)} (2000),  325--338.

\bibitem{DGP} A. Dall'Aglio, D. Giachetti, J.-P. Puel: Nonlinear
elliptic equations with natural
growth on general domains, to appear on {\it Ann. Mat. Pura Appl.}

\bibitem{DV-P}  T. Del Vecchio, M.M. Porzio:
Existence results for a class of non-coercive Dirichlet problems,
{\it Ricerche Mat.} {\bf 44} (1995), no. 2, 421--438.

\bibitem{D-G}  P. Donato, D. Giachetti:
Quasilinear elliptic equations with quadratic growth on unbounded domains,
{\it Nonlinear Anal.} {\bf 10} (1986),  791--804.

\bibitem{G-T} D. Gilbarg, N.S. Trudinger: {\it Elliptic Partial
Differential
Equations of Second Order, 2nd edition}, Springer-Verlag (1983).

\bibitem{G} E. Giusti: {\it Metodi Diretti del Calcolo delle Variazioni},
    Unione Matematica Italiana, Bologna, (1994).

\bibitem{LL} J. Leray, J.L. Lions: Quelques r\'esultats de Vi\v sik sur les
probl\`emes elliptiques non lin\'eaires par les m\'ethodes de
Minty-Browder, {\sl
Bull. Soc. Math. France}, {\bf 93} (1965), 97--107.

\bibitem{JLLions} J.-L. Lions: {\it Quelques m\'ethodes de
r\'esolution des probl\`emes aux limites non
lin\'eaires}, Dunod, Gauthier-Villars, Paris, 1969.


\bibitem{L}  P. L. Lions:
A remark on some elliptic second-order problem,
{\it Boll. U.M.I.} {\bf (17)} (1980), no. 2, 267--270.

\bibitem{L1}  P. L. Lions:
Remarques sur les \'equations lin\'eaires elliptiques du
second ordre sous forme divergence dans les domaines non born\'es,
{\it Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8)}
{\bf (78)} (1985),
    no. 5, 205--212.

\bibitem{L2}  P. L. Lions:
Remarques sur les \'equations lin\'eaires elliptiques du
second ordre sous forme divergence dans les domaines non born\'es. II,
{\it Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8)}
{\bf (79)} (1985),
    no. 6, 178--183.


\bibitem{Ni} L. Nirenberg: On elliptic partial differential
equations, {\sl Ann.
Scuola Norm. Sup. Pisa\/}, {\bf 13} (1959), 116--162.

\bibitem{St} G. Stampacchia:
Equations elliptiques du second ordre \`a coefficients
discontinus. S\'eminaire de Math\'ematiques Sup\'erieures, No. 16,
Les Presses de
l'Universit\'e de Montr\'eal, Montr\'eal, Que. (1966).

\end{thebibliography}

Addresses of the authors:\par\medskip
A. Dall'Aglio (aglio@dmmm.uniroma1.it)\par
V. De Cicco   (decicco@dmmm.uniroma1.it) \par
D. Giachetti  (giachett@dmmm.uniroma1.it)\par
Dipartimento di Metodi e Modelli Matematici per le Scienze Applicate, \par
Universit\`a di Roma I ``La Sapienza'', Via Antonio Scarpa 16, \par
00161 Roma, Italy  \par
\bigskip
J.-P. Puel (jppuel@cmapx.polytechnique.fr)\par
Laboratoire De Math\'ematiques Appliqu\'ees,\par
Universit\'e De Versailles St. Quentin, 45 Avenue Des Etats Unis,\par
78035, Versailles Cedex, France\par


\end{document}


