\documentclass[twoside]{article}
\usepackage{amsfonts, amsmath} % used for R in Real numbers
\pagestyle{myheadings}
\markboth{Nonlinear equations with natural growth terms and measure data }
{ Alessio Porretta }
\begin{document}
\setcounter{page}{183}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
2002-Fez conference on Partial Differential Equations,\newline
Electronic Journal of Differential Equations,
Conference 09, 2002, pp 183--202. \newline
http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.swt.edu (login: ftp)}
\vspace{\bigskipamount} \\
%
Nonlinear equations with natural growth terms and measure data
%
\thanks{ {\em Mathematics Subject Classifications:} 35J60, 35J65, 35R05.
\hfil\break\indent
{\em Key words:} Nonlinear elliptic equations, natural growth terms, measure data.
\hfil\break\indent
\copyright 2002 Southwest Texas State University. \hfil\break\indent
Published December 28, 2002. } }
\date{}
\author{Alessio Porretta}
\maketitle
\begin{abstract}
We consider a class of nonlinear elliptic equations containing
a $p$-Laplacian type operator, lower order terms having natural
growth with respect to the gradient, and bounded measures as data.
The model example is the equation
$$ -\Delta_p(u) + g(u)|\nabla u|^p=\mu
$$
in a bounded set $\Omega\subset \mathbb{R}^N$, coupled with a
Dirichlet boundary condition. We provide a review of the
results recently obtained in the absorption case (when $g(s)s\geq0$)
and prove a new existence result without any sign condition on $g$,
assuming only that $g\in L^1({\bf R})$. This latter assumption is
proved to be optimal for existence of solutions for any measure $\mu$.
\end{abstract}
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{example}[theorem]{Example}
\newtheorem{remark}[theorem]{Remark}
\section{Introduction}
In this work we focus our attention on nonlinear Dirichlet problems whose
model is
\begin{equation}\label{un}
\begin{gathered}
-\Delta_p(u) + g(u)|\nabla u|^p=\mu \quad\mbox{in }\Omega,\\
u=0 \quad\mbox{on }\partial\Omega,
\end{gathered}
\end{equation}
where $p>1$, $g:\mathbb{R}\to \mathbb{R}$ is a continuous function,
and $\mu$ is a bounded Radon measure on $\Omega$ which is a bounded subset
of $\mathbb{R}^N$.
Recently, many researchers have investigated the possibility to find solutions
of \eqref{un} under the assumption that $g(s)s\geq 0$, in which case the term
$g(u)|\nabla u|^p$ is said to be an absorption term. In this case
a detailed picture of what happens is now available, according to the growth
at infinity of
$g(s)$ and to whether the measure $\mu$ charges or not sets
of zero $p$-capacity (the capacity defined in $W^{1,p}_0(\Omega)$).
In the next section, we try to give a quick review of these
results and explain the main features of the problem in the absorption case,
both for elliptic and for parabolic equations.
No results for general measures $\mu$ are known to our knowledge if the sign
condition is not assumed to hold, possibly including the reaction case
in which $g(s)s\leq 0$. It is the purpose of the third section
of this paper to give new results in this situation. Eventually, these new results seem to fit
perfectly those proved in the absorption case, and we will prove (stated in more
generality in Section 3) the following theorem, which extends that proved in \cite{Se} (under the same assumptions) for data
$\mu\in L^1(\Omega)$.
\begin{theorem}\label{tun}
Let $\mu$ be a nonnegative bounded Radon measure on $\Omega$. Assume that
$g\in L^1(\mathbb{R})$. Then there exists a distributional solution $u$
of \eqref{un}.
\end{theorem}
Next we will give an example which somehow expresses that the assumption
$g\in L^1(\mathbb{R})$ in Theorem \ref{tun} is optimal; if $\mu$ is
the Dirac mass, we prove that no solution can be obtained by approximation. In particular,
in the reaction case ($g(s)s\leq 0$),
if $\mu$ is approximated by a sequence of smooth functions, the sequence
of approximating solutions converges to a solution of \eqref{un} if $g\in L^1(\mathbb{R})$,
while it blows up everywhere in
$\Omega$ if $g\not\in L^1(\mathbb{R})$. We recall that in \cite{MuPo} the absorption case
$g(s)s\geq 0$ had already been studied; in that situation if the Dirac mass is approximated by
smooth functions, the approximated solutions still converge to a solution of the problem if
$g\in L^1(\mathbb{R})$, while they converge to zero if $g\not \in L^1(\mathbb{R})$. Thus, even if for different reasons,
in both cases the assumption $g\in L^1(\mathbb{R})$ turns out to be optimal.
\section{The absorption case: a quick review}
A wide literature has dealt with elliptic and parabolic equations with measure
data in the last decades. In particular, the techniques of a priori estimates and
compactness of approximating solutions, firstly introduced in \cite{BG1}, have been
proved to work well enough for pseudomonotone operators of Leray-Lions type (\cite{LL}),
providing several existence results in case of
$L^1$ data. The presence of absorbing lower order terms (i.e.
satisfying a sign condition) often brings in this kind of
problems new features; for instance, as in
\cite{BGV}, \cite{BG2}, lower order terms may have a regularizing effect on
solutions of problems with $L^1$ data. The two main examples are the following
problems:
\begin{equation}\label{semil}
\begin{gathered}
-\Delta_p u+|u|^{r-1}u=\mu \quad\mbox{in }\Omega,\\
u=0 \quad\mbox{on }\partial\Omega\,,
\end{gathered}
\end{equation}
and
\begin{equation}\label{ng}
\begin{gathered}
-\Delta_p u+u|\nabla u|^p=\mu \quad\mbox{in }\Omega,\\
u=0\quad\mbox{on }\partial\Omega\,.
\end{gathered}
\end{equation}
If $\mu\in L^1(\Omega)$, problem \eqref{semil} has a solution in $W^{1,q}_0(\Omega)$
for any $q< {\frac{pr}{r+1}}$, while problem \eqref{ng} has a finite energy solution $u$, which
belongs to $W^{1,p}_0(\Omega)$. In general, if the lower order term is absorbing, one can prove the
existence of a solution with $L^1(\Omega)$ data; for instance, the
problem:
\begin{equation}\label{mp}
\begin{gathered}
-\Delta_p(u)+H(x,u,\nabla u)=f \quad\mbox{in }\Omega,\\
u=0 \quad\mbox{on }\partial\Omega\,,
\end{gathered}
\end{equation}
with $\xi\mapsto H(x,s,\xi)$ growing at most like $|\xi|^p$ (the
so-called natural growth), always admits a solution if $f\in L^1(\Omega)$ (see
\cite{Po2}). In fact, dealing with the limit growth for $H(x,s,\xi)$ is not
that easy and requires the strong compactness of truncations in
the energy space; on the other hand, these truncation methods can be
adapted to several different contexts if still dealing with $L^1(\Omega)$
data, as obstacle problems or more general operators (see \cite{BeE}, \cite{BeEM}, \cite{EM}).
When trying to extend the previous results to measure data, it
turns out that precisely the regularizing effect mentioned above may be
responsible for nonexistence of solutions. Actually, this fact was first observed
in the pioneering works of H. Brezis (\cite{B,B1}) and in a whole series of
papers (see \cite{BaPi,BV,GM,Ve,VV,VV1} and the references therein) concerning problem
\eqref{semil} in the linear case $p=2$.
More recently, the nonlinear case $p\neq 2$ has been dealt with in
\cite{OP1,OP2,Bi-Ve}. Summing up these results, it is proved that
problem \eqref{semil} has a solution
for every given bounded measure $\mu$ only if $r<\frac{N(p-1)}{N-p}$, while if
$r\geq \frac{N(p-1)}{N-p}$ then no solution exists if $\mu$ charges sets of zero
$q$-capacity with $\frac{q(p-1)}{q-p}1$,
then any compact set of zero capacity (the standard Newtonnian capacity) is
removable. In \cite{BGO2}, \cite{Po0}, \cite{MuPo}, the behaviour of sequences of
approximating solutions was studied if $\mu$ is approximated in the narrow topology,
say by a standard convolution. It is proved that if
$g\in L^1(\mathbb{R})$, then there exists a solution $u$ of \eqref{quad} for any measure
$\mu$, while if $g\not\in L^1(\mathbb{R})$ approximating solutions converge to a solution
$u$ of the same problem but with datum $\mu_0$, the absolutely continuous part of
$\mu$ with respect to $p$-capacity. Here $p$-capacity denotes the capacity defined in $W^{1,p}_0(\Omega)$ and we recall
(see \cite{FST}) that any measure $\mu$ admits a unique decomposition as $\mu=\mu_0+\lambda$, where $\lambda$ is concentrated on a set
of zero $p$-capacity and $\mu_0(E)=0$ for any set $E$ of zero $p$-capacity. In
other words, in the approximation method, one looses the singular part of the measure which is concentrated on sets of zero
$p$-capacity; if $\mu$ does not charge sets of zero $p$-capacity then existence is proved for any function $g(s)$.
Removability properties in the stability approach are investigated in
\cite{OPo}, where the approximating equations of \eqref{quad} are considered
with data $f_n$ only converging to a function
$f$ in $L^1_{\rm loc}(\Omega\setminus K)$, where $K$ has $p$-capacity zero. It is proved
that, setting $ G(s)=\int_0^s g(t)dt$, if, roughly speaking,
$\exp(-G(s)/(p-1))\in L^1(\mathbb{R})$ then $u_n$ still converges to a solution with datum
$f$; thus, whatever singular perturbations, provided they are localized on sets of zero $p$-capacity, are
not seen by the equation. This result somehow includes the removable
singularity point of view, and extends the result in
\cite{BN} since the assumption that $\exp(-G(s)/(p-1))\in L^1(\mathbb{R})$, in the case $p=2$, is weaker than
assuming that $sg(s)\geq \frac\gamma{s^2}$ with $\gamma>1$.
This kind of phenomena due to absorption terms has been investigated for
parabolic equations as well. As it happens
for the stationary case, the semilinear evolution problem
\begin{equation}\label{psem}
\begin{gathered}
u_t-\Delta u+|u|^{r-1}u=\mu \quad\mbox{in }Q:=\Omega\times (0,T)\\
u=0 \quad \mbox{on } \Sigma:=\partial \Omega\times (0,T)\\
u(0)=u_0 \quad\mbox{in }\Omega,
\end{gathered}
\end{equation}
does not always have a solution for any
measure $\mu$ on $Q$ and any measure initial datum $u_0$. In
\cite{BF}, the authors study the problem with $\mu=0$ concentrating the
attention on the initial measure $u_0$. They point out that, ir $r$ is large enough,
nonexistence phenomena may occurr, and can appear as initial layer phenomena. In
fact, a singular measure as initial condition may be lost
while approximating the problem with smooth approximating problems. Subsequently,
in
\cite{BaPi1}, necessary and sufficient conditions are given on the measures $\mu$
and $u_0$ in order to have a solution of \eqref{psem}; as expected, these conditions
involve some notions of space-time dependent capacity. Further results on nonlinear analogue of \eqref{psem}
are proved in \cite{And}, \cite{MV}, \cite{BCV} (see also the references in these papers).
In view of the results mentioned above for elliptic equations, recent study has
been devoted to evolution problems as the following:
\begin{equation}\label{parquad}
\begin{gathered}
u_t-\Delta_p u+g(u)|\nabla u|^p=0 \quad\mbox{in }Q:=\Omega\times (0,T)\\
u=0 \quad \mbox{on } \Sigma:=\partial \Omega\times (0,T)\\
u(0)=u_0 \quad\mbox{in }\Omega,
\end{gathered}
\end{equation}
in case $u_0$ is a bounded measure. The existence of a
solution in case
$u_0\in L^1(\Omega)$ is proved in \cite{Po1}. The
possibility to extend this result to a general measure initial datum is studied in
\cite{BlPo}. Again, under the assumption that
$g\in L^1(\mathbb{R})$, it is proved the existence of a solution for any measure
$u_0$. On the other hand, if $g\not\in L^1(\mathbb{R})$, then initial layer phenomena
occur; in particular, if $u_{0n}$ is a convolution approximation of the measure
$u_0$, the sequence of approximating solutions
$u_n$ of the same problem, with initial datum $u_{0n}$, converges to a solution
$u$ of the problem having, as initial value, the absolutely continuous part of
$u_0$ with respect to Lebesgue measure. Sharp removable singularity type results,
which in a stronger way express the nonexistence of solution, still depend on the
growth at infinity of $g(s)$ and are obtained in \cite{Po3}.
Eventually, one obtains for the evolution problem \eqref{parquad} the same type
of results obtained for the elliptic problem \eqref{quad} replacing the role of
$\mu$ with $u_0$ and the $p$-capacity (capacity in $W^{1,p}_0(\Omega)$) with the Lebesgue
measure in
$\Omega$. Are then these results consistent? The answer has to be found in the
study of the notion of capacity for parabolic equations. A functional type
presentation and construction of the parabolic $p$-capacity (capacity defined in
the space $W=\{u\in L^p(0,T;W^{1,p}_0(\Omega))\,, u_t\in L^{p'}(0,T;W^{-1,p'}(\Omega))\}$) is given in \cite{Pi}
for $p=2$ and in \cite{DPP} for $p\neq 2$. In this last paper, it is proved that
given
$B\subset \Omega$, the set
$\{t=0\}\times B$ has zero parabolic capacity in $(0,T)\times \Omega$ if and only
if $B$ has zero Lebesgue measure. Thus, if one looks at singularities at
initial time as singularities on $Q$ concentrated at
$t=0$, the results obtained on \eqref{parquad} reflect perfectly those on
\eqref{quad}. Moreover, it becomes clear that in order to deal with problem
\eqref{parquad} with interior space-time dependent measures as data, one has to
follow the outlines of the stationary case and use a decomposition theorem for
measures with respect to parabolic $p$-capacity. This latter result, which extends the
stationary one given in \cite{BGO1}, is proved in \cite{DPP} and states that
any measure $\mu$ on $(0,T)\times \Omega$ which does not charge sets of zero
parabolic $p$-capacity admits the decomposition (as a distribution)
$$
\mu=f+g_1-(g_2)_t
$$
with $f\in L^1(Q)$, $g_1\in L^{p'}(0,T;W^{-1,p'}(\Omega))$ and $g_2\in L^p(0,T;W^{1,p}_0(\Omega))$.
Finally, let us mention that, in the linear case ($p=2$),
other existence and nonexistence results with gradient dependent
lower order terms (absorbing or repulsive) and measure data are obtained in
\cite{A,APi,BeLa,BSW} (see also the references cited therein).
We point out that the techniques used in these papers are mainly based on
a linear operator and on the concept of
distributional solution (with two integration by parts), or on semigroup theory
and the concept of integral solution.
These approaches allow to have sharper nonexistence results especially for the case
of subcritical growth, on the other hand their study is mostly restricted to the case $g(u)\equiv 1$.
\subsection{Natural growth reaction terms and measure data}
As explained in the previous section, if the term $H(x,u,\nabla u)$ is an
absorption term and has natural growth, the borderline case which allows to have
solutions of \eqref{mpmis} for all measures $\mu$ is the case in which
$$
|H(x,u,\nabla u)|\leq g(u)|\nabla u|^p\,,\quad\mbox{with $g\in L^1(\mathbb{R})$.}
$$
Our aim is now to show that, somehow surprisingly, the same assumption is
necessary and sufficient to have solutions for any measure even in the reaction
case, that is without assuming any sign condition on $H(x,s,\xi)$. In particular,
if we aim to have solutions of \eqref{quad} for {\emph {any}} given measure data, there is no
difference between the reaction and the absorption case.
Heuristically, this feature can be easily explained. In fact, the
model equation
\begin{equation}\label{mc}
-\Delta u=g(u)|\nabla u|^2 +\mu\,,
\end{equation}
can be transformed, through a change of unknown, into the equation
\begin{equation}\label{eqmp}
-\Delta v=\exp(G(u)) \mu\,,
\end{equation}
with $v=\int_0^{u}\exp(G(s))ds$ and
$G(s)=\int_0^s g(r)dr$.
In \cite{MuPo} we proved that equation \eqref{eqmp} has a solution if $\exp(G(u))$
has a finite limit at infinity, which is the case whenever $g\in L^1(\mathbb{R})$, so that
in this case \eqref{mc} is also expected to have a solution. On the other hand, if
$g\not\in L^1(\mathbb{R})$, then the right hand side of \eqref{eqmp} can be hardly handled
since $\exp(G(u))$ is not bounded. We are going to provide an example where
$\mu$ is the Dirac mass and no solution of
\eqref{eqmp} can be found by approximation, precisely proving that approximated
solutions of \eqref{mc} in this case blow up completely (i.e. at every point of
$\Omega$).
We will prove our result in a more general situation. Assume that $a(x,s,\xi)$ and $H(x,s,\xi)$ are Carath\'eodory
functions satisfying, for almost every $x\in \Omega$, for every $s\in \mathbb{R}$, $\xi$, $\eta\in \mathbb{R}^N$
($\xi\neq \eta$):
\begin{gather}
\label{coerc}
a(x,s,\xi) \cdot \xi \geq \alpha |\xi |^p\,,\quad
\alpha>0\,,\; p>1\,,\\
\label{gro}
|a(x,s,\xi) |\leq \beta (k(x)+|s|^{p-1}+|\xi|^{p-1})\quad k(x)\in L^{p'}(\Omega),
\;\beta>0\,, \\
\label{mon}
(a(x,s,\xi)-a(x,s,\eta))\cdot (\xi-\eta)>0\,,
\end{gather}
and
\begin{equation}\label{creh}
\begin{gathered}
|H(x,s,\xi)|\leq \gamma(x)+g(s)|\xi|^p\,,\;\gamma(x)\in L^1(\Omega)\\
\mbox{and }\quad g\,:{\bf R}\to {\bf R}^+\quad \mbox{continuous}\,,\quad g\geq 0\,,\quad g\in
L^1(\mathbb{R})\,.
\end{gathered}
\end{equation}
In the following we denote by $\mathop{\rm cap}_p(B)$ the $p$-capacity of a borelian set
$B\subset\Omega$, where the $p$-capacity is the standard notion of capacity
defined in the Sobolev space $W^{1,p}_0(\Omega)$.
Let us recall (see \cite{FST}) that any bounded Radon measure $\mu$ has a unique
decomposition as
\begin{equation}\label{spl}
\mu=\mu_0+\lambda\,,
\end{equation}
where $\mu_0$, $\lambda$ are bounded measures such that $\mu_0$ does not charge sets of
zero $p$-capacity (i.e. $\mu_0(B)=0$ for every $B$ with $\mathop{\rm cap}_p(B)=0$) and $\lambda$ is
concentrated on a set $E\subset
\Omega$ such that $\mathop{\rm cap}_p(E)=0$. Moreover, if $\mu$ is nonnegative, then both $\mu_0$
and $\lambda$ are nonnegative. For a presentation of the basic notions concerning
measures and capacity the reader may refer to \cite{HKM}, \cite{DMOP}. We also have,
from \cite{BGO1}, that $\mu_0$ furtherly admits a decomposition (in distributional
sense) as
\begin{equation}\label{spli}
\mu_0=f-\mathop{\rm div}(F)\,,\quad \mbox{$f\in L^1(\Omega)$, $F\in L^{p'}(\Omega)^N$.}
\end{equation}
Hereafter, let $\mu$ be a bounded nonnegative Radon measure on $\Omega$.
Referring to the previous decomposition of $\mu$ and $\mu_0$ in \eqref{spl}, \eqref{spli},
there exists a sequence $\mu_n$ of bounded functions such that
\begin{equation}\label{mun}
\begin{gathered}
\mu_n=\mu_{0n}+\lambda_n\,,\quad \mbox{$\mu_{0n}\geq 0$, $\lambda_n\geq 0$,}\\
\mu_{0n}=f_n-\mathop{\rm div}(F_n)\,,\quad f_n\in L^\infty(\Omega)\,,\;
F_n\in L^\infty(\Omega)^N\,,\\
f_n\to f\quad \mbox{strongly in $L^1(\Omega)$,}\\
F_n \to F\quad \mbox{strongly in $L^{p'}(\Omega)^N$,}\\
\int_\Omega\varphi \lambda_n dx\to \int_\Omega \varphi d\lambda\quad \forall \varphi\in C_b(\Omega)\,,
\end{gathered}
\end{equation}
where $C_b(\Omega)$ denotes the space of bounded continuous functions in $\Omega$.
Such a sequence $\mu_n$ can be constructed using convolution and a suitable
compactly supported approximation of $\mu$.
For fixed $n\in \mathbb{N}$, since $\mu_n\in L^\infty(\Omega)$, under the previous
assumptions it is proved in \cite{BST} that there exists a weak solution $u_n\in
W^{1,p}_0(\Omega)\cap L^\infty(\Omega)$
of the
problem:
\begin{equation}\label{appell}
\begin{gathered}
-\mathop{\rm div}(a(x,u_n,\nabla u_n))=H(x, u_n,\nabla u_n)+\mu_n \quad
\mbox{in } \Omega\,,\\
u_n=0 \quad\mbox{on } \partial \Omega \,.
\end{gathered}
\end{equation}
Our main result is the following.
\begin{theorem}\label{primo}
Let $a(x,s,\xi)$ and $H(x,s,\xi)$ satisfy assumptions \eqref{coerc}--\eqref{creh}.
Let $\mu$ be a nonnegative bounded Radon measure on $\Omega$. Then there exists a
solution $u$ of the problem
\begin{equation}\label{reac}
\begin{gathered}
-\mathop{\rm div}(a(x,u,\nabla u))=H(x, u,\nabla u)+\mu \quad\mbox{in } \Omega \,,\\
u=0 \quad\mbox{on }\partial \Omega\,.
\end{gathered}
\end{equation}
\end{theorem}
\paragraph{Proof.}
We essentially follow the method used in \cite{Se}, which consists in
multiplying the equation \eqref{appell} by $\exp(G(u_n))$ or by
$\exp(-G(u_n))$,
where
$G(s)=\int_0^s g(t)/\alpha dt$ (the function $g$ appears in \eqref{creh}). In
other words this replaces the idea of the change of unknown which transforms
the model problem
\eqref{mc} into \eqref{eqmp}. After this multiplication, we will apply the
techniques fully developed in \cite{Po0}, \cite{MuPo} to obtain the strong
convergence of truncations.
In the following, we omit for shortness the dependence on $x$ in the integrals,
and we denote by $c$ any positive constant
independent on $n$. Let $\varphi\in W^{1,p}_0(\Omega)\cap L^\infty(\Omega)$; choosing
$\exp(G(u_n)) \varphi$ as test function in
\eqref{appell} we have
\begin{align*}
\int_\Omega &\exp(G(u_n))a(u_n,\nabla u_n)\nabla \varphi+
\int_\Omega \frac{g(u_n)}{\alpha}\exp(G(u_n))a(u_n,\nabla u_n)\nabla u_n \varphi\\
&= \int_\Omega H(u_n,\nabla u_n) \exp(G(u_n)) \varphi + \int_\Omega \varphi\exp(G(u_n)) \mu_n\,.
\end{align*}
For any $\varphi\geq 0$, thanks to \eqref{coerc} and \eqref{creh} we obtain
\begin{equation}\label{poseq}
\begin{gathered}
\int_\Omega \exp(G(u_n))a(u_n,\nabla u_n)\nabla \varphi\leq \int_\Omega
\gamma(x) \varphi\exp(G(u_n))+ \int_\Omega
\varphi\exp(G(u_n)) \mu_n \\
\forall \varphi\in W^{1,p}_0(\Omega)\cap L^\infty(\Omega)\,,\varphi\geq 0\,.
\end{gathered}
\end{equation}
Similarly, taking $\exp(-G(u_n)) \varphi$ as test function in
\eqref{appell} we obtain
\begin{multline}\label{negeq}
\int_\Omega \exp(-G(u_n))a(u_n,\nabla u_n)\nabla \varphi
+ \int_\Omega \gamma(x) \varphi\exp(-G(u_n)) \\
\geq \int_\Omega \varphi\exp(-G(u_n)) \mu_n
\quad \forall \varphi\in W^{1,p}_0(\Omega)\cap
L^\infty(\Omega)\,,\varphi\geq 0\,.
\end{multline}
Let $\varphi=T_k(u_n)^+$ in \eqref{poseq} and
$\varphi=T_k(u_n)^-$ in \eqref{negeq}.
Also let $G(\pm \infty)=\frac1\alpha\int_0^{\pm\infty}g(s)ds$ which are well
defined since $g\in L^1(\mathbb{R})$. Since
$ \exp(G(-\infty))\leq \exp(G(s))\leq
\exp(G(+\infty))$ and $\exp(|G(\pm \infty)|)\leq \exp
(\|g\|_{L^1(\mathbb{R})}/\alpha)$, using
\eqref{coerc}, we obtain
\begin{equation}\label{stitr}
\|T_k(u_n)\|_{W^{1,p}_0(\Omega)}^p\leq
\frac1\alpha \exp\big(\frac{\|g\|_{L^1(\mathbb{R})}}\alpha\big) k(\|\gamma\|_{L^1(\Omega)}+
\|\mu_n\|_{L^1(\Omega)})\leq c k\,.
\end{equation}
Standard estimates (see \cite{BBGGPV}) imply that $u_n$ is bounded in
the Marcinkiewicz space $M^{\frac{N(p-1)}{N-p}}(\Omega)$ and $|\nabla u_n|$
is bounded in
the Marcinkiewicz space $M^{\frac{N(p-1)}{N-1}}(\Omega)$. In particular we have
from \eqref{gro} that $a(x,u_n,\nabla u_n)$ is bounded in $L^q(\Omega)^N$ for any
$q<\frac N{N-1}$. Furthermore, there exist a function $u$ and a subsequence such
that
\begin{align*}
& u_n\to u\quad \mbox{a.e. in $\Omega$,}\\
&T_k(u_n)\to T_k(u)\quad \mbox{weakly in $W^{1,p}_0(\Omega)$ and a.e. in $\Omega$ for any
$k>0$.}
\end{align*}
Let us take $\varphi=T_1(u_n-T_j(u_n))^-$ in \eqref{negeq}; we obtain
\begin{multline}\label{entry}
\int_{\{-(j+1)\leq u_n\leq -j\}}
a(u_n,\nabla u_n)\nabla u_n+
\int_\Omega \exp(-G(u_n)) T_1(u_n-T_j(u_n))^- \mu_n\\
\leq \gamma \int_\Omega \exp(-G(u_n))T_1(u_n-T_j(u_n))^- \,.
\end{multline}
The term with $\mu_n$ can be neglected since it is
nonnegative. In the right hand side we can pass to the limit in $n$ and in $j$ by
Lebesgue's theorem, using that $G$ is bounded; indeed, since
$$
\exp(-G(u))T_1(u -T_j(u ))^- \leq \exp\big(\frac{\|g\|_{L^1(\mathbb{R})}}\alpha\big)
\chi_{\{u <-j\}}
$$
we have
$$
\int_\Omega \exp(-G(u_n))T_1(u_n-T_j(u_n))^- \;
\mathop{\to}\limits^{n\to \infty} \;
\int_\Omega \exp(-G(u ))T_1(u -T_j(u ))^-\;
\mathop{\to}\limits^{j\to \infty} \; 0\,,
$$
so that we deduce from \eqref{entry}
\begin{equation}\label{dmop}
\lim_{j\to \infty}\limsup_{n\to \infty}\quad
\int_{\{-(j+1)\leq u_n\leq -j\}} a(u_n,\nabla u_n)\nabla u_n=0\,.
\end{equation}
We are going now to prove that the truncations strongly converge in $W^{1,p}_0(\Omega)$.
Following the idea introduced in \cite{DMOP}, this is done by using a
suitable sequence of cut-off functions. Indeed, let $\delta>0$;
since $\lambda$ is a regular measure concentrated on $E$ and since $E$ has zero
$p$-capacity, there exist a compact set
$K_\delta\subset E$ and a sequence $\{\psi_\delta\}$ of functions
in $C^{\infty}_c(\Omega)$ with the properties that
\begin{equation}\label{pside}
\begin{gathered}
\lambda(E\setminus K_\delta)<\delta\,,\quad
0\leq \psi_\delta\leq 1\,,\\
\mbox{$\psi_\delta\equiv 1$ on an open neighbourhood $A_\delta$ of $K_\delta$}\\
\psi_\delta\mathop{\to}\limits^{\delta\to 0}\; 0 \quad \mbox{strongly in $W^{1,p}_0(\Omega)$.}
\end{gathered}
\end{equation}
Take now
$\varphi=(k-T_k(u_n)) (1-|T_1(u_n-T_j(u_n)|) \psi_\delta$ in
\eqref{negeq}, with $j>k$. Observe that $\varphi=(k-u_n)\psi_\delta$ if $|u_n|k$.
Thus we get, using also that $\exp(-G(u_n))\leq
\exp\big(\frac{\|g\|_{L^1(\mathbb{R})}}\alpha\big)$ and $\psi_\delta\leq 1$,
\begin{align}\label{near}
& \int_\Omega \exp(-G(u_n))a(u_n,\nabla u_n)\nabla
\psi_\delta (k-T_k(u_n)) (1-|T_1(u_n-T_j(u_n)|) \nonumber\\
&+ 2k\exp\big(\frac{\|g\|_{L^1(\mathbb{R})}}\alpha\big)
\int_{\{-(j+1)\leq u_n\leq -j\}} \hskip -5mm
a(u_n,\nabla u_n)\nabla u_n +2k\exp\big(\frac{\|g\|_{L^1(\mathbb{R})}}
\alpha\big)\int_\Omega \gamma \psi_\delta \nonumber\\
&\geq \exp\big(-\frac{\|g\|_{L^1(\mathbb{R})}}\alpha\big)
\int_\Omega a(T_k(u_n),\nabla T_k(u_n))\nabla T_k(u_n) \psi_\delta \\
&\quad +\int_\Omega
(k-T_k(u_n))\exp(-G(u_n)) (1-|T_1(u_n-T_h(u_n)|) \psi_\delta \mu_n\,.
\nonumber
\end{align}
Since
$$
|a(u_n,\nabla u_n) (1-|T_1(u_n-T_j(u_n)|)|\leq
|a(T_{j+1}(u_n),\nabla T_{j+1}(u_n))|\,,
$$
and since last term is bounded in $L^{p'}(\Omega)$ and $G$ is bounded, we have
that there exists $\Lambda_j\in L^{p'}(\Omega)^N$ such that
$$
\exp(-G(u_n))a(u_n,\nabla u_n)(k-T_k(u_n)) (1-|T_1(u_n-T_j(u_n)|)\to
\Lambda_j
$$
weakly in $L^{p'}(\Omega)^N$.
Thus we get
\begin{multline*}
\lim_{n\to \infty}
\int_\Omega \exp(-G(u_n))a(u_n,\nabla u_n)\nabla
\psi_\delta (k-T_k(u_n)) (1-|T_1(u_n-T_j(u_n)|)\\
= \int_\Omega \Lambda_j\nabla \psi_\delta \,,
\end{multline*}
and then, as $\delta$ tends to zero, thanks to \eqref{pside} we have
$$
\lim_{\delta\to 0}\lim_{n\to \infty}
\int_\Omega \exp(-G(u_n))a(u_n,\nabla u_n)\nabla
\psi_\delta (k-T_k(u_n)) (1-|T_1(u_n-T_j(u_n)|)=0\,.
$$
The third integral in \eqref{near} easily goes to zero since $\psi_\delta$ converges to zero
and $\gamma\in L^1(\Omega)$.
Furthermore, the term with $\mu_n$ can again be neglected since it is
nonnegative. Therefore, passing to the limit first in
$n$, then in
$\delta$ we obtain from
\eqref{near}
\begin{align*}
& \lim_{\delta\to 0}\limsup_{n\to \infty}
\int_\Omega a(T_k(u_n),\nabla T_k(u_n))\nabla T_k(u_n) \psi_\delta\\
&\quad \leq \limsup_{n\to \infty}
2k\exp\big(\frac{2\|g\|_{L^1(\mathbb{R})}}\alpha\big)\int_{\{-(j+1)\leq u_n\leq
-j\}} a(u_n,\nabla u_n)\nabla u_n\,.
\end{align*}
Then, as $j$ goes to infinity, using \eqref{dmop} and since
$a(x,s,\xi)\cdot \xi\geq 0$, we get
\begin{equation}\label{half}
\lim_{\delta\to 0}\limsup_{n\to \infty}
\int_\Omega a(T_k(u_n),\nabla T_k(u_n))\nabla T_k(u_n) \psi_\delta = 0\,.
\end{equation}
Let now $w_n=T_{2k}(u_n-T_h(u_n)+T_k(u_n)-T_k(u))$, we take
$\varphi=w_n^+(1-\psi_\delta)$ in \eqref{poseq} and $\varphi=w_n^-(1-\psi_\delta)$ in
\eqref{negeq} to obtain
\begin{align*}
& \int_{\{w_n\geq 0\}} \exp(G(u_n))a(u_n,\nabla u_n)\nabla w_n
(1-\psi_\delta)\\
&\leq \int_\Omega \gamma w_n^+\exp(G(u_n)) (1-\psi_\delta)
+ \int_\Omega w_n^+\exp(G(u_n)) (1-\psi_\delta)\mu_n \\
&\quad +\int_\Omega \exp(G(u_n))a(u_n,\nabla u_n)\nabla \psi_\delta w_n^+
\end{align*}
and
\begin{align*}
& \int_{\{w_n\leq 0\}} \exp(-G(u_n))a(u_n,\nabla u_n)\nabla w_n
(1-\psi_\delta) \\
&\leq \int_\Omega\gamma w_n^-\exp(-G(u_n)) (1-\psi_\delta)
- \int_\Omega w_n^-\exp(-G(u_n)) (1-\psi_\delta)\mu_n \\
&\quad - \int_\Omega \exp(-G(u_n))a(u_n,\nabla u_n)\nabla \psi_\delta w_n^-\,.
\end{align*}
Setting $M=h+4k$ and using $a(x,s,\xi)\cdot\xi\geq 0$, we have
\begin{align*}
a(u_n,\nabla u_n)\nabla w_n\geq&
a(T_k(u_n),\nabla T_k(u_n))\nabla (T_k(u_n)-T_k(u))\\
&-|a(x,{T_M(u_n)},{\nabla T_M(u_n)})| |\nabla T_k(u)| \chi_{\{|u_n|>k\}}\,.
\end{align*}
Then
\begin{equation}\label{una}
\begin{aligned}
& \int_{\{w_n\geq 0\}} \exp(G(u_n))a(T_k(u_n),\nabla T_k(u_n))\nabla
(T_k(u_n)-T_k(u)) (1-\psi_\delta)\\
&\leq \int_\Omega \gamma w_n^+\exp(G(u_n)) (1-\psi_\delta)
+\int_\Omega w_n^+\exp(G(u_n)) (1-\psi_\delta)\mu_n \\
&\quad +
\int_\Omega \exp(G(u_n))a(u_n,\nabla u_n)\nabla \psi_\delta w_n^+\cr &\quad +
\int_\Omega \exp(G(u_n))|a(x,{T_M(u_n)},{\nabla T_M(u_n)})| |\nabla
T_k(u)| \chi_{\{|u_n|>k\}} (1-\psi_\delta)
\end{aligned}
\end{equation}
and
\begin{equation}\label{altra}
\begin{aligned}
&\int_{\{w_n\leq 0\}}
\exp(-G(u_n))a(T_k(u_n),\nabla T_k(u_n))\nabla (T_k(u_n)-T_k(u)) (1-\psi_\delta)\\
&\leq \int_\Omega \gamma w_n^-\exp(-G(u_n)) (1-\psi_\delta)
- \int_\Omega w_n^-\exp(-G(u_n)) (1-\psi_\delta)\mu_n \\
&\quad -\int_\Omega \exp(-G(u_n))a(u_n,\nabla u_n)\nabla \psi_\delta w_n^-\\
&\quad +\int_\Omega\exp(-G(u_n)) |a(x,{T_M(u_n)},{\nabla T_M(u_n)})|
|\nabla T_k(u)| \chi_{\{|u_n|>k\}}(1-\psi_\delta)\,.
\end{aligned}
\end{equation}
Since $a(u_n,\nabla u_n)$ is bounded in $L^q(\Omega)^N$ for any
$q<\frac N{N-1}$, there exists $\nu\in L^q(\Omega)^N$ such that
$a(u_n,\nabla u_n)$ weakly converges to
$\nu$ in $L^q(\Omega)^N$. Since $\psi_\delta\in W^{1,\infty}_0(\Omega)$, and $G$ is bounded,
we get
\begin{equation}\label{p}
\begin{aligned}
&\int_\Omega \exp(G(u_n))a(u_n,\nabla u_n)\nabla \psi_\delta w_n^+\\
&\mathop{\to}\limits^{n\to \infty}
\; \int_\Omega \exp(G(u))\nu\nabla \psi_\delta T_{2k}(u-T_h(u))
\; \mathop{\to}\limits^{h\to \infty}\; 0\,.
\end{aligned}
\end{equation}
Using that $|\nabla T_k(u)| \chi_{\{|u_n|>k\}}$ strongly converges to zero in
$L^p(\Omega)$ and that $\nabla T_M(u_n)$ is bounded in $L^{p'}(\Omega)^N$ we also have that
\begin{equation}\label{o}
\int_\Omega \exp(G(u_n))|a(x,{T_M(u_n)},{\nabla T_M(u_n)})| |\nabla
T_k(u)| \chi_{\{|u_n|>k\}} (1-\psi_\delta)\; \mathop{\to}\limits^{n\to \infty}
\; 0\,.
\end{equation}
Similarly, using
the weak convergence of $T_k(u_n)$ to $T_k(u)$ in $W^{1,p}_0(\Omega)$,
we have
\begin{equation}\label{r}
\int_\Omega \exp(-G(u_n))a(T_k(u_n),\nabla T_k(u))\nabla (T_k(u_n)-T_k(u))
(1-\psi_\delta)\; \mathop{\to}\limits^{n\to \infty}
\; 0\,,
\end{equation}
and, since $\gamma\in L^1(\Omega)$,
\begin{equation}\label{e}
\begin{aligned}
&\int_\Omega
\gamma w_n^+\exp(G(u_n)) (1-\psi_\delta)\\
&\mathop{\to}\limits^{n\to \infty}
\; \int_\Omega \gamma\exp(G(u))(1- \psi_\delta ) T_{2k}(u-T_h(u))^+
\;\mathop{\to}\limits^{h\to \infty} \; 0\,.
\end{aligned}
\end{equation}
Moreover, we have, using the decomposition of $\mu_n$ in \eqref{mun},
\begin{align*}
&\int_\Omega w_n^+\exp(G(u_n)) (1-\psi_\delta)\mu_n \\
&=\int_\Omega w_n^+\exp(G(u_n))(1-\psi_\delta) d \mu_{0n}+ \int_\Omega w_n^+\exp(G(u_n))
(1-\psi_\delta)\lambda_n \\
&\leq \exp\big(\frac{\|g\|_{L^1(\mathbb{R})}}\alpha\big)
\int_\Omega w_n^+(1-\psi_\delta) d \mu_{0n}+
2k\exp\big(\frac{\|g\|_{L^1(\mathbb{R})}}\alpha\big) \int_\Omega (1-\psi_\delta)\lambda_n\,.
\end{align*}
Since $w_n^+$ converges to $T_{2k}(u-T_h(u))^+$ weakly-$*$ in $L^\infty(\Omega)$
and weakly in
$W^{1,p}_0(\Omega)$, using the convergence of $ \mu_{0n}$ (which is strong in
$L^1(\Omega)+W^{-1,p'}(\Omega)$) and $\lambda_n$ we obtain
\begin{equation}\label{majo}
\begin{aligned}
&\limsup_{n\to \infty}\int_\Omega w_n^+\exp(G(u_n)) (1-\psi_\delta)\mu_n \\
&\leq \exp\big(\frac{\|g\|_{L^1(\mathbb{R})}}\alpha\big)
\int_\Omega T_{2k}(u-T_h(u))^+(1-\psi_\delta) d\mu_{0}\\
&\quad + 2k\exp\big(\frac{\|g\|_{L^1(\mathbb{R})}}\alpha\big)
\int_\Omega(1-\psi_\delta) \,d\lambda\,.
\end{aligned}
\end{equation}
Since $T_k(u)\in W^{1,p}_0(\Omega)$ for any $k>0$ and
\eqref{stitr} holds true, we have (see e.g. Remark 2.11 in \cite{DMOP}) that $u$ has
a cap-quasi continuous representative which is cap-quasi everywhere finite, that
is there exists a function $\tilde u$ such that $\tilde u=u$ almost everywhere
and ${\rm cap} \{|\tilde u|=+\infty\}=0$. In particular, since $\mu_0$ does not charge sets of zero capacity, we have that
$\tilde u$ is finite $\mu_0$-quasi everywhere, hence $T_{2k}(\tilde u-T_h(\tilde u))$
converges to zero $\mu_0$-quasi everywhere. Letting $h$ go to infinity we deduce that
$$
\lim_{h\to \infty}\int_\Omega T_{2k}(u-T_h(u))^+(1-\psi_\delta)\,d\mu_{0}=0\,,
$$
so that \eqref{majo} implies
\begin{equation}\label{t}
\lim_{h\to \infty} \limsup_{n\to \infty}\int_\Omega w_n^+\exp(G(u_n))
(1-\psi_\delta)\mu_n \leq
2k\exp\big(\frac{\|g\|_{L^1(\mathbb{R})}}\alpha\big) \int_\Omega(1-\psi_\delta)\, d\lambda
\end{equation}
Then, as $n$ and then $h$ go to infinity, using \eqref{p}, \eqref{o},
\eqref{r}, \eqref{e}, \eqref{t}, we obtain from \eqref{una},
\begin{align*}
&\limsup_{h\to \infty}\limsup_{n\to \infty}
\int_{\{w_n\geq 0\}}
\exp(G(u_n))\big[a(T_k(u_n),\nabla T_k(u_n))\\
&-a(T_k(u_n),\nabla T_k(u))\big]\nabla(T_k(u_n)-T_k(u)) (1-\psi_\delta)\\
&\quad \leq 2k\exp(\frac{\|g\|_{L^1(\mathbb{R})}}\alpha) \int_\Omega
(1-\psi_\delta)d \lambda \leq
2k\exp\big(\frac{\|g\|_{L^1(\mathbb{R})}}\alpha\big)
\lambda(\Omega\setminus K_\delta)\,.
\end{align*}
By means of \eqref{mon} and recalling \eqref{pside} we deduce
\begin{multline*}
\limsup_{\delta \to 0}\limsup_{h\to \infty}\limsup_{n\to \infty}
\int_{\{w_n\geq 0\}} \big[a(T_k(u_n),\nabla T_k(u_n))\\
-a(T_k(u_n),\nabla T_k(u))\big]\nabla(T_k(u_n)-T_k(u)) (1-\psi_\delta)\leq 0.
\end{multline*}
In the same way we work on \eqref{altra}, obtaining
\begin{multline*}
\limsup_{\delta \to 0}\limsup_{h\to \infty}\limsup_{n\to \infty}
\int_{\{w_n\leq 0\}} \big[a(T_k(u_n),\nabla T_k(u_n))\\
- a(T_k(u_n),\nabla T_k(u))\big]\nabla (T_k(u_n)-T_k(u)) (1-\psi_\delta)\leq 0.
\end{multline*}
Adding the two inequalities we conclude
\begin{multline}\label{shalf}
\limsup_{\delta\to 0}\limsup_{n\to \infty}
\int_\Omega \big[a(T_k(u_n),\nabla T_k(u_n))\\
-a(T_k(u_n),\nabla T_k(u))\big]\nabla (T_k(u_n)-T_k(u)) (1-\psi_\delta)=0\,.
\end{multline}
Now, we have
\begin{align*}
& \int_\Omega [a(T_k(u_n),\nabla T_k(u_n))-
a(T_k(u_n),\nabla T_k(u))]\nabla (T_k(u_n)-T_k(u))\\
&= \int_\Omega [a(T_k(u_n),\nabla T_k(u_n))-
a(T_k(u_n),\nabla T_k(u))]\nabla (T_k(u_n)-T_k(u))(1-\psi_\delta)\\
&\quad + \int_\Omega a(T_k(u_n),\nabla T_k(u_n))\nabla T_k(u_n) \psi_\delta
-\int_\Omega a(T_k(u_n),\nabla T_k(u_n))\nabla T_k(u) \psi_\delta\\
&\quad -\int_\Omega a(T_k(u_n),\nabla T_k(u))\nabla (T_k(u_n)-T_k(u)) \psi_\delta\,.
\end{align*}
Using the weak convergence of $T_k(u_n)$ to $T_k(u)$ last term converges to
zero as $n$ goes to infinity. Similarly, we have that
$a(T_k(u_n),\nabla T_k(u_n))$ is bounded in $L^{p'}(\Omega)^N$
uniformly on $n$ while $\nabla T_k(u)\psi_\delta$ converges to zero in
$L^p(\Omega)^N$ as $\delta$ tends to zero. Using also
\eqref{shalf} and \eqref{half} we finally get, letting first $n$ go
to infinity and then $\delta$ to zero,
$$
\lim_{n\to \infty}
\int_\Omega [a(T_k(u_n),\nabla T_k(u_n))-
a(T_k(u_n),\nabla T_k(u))]\nabla
(T_k(u_n)-T_k(u))=0.
$$
Under assumptions \eqref{coerc}--\eqref{mon}, it is well known that this
implies
\begin{equation}\label{sconv}
T_k(u_n)\to T_k(u)\quad \mbox{strongly in $W^{1,p}_0(\Omega)$ for any $k>0$.}
\end{equation}
Moreover, using that $\mathop{\rm meas}\{|u_n|>k\}$ goes to zero as $k$ goes to infinity
uniformly on $n$, as a consequence of \eqref{sconv} we also have that, up to
subsequences, $\nabla u_n$ almost everywhere converges to $\nabla u$ in $\Omega$.
In turns, this implies that
\begin{equation}\label{aun}
a(x,{u_n},{\nabla u_n})\to a(x,u,\nabla u)\quad
\mbox{strongly in $L^q(\Omega)^N$ for any $q<\frac N{N-1}$.}
\end{equation}
Let $\varphi=\int_0^{u_n} g(s)\chi_{\{s>h\}}ds$ in
\eqref{poseq}; since $|\varphi|\leq \int_h^\infty g(s)ds$ we have
\begin{align*}
&\int_\Omega a(u_n,\nabla u_n)\nabla u_n g(u_n)\chi_{\{u_n>h\}}\\
&\leq \exp\big(\frac{\|g\|_{L^1(\mathbb{R})}}\alpha\big)
\Big(\int_h^\infty g(s)ds\Big) (\|\gamma\|_{L^1(\Omega)}+ \|\mu_n\|_{L^1(\Omega)})\,.
\end{align*}
Using \eqref{coerc} and the fact that $\mu_n$ is bounded in $L^1(\Omega)$ gives
$$
\alpha \int_{\{u_n>h\}}g(u_n)|\nabla u_n|^p\leq c\Big(\int_h^\infty
g(s)ds\Big)\,,
$$
and then since $g\in L^1(\mathbb{R})$ we obtain
$$
\lim_{h\to \infty}\quad \sup_{n\in \mathbb{N}}
\int_{\{u_n>h\}}g(u_n)|\nabla u_n|^p=0\,.
$$
Similarly, taking $\varphi=\int_{u_n}^0 g(s)\chi_{\{s<-h\}}ds$ in \eqref{negeq}
we obtain the corresponding result on the set $\{u_n<-h\}$, hence
\begin{equation}\label{equ}
\lim_{h\to \infty}\quad \sup\limits_{n\in \mathbb{N}}
\int_{\{|u_n|>h\}}g(u_n)|\nabla u_n|^p=0\,.
\end{equation}
A standard argument
allows to conclude from
\eqref{sconv} and
\eqref{equ} that $g(u_n)|\nabla u_n|^p$ strongly converges in $L^1(\Omega)$ to
$g(u)|\nabla u|^p$. Then from \eqref{creh}, the almost everywhere convergence of
$u_n$ and $\nabla u_n$ and Lebesgue's theorem we conclude that
\begin{equation}\label{hn}
H(x,u_n,\nabla u_n)\to H(x,u,\nabla u)\quad \mbox{strongly in $L^1(\Omega)$.}
\end{equation}
Thanks to \eqref{aun} and \eqref{hn} we can pass to the limit in \eqref{appell}
and we obtain that $u$ is a distributional solution of \eqref{reac}.
\hfill$\square$
\begin{remark} \rm
The assumption that $\mu$ is nonnegative is not essential in Theorem \ref{primo}.
In order to deal with changing sign measures it is enough to follow the same lines
of the previous proof with suitable modifications while proving the strong
convergence of truncations similar to those developed in \cite{DMOP}.
\end{remark}
\begin{example} \rm
Let $\mu=\delta_{0}$ be the Dirac mass at the origin and let $\Omega=B(0,1)$ be the unit
ball in $\mathbb{R}^N$, with $N\geq 3$. Let $\mu_n=n^N\chi_{B(0,\frac1n)}$; clearly $\mu_n$ converges, in the narrow topology, to
$\lambda\delta_0$ for some constant $ \lambda>0$. Note that in particular $\mu_n$ satisfies \eqref{mun}
(with $f_n=F_n=0$).
Let $u_n$ be any sequence of solutions of
\begin{equation}\label{apex}
\begin{gathered}
-\Delta u_n=g(u_n)|\nabla u_n|^2+\mu_n\quad \mbox{in }\Omega\,,\\
u_n=0 \quad \mbox{on }\partial\Omega.
\end{gathered}
\end{equation}
We claim that if the following assumption holds:
\begin{equation}\label{assex}
\begin{gathered}
\mbox{$\exists h\in C(\mathbb{R},\mathbb{R}^+)$: $g(s)\geq h(s)$ for every $s\in \mathbb{R}^+$,}\\
\mbox{$h$ is nonincreasing, $\lim_{s\to +\infty}h(s)=0$ and
$h\not\in L^1(\mathbb{R}^+)$,}
\end{gathered}
\end{equation}
then the sequence $u_n$ blows up completely, namely $u_n(x)\to +\infty$ for
every $x\in \Omega$.
As far as assumption \eqref{assex} is concerned,
observe that if $g$ is nonincreasing, converges to zero at infinity and $g\not \in
L^1(\mathbb{R})$, we can clearly take
$h=g$ in \eqref{assex}; this includes the main examples of $g$ around the borderline
case $g\in L^1(\mathbb{R})$, as $g(s)=1/(|s|+1)$ or
$g(s)=1/((1+|s|)\log(1+|s|))$.
Anyway, assumption
\eqref{assex} is stated in this generality to include most examples of $g$; in
particular, note that the it requires $g$ to be {\bf larger} than a
nonincreasing function which is not integrable, so that $g$ itself may also
be unbounded.
In order to prove our claim, we adapt an idea
used in a context of sublinear equations by L. Orsina (\cite{Oper}).
Let us set $H(s)=\int_0^s h(\xi)d\xi$, $\psi(s)=\int_0^s \exp(H(\xi))d\xi$
and define
$v_n :=\psi(u_n)$ (the function $h$ is defined in \eqref{assex}).
Observe that $\psi$ is an increasing unbounded function, so that $v_n$
goes to infinity if and only if $u_n$ goes to infinity.
Since $g(u_n)\geq h(u_n)$, $v_n$ satisfies
\begin{equation}\label{eqv}
\begin{gathered}
-\Delta v_n \geq \exp(H(u_n)) \mu_n \quad \mbox{in } \Omega\,,\\
v_n=0\quad \mbox{on }\partial \Omega.
\end{gathered}
\end{equation}
In particular, by definition of $\mu_n$, we have that $v_n$ is a supersolution
of the problem
\begin{equation}\label{auco}
\begin{gathered}
-\Delta z=\exp(H(\psi^{-1}(z))) n^N \quad\mbox{in } B(0,\frac1n),\\
z=0\quad \mbox{on } \partial B(0,\frac1n)\,.
\end{gathered}
\end{equation}
Let $\varphi_{1,n}$ be the first eigenfunction of the Laplacian on $B(0,\frac1n)$,
normalized so that $\|\varphi_{1,n}\|_{L^\infty(\Omega)}=1$, and let $\lambda_{1,n}$ be the
first eigenvalue. Let us set
$$
B(s) :=\frac{\exp(H(\psi^{-1}(s)))}{s}.
$$
Since $h$ is nonincreasing we have
\begin{align*}
\frac{d}{dr}\Big(\frac{\exp(H(r))}{\psi(r)}\Big)=&
\frac{\exp(H(r))}{\psi(r)^2}\Big( h(r)\int_0^r
\exp(H(\xi))d\xi-\exp(H(r))\Big)\\
\leq& \frac{\exp(H(r))}{\psi(r)^2}\Big( \int_0^r
\exp(H(\xi)) h(\xi)d\xi-\exp(H(r))\Big)<0\,,
\end{align*}
so that $B(\psi(s))$ is decreasing. Since $\psi$ is increasing, we deduce that
$B$ is a decreasing function. Let us set $T_n=B^{-1}(\frac{\lambda_{1,n}}{n^N})$.
Since $B$ is decreasing, we deduce that
$$
\frac{\lambda_{1,n}}{n^N}=B(T_n)=B(T_n\|\varphi_{1,n}\|_{L^\infty(\Omega)})\leq
B(T_n \varphi_{1,n}(x))\quad
\forall x\in B(0,\frac1n)\,,
$$
which implies, by definition of $B$,
$$
\lambda_{1,n} T_n \varphi_{1,n}(x)\leq \exp(H(\psi^{-1}(T_n \varphi_{1,n}(x)))) n^N \quad
\forall x\in B(0,\frac1n)\,.
$$
Since $\lambda_{1,n} T_n \varphi_{1,n}=-\Delta (T_n \varphi_{1,n})$ we conclude that
$T_n \varphi_{1,n}$ is a subsolution of \eqref{auco}. Since
$\exp\big(H(\psi^{-1}(z))\big)/z=B(z)$ is decreasing, a well-known
comparison principle holds
for positive sub-super solutions of \eqref{auco} (see for example \cite{BK}),
so that we get $v_n\geq T_n \varphi_{1,n}$ in $B(0,\frac1n)$.
By scaling arguments we know that
$$
\varphi_{1,n}(x)=\varphi_{1,1}(nx)\,,\quad \lambda_{1,n}=\lambda_{1,1} n^2\,,
$$
hence we obtain
$$
\forall x\in B(0,\frac1{2n}) : \quad v_n(x)\geq
B^{-1}(\frac{\lambda_{1,1}}{n^{N-2}})\min_{B(0,\frac1{2})}\varphi_{1,1}\,.
$$
Since $\varphi_{1,1}$ is radial, we have
$\min_{B(0,\frac1{2})}\varphi_{1,1}=\varphi_{1,1}(\frac12)$, so that
\begin{equation}\label{uf}
\min_{B(0,\frac1{2n})} v_n\geq
B^{-1}(\frac{\lambda_{1,1}}{n^{N-2}}) \varphi_{1,1}(\frac12)\,.
\end{equation}
Now observe that, using De L'Hospital's theorem and the fact that $h(s)$ goes to
zero at infinity, we have
$\lim_{s\to +\infty} B(s)=0$.
Since $\lambda_{1,1}/n^{N-2}$ converges to zero as $n$ tends to infinity,
we end up with
$$
\lim_{n\to +\infty}B^{-1}(\frac{\lambda_{1,1}}{n^{N-2}})=+\infty\,,
$$
and then from \eqref{uf}
\begin{equation}\label{minex}
\lim_{n\to +\infty} \min_{B(0,\frac1{2n})} v_n=+\infty\,.
\end{equation}
Let now $G(x,y)$ be the kernel of the Laplacian with zero boundary condition;
we have from \eqref{eqv}
\begin{equation}\label{pref}
\begin{aligned}
v_n(x)&\geq \int_\Omega G(x,y)\exp(H(u_n))(y) \mu_n(y) dy\\
&\geq \min_{B(0,\frac1{2n})}\left( \exp(H(\psi^{-1}(v_n)))\right)
\int_{B(0,\frac1{2n})} G(x,y) n^N\, dy\,.
\end{aligned}
\end{equation}
Since there exists a constant $c>0$ such that
$$
\int_{B(0,\frac1{2n})} G(x,y) n^N dy\to c \int_\Omega G(x,y)d\delta_0(y)>0\,,
$$
and since both $\psi^{-1}$ and $H$ go to infinity at infinity (because $h\not\in
L^1(\mathbb{R}^+)$), we deduce using \eqref{minex} that the right hand side of \eqref{pref}
goes to infinity as $n$ goes to infinity. We then conclude
$$
\lim_{n\to +\infty} v_n(x)=+\infty\quad \forall x\in \Omega\,.
$$
Since $\psi$ is unbounded and $u_n=\psi^{-1}(v_n)$, we have proved that the
solutions $u_n$ of \eqref{apex} blow up completely in $\Omega$.
This is in sharp contrast with what proved in Theorem \ref{primo} when $g\in
L^1(\mathbb{R})$, so that this assumption is optimal in the existence result above.
\end{example}
\begin{thebibliography}{99} \frenchspacing
\bibitem{A} N. Alaa, Solutions faibles d'\'equations paraboliques quasilinéaires avec donn\'ees initiales mesures, {\sl Ann.
Math. Blaise Pascal} {\bf 3} (1996), no. 2, 1--15.
\bibitem{APi} N. Alaa, M. Pierre, Weak solutions of some quasilinear elliptic equations with data measures, {\sl SIAM J. Math.
Anal.} {\bf 24} (1993), n. 1, 23--35.
\bibitem{And} D. Andreucci, Degenerate parabolic equations with initial data measures, {Trans. Amer. Math. Soc.} {\bf 349}
(1997), 3911--3923.
\bibitem{BaPi} P. Baras, M. Pierre, Singularit\'es \'eliminables pour des \'equations
semi-lin\'eaires, {\sl Ann. Inst. Fourier (Grenoble)}, {\bf 34} (1984), 185--206.
\bibitem{BaPi1} P. Baras, M. Pierre, Problemes paraboliques semi-lin\'eaires avec donn\'ees mesures,
{\sl Applicable Anal.} {\bf 18} (1984), no.
1-2, 111--149.
\bibitem{BeLa} S. Benachour, Ph. Laurencot, Global solutions to viscous Hamilton--Jacobi equation with irregular data, {Comm.
P.D.E.} {\bf 24} (1999), 1999--2021.
\bibitem{BSW} M. Ben--Artzi, P. Souplet, F. Weissler, The local theory for viscous Hamilton--Jacobi equations in Lebesgue
spaces, {Journ. Math. Pures et Appl.} {\bf 81} (2002).
\bibitem{BBGGPV} P. Benilan, L. Boccardo, T. Gallou\"et, R. Gariepy,
M. Pierre, J. L. V\'azquez, An $L^1$ theory of existence and
uniqueness of nonlinear elliptic equations, {\sl Ann. Scuola Norm. Sup. Pisa Cl.
Sci.}, {\bf 22} (1995), 240--273.
\bibitem{BeE} A. Benkirane, A. Elmahi, A strongly nonlinear elliptic equation having natural growth terms
and $L^1$ data, {Nonlin. Anal. T.M.A.}{\bf 39} (2000), 403--411.
\bibitem{BeEM} A. Benkirane, A. Elmahi, D. Meskine, An existence theorem for a class of elliptic problems in $L^1$,
submitted.
\bibitem{Bi-Ve} M-F. Bidaut--V\'eron, Removable singularities and existence for a quasilinear equation with absorption or
source term and measure data, preprint.
\bibitem{BCV} M-F. Bidaut--V\'eron, E. Chasseigne, L. V\'eron, Initial trace of solutions of some quasilinear parabolic
equations with absorption, {\sl J. Funct. Anal.\/} {\bf 193\/} (2002), no. 1, 140--205.
\bibitem{BlPo} D. Blanchard, A. Porretta, Nonlinear parabolic equations with natural growth terms and
measure initial data, {\sl Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)} {\bf 30} (2001), no. 3-4, 583--622.
\bibitem{BG1} L. Boccardo, T. Gallou\"et, Nonlinear elliptic and parabolic
equations involving
measure data, {\sl J. Funct. Anal.\/}, {\bf 87} (1989), 149--169.
\bibitem{BG2} L. Boccardo, T. Gallou\"et, Strongly nonlinear elliptic
equations having
natural growth terms and $L^1$ data, {\sl Nonlinear Anal. T.M.A.\/},
{\bf 19\/} (1992), 573--579.
\bibitem{BGO1} L. Boccardo, T. Gallou\"{e}t, L. Orsina, Existence and
uniqueness of
entropy solutions for nonlinear elliptic equations with measure data. {\sl
Ann. Inst. H.
Poincar\'e Anal. Non Lin\'eaire}, {\bf 13} (1996), 539--551.
\bibitem{BGO2} L. Boccardo, T. Gallou\"et, L. Orsina, Existence and nonexistence of solutions
for some nonlinear elliptic equations, {\sl Journal d'Analyse Math.\/} {\bf 73\/}
(1997), 203--223.
\bibitem{BGV} L. Boccardo, T. Gallou\"et, J.L. Vazquez, Nonlinear elliptic equations in $R\sp N$ without growth restrictions
on the data, {\sl J. Differential Equations\/} {\bf 105} (1993), no. 2, 334--363.
\bibitem{BST} L. Boccardo, S. Segura, C. Trombetti, Bounded and unbounded solutions for a class of quasi-linear elliptic
problems with a quadratic gradient term, {\sl J. Math. Pures Appl.\/}, (9) {\bf 80\/} (2001), 919--940.
\bibitem{B} H. Brezis, Nonlinear elliptic equations involving measures, in {\sl Variational Inequalities},
Cottle, Gianessi, Lions ed., Wiley, 1980, 53--73.
\bibitem{B1} H. Brezis, Some Variational Problems of the Thomas--Fermi type, in {\sl Contributions to
nonlinear partial differential equations (Madrid, 1981)}, 82--89, Res. Notes in Math., 89,
Pitman, Boston, Mass.-London, 1983.
\bibitem{BF} H. Brezis, A. Friedman, Nonlinear parabolic equations involving measures as initial data,
{\sl J. Math. Pures et Appl.} {\bf 62} (1983), 73--97.
\bibitem{BK} H. Brezis, S. Kamin, Sublinear elliptic equations in $R\sp n$, {\sl Manuscripta Math.\/} {\bf 74} (1992), no. 1,
87--106.
\bibitem{BN} H. Brezis, L. Nirenberg, Removable singularities for some nonlinear elliptic equations, {\sl
Topological Methods for Nonlin. Anal.\/}, {\bf 9\/} (1997), 201--219.
\bibitem{BV} H. Brezis, L. V\'eron, Removable singularities for nonlinear elliptic equations, {\sl
Arch. Rational Mech. Anal.\/}, {\bf 75\/} (1980), 1--6.
\bibitem{DMOP} G. Dal Maso, F. Murat, L. Orsina, A. Prignet, Renormalized solutions of elliptic
equations with general measure
data, {\sl Ann. Scuola Norm. Sup. Pisa Cl. Sci.} {\bf 28}, 4 (1999),
741--808.
\bibitem{DPP} J. Droniou, A. Porretta, A. Prignet, Parabolic
capacity and soft measures for nonlinear
equations, {\sl Potential Analysis\/}, to appear.
\bibitem{EM} A. Elmahi, D. Meskine, Unilateral problems in $L^1$ having natural growth terms, submitted.
\bibitem{FST} M. Fukushima, K. Sato, S. Taniguchi, On the closable part of pre-Dirichlet forms and the fine support
of the underlying measures, {\sl Osaka J. Math.\/} {\bf 1991}, $28$, 517--535.
\bibitem{GM} T. Gallou\"{e}t, J.M. Morel, Resolution of a semilinear equation in
$L^1$, {\sl Proc. Roy. Soc. Edinburgh}, {\bf 96} (1984), 275--288.
\bibitem{HKM} J. Heinonen, T. Kilpel\"ainen, O. Martio, {\it Nonlinear
potential theory of
degenerate elliptic equations}, Oxford University Press (1993).
\bibitem{LL} J.~Leray, J.-L. Lions, Quelques r\'esultats de Vi\v sik sur
les probl\`emes ellip\-tiques
non lin\'eaires par les m\'ethodes de Minty-Browder, {\sl Bull. Soc.
Math. France\/}, {\bf 93} (1965),
97--107.
\bibitem{MV} M. Marcus, L. V\'eron, Initial trace of positive solutions of some nonlinear parabolic equations, {\sl
Comm. P.D.E.\/}, {\bf 24\/} (1999), 1445--1499.
\bibitem{MuPo} F. Murat, A. Porretta, Stability properties, existence and nonexistence
of renormalized solutions
for elliptic equations with measure data, {\sl Comm. P.D.E.}, to appear.
\bibitem{Oper} L. Orsina, personal communication.
\bibitem{OPo} L. Orsina, A. Porretta, Strong stability results for nonlinear elliptic equations
with respect to
very singular perturbation of the data, {\sl Comm. in Contemporary Mathematics\/} {\bf 3} (2001), pp. 259--285.
\bibitem{OP1} L. Orsina, A. Prignet, Non-existence of solutions for some nonlinear elliptic equations involving measures.
{\sl Proc. Roy. Soc. Edinburgh Sect. A} {\bf 130} (2000), no. 1, 167--187.
\bibitem{OP2} L. Orsina, A. Prignet, Strong stability results for solutions of elliptic equations with power-like lower order
terms and measure data {\sl J. Funct. Anal.} {\bf 189} (2002), no. 2, 549--566.
\bibitem{Pi} M.~Pierre, Parabolic capacity and Sobolev spaces, {\sl Siam J.
Math. Anal.}, {\bf 14} (1983), 522--533.
\bibitem{Po0} A. Porretta, Some remarks on the regularity of solutions for a class of
elliptic equations with measure data, {\sl Houston
Journ. of Math.\/} {\bf 26} (2000), pp. 183--213.
\bibitem{Po1} A. Porretta, Existence results for nonlinear parabolic equations via
strong convergence of truncations, {\sl Ann.
Mat. Pura ed Applicata (IV)\/} {\bf 177} (1999), pp. 143--172.
\bibitem{Po2} A. Porretta, Existence for elliptic equations in
$L^1$ having lower order terms with natural growth,
{\sl Portugaliae
Mathematica \/} {\bf 57} (2000), pp. 179--190.
\bibitem{Po3} A. Porretta, Removable singularities and strong stability results for some nonlinear parabolic equations,
in preparation.
\bibitem{Se} S. Segura de Leon, Existence and uniqueness for $L^1$ data of some elliptic equations with natural growth,
{\sl Advances in Diff. Eq.}, to appear.
\bibitem{VV} J.L. Vazquez, L. Veron, Isolated singularities of some semilinear
elliptic equations, {\sl J. Differential Equations}, {\bf 60} (1985), 301--321.
\bibitem{VV1} J.L. Vazquez, L. Veron, Removable singularities of some strongly nonlinear elliptic
equations, {\sl Manuscripta Math.}, {\bf 33} (1980/81), 129--144.
\bibitem{Ve} L. Veron, Singularit\'es \'eliminables d'\'equations elliptiques non lin\'eaires, {\sl
J. Differential Equations}, {\bf 41} (1981), 87--95.
\end{thebibliography}
\noindent\textsc{Alessio Porretta}\\
Dipartimento di Matematica,
Universit\`a di Roma \lq\lq Tor Vergata\rq\rq,\\
Via della Ricerca Scientifica 1, 00133, Roma, Italia.\\
email: porretta@mat.uniroma2.it
\end{document}