\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2016 (2016), No. 196, pp. 1--22.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2016 Texas State University.}
\vspace{7mm}}

\begin{document}
\title[\hfilneg EJDE-2016/196\hfil Stable solutions]
{Stable solutions for equations with a quadratic gradient term}

\author[J. Terra \hfil EJDE-2016/196\hfilneg]
{Joana Terra}

\address{Joana Terra \newline
Departamento de Matem\'atica, FCEyN, 
and IMAS-CONICET, UBA (1428),
Buenos Aires, Argentina}
\email{joanamterra@gmail.com} 

\thanks{Submitted July 30, 2015. Published July 21, 2016.}
\subjclass[2010]{35B35, 35J61}
\keywords{Non-variational problem; stable solution}

\begin{abstract}
 We consider positive solutions to the non-variational family of equations 
 \[
 -\Delta u -b(x)|\nabla u|^2= \lambda g(u)\quad \text{in }\Omega,
 \]
 where $\lambda\geq 0$, $b(x)$ is a given function, $g$ is an increasing
 nonlinearity with $g(0)>0$ and $\Omega\in\mathbb{R}^n$ is a bounded smooth domain.
 We introduce the definition of stability for non-variational problems and
 establish existence and regularity results for stable solutions.
 These results generalize the classical results obtained when $b(x)=b$
 is a constant function making the problem variational after a suitable
 transformation.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{definition}[theorem]{Definition}
\allowdisplaybreaks

\section{Introduction} \label{Intro}

In this article we are interested in the existence and qualitative
properties of positive solutions to equations of the form
\[
-\Delta u -b(x)|\nabla u|^2= \lambda g(u)
\]
in a bounded smooth domain $\Omega$ of $\mathbb{R}^n$, for $\lambda\geq 0$, $b=b(x)$
a given function and $g$ an increasing nonlinearity with $g(0)>0$. This type of
equations arise in different contexts from physics to stochastic
processes. Equations with the
quadratic gradient term
$-\Delta -b(x)|\nabla u|^2$ appear in relation to different contexts within the
literature. If $b=b(x)$ is constant and positive, the equation can be
thought as the stationary part of the parabolic equation
$u_t-\epsilon\Delta u=|\nabla u|^2$ which in turn may be seen as the
viscosity approximation, as $\epsilon$ tends to $0^+$, of
Hamilton-Jacobi equations from stochastic control theory \cite{L}. In
\cite{KPZ} the same equation (known in this context as Kardar-Parisi-Zhang equation)
 arises related to the physical theory of
growth and roughening on surfaces. Also classical are the existence
results for equations involving a quadratic gradient term and such
that $b=b(u)$ (see for instance \cite{LU, LL}). For more on such
equations with $b=b(u)$ see for example \cite{ADP}.


If the coefficient function $b$ is constant, the above equation can
be transformed, using the Hopf-Cole transformation, into the
equation $-\Delta v = \lambda f(v)$, where $f$ satisfies
the same hypothesis as $g$. This simpler equation for $v$ appears in
many different contexts and has been extensively studied. This family
of equations includes, for example, the Gelfand problem, where
$f(v)=e^v$ with zero Dirichlet boundary conditions on the boundary of
$\Omega=B_1$, the unit ball.
Some first results concerning this problem involved the construction of
 explicit radial solutions in dimensions $2$ and $3$, and in the special
case where $\lambda=2$ and $n=3$ it was established that there are infinitely many solutions.


The natural question that arises regarding the equation for $v$ is the
study of the solutions $(\lambda, v)$, their existence and
properties. The classical existence result says there exists a finite extremal parameter
$\lambda^*$ such that for $\lambda>\lambda ^*$ there exist no bounded
solutions $v$, whereas for $0<\lambda<\lambda^*$ there exists a minimal
(i.e., smallest) bounded solution $v_{\lambda}$. Moreover, the branch
$\{v_{\lambda}\}$ is increasing in $\lambda$ and each solution
$v_{\lambda}$ is stable. A more
delicate problem is the study of the increasing limit $v^*=\lim_{\lambda\uparrow\lambda^*}
v_{\lambda}$, which turns out to be a weak solution of the problem with
parameter $\lambda^*$. However $v^*$ may be either bounded or
singular, depending on the domain $\Omega$ and the nonlinearity $f$.

In the case where $\Omega$
is the unit ball of $\mathbb{R}^n$ and $f(v)=e^v$, Joseph and Lundgren
\cite{JL} completely described the
existence and regularity of solutions in terms of $\lambda$. Their
result also applies to the other classical model, that is, when
$f(v)=(1+v)^p$ and $p>1$. For general domains, Crandall and Rabinowitz
\cite{CR} and Mignot and Puel \cite{MP} gave sufficient conditions for
the extremal solution $v^*$ to be classical, when the nonlinearity $f$
is either exponential or power like. Regarding more general $f$, namely $f$
convex, nonnegative and asymptotically linear, the existence of an extremal
parameter was known. However, the case where $\lambda=\lambda^*$ was first
studied by Mironescu and Radulescu in \cite{MR0,MR}. They establish two
different scenarios, depending whether $f$ obeyed the monotone case or the
non-monotone case (see \cite{MR} for appropriate definitions).
These cases implied non existence of extremal solution and uniqueness of
extremal solution respectively. Brezis and V\'azquez \cite{BV}
raised the question of studying when is the extremal solution bounded for general
$f$ convex, depending on the dimension $n$ and the domain $\Omega$.
For $n\leq 3$ Nedev \cite{Ne} proved that $v^*$ is bounded
for any domain $\Omega$. More recently in \cite{C} Cabr\'e proves that
the extremal solution is bounded if the domain
$\Omega$ is convex and $n\leq 4$. Then, Villegas \cite{V} established a similar
result for convex $f$ rather than convex $\Omega$. For higher dimensions the
only known result so far for general $f$ concerns radial solutions.
Namely, Cabr\'e and Capella \cite{CC2} prove that if $\Omega$ is the unit ball and
$n\leq 9$ then $v^*$ is bounded for every $f$. Boundedness for general domains
and $n\leq 9$ is still open. Note that, for $n\geq 10$ there exist unbounded
solutions.

Assuming some extra conditions on $f$, on a recent paper by Cabr\'e
and Sanch\'on \cite{CSS}, $L^{\infty}$ bounds were obtained for dimensions
$n=5$ and $n=6$. They also obtain very interesting $L^p$ estimates for $f'(v)$.

In the case where $\Omega=\mathbb{R}^N$ and $f$ is a general convex non-decreasing
functions, Dupaigne and Farina \cite {DuF} established Liouville type results
for stable solutions.

Other lines of research have included equations that can be written as
$-\Delta u=a(x)g(u)+b(x)$ where $g$ is a continuous nondecreasing non-negative
function satisfying some growth conditions at infinity.
The power functions $g(u)=u^p$ for $p>1$ appear as the natural example.
Under some conditions on $a$ and $b$ Brezis and Cabr\'e \cite{BC}
establish non-existence of weak solutions. If instead we assume $g$ is
non-increasing and unbounded near the origin and furthermore we consider
$b=b(u)=\lambda f(u)$ where $f$ is positive, non-decreasing and such that
$f(s)/s$ in non-increasing, converging to some $m$, then Cirstea, Ghergu
and Radulescu established in \cite{CGR} that if $m=0$ there is uniqueness
of solution (for some range of $\lambda$ depending on $a$) whereas if $m>0$
then there exists an extremal parameter $\lambda^*$ which is related to $m$
and the first Dirichlet eigenvalue of the Laplacian in $\Omega$.
For a more comprehensive reading we recommend the book on singular elliptic
problems by Ghergu and Radulescu \cite{GR}.
In another direction we point out the results of D\'avila and Dupaigne \cite{DD}
regarding the classical Gelfand problem but stated in domain obtained by a
perturbation of a ball. They establish existence of singular solutions
for dimensions $N\geq 4$ which correspond to the extremal solution in case
$N\geq 11$. We also refer to Castorina and Sanch\'on \cite{casto} for
results concerning the regularity of the stable solutions.


In this article we derive results similar to the classical ones but for the
case where $b(x)$ is non-constant, and therefore the problem is not variational.
Although there is no energy functional associated to our problem, and hence there
is no quadratic form, we are still interested in ``stable'' solutions. To
define stability of a solution to a non-variational problem we will use a
different condition than the one used in the variational setting (see section $2$).

The article is organized as follows. In section $2$ we introduce the definition
of stability for non-variational problems.
In section 3, for some special nonlinearities $g$, we
derive the stability of the classical solutions.
In addition, for the class of stable solutions, we prove new
regularity results involving conditions on the function $b(x)$ and the dimension $n$.

In the following section we establish an existence theorem in terms
of $\lambda$. The result is similar to
the one in the classical context with $b\equiv 0$. Namely we prove the existence
of an extremal parameter $\lambda^*$ such that for $\lambda>\lambda^*$ there
is no solution, whereas for $0<\lambda<\lambda^*$ there is a minimal
classical solution $u_{\lambda}$. Moreover, for $g(u)=e^u$ and some dimensions $n$
we are able to prove that the minimal classical solutions
$u_{\lambda}$ are stable and that the extremal function
$u^*=\lim_{\lambda\to\lambda^*}u_{\lambda}$ is a weak solution
for $\lambda=\lambda^*$. As before, in the case where $b(x)=b$ is constant,
the existence result coincides with the classical one.

Finally in the last section we establish some sufficient conditions for stable
solutions $u$ to be in $H^1(\Omega)$. Once again we
consider two cases separately: $b$ positive and $b$ negative.
On the one hand, we have the case where
$b(x)=b>0$ is constant and positive. In this setting we are able to
prove an $H^1(\Omega)$ result following the similar technique of the
classical case (see Brezis-V\'azquez \cite{BV}) which requires an extra condition on
$g$. We note here that via the Hopf-Cole transformation, one could use
the classical result to obtain a condition for $e^u$ to be in
$H^1(\Omega)$. This would imply, of course, that $u$ is also in
$H^1(\Omega)$ but this gives a
much stronger condition on $g$ than the more optimal one that we prove.
On the other hand, using different techniques, namely truncations as introduced by
Boccardo \cite{B}, we prove the $H^1(\Omega)$ result for every solution
(not necessarily stable)
with $b(x)$ strictly negative, and any $L^1$ nonlinearity $g$.


\section{Preliminaries}

We are interested in nonnegative solutions of the problem
\begin{equation}\label{eqg}
\begin{gathered}
 -\Delta u -b(x)|\nabla u|^2 = \lambda g(u) \quad\text{in } \Omega\\
 u = 0 \quad\text{on }\partial\Omega,
 \end{gathered}
\end{equation}
where $\Omega\subset\mathbb{R}^n$ is a bounded smooth domain,
$g:[0,+\infty)\to \mathbb{R}$ is a given nonlinearity, $\lambda\geq0$ is
 a parameter and $b=b(x)$ is a  bounded function. The properties of
 the solutions $u$ of \eqref{eqg} will depend on the coefficient function
 $b$ and hence we will distinguish different cases.

As was discussed in the introduction, in the classical variational setting,
the appropriate class of solutions to work with is that of stable solutions,
in the sense that the second variation of the energy functional associated
to the equation is positive. In our current setting, equation \eqref{eqg} is not,
in general, of variational nature and therefore we need to define what we mean
by stable solutions. In order to define stability for a wider family of problems
we use the linearized equation.

\begin{definition}\label{stable5} \rm
Let $u$ be a classical solution of problem \eqref{eqg}. We say that $u$ is
\emph{stable} if there exists a function
 $\phi\in W^{2,p}(\Omega)$ for some $p>n$ such that $\phi>0$ in
 $\overline{\Omega}$ and
\begin{equation}\label{phistable}
-\Delta \phi -2b(x)\nabla u\nabla\phi \geq \lambda g'(u)\phi \quad
\text{in }\Omega.
\end{equation}
\end{definition}
In the variational setting (for example \eqref{eqg} with
$b\equiv 0$), the existence of such
a supersolution $\phi$, positive in $\overline{\Omega}$, is equivalent to
saying that $u$ is stable
in the variational sense, i.e., the quadratic form
defined defined obtained from the second variation of the energy functional
is positive for every test function $\xi\not\equiv 0$. Equivalently,
 the first eigenvalue of the linearized problem is positive (see \cite{BNV}).

However, for a general function $b$, problem
\eqref{eqg} is not self-adjoint and therefore we bypass this
difficulty by considering the existence of $\phi$ instead of working
with the quadratic form, which makes no sense (or does
not exist) for non self-adjoint problems.

\section{Case $b(x)\equiv b$ is constant}

In this section we consider the case where the coefficient function
$b$ is constant, that is, we study nonnegative solutions to
\begin{equation}\label{equb}
\begin{gathered}
-\Delta u-b|\nabla u|^2  = \lambda g(u)\quad \text{in }\Omega\\
u  =  0 \quad \text{on }\partial\Omega.
\end{gathered}
\end{equation}

This problem can be easily transformed
into a classical semilinear elliptic equation for a new
function $v$ and a new nonlinearity $f=f(v)$ that depends on
$g$. Since the transformation, called Hopf-Cole transformation,
 depends on the sign of the constant
$b$, we will treat both cases separately in the next two
subsections. The nonlinearities $f$ that arise in these two cases
are quite different. Nevertheless, if $g(u)=e^{\beta u}$ for some
constant $\beta$, the
classical regularity results for $v$ and our regularity results for
$u$ (that we later generalize to $b=b(x)$) agree regardless of the sign of
$b$.

\subsection{Case $b=ctt>0$}

Let $b$ be constant and positive, that is, $b(x)\equiv b>0$.
In this special case we can perform the
Hopf-Cole transformation $v=e^{bu}-1$. The new nonnegative function $v$
satisfies
\begin{equation}\label{eqv}
\begin{gathered}
 -\Delta v = \lambda b(v+1)g(\frac{1}{b}\log(v+1)) \quad\text{in } \Omega\\
 v  =  0 \quad\text{on }\partial\Omega.
 \end{gathered}
\end{equation}
We will denote by $f$ the nonlinearity appearing on the right-hand
side of the equation above, that is, $v$ satisfies
\begin{equation}\label{expf}
-\Delta v =\lambda f(v) \quad \text{in $\Omega$, where }
f(v)=b(v+1)g\big(\frac{1}{b}\log(v+1)\big).
\end{equation}

A first example is the one we obtain letting $g(u)=e^{e^{b u}-1-bu}$ ,
i.e., considering the equation
$$
-\Delta u -b|\nabla u|^2 = \lambda e^{e^{b u}-1-bu}.
$$
For this choice of $g$ the equation for $v$ becomes
$-\Delta v = \lambda be^v,$ the classical exponential
nonlinearity. We know (see \cite{CR}) that every stable weak
solution satisfies $v\in L^{\infty}(\Omega)$ if $n\leq 9$.

Another example is the one we obtain letting $g(u)=e^{\beta u}$ for
some constant $\beta>0$, i.e.,
\begin{equation}\label{522}
\begin{gathered}
-\Delta u -b|\nabla u|^2 = \lambda e^{\beta u}\quad \text{in } \Omega\\
u  =  0 \quad \text{on }\partial\Omega.
\end{gathered}
\end{equation}
Then, the equation for $v$ becomes
$ -\Delta v = \lambda b(v+1)^p,$
where $p=1+\frac{\beta}{b}>1$. This is the classical power nonlinearity
case for $v$. For this equation it is known (see \cite{BV}) that, if
$v$ is a $H_0^1(\Omega)$ semi-stable solution then
\[
v\in L^{\infty}(\Omega) \quad \text{if } \quad
\begin{cases}
n\leq 10 & \text{or} \\
 n>10 & \text{and }  { p< \frac{n-2\sqrt{n-1}}{n-4-2\sqrt{n-1}}};
\end{cases}
\]
that is, if $n\leq 10$ or $10<n<2+\frac{4p}{p-1} + 4\sqrt{\frac{p}{p-1}}$.
In our case, for $p=1+\beta/b$ we have
\begin{equation}\label{vbctt}
v\in L^{\infty}(\Omega) \quad \text{if} \quad n\leq 10\; \text{ or
}\;10<n< 6+ 4\frac{b}{\beta}+4\sqrt{1+\frac{b}{\beta}}.
\end{equation}

Note that our stability assumption on $u$, that is,
the existence of a function $\phi$, positive in $\overline{\Omega}$,
satisfying \eqref{phistable}
is equivalent to the existence of a function $\psi= e^{b u}\phi$,
positive in $\overline{\Omega}$,
satisfying
$$
-\Delta \psi \geq \lambda f'(v)\psi,
$$
where $v=e^{bu}-1$ and $f$ is
given by \eqref{expf}, which is
in turn equivalent to the stability of $v$.

Now, since $v=e^{bu}-1$ we may conclude that, for every stable classical
solution $u$ of \eqref{522},
$$
u\in L^{\infty}(\Omega) \quad\text{if}\quad n\leq 10\; \text{ or
 }\; 10<n<
6+4\frac{b}{\beta}+4\sqrt{1+\frac{b}{\beta}},
$$
and that this is a uniform $L^{\infty}$ estimate for all stable
 solutions (as the one for $v$ in \eqref{vbctt}). In particular this
 establishes a uniform bound for all minimal solutions $u_{\lambda}$
 and therefore yields a sufficient condition for the extremal weak solution $u^*$
 to be in $L^{\infty}(\Omega)$.
That is, we have the following

\begin{proposition}\label{propb+}
Let $b>0$ and $u$ a positive classical stable solution to
\begin{gather*}
-\Delta u -b|\nabla u|^2 =  \lambda e^{\beta u}\quad \text{in } \Omega\\
u  =  0 \quad\text{on }\partial\Omega,
\end{gather*}
where $\lambda>0$ is a parameter. Then
$$
\|u\|_{L^{\infty}(\Omega)}\leq C \quad \text{if} \quad n\leq 10\; \text{ or
 }\; 10<n<
6+4\frac{b}{\beta}+4\sqrt{1+\frac{b}{\beta}},
$$
where $C$ is a
 constant depending only on $n, b, \beta$ and $\Omega$ (in particular
 is independent of $\lambda$).
\end{proposition}


Let us now prove directly this result, in the case where $\beta>b/8$, by
using the equation for $u$ and
the fact that we are assuming $u$ stable. As we will
see, for such $\beta$, we reach the same optimal result. The motivation for the
following calculations is that in the case where $b(x)$ is
non-constant, we are forced to work with the equation for $u$, since
there is, in principle, no transformation to a classical semilinear problem
without terms involving the square of the gradient.
We begin by establishing a technical lemma.

\begin{lemma}\label{lemmaprop}
Let $b>0$ and $u$ be a positive classical solution to
\begin{gather*}
-\Delta u -b|\nabla u|^2 =  \lambda e^{\beta u}\quad \text{in } \Omega\\
u =  0 \quad \text{on }\partial\Omega,
\end{gather*}
where $\lambda>0$ is a parameter and
$\beta>0$. For $\gamma\in {\mathbb N}$ satisfying $\gamma\geq 2$, we have
\begin{equation}\label{intlemma}
\int_{\Omega}|\nabla u|^2e^{\gamma
 bu}(e^{bu}-1)^{2\alpha-2}dx
\leq\frac{\lambda}{b(2\alpha+\gamma-3)}\int_{\Omega}
 e^{(\beta+(2\alpha+\gamma-2)b)u}dx+\lambda L_{\gamma},
\end{equation}
where $\alpha>\frac{3-\gamma}{2}$ is a parameter and $L_{\gamma}$
is a linear combination of the
$\gamma-2$ integrals $\int_{\Omega}e^{(\beta+(2\alpha+k)b)u}dx$,
$k=0,1,\dots, \gamma-3$, with coefficients depending only on
$b,\alpha$ and $\gamma$.
\end{lemma}

\begin{proof}
Let $2\alpha>1$ and $\gamma\geq 2$ an integer. We have
\begin{align*}
 { \int_{\Omega}|\nabla u|^2e^{\gamma bu}(e^{bu}-1)^{2\alpha-2}}
&=  { \int_{\Omega} \nabla u e^{(\gamma-1)bu}\nabla ue^{bu}
 (e^{bu}-1)^{2\alpha-2}}\\
&=   { \int_{\Omega} \nabla u e^{(\gamma-1)bu}
\frac{\nabla (e^{bu}-1)^{2\alpha-1}}{b(2\alpha-1)}}\\
&=   {\int_{\Omega} \frac{
 e^{(\gamma-1)bu}(e^{bu}-1)^{2\alpha-1}}{b(2\alpha-1)}(-\Delta
u-(\gamma-1)b|\nabla u|^2).}
\end{align*}
Using the equation for $u$ we have
\begin{align*}
& \int_{\Omega}|\nabla u|^2e^{\gamma bu}(e^{bu}-1)^{2\alpha-2} \\
&=  { \int_{\Omega} \frac{
 e^{(\gamma-1)bu}(e^{bu}-1)^{2\alpha-1}}{b(2\alpha-1)}(\lambda e^{\beta u}
-(\gamma-2)b|\nabla u|^2)}\\
&\leq   \frac{\lambda}{b(2\alpha-1)}\int_{\Omega}
e^{(\beta+(2\alpha+\gamma-2)b)u}
-\frac{\gamma-2}{2\alpha-1}\int_{\Omega}|\nabla u|^2e^{\gamma
 bu}(e^{bu}-1)^{2\alpha-2}\\
&\quad + \frac{\gamma-2}{2\alpha-1}
\int_{\Omega}|\nabla u|^2e^{(\gamma-1)bu}(e^{bu}-1)^{2\alpha-2}.
 \end{align*}
This yields, adding the left hand side to the second term on the right
hand side, and since $2\alpha+\gamma-3>0$,
\begin{equation} \label{gamma}
\begin{aligned}
\int_{\Omega}|\nabla u|^2e^{\gamma bu}(e^{bu}-1)^{2\alpha-2}
&\leq  \frac{\lambda}{b(2\alpha+\gamma-3)}\int_{\Omega}
e^{(\beta+(2\alpha+\gamma-2)b)u} \\
&\quad + \frac{\gamma-2}{2\alpha+\gamma-3}
 \int_{\Omega}|\nabla u|^2e^{(\gamma-1)bu}(e^{bu}-1)^{2\alpha-2}.
\end{aligned}
\end{equation}
If $\gamma=2$ the second term on the right hand side of \eqref{gamma}
is zero and we conclude \eqref{intlemma} (as desired) with
$L_{\gamma}=0$.
 Otherwise, for $\gamma\in{\mathbb N}$, $\gamma\geq 3$, we may repeat
the computations above with $\gamma$ replaced by $\gamma-1$ to obtain
\begin{align*}
\int_{\Omega}|\nabla u|^2e^{(\gamma-1)bu}(e^{bu}-1)^{2\alpha-2}
&\leq  \frac{\lambda}{b(2\alpha+\gamma-4)}\int_{\Omega}
e^{(\beta+(2\alpha+\gamma-3)b)u} \\
&\quad + \frac{\gamma-3}{2\alpha+\gamma-4}\int_{\Omega}|\nabla u|^2
e^{(\gamma-2)bu}(e^{bu}-1)^{2\alpha-2}.
\end{align*}
Note that the left hand side is the second integral on the right hand
side of \eqref{gamma}, which remains to be controlled. Note also that
the exponent of the exponential function in the first
integral on the right hand side decreases on each iteration. We may
continue this process until we are left with the integral
 of $|\nabla u|^2e^{2bu}(e^{bu}-1)^{2\alpha-2}$. For this integral,
 $$
\int_{\Omega} |\nabla
 u|^2e^{2bu}(e^{bu}-1)^{2\alpha-2}\leq\frac{\lambda}{b(2\alpha-1)}\int_{\Omega}
 e^{(\beta+2\alpha b)u}.
$$
Hence,
\begin{equation*}
\int_{\Omega}|\nabla u|^2e^{\gamma
 bu}(e^{bu}-1)^{2\alpha-2}dx
\leq\frac{\lambda}{b(2\alpha+\gamma-3)}\int_{\Omega}
 e^{(\beta+(2\alpha+\gamma-2)b)u}dx+\lambda L_{\gamma},
\end{equation*}
where $L_{\gamma}$ is a linear combination of the
$\gamma-2$ integrals $\int_{\Omega}e^{(\beta+(2\alpha+k)b)u}dx$,
$k=0,1,\dots, \gamma-3$, with coefficients depending only on
$b,\alpha$ and $\gamma$.
\end{proof}

Next we use the assumption that $u$ is stable according
to Definition \ref{stable5}, that is, there exists a positive
function $\phi$ in $\overline{\Omega}$ such that
$$
-\Delta \phi -2b\nabla u\nabla\phi\geq \lambda \beta e^{\beta u}\phi.
$$
We multiply the previous inequality by $(e^{bu}-1)^{2\alpha}e^{2bu}/\phi$
for $\alpha>0$ and integrate by parts to obtain
\begin{equation} \label{exp}
\begin{aligned}
&\lambda \beta \int_{\Omega} e^{(\beta +2b)u}(e^{bu}-1)^{2\alpha} &\leq \\
& \leq
\int_{\Omega}\nabla\phi\nabla\Big(\frac{e^{2bu}(e^{bu}-1)^{2\alpha}}{\phi}\Big)
- \int_{\Omega}2b\frac{\nabla u\nabla\phi}{\phi}e^{2bu}(e^{bu}-1)^{2\alpha}\\
& =  - \int_{\Omega}
\frac{|\nabla\phi|^2}{\phi^2}e^{2bu}(e^{bu}-1)^{2\alpha}
  +\int_{\Omega} 2b\frac{\nabla u\nabla\phi}{\phi}e^{2bu}(e^{bu}-1)^{2\alpha} \\
&\quad + { \int_{\Omega} 2\alpha b\frac{\nabla
 u\nabla\phi}{\phi}e^{bu}e^{2bu}(e^{bu}-1)^{2\alpha-1}- \int_{\Omega}
2b\frac{\nabla u\nabla\phi}{\phi}e^{2bu}(e^{bu}-1)^{2\alpha}}\\
& =  - \int_{\Omega} \frac{|\nabla\phi|^2}{\phi^2}e^{2bu}(e^{bu}-1)^{2\alpha} +
\int_{\Omega} 2\alpha b\frac{\nabla u\nabla\phi}{\phi}e^{2bu}
 e^{bu}(e^{bu}-1)^{2\alpha-1}\\
& \leq  - \int_{\Omega}
 \frac{|\nabla\phi|^2}{\phi^2}e^{2bu}(e^{bu}-1)^{2\alpha}+
 \alpha^2b^2\int_{\Omega} |\nabla u|^2
 e^{4bu}(e^{bu}-1)^{2\alpha-2} \\
&\quad + \int_{\Omega}\frac{|\nabla\phi|^2}{\phi^2}e^{2bu}(e^{bu}-1)^{2\alpha}\\
  &=  \alpha^2b^2\int_{\Omega} |\nabla u|^2
e^{4bu}(e^{bu}-1)^{2\alpha-2}.
\end{aligned}
\end{equation}
Using Lemma \ref{lemmaprop} with $\gamma=4$ we obtain, if $\alpha>0$,
$$
\int_{\Omega}|\nabla u|^2e^{4bu}(e^{bu}-1)^{2\alpha-2}dx
\leq\frac{\lambda}{b(2\alpha+1)}\int_{\Omega}
 e^{(\beta+(2\alpha+2)b)u}dx+\lambda L_4,
$$
where $L_4$ is a linear combination of 
 $\int_{\Omega}e^{(\beta+2\alpha b)u}dx$ and
 $\int_{\Omega}e^{(\beta+(2\alpha+1)b)u}dx$ with coefficients depending only on
$b$ and $\alpha$.


Therefore, replacing $(e^{bu}-1)^{2\alpha}$ by
$e^{2\alpha bu}$ on the left hand side of \eqref{exp} and combining
all the remaining terms with $L_4$ from above and denoting it by $L$, we obtain
\begin{equation}\label{contab+}
\lambda\beta\int_{\Omega} e^{(\beta+(2\alpha+2)b)u}
\leq \lambda\frac{\alpha^2 b}{2\alpha+1}\int_{\Omega}
 e^{(\beta+(2\alpha+2)b)u}+\lambda L.
\end{equation}
Note that $L$ represents a linear combination of integrals involving the exponential
function $e^{\delta u}$ with exponent $\delta<\beta+(2\alpha+2)b$. Such terms
can be absorbed into the left hand side of \eqref{contab+}. In fact,
if $0<a_1<a_2$ then, for every $\epsilon>0$ there exists a constant
$C_{\epsilon}>0$ such that $e^{a_1u}\leq\epsilon
e^{a_2u}+C_{\epsilon}$ for all $u\in(0,+\infty)$. Hence, for every
$\delta$ as above and every $\epsilon$ there exists $C_{\epsilon,\delta}$ such that
$$
\int_{\Omega}e^{\delta u}\leq \epsilon\int_{\Omega}e^{(\beta+(2\alpha+2)b)u}
+C_{\epsilon,\delta}|\Omega|
$$
Thus, if $\alpha>0$ satisfies
$$
\frac{\alpha^2 b}{2\alpha+1}<\beta,
$$
then $e^{(\beta+(2\alpha+2)b)u}\in L^1(\Omega)$. Solving for $\alpha$
we obtain
\begin{equation}\label{alphabctt}
\alpha<\frac{\beta+\sqrt{\beta(\beta+b)}}{b}.
\end{equation}
 Therefore
$$
e^{\beta u} \in L^q(\Omega) \quad \text{for}\quad
1+3\frac{b}{\beta}<q=2\frac{(\alpha+1)b}{\beta}+1 <3 +
2\frac{b}{\beta}+2\sqrt{1+ \frac{b}{\beta}}.
$$
Note that the function $v=e^{bu}-1$ defined at the beginning of this section
is thus
in $L^{r_1=q\beta/b}(\Omega)$ and $v$ satisfies
$-\Delta v=\lambda b(v+1)^p$ where
$p=1+\beta/b$. Therefore $(v+1)^p\in L^{r_1/p}$ and hence, using the
 equation for $v$, we have that $v\in W^{2,r_1/p}$. Since
 $W^{2,r}\subset L^s$ if $1/s=1/r-2/n$ we obtain that $v\in L^s$ for
 $s=(nr_1)/(pn-2r_1)$. If $s>r_1$, that is,
$n< 2q$, then $v$ is bounded by an iterative procedure.
Hence, $v\in L^{\infty}(\Omega)$ if
$$
n < 6 + 4\frac{b}{\beta} +4\sqrt{1+\frac{b}{\beta}},
$$
as we already knew from \eqref{vbctt}. This was totally expected since
both results are achieved using equivalent assumptions.

Finally, as an example, consider the case $\beta=1$ and $b\equiv 1$ in the
expression above. The equation for $u$ becomes
$$
-\Delta u -|\nabla u|^2=e^u.
$$
The stable solutions $u$ of this equation satisfy
$$
u\in L^{\infty}(\Omega) \quad\text{if}\quad n<
10+4\sqrt{2},\quad \text{that is }\,n\leq 15,
$$
with a uniform $L^{\infty}$ bound as in Proposition \ref{propb+}.

For another example let $\beta= 1$ and $b$ tend to $0$. The equation becomes
$$
-\Delta u = e^u
$$
and the result above yields
$u\in L^{\infty}(\Omega)$ if and only if $n<10$,
which coincides with the result of \cite{CR}.

\begin{remark}\rm
We note here that by perturbing the equation
 $-\Delta u=e^u$ with a quadratic gradient term we actually obtain
 more regularity for stable solutions $u$.
\end{remark}


\subsection{Case $b=ctt<0$}

In this case we use a modified
Hopf-Cole transformation $v=1-e^{bu}$. If $u$ is bounded, the new function
$v$ is positive and bounded by $1$, that is, $0<v<1$ for $u$ bounded.
 We note here that $v=1$ corresponds to $u=\infty$. Moreover, $v$ satisfies
\begin{equation}\label{eqv-}
\begin{gathered}
 -\Delta v =  \lambda|b|(1-v)g(\frac{1}{b}\log(1-v)) \quad\text{in } \Omega\\
 v \geq  0 \quad\text{in } \Omega\\
 v =  0 \quad\text{on }\partial\Omega,
 \end{gathered}
\end{equation}
We will again denote by $f$ the nonlinearity appearing on the right-hand
side of the equation above, that is, $v$ satisfies
$$
-\Delta v =\lambda f(v) \text{ in $\Omega$,  where }
 f(v)=|b|(1-v)g(\frac{1}{b}\log(1-v)) .
$$
Considering the same typical example as in the previous section, we
let $g(u)=e^{\beta u}$ for some constant $\beta>0$,
$$
-\Delta u -b|\nabla u|^2 = e^{\beta u}.
$$
The equation for $v$ becomes
$$
 -\Delta v = \lambda|b|(1-v)^p,
$$
where $p=1+\frac{\beta}{b}$. If $\beta> -b=|b|$ then $p<0$. This is the
case studied by Mignot and Puel \cite{MP} and more recently by
Esposito in \cite{E}. They prove that stable solutions $v$ satisfy
$$
v<1\text{ in }\Omega \quad\text{if}\quad n< 2+\frac{4p}{p-1} + 4\sqrt{\frac{p}{p-1}},
$$
 with a bound for $v$ away from $1$, uniform in $v$. In our case, for
$p=1+ \beta/b$ and $\beta>-b$ we have that
$$
v<1 \text{ in }\Omega \quad\text{if}\quad n<
6+ 4\frac{b}{\beta}+4\sqrt{1+\frac{b}{\beta}}.
$$

\begin{proposition}\label{propb-}
Let $b<0$ be a constant, $\beta>-b$ and $u$ a positive classical stable solution
to
\begin{gather*}
-\Delta u -b|\nabla u|^2 =  \lambda e^{\beta u} \quad \text{in } \Omega\\
u =  0 \quad \text{on }\partial\Omega,
\end{gather*}
 where $\lambda>0$ a parameter. Then
$$
\|u\|_{L^{\infty}(\Omega)}\leq C \quad\text{if}\quad
n<6+4\frac{b}{\beta}+4\sqrt{1+\frac{b}{\beta}},
$$
where $C$ is a
constant depending only on $n, b, \beta$ and $\Omega$ (in particular
is independent of $\lambda$).
\end{proposition}

It is a nontrivial fact to note that, since $b<0$, this result yields
less regularity
for stable solutions
than the one obtained for $b>0$ (recall Proposition \ref{propb+}).
Note that the condition on
the exponent $\beta$ is more restrictive than the one we have in the
case of $b>0$. For example,
if we consider the case $b=-1$,
that is, if $u$ satisfies the equation
$$
-\Delta u +|\nabla u|^2 = e^{\beta u},
$$
we find the assumption $\beta >1$ in Proposition \ref{propb-},
which means that we can not apply the previous result to the equation
$-\Delta u + |\nabla u|^2=e^u$. Nevertheless, for this particular
case, the equation for $v$ would be a linear Poisson equation $-\Delta
v=\lambda |b|$, and therefore $u$ would be bounded for all dimensions.

\section{General $b(x)$}

In this section we study the case of a general bounded function
$b=b(x)$. Let $u$ be a positive solution to
the equation
\begin{equation}\label{eqb}
-\Delta u -b(x)|\nabla u|^2= \lambda g(u)
\end{equation}
with Dirichlet boundary conditions.
We denote by $\underline{b}$ and $\overline{b}$ the infimum and the supremum of
$b(x)$ respectively, that is,
$$
\underline{b}\leq b(x)\leq \overline{b} \quad \text{for every } x\in\Omega.
$$
Equation \eqref{eqb} can no longer be transformed into a classical one, and
there are no known regularity results for stable solutions. Following the
computations we introduced in the previous sections we will study this
equation directly, only with the assumptions on $u$, that is, $u$ is
stable as defined in Definition \ref{stable5}.

We consider the special case where $g(u)=e^{\beta u}$. Then, there exists
$\phi>0$ in $\overline{\Omega}$ such that
\begin{equation}\label{propphi}
-\Delta\phi -2b(x)\nabla u\nabla\phi \geq \lambda e^{\beta u}\phi.
\end{equation}
The first result that we prove is the following.

\begin{proposition}\label{propgenb}
Let $b=b(x)$ be a bounded function such that $\underline{b}\leq b(x)\leq\overline{b}$
for some constants $\underline{b}$ and $\overline{b}$ with $\overline{b}>0$
and $u$ a positive classical
stable solution to
\begin{gather*}
-\Delta u -b(x)|\nabla u|^2  = \lambda e^{\beta u} \quad \text{in }\Omega\\
u =  0 \quad \text{on }\partial\Omega,
\end{gather*}
 where $\Omega\subset\mathbb{R}^n$ is a smooth bounded domain
and $\lambda>0$ is a parameter.
Then, for every positive constants $\delta$ and $\eta$ with
$\delta^2+\eta^2\leq 1$, if
$(\overline{b}-\underline{b})<\frac{\delta^2}{\eta^2}(\eta^2-\frac{\overline{b}}{8})$,
$$
\|e^{u}\|_{L^q(\Omega)}\leq C \quad {\rm for }\quad q <\beta + 2\beta\eta^2+
2\overline{b} +2\sqrt{\beta\eta^2(\beta\eta^2+\overline{b})-2\overline{b}(\overline{b}-\underline{b})
\frac{\eta^2}{\delta^2}},
$$
where $C$ depends only on $n, b$ and $\Omega$ (in particular is
independent of $\lambda$).
 \end{proposition}

We note that we have made no assumptions on the sign of the function $b(x)$.
In fact, the only condition we have on $b(x)$ is the oscillation condition
 involving $\underline{b}$ and $\overline{b}$,
$(\overline{b}-\underline{b})<\frac{\delta^2}{\eta^2}(\eta^2-\frac{\overline{b}}{8})$.
This condition guarantees that the expression
inside the square root above is nonnegative.

In the case that $b>0$ is constant we have that $b\equiv\overline{b}=\underline{b}>0$
and hence $\overline{b}-\underline{b}=0$ and we may choose $\eta\uparrow 1$ and
$\delta\downarrow 0$ to obtain
$$
e^{\beta u}\in L^q(\Omega),\quad \text{for }
 q< 3+2\frac{b}{\beta}+2\sqrt{\frac{b}{\beta}+1},
$$
if $b<8$, which coincides with the result of Proposition \ref{propb+}.

The same regularity result still holds if $b$ has oscillation of order
$\epsilon$, since we may choose $\delta^2=2\overline{b}\sqrt{\overline{b}-\underline{b}}$
which is again of order $\epsilon$ and therefore we can let $\eta$
tend to $1$ and $\delta$ tend to $0$.
As before, we begin by establishing the following estimate:

\begin{lemma}\label{lemmapropgenb}
Let $b(x)\leq \overline{b}$ with $\overline{b}>0$ and $u$ be a positive solution to
\begin{gather*}
-\Delta u -b(x)|\nabla u|^2=  \lambda e^{\beta u}\quad \text{in } \Omega\\
u =  0\quad \text{on }\partial\Omega,
\end{gather*}
 where $\lambda>0$ is a parameter and
$\beta>0$. For $\gamma\in {\mathbb N}$ satisfying $\gamma\geq 2$ have
\begin{equation}\label{intlemmagenb}
\int_{\Omega}|\nabla u|^2e^{\gamma
 \overline{b} u}(e^{\overline{b} u}-1)^{2\alpha-2}dx
\leq\frac{\lambda}{b(2\alpha+\gamma-3)}\int_{\Omega}
 e^{(\beta +(2\alpha+\gamma-2)\overline{b} )u}dx+\lambda L_{\gamma},
\end{equation}
where $\alpha>\frac{1}{2}$ is a parameter and $L_{\gamma}$ is a
linear combination of the
$\gamma-2$ integrals $\int_{\Omega}e^{(\beta+(2\alpha+k)b)u}dx$,
$k=0,1,\dots, \gamma-3$, with coefficients depending only on
$b,\alpha$ and $\gamma$.
\end{lemma}

\begin{proof}
Let $2\alpha>1$ and $\gamma\geq 2$ an integer. We have
\begin{align*}
 { \int_{\Omega}|\nabla u|^2e^{\gamma \overline{b} u}(e^{\overline{b} u}-1)^{2\alpha-2}}
&=  { \int_{\Omega} \nabla u e^{(\gamma-1)\overline{b} u}\nabla ue^{\overline{b} u}
(e^{\overline{b} u}-1)^{2\alpha-2}}\\
& =  \int_{\Omega} \nabla u e^{(\gamma-1)\overline{b} u}
\frac{\nabla (e^{\overline{b} u}-1)^{2\alpha-1}}{b(2\alpha-1)}\\
&=  \int_{\Omega} \frac{
 e^{(\gamma-1)\overline{b} u}(e^{\overline{b} u}-1)^{2\alpha-1}}{\overline{b}(2\alpha-1)}(-\Delta
u-(\gamma-1)\overline{b}|\nabla u|^2).
\end{align*}
Using the equation for $u$ and the fact that $b(x)\leq \overline{b}$ with $\overline{b}>0$
we have
\begin{align*}
& \int_{\Omega}|\nabla u|^2e^{\gamma \overline{b} u}(e^{\overline{b} u}-1)^{2\alpha-2}\\
&=  \int_{\Omega} \frac{
 e^{(\gamma-1)\overline{b} u}(e^{\overline{b} u}-1)^{2\alpha-1}}{\overline{b}(2\alpha-1)}
 (\lambda e^{\beta u}-(\gamma-2)\overline{b}|\nabla u|^2) \\
&\quad + \int_{\Omega} \frac{
 e^{(\gamma-1)\overline{b} u}(e^{\overline{b} u}-1)^{2\alpha-1}}
{\overline{b}(2\alpha-1)}(b-\overline{b})|\nabla u|^2\\
&\leq  { \frac{\lambda}{\overline{b}(2\alpha-1)}\int_{\Omega}
e^{(\beta+(2\alpha+\gamma-2)\overline{b}) u}-}\\
& \quad-\frac{\gamma-2}{2\alpha-1}\int_{\Omega}|\nabla u|^2e^{\gamma
 \overline{b} u}(e^{\overline{b} u}-1)^{2\alpha-2}+ \frac{\gamma-2}{2\alpha-1}
\int_{\Omega}|\nabla u|^2e^{(\gamma-1)\overline{b} u}(e^{\overline{b} u}-1)^{2\alpha-2}.
 \end{align*}
This yields, adding the left hand side to the second term on the right
hand side, and since
$2\alpha+\gamma-3>0$,
\begin{equation} \label{gammagenb}
\begin{aligned}
\int_{\Omega}|\nabla u|^2e^{\gamma \overline{b} u}(e^{\overline{b} u}-1)^{2\alpha-2} 
&\leq  \frac{\lambda}{\overline{b}(2\alpha+\gamma-3)}\int_{\Omega}
e^{(\beta +(2\alpha+\gamma-2)\overline{b})u} \\
&\quad + \frac{\gamma-2}{2\alpha+\gamma-3}\int_{\Omega}|\nabla u|^2
 e^{(\gamma-1)\overline{b} u}(e^{\overline{b} u}-1)^{2\alpha-2}.
\end{aligned}
\end{equation}
If $\gamma=2$ the second term on the right hand side of
\eqref{gammagenb} is zero and we conclude \eqref{intlemmagenb}
(as desired) with $L=0$. Otherwise, for $\gamma\in{\mathbb N}$,
$\gamma\geq 3$, we may repeat the computations
above with $\gamma$ replaced by $\gamma-1$ to obtain
\begin{align*}
&\int_{\Omega}|\nabla u|^2e^{(\gamma-1)\overline{b} u}(e^{\overline{b} u}-1)^{2\alpha-2}\\
&\leq  \frac{\lambda}{\overline{b}(2\alpha+\gamma-4)}\int_{\Omega}
e^{(\beta +(2\alpha+\gamma-3)\overline{b})u}
+ \frac{\gamma-3}{2\alpha+\gamma-4}\int_{\Omega}
|\nabla u|^2e^{(\gamma-2)\overline{b} u}(e^{\overline{b} u}-1)^{2\alpha-2}.
\end{align*}
Note that the left hand side is the second integral on the right hand
side of \eqref{gammagenb}, which remains to be controlled.
Note also that the exponent of the exponential function in the first
integral on the right hand side decreases on each iteration. We may
continue this process until we are left with the integral
 of $|\nabla u|^2e^{2\overline{b} u}(e^{\overline{b} u}-1)^{2\alpha-2}$. For this integral,
 $$
\int_{\Omega} |\nabla
 u|^2e^{2\overline{b} u}(e^{\overline{b} u}-1)^{2\alpha-2}
\leq\frac{\lambda}{\overline{b}(2\alpha-1)}\int_{\Omega}
 e^{(1+2\alpha \overline{b})u}.
$$
Hence,
\begin{equation*}
\int_{\Omega}|\nabla u|^2e^{\gamma
 \overline{b} u}(e^{\overline{b} u}-1)^{2\alpha-2}dx
\leq\frac{\lambda}{\overline{b}(2\alpha+\gamma-3)}\int_{\Omega}
 e^{(\beta +(2\alpha+\gamma-2)\overline{b})u}dx+\lambda L_{\gamma},
\end{equation*}
where $L_{\gamma}$ is a linear combination of the
$\gamma-2$ integrals $\int_{\Omega}e^{(\beta+(2\alpha+k)b)u}dx$ for
$k=0,1,\dots, \gamma-3$, with coefficients depending only on
$b,\alpha$ and $\gamma$.
\end{proof}

We now prove the proposition. We follow the computations as in the
proof of Proposition \ref{propb+}.

\begin{proof}
The assumption we have made on $u$ is that it is stable according
to Definition \ref{stable5}, that is, there exists a positive
function $\phi$ in $\overline{\Omega}$ such that
$$
-\Delta \phi -2b(x)\nabla u\nabla\phi\geq \lambda\beta e^{\beta u}\phi.
$$
We multiply the previous inequality by
$(e^{\overline{b} u}-1)^{2\alpha}e^{2\overline{b} u}/\phi$ for $\alpha>0$
and integrate by parts to obtain
\begin{align*} %\label{expgenb}
&\lambda \beta \int_{\Omega} e^{(\beta +2\overline{b})u}(e^{\overline{b} u}-1)^{2\alpha}\\
&\leq  \int_{\Omega}\nabla\phi\nabla
\Big(\frac{e^{2\overline{b} u}(e^{\overline{b} u}-1)^{2\alpha}}{\phi}\Big)
- \int_{\Omega} 2b\frac{\nabla u\nabla\phi}{\phi}e^{2\overline{b} u}
 (e^{\overline{b} u}-1)^{2\alpha}\\
&=  - \int_{\Omega}\frac{|\nabla\phi|^2}{\phi^2}e^{2\overline{b} u}(e^{\overline{b} u}-1)^{2\alpha}
 +\int_{\Omega} 2\overline{b}\frac{\nabla
 u\nabla\phi}{\phi}e^{2\overline{b} u}(e^{\overline{b} u}-1)^{2\alpha} \\
& \quad +{ \int_{\Omega} 2\alpha \overline{b}\frac{\nabla
 u\nabla\phi}{\phi}e^{\overline{b} u}e^{2\overline{b} u}(e^{\overline{b} u}-1)^{2\alpha-1}
 - \int_{\Omega}2b\frac{\nabla u\nabla\phi}{\phi}e^{2\overline{b} u}
 (e^{\overline{b} u}-1)^{2\alpha}}\\
& =  - \int_{\Omega}
\frac{|\nabla\phi|^2}{\phi^2}e^{2\overline{b} u}(e^{\overline{b} u}-1)^{2\alpha}
 + \int_{\Omega} 2\alpha \overline{b}\frac{\nabla u\nabla\phi}{\phi}
 e^{2\overline{b} u}e^{\overline{b} u}(e^{\overline{b} u}-1)^{2\alpha-1}\\
&\quad + \int_{\Omega} 2(\overline{b}-b)\frac{\nabla u\nabla\phi}
 {\phi}e^{2\overline{b} u}(e^{\overline{b} u}-1)^{2\alpha}\\
& \leq  (\delta^2+\eta^2- 1)\int_{\Omega}
\frac{|\nabla\phi|^2}{\phi^2}e^{2\overline{b} u}(e^{\overline{b} u}-1)^{2\alpha} \\
 & \quad +\frac{2\overline{b}(\overline{b}-\underline{b})}{\delta^2}\int_{\Omega} |\nabla u|^2
e^{2\overline{b} u}(e^{\overline{b} u}-1)^{2\alpha}
 + \frac{\alpha^2\overline{b}^2}{\eta^2}\int_{\Omega} |\nabla u|^2
e^{4\overline{b} u}(e^{\overline{b} u}-1)^{2\alpha-2} \\
&=  (\delta^2+\eta^2- 1)\int_{\Omega}
\frac{|\nabla\phi|^2}{\phi^2}e^{2\overline{b} u}(e^{\overline{b} u}-1)^{2\alpha}+\\
 & \quad + (\frac{2\overline{b}(\overline{b}-\underline{b})}{\delta^2}
 +\frac{\alpha^2\overline{b}^2}{\eta^2})\int_{\Omega} |\nabla u|^2
 e^{4\overline{b} u}(e^{\overline{b} u}-1)^{2\alpha-2},
\end{align*}
where $\delta>0$ and $\eta>0$ are constants.
Using Lemma \ref{lemmapropgenb} with $\gamma=4$ we obtain, for $\alpha>0$,
$$
\int|\nabla u|^2e^{4\overline{b} u}(e^{\overline{b} u}-1)^{2\alpha-2}dx
\leq\frac{\lambda}{\overline{b}(2\alpha+1)}\int
 e^{(1+(2\alpha+2)\overline{b})u}dx+\lambda L_4,
$$
where $L_4$ is a linear combination of 
$\int_{\Omega}e^{(\beta+(2\alpha)b)u}dx$ and
 $\int_{\Omega}e^{(\beta+(2\alpha+1)b)u}dx$ with coefficients depending only on
$b$ and $\alpha$.


Therefore, replacing $(e^{\overline{b} u}-1)^{2\alpha}$ by
$e^{2\alpha \overline{b} u}$ on the left hand side of \eqref{exp} and combining
all the remaining terms with $L_4$ from above and denoting it by $L$, we obtain
\begin{equation} \label{contagenb}
\begin{aligned}
\lambda\beta\int e^{(\beta+(2\alpha+2)\overline{b})u}
&\leq (\delta^2+\eta^2- 1)\int
\frac{|\nabla\phi|^2}{\phi^2}e^{2\overline{b} u}(e^{\overline{b} u}-1)^{2\alpha} \\
&\quad + (\frac{2\overline{b}(\overline{b}-\underline{b})}{\delta^2}+\frac{\alpha^2\overline{b}^2}{\eta^2})
\frac{\lambda}{\overline{b}(2\alpha+1)}\int e^{(\beta+(2\alpha+2)\overline{b})u}+\lambda L.
\end{aligned}
\end{equation}
Note that $L$ represents a linear combination of integrals involving the exponential
function $e^{\delta u}$ with exponent $\delta<\beta+(2\alpha+2)\overline{b}$. Such terms
can be absorbed into the left hand side of \eqref{contagenb}. In fact,
if $0<a_1<a_2$ then, for every $\epsilon>0$ there exists a constant
$C_{\epsilon}>0$ such that $e^{a_1u}\leq\epsilon
e^{a_2u}+C_{\epsilon}$ for all $u\in(0,+\infty)$. Hence, for every
$\delta$ as above and every $\epsilon$ there exists $C_{\epsilon,\delta}$ such that
$$
\int_{\Omega}e^{\delta u}\leq \epsilon\int_{\Omega}e^{(\beta+(2\alpha+2)\overline{b})u}
+C_{\epsilon,\delta}|\Omega|
$$
Thus, if $\delta^2+\eta^2\leq 1$ and $\alpha>1/2$ satisfies
$$
(\frac{2\overline{b}(\overline{b}-\underline{b})}{\delta^2}+\frac{\alpha^2\overline{b}^2}{\eta^2})
\frac{1}{\overline{b}(2\alpha+1)}<\beta,
$$
then $e^{(\beta+(2\alpha+2)\overline{b})u}\in L^1(\Omega)$. Solving for $\alpha$
we obtain
\begin{equation}\label{alphagenb}
\frac{1}{2}<\alpha<\frac{\beta\eta^2+\sqrt{\beta\eta^2(\beta\eta^2+\overline{b})
-2\overline{b}(\overline{b}-\underline{b})\frac{\eta^2}{\delta^2}}}{\overline{b}}.
\end{equation}
This inequality is satisfied for some $\alpha$ since
$(\overline{b}-\underline{b})<\frac{\delta^2}{\eta^2}(\eta^2-\frac{\overline{b}}{8})$. Therefore
\begin{equation}\label{qbx}
\|e^{u}\|_{L^q(\Omega)}\leq C \quad \text{for }
 q<\beta + 2\beta\eta^2+
2\overline{b} +2\sqrt{\beta\eta^2(\beta\eta^2+\overline{b})
-2\overline{b}(\overline{b}-\underline{b})\frac{\eta^2}{\delta^2}},
\end{equation}
where $C$ is independent of $\lambda$.
\end{proof}


\begin{remark}\label{notestable} \rm
We note that we can perform all the computations if we assume only that
there exists a function $\phi_{\epsilon}$, positive in $\overline{\Omega}$,
such that
$$
-\Delta\phi_{\epsilon} -2b(x)\nabla u\nabla\phi_{\epsilon}\geq
(\lambda -\epsilon)e^u\phi_{\epsilon},
$$
for some small $\epsilon>0$. Letting $\epsilon$ tend to $0$ we obtain
the result above with the constant $C$ independent of $\epsilon$.
\end{remark}

\section{Further regularity for $b(x)\geq 0$}\label{sctreg}

In the case where the function $b$ is non-negative, we can reach
further regularity and prove a similar result to the one where $b$ is
constant, even though we are not able to transform the equation into a
classical one. Using a well chosen Hopf-Cole
transformation we can construct a subsolution of the
classical equation with a power nonlinearity. Using a bootstrap
argument and Proposition \ref{propgenb}
 this is enough to
conclude about the regularity of $u$.


\begin{proposition}\label{propreg}
Let $b(x)\geq 0$ and $0\leq \underline{b}\leq b(x)\leq\overline{b}$ in $\Omega$ for
some constants $\underline{b}$ and $\overline{b}$, and $u$ a positive classical stable
solution of
\begin{gather*}
-\Delta u-b(x)|\nabla u|^2=  \lambda e^u \quad \text{ in }\Omega\\
u= 0 \quad \text{ on }\partial\Omega,
\end{gather*}
 and $\lambda>0$ a parameter.
For every positive constants $\delta$ and $\eta$ with
$\delta^2+\eta^2\leq 1$, let
$(\overline{b}-\underline{b})<\frac{\delta^2}{\eta^2}(\eta^2-\frac{\overline{b}}{8})$ and
$n <2 + 4\eta^2+4\overline{b} +4\sqrt{\eta^2(\eta^2+\overline{b})-2\overline{b}(\overline{b}-\underline{b})
\frac{\eta^2}{\delta^2}}$.
Then, $\|u\|_{L^{\infty}(\Omega)}\leq C$, where $C$ depends only on $n, b$
and $\Omega$.
\end{proposition}

\begin{proof}
Consider the Hopf-Cole transformation $v=e^{\overline{b} u}-1$. We have that
\begin{align*}
-\Delta v
&=  \overline{b} e^{\overline{b} u}(-\Delta u - \overline{b}|\nabla u|^2)\\
&=  \overline{b} e^{\overline{b} u}(\lambda e^u+ (b(x)-\overline{b} )|\nabla u|^2)\\
&\leq  \lambda\overline{b} e^{\overline{b} u}e^u\\
&=  \lambda\overline{b} (v+1)^{\frac{\overline{b} +1}{\overline{b} }}
\end{align*}
which means that
 $v$ is a positive subsolution of the classical equation, i.e.,
$$
-\Delta v \leq \lambda\overline{b} (v+1)^p \quad \text{ in } \Omega,
$$
for $p=(\overline{b} +1)/\overline{b} $. Let $w$ be the solution to the linear problem
\begin{gather*}
-\Delta w =  \lambda\overline{b} (v+1)^p \quad \text{in }\Omega\\
w =  v \quad \text{in }\partial\Omega.
\end{gather*}
Then, trivially $-\Delta v\leq-\Delta w$ and hence, by the maximum principle
$$
0\leq v\leq w.
$$
If $v\in L^s(\Omega)$
then, using the equation for $w$ we obtain that
$w\in W^{2,s/p}(\Omega)\subset L^r(\Omega)$ for $r=(ns)/(np-2q)$. Now, $r>s$ if
$n<2s/(p-1)$. Therefore, by a bootstrap argument, $w$ and hence $v$ is in
$L^{\infty}(\Omega)$ if $n<2s/(p-1)$, that is, $n< 2\overline{b} s$.


From the previous section we know that $e^u\in L^q(\Omega)$ for $q$ given by
\eqref{qbx}. Given the definition of $v$ we obtain that
$v\in L^{q/\overline{b}}(\Omega)$,
i.e., we can replace $s=q/\overline{b}$ in the discussion above. Thus we obtain
that $v$ and hence $u$ are in $L^{\infty}(\Omega)$ if $n< 2q$, that is,
$$
u\in L^{\infty}(\Omega)\quad \text{if}\quad n <2 + 4\eta^2+
4\overline{b}+4\sqrt{\eta^2(\eta^2+\overline{b})-2\overline{b}(\overline{b}-\underline{b})\frac{\eta^2}{\delta^2}},$$
with $\delta^2+\eta^2\leq 1$.
\end{proof}

\section{Existence for $b(x)\geq 0$}

In this section we prove an existence theorem,
in terms of $\lambda$, for solutions to the problem
\begin{equation}\label{lambdau}
\begin{gathered}
 -\Delta u-b(x)|\nabla u|^2=  \lambda g(u) \quad\text{in } \Omega \\
 u  \geq  0 \quad\text{in } \Omega\\
 u =  0 \quad \text{on }\partial\Omega,
\end{gathered}
\end{equation}
where $g$ is a nonlinearity with assumptions to be detailed later,
 $b(x)\geq 0$ and $\Omega\subset\mathbb{R}^n$ is a smooth bounded domain with $n\geq 2$.

If $b(x)=b$ is constant, using the Hopf-Cole transformation we
reach an equation for $v$ of the form
\begin{equation}\label{lambdav}
\begin{gathered}
-\Delta v =  \lambda f(v) \quad \text{in } \Omega\\
v =  0 \quad \text{in }\partial\Omega.
\end{gathered}
\end{equation}
 where $f(v)=b(v+1)g(\frac{1}{b}\ln(v+1))$.
Equations of the type \eqref{lambdav} have been extensively studied.
Under the following conditions on $f$:
\begin{equation}\label{condf}
\text{$f$ is $C^1$, convex, nondecreasing $f(0)>0$  and }
\lim_{v\to +\infty}\frac{f(v)}{v}=+\infty,
\end{equation} 
there exists a finite parameter $\lambda^*>0$ such
that, for $\lambda>\lambda^*$ there is no bounded solution to
\eqref{lambdav}. On the other hand, for $0<\lambda<\lambda^*$ there
exists a minimal bounded solution $v_{\lambda}$, where minimal means
smallest.

These conditions hold for $f$ if we assume that $g$ satisfies:
\begin{equation}\label{condg2}
\text{$g$ is $C^1$ convex, nondecreasing, $g(0)>0$ and }
\lim_{u\to +\infty}g(u)=+\infty.
\end{equation}
In the general case for a non-negative function $b$ we prove the
following theorem.

 \begin{theorem}\label{prop1}
Let $b=b(x)\geq 0$ be a $C^{\alpha}(\overline{\Omega})$ function defined 
in a smooth bounded domain $\Omega\subset\mathbb{R}^n$, and let
$g$ be a nondecreasing $C^1$ function with $g(0)>0$ and
$\lim_{u\to+\infty} g(u)/u=+\infty$. Then, there
exists a parameter $0<\lambda^{*}<\infty $ such that:
\begin{itemize}
\item[(a)] If $\lambda > \lambda^{*}$ then there is no
classical solution of
\eqref{lambdau}.
\item[(b)] If $0\leq \lambda < \lambda^{*}$ then there exists a minimal
classical solution $u_\lambda$ of \eqref{lambdau}.
Moreover, $u_\lambda<u_\mu$ if $\lambda<\mu<\lambda^*$.
\end{itemize}
In addition, if $g(u)=e^u$ and for every positive constants $\delta$
and $\eta$ with $\delta^2+\eta^2\leq 1$, the function $b$ satisfies 
$0\leq\underline{b}\leq b \leq\overline{b}$ in $\Omega$ for constants $\underline{b}$ and 
$\overline{b}$ such that
$(\overline{b}-\underline{b})<\frac{\delta^2}{\eta^2}(\eta^2-\frac{\overline{b}}{8})$ and
$n <2 + 4\eta^2+4\overline{b}
+4\sqrt{\eta^2(\eta^2+\overline{b})-2\overline{b}(\overline{b}-\underline{b})\frac{\eta^2}{\delta^2}}$,
then $u_\lambda$ is semi-stable. Moreover, the limit
$u^*=\lim_{\lambda\to\lambda^*}u_{\lambda}$ is a weak solution
of \eqref{lambdau} for $\lambda=\lambda^*$. That is, it satisfies
$$
-\int_{\Omega}u^*\Delta\xi -\int_{\Omega}b(x)|\nabla u^*|^2 \xi
=\lambda^*\int_{\Omega}e^{u^*}\xi,
$$
for every $\xi\in C^2(\overline{\Omega})$ with $\xi= 0$ on
$\partial\Omega$. In addition, the estimates of Proposition
\ref{propreg} apply to $u^*$.
\end{theorem}

\begin{proof}
First, we prove that there is no classical solution for large
$\lambda$. Let $u_{\lambda}$ be a bounded solution corresponding
to $\lambda$. Then, since $b\geq 0$ this function $u_{\lambda}$
is a supersolution of the classical problem
\begin{gather*}
-\Delta u \geq \lambda g(u) \quad \text{in }\Omega\\
u =  0 \quad \text{ on }\partial\Omega.
\end{gather*}
Since $g(0)>0$, $\underline{u}=0$ is a strict subsolution for every
$\lambda>0$. This would imply the existence of a classical solution
corresponding to $\lambda$ between $0$ and our supersolution
$u_{\lambda}$. We know this is only possible for $\lambda$ smaller
than a finite extremal parameter,
hence the same applies for the solutions to our problem
\eqref{lambdau}.

Next, we prove the existence of a classical solution of \eqref{lambdau}
for small $\lambda$. For general $\lambda$, the existence of a bounded supersolution
implies the existence of
a minimal (smallest) classical solution $u_\lambda$. 
This solution is obtained by monotone iteration
starting from $0$. That is, $u_\lambda$ is the increasing limit of
$u_m$ where the functions $u_m$ are defined as $u_0\equiv 0$ and, for
$m\geq 1$
\begin{gather*}
-\Delta u_m-b(x)|\nabla u_m|^2 =  \lambda g(u_{m-1}) \quad\text{in }
\Omega\\
u_m = 0 \quad \text{on }\partial\Omega.
\end{gather*}
The equation for $u_m$ may be written as
$$
-\Delta u_m= F(x,u_m,\nabla u_m)
$$ 
where $F$ satisfies
$$
|F(x,u_m,\xi)|\leq K(1+|\xi|^2),
$$ 
for some constant $K$, since $b$
is a bounded function and, at step $m$, the function $u_{m-1}$ is
known and bounded. For this equation and under such conditions on $F$ we have
existence of solution $u_m\in W^{2,p}(\Omega)$ for every $p>1$ 
(see \cite{ACr}) and this
implies, for $p$ large, that $u_m\in C^{1,\alpha}(\overline{\Omega})$. 
Moreover $C^{1,\alpha}(\overline{\Omega})$ is compactly
embedded in $C^1(\overline{\Omega})$.

We will prove by induction that this sequence $u_m$ is increasing. For
$m=1$ we have that
$$
-\Delta u_1-b(x)|\nabla u_1|^2=\lambda g(0)>0=-\Delta
u_0-b(x)|\nabla u_0|^2,
$$ 
which implies, for $b(x)\geq 0$ and since
$u_0\equiv 0$,
$$
-\Delta u_1> -\Delta u_0.
$$
By the classical maximum principle we have $u_1\geq u_0$.

Now assume $u_m\geq u_{m-1}$. Then
\begin{align*}
-\Delta u_{m+1}-b(x)|\nabla u_{m+1}|^2 
&=  \lambda g(u_m) \\
& \geq  \lambda g(u_{m-1})\\
&=  -\Delta u_m-b(x)|\nabla u_m|^2,
\end{align*}
where we have used that $g$ is nondecreasing. Let
$w=u_{m+1}-u_m$. From the previous inequality we derive an inequality
satisfied by $w$. Namely,
$$
-\Delta w - \vec{B}(x)\cdot\nabla w\geq 0,
$$
where $\vec{B}(x)=b(x)\nabla (u_{m+1}+u_m)$. 
By the maximum principle we have that $w\geq 0$, that is, $u_{m+1}\geq u_m$.
Therefore we have constructed an increasing sequence $u_m$.

Let now $\overline{u}$ be the solution of
\begin{gather*}
 -\Delta \overline{u} -b(x)|\nabla\overline{u}|^2=1 \quad \text{in } B_{1} \\
 \overline{u}=0 \quad \text{on }\partial B_{1}.
\end{gather*}
This $\overline{u}$ is a bounded supersolution of \eqref{lambdau}
for small $\lambda$,
whenever $\lambda g(\max \overline{u}) < 1$.

Using induction and the maximum principle as above we can prove that
the sequence is bounded by $\overline{u}$, i.e.,
$$
u_0\leq u_1\leq \dots \leq u_m\leq u_{m+1}\leq \dots \leq \overline{u}.
$$
This implies there exists a limit,
$$
u_{\lambda}:=\lim_{m\to\infty} u_m,
$$
and moreover, $u_{\lambda}$ is a solution to \eqref{lambdau}. 
In fact, since $u_m\in W^{2,p}(\Omega)$ we obtain that, for $p$ 
large, $u_m\in C^{1,\alpha}(\overline{\Omega})$. Using the
equation and the fact that $b\in C^{\alpha}(\overline{\Omega})$, 
we obtain that $u_m\in C^{2,\alpha}(\overline{\Omega})$ and hence 
converges to a solution of \eqref{lambdau}.

The extremal parameter $\lambda^*$ is now defined as the supremum of
all $\lambda>0$ for which \eqref{lambdau} admits a classical solution. Hence,
both $0<\lambda^*<\infty$ and part (a) of the proposition holds.


(b) Next, if $\lambda<\lambda^*$ there exists $\mu$ with
$\lambda<\mu<\lambda^*$ and such that \eqref{lambdau} admits a classical
solution $u_{\mu}$. Since $g>0$, $u_{\mu}$ is a bounded supersolution of
\eqref{lambdau}, and hence the same monotone iteration argument used above
shows that \eqref{lambdau} admits a classical solution $u_\lambda$ with
$u_\lambda\le u$.
In addition, we have shown that
$u_\lambda$ is smaller than any classical supersolution of \eqref{lambdau}.
It follows that $u_\lambda$ is minimal (i.e., the smallest solution)
and that $u_\lambda<u_\mu$.

Consider now the case where $g(u)=e^u$, and assume that for every positive
constants $\delta$ and $\eta$ with $\delta^2+\eta^2\leq 1$ we have
 $$
(\overline{b}-\underline{b})<\frac{\delta^2}{\eta^2}(\eta^2-\frac{\overline{b}}{8})\;
\text{ and }\; n <2 + 4\eta^2+4\overline{b}
+4\sqrt{\eta^2(\eta^2+\overline{b})-2\overline{b}(\overline{b}-\underline{b})\frac{\eta^2}{\delta^2}}.
$$
First we prove that
$u_{\lambda}$ is semi-stable, meaning by semi-stable that the first
eigenvalue $\lambda_1$ of the linearized operator $L_{\lambda}$ is
non-negative. That is,
$$
\lambda_1(L_{\lambda})\geq 0\quad \text{ where } \; L_{\lambda}:=-\Delta -2b(x)\nabla u_{\lambda}\nabla -\lambda
e^{u_{\lambda}}.
$$
We have seen that $u_{\lambda}$ forms an increasing sequence with
respect to $\lambda$. For $\delta>0$ let
$v_{\delta}=u_{\lambda+\delta}-u_{\lambda}>0$. Using the equations for
$u_{\lambda+\delta}$ and $u_{\lambda}$ we have that $v_{\delta}$
satisfies
$ -\Delta v_{\delta} -2b(x)\nabla
(\frac{u_{\lambda+\delta}+u_{\lambda}}{2})\nabla v_{\delta}
- \lambda e^{u_{\lambda}}v_{\delta} >0,$
where $\eta$ is between $u_{\lambda}$ and $u_{\lambda+\delta}$ and we
have used that $\delta>0$. Therefore, if we define the linear operator
$$
L_{\lambda,\delta}:=-\Delta
-2b(x)\nabla\Big(\frac{u_{\lambda+\delta}+u_{\lambda}}{2}\Big)\nabla
-\lambda e^{u_{\lambda}},
$$
we have that, at $v_{\delta}$,
$$
L_{\lambda,\delta} v_{\delta} = -\Delta v_{\delta} -2b(x)\nabla
\Big(\frac{u_{\lambda+\delta}+u_{\lambda}}{2}\Big)\nabla v_{\delta}
- \lambda e^{u_{\lambda}}v_{\delta} >0 \;\text{ in }\;\Omega,
$$
and thus $v_{\delta}$ is a strict supersolution positive in $\Omega$ of
$L_{\lambda,\delta} = 0$ in $\Omega$
and hence $\lambda_1(L_{\lambda,\delta})>0$.

Now we pass to the limit in $\delta$ and obtain that
$\lambda_1(L_{\lambda})\geq 0$, that is, $u_{\lambda}$ is semi-stable
as defined above. This can
be done using Propositions 2.1 and 5.1 of \cite{BNV} which establishes that, for
bounded coefficients,
$\lambda_1$ is Lispchitz continuous with respect to both the first and
the zeroth order coefficients.

For every $\epsilon>0$, since $\lambda_1(L_{\lambda})\geq 0$ we have that
$\lambda_1(L_{\lambda}-\epsilon)>0$. This implies there exists a
function $\phi_{\epsilon}$, positive in $\overline{\Omega}$, as in
Remark \ref{notestable}. Hence we have that
$\|u_{\lambda}\|_{ L^{\infty}(\Omega)}\leq C$, where $C$ is 
independent of $\lambda$.

Under the same conditions as above, we can establish that the limiting
function $u^*=\lim_{\lambda\to\lambda^*} u_{\lambda}$ is a
weak solution to \eqref{lambdau} with $\lambda=\lambda^*$. Just use
the weak formulation for $u_{\lambda}$ and the fact that
$u_{\lambda}\in L^{\infty}(\Omega)$, so that we can take limits in $\lambda$
and obtain that $u^*$ is a weak solution.
Therefore using the $L^{\infty}$ uniform bound on $u_{\lambda}$ we 
have $\|u^*\|_{L^{\infty}}\leq C$.
 \end{proof}

\section{$H^1$ regularity}

In this section we study the $H^1$ regularity of positive solutions to the equation
\begin{equation}\label{equ}
-\Delta u -b(x)|\nabla u|^2=\lambda g(u)\quad {\rm in} \; \Omega,
\end{equation}
such that $u\equiv 0$ on $\partial\Omega$, $\Omega\subset\mathbb{R}^n$ is a bounded
domain, $g\geq 0$ and $g'>0$ in $\Omega$.

We consider two cases.
\smallskip

\noindent\textbf{Case 1: $b(x)=b>0$ is constant.}
In this setting one can use the Hopf-Cole transformation and study
the resulting equation for the new function $v$. Then, using the
results of \cite{BV}, we have that $v$ is in $H^1$ if it is stable
and the nonlinearity $f$ satisfies
\begin{equation}\label{condfH1}
\liminf_{s\to\infty}\frac{f'(s)s}{f(s)}>1.
\end{equation}
This condition could be rewritten in terms of the nonlinearity $g(u)$ allowing
us to conclude that $e^u$, and hence $u$, is in $H^1$. It is natural
to expect that such assumptions on $g$ will be too restrictive, since they
give a condition for $e^u$, and not just $u$, to be in $H^1$. 
In what follows we study directly the problem for $u$ and find the 
natural conditions to impose on $g$.


\begin{proposition}\label{teoH1b}
Let $b>0$ be a constant and $u$ a positive classical solution of the problem
$-\Delta u -b|\nabla u|^2 = \lambda g(u)$ with zero Dirichlet boundary
conditions, $g\geq 0$ and $g'>0$ in $\Omega$ and $\lambda>0$ a parameter.
Assume that $u$ is a stable solution
Then, if
$$
\liminf_{s\to\infty}\frac{g'(s)(e^{bs}-1)}{bg(s)}>1,
$$
we have that $\|u\\|_{H^1(\Omega)}\leq C$ where $C$ is independent of $\lambda$.
\end{proposition}

To better understand the above condition on $g$, let us consider the case
where equality holds, i.e.,
$$
\frac{g'(s)(e^{bs}-1)}{bg(s)}=1.
$$
Integrating we obtain $\log g(s)=\log (e^{bs} -1) -bs +C$ for some constant
$C$ and hence,
$g(s)=C(1-e^{-bs})$.
Recall that $b>0$ so  $g$ is bounded.

As we mentioned before, this condition on $g$ is less restrictive than
the one imposed via $f$. In fact, if $g(u)=e^u$ then $u$ is in
$H^1$ by the previous theorem. However, if we pass to the equation for 
$v=e^{bu}-1$ we have that
$-\Delta v=\lambda f(v)$ with $f(v)=b(v+1)^p$, $p=1+1/b$ and $f$ does
not satisfy condition \eqref{condfH1} of \cite{BV}.

\begin{proof}[Proof of Proposition \ref{teoH1b}]
Since $u$ is stable there exists a positive function $\phi$ on
$\overline{\Omega}$ such that
$$
-\Delta\phi-2b\nabla u\nabla\phi\geq \lambda g'(u)\phi.
$$
Multiplying by $(e^{bu}-1)^2/\phi$ and integrate in $\Omega$.
\begin{align*}
&\lambda\int_{\Omega} g'(u)(e^{bu}-1)^2\\
&\leq  \int_{\Omega}  -\frac{|\nabla\phi|^2}{\phi^2}(e^{b u}-1)^2 
 + \int_{\Omega} 2b\frac{\nabla\phi}{\phi}\nabla u e^{b u}(e^{b u}-1) 
 - \int_{\Omega}  2b\frac{\nabla\phi}{\phi}\nabla u (e^{b u}-1)^2 \\
&=  \int_{\Omega} -\frac{|\nabla\phi|^2}{\phi^2}(e^{b u}-1)^2 +
\int_{\Omega} 2b\frac{\nabla\phi}{\phi}\nabla u (e^{b u}-1) \\
&\leq  \int_{\Omega} b^2|\nabla u|^2.
\end{align*}
On the other hand, multiplying \eqref{equ} by $e^{bu}-1$ and
integrating we obtain
 \begin{equation}\label{demo1}
 { \lambda\int_{\Omega} g(u)(e^{bu}-1)} 
=  { \int_{\Omega} \nabla u\nabla(e^{bu}-1) -
 b|\nabla u|^2(e^{bu}-1)} 
= { b\int_{\Omega}|\nabla u|^2.}
\end{equation}
Thus, we have 
\begin{equation}\label{intg}
\lambda \int_{\Omega} g'(u)(e^{bu}-1)^2 \leq \lambda b\int_{\Omega} g(u)(e^{bu}-1).
\end{equation}
Assume that
\begin{equation}\label{condg}
bg(s)(e^{bs}-1)\leq \delta g'(s)(e^{bs}-1)^2 + C
\end{equation}
for some constant $\delta<1$ and some constant $C$.
Then, from \eqref{intg} we obtain that
$$
\int_{\Omega} g'(u)(e^{bu}-1)^2 \leq C,
$$
which implies both
$$
\int_{\Omega} g(u)(e^{bu}-1) \leq C \quad 
\text{and}\quad \int_{\Omega} |\nabla
u|^2\leq C, \quad \text{by \eqref{demo1}}.
$$

Now, going back to \eqref{condg} we see that, for $s$ small it is
always possible to find $\delta$ and $C$. The problem occurs when $s$
tends to infinity (that is, when $u$ is unbounded).
It is easy to see that \eqref{condg} holds if
\begin{equation}\label{cond}
\liminf_{s\to\infty} \frac{g'(s)(e^{bs}-1)}{bg(s)}\geq\frac{1}{\delta}>1.
\end{equation}
\end{proof}



\noindent\textbf{Case 2: $b(x)\leq -\epsilon<0$.}
This case is, in some sense, more general than the previous one since
we do not need to assume that $u$ is a stable solution.
The proof uses a technique due to Boccardo (see \cite{B})
involving truncations. For a function $u$ we define the truncation
$T_1u$ as
\begin{equation}\label{T1}
T_1u= \begin{cases}
 1, & u>1 \\
 u, & |u|\leq 1 \\
 -1, & u<-1.
 \end{cases}
\end{equation}
We have $\nabla T_1u=\nabla u$ where $|u|\leq 1$ and $\nabla T_1=0$ otherwise.

\begin{proposition}\label{teoH1b-}
Let $b(x)\leq -\epsilon<0$ for some $\epsilon>0$ and $u$ a positive classical
solution to the problem $-\Delta u - b(x)|\nabla u|^2=\lambda g(u)$
with zero Dirichlet boundary conditions, $\lambda>0$ a parameter, and
assume that $g(u)\in L^1(\Omega)$.
Then, $\|u\|_{H^1(\Omega)}\leq C$, where $C$ is independent of $\lambda$.
\end{proposition}

\begin{proof}
 We multiply equation
\eqref{equ} by $T_1u$ and integrate by parts
$$
\int_{\Omega} \nabla u\nabla T_1u 
-\int_{\Omega} b(x)|\nabla u|^2 T_1u=\lambda \int_{\Omega} g(u)T_1u.
$$
Given the definition of $T_1u$ this yields
$$
\int_{\{|u|\leq 1\}} |\nabla u|^2 = \lambda \int_{\Omega} g(u)T_1u + \int_{\Omega}
b(x)T_1u|\nabla u|^2.
$$
Since $u$ is assumed to be positive, $b(x)\leq -\epsilon<0$ for some
$\epsilon>0$ and
$0\leq T_1u\leq 1$ we obtain
$$
\int_{\{u\leq 1\}} |\nabla u|^2 + \epsilon\int_{\{u>1\}}|\nabla u|^2
\leq \lambda \int_{\Omega} |g(u)| .
$$
\end{proof}

\subsection*{Acknowledgements} 
I would like to express my gratitude to Prof. Xavier Cabr\'e for proposing 
the problem studied in this paper and for the endless discussions we 
held regarding the results obtained.

This research was supported by
CONICET PIP625 Res. 960/12, ANPCyT PICT-2012-0153
and UBACYT X117.

\begin{thebibliography}{00}

\bibitem{ADP} B. Abdellaoui, A. Dall'Aglio, I. Peral; 
\emph{Some remarks on  elliptic problems with critical growth in the gradient},
J. Diff. Equations, \textbf{222}, No 1 (2006), 21--62.

\bibitem{ABG} S. Alama, L. Bronsard, and C. Gui, \emph{Stationary layered solutions in
$\mathbb{R}^2$ for an Allen-Cahn system with multiple well potential,}
Calc. Var. Partial Differential Equations \textbf{5} (1997), 359--390.


\bibitem{AAC} G. Alberti, L. Ambrosio and X. Cabr\'e;
\emph{On a long-standing conjecture of E. De Giorgi: symmetry in 3D for general
nonlinearities and a local minimality property,} Acta Appl. Math.
\textbf{65} (2001), 9--33.


\bibitem{A2} H. Alencar, A. Barros, O. Palmas, J. G. Reyes, W. Santos;
\emph{$\rm {O} (m) \times \rm {O} (n)$-invariant minimal hypersurfaces in
${\mathbb{R}}^{m + n}$,} Ann. Global Anal. Geom. \textbf{27} (2005), 179--199.

\bibitem{ACM} F. Alessio, A. Calamai, P. Montecchiari;
\emph{Saddle-type solutions for a class of semilinear elliptic equations,}
Adv. Differential Equations \textbf{12} (2007), 361--380.


\bibitem{A} F. J. Almgren;
\emph{Some interior regularity theorems for minimal
 surfaces and an extension of Bernstein's theorem}, Ann. of Math. 
\textbf{  85}, (1966), 277--292.

\bibitem{ACr} H. Amann and M. G. Crandall;
\emph{On Some Existence Theorems for  Semi-linear Elliptic Equations,} 
Ind. Univ. Math. J., \textbf{27}, No 5 (1978), 779--790.

\bibitem{AC} L. Ambrosio, X. Cabr\'e; 
\emph{Entire Solutions of Semilinear
Elliptic Equations in $\mathbb{R}^3$ and a Conjecture of De Giorgi,}
Journal Amer. Math. Soc. \textbf{13} (2000), 725--739.

\bibitem{BCR} H. Berestycki, X. Cabr\'e, L. Ryzhik;
\emph{Bounds for the explosion  problem in a flow}, in preparation.

\bibitem{BCN} H. Berestycki, L. Caffarelli, L. Nirenberg;
\emph{Further qualitative properties for elliptic equations in unbounded
domains,} Ann. Scuola Norm. Sup. Pisa Cl. Sci. \textbf{25} (1997),
69--94.

\bibitem{BHN}  H. Berestycki, F. Hamel, N. Nadirashvili;
\emph{The speed of propagation for {KPP} type problems. {I}.
{P}eriodic framework}, J. Eur. Math. Soc.,\textbf{7},(2005), no 2, 173--21.

\bibitem{BHR} H. Berestycki, F. Hamel, L. Rossi;
\emph{Louville type results for semilinear elliptic equations in unbounded domains}, 
to appear Ann. Mat. Pura Appl.

\bibitem{BNV} H. Berestycki, L. Nirenberg, S. R. S. Varadhan;
\emph{The Principal Eigenvalue and Maximum Principle for Second-Order Elliptic 
Operators in General Domains}, Comm. Pure Appl. Math, \textbf{47}, (1994), 47--92.

\bibitem{B} L. Boccardo;
 \emph{T-minima: An approach to minimization problems in
 $L^1$}, Contributi dedicati alla memoria di Ennio De Giorgi,
Rich. Mat. \textbf{49} (2000), 135--154.

\bibitem{BGG} E. Bombieri, E. De Giorgi, E. Giusti;
\emph{Minimal cones and the Bernstein problem,} Inv. Math. \textbf{7} (1969), 243--268.

\bibitem{BC} H. Brezis, X. Cabr\'e;
\emph{Some simple nonlinear PDE's without solutions}, Boll.
Unione Mat. Ital. Sez. B Artic. Ric. Mat. \textbf{8} 1 (1998), no. 2, 223--262.

\bibitem{BV} H. Brezis, J. L. V\'azquez;
 \emph{Blow-up solutions of some  nonlinear elliptic problems},
Rev. Mat. Univ. Comp. Madrid, \textbf{  10}, No 2 (1997), 443--468.

\bibitem{C} X. Cabr\'e;
\emph{Boundedness of minimizers of semilinear problems up to
 dimension four}, in preparation.

\bibitem{CC} X. Cabr\'e, A. Capella;
\emph{On the stability of radial solutions of semilinear elliptic equations 
in all of $\mathbb{R}^n$,} C. R. Acad. Sci. Paris, Ser. I \textbf{338} (2004), 769--774.


\bibitem{CC2} X. Cabr\'e, A. Capella; \emph{Regularity of radial minimizers and
 extremal solutions of semilinear elliptic equations}, J. Functional
Analysis \textbf{238} (2006), 709--733.

\bibitem{CSS} X. Cabr\'e, M. Sanch\'on, J. Spruck;
\emph{A priori estimates for semistable solutions of semilinear elliptic equations},
Dis. Cont. Dyn. Systems, \textbf{36} 2 (2016), 601--609.

\bibitem{CT}X. Cabr\'e, J. Terra;
 \emph{Saddle-shaped solutions of bistable diffusion equations in all 
of $\mathbb{R}^{2m}$}, Jour. of the European Math. Society \textbf{11} (2009), 819--843.

\bibitem{CT2} X. Cabr{\'e}, J. Terra;
 \emph{Qualitative properties of saddle-shaped solutions to bistable 
diffusion equations}, Comm. in Partial Differential Equations 
\textbf{35} (2010), 1923--1957.

\bibitem{CGS} L. Caffarelli, N. Garofalo, F. Seg{\`a}la;
 \emph{A gradient bound for entire solutions of quasi-linear equations and its
consequences,} Comm. Pure and Appl. Math. \textbf{47} (1994), no 11,
1457--1473.

\bibitem{casto} D. Castorina, M. Sanch\'on;
\emph{Regularity of stable solutions to semilinear elliptic equations 
on Riemannian models}, Adv. Nonlinear Anal. \textbf{4} (2015), no. 4, 295--309.

\bibitem{CGR} F. Cirstea, M. Ghergu, V. Radulescu;
\emph{Combined effects of asymptotically
linear and singular nonlinearities in bifurcation problems of
Lane-Emden-Fowler type}, J. Math. Pures Appliqu\'ees \textbf{84} (2005), 493--508.

\bibitem{CR} M. G. Crandall, P. H. Rabinowitz;
 \emph{Some continuation and  variational methods for positive solutions of 
nonlinear elliptic  eigenvalue problems}, Arch. Rat. Mech. Anal., 
\textbf{58} (1975), 207--218.

\bibitem{DF} E. N. Dancer, A. Farina;
 \emph{On the classification of solutions of $\Delta u = e^u$ on $\mathbb{R}^N$:
stability outside a compact set and applications,}
Proc. Amer. Math. Soc. \textbf{137} (2009), 1333--1338 .

\bibitem{DFP} H. Dang, P. C. Fife, L. A. Peletier;
\emph{Saddle solutions of the bistable diffusion equation,} 
Z. Angew Math. Phys. \textbf{43} (1992), no 6, 984--998.

\bibitem{DD} J. D\'avila, L. Dupaigne;
 \emph{Perturbing singular solutions of the Gelfand
problem}, Commun. Contemp. Math. \textbf{9} (2007), no. 5, 639--680.

\bibitem{DG1} E. De Giorgi; 
\emph{Una estensione del teorema di Bernstein,} Ann.
Sc. Norm. Sup. Pisa \textbf{19} (1965), 79--85.

\bibitem{DG2} E. De Giorgi; 
\emph{Convergence problems for functionals and
operators,} Proc. Int. Meeting on Recent Methods in Nonlinear
Analysis (Rome, 1978), 131--188, Pitagora, Bologna (1979).

\bibitem{DuF} L. Dupaigne, A. Farina;
 \emph{Liouville theorems for stable solutions of
semilinear elliptic equations with convex nonlinearities}, Nonlinear Anal.
\textbf{70} (2009), no. 8, 2882--2888.

\bibitem{E} P. Esposito;
 \emph{Linear unstability of entire solutions for a class
 ot non-autonomous elliptic equations}.

\bibitem{F1} A. Farina;
 \emph{On the classification of solutions of the
Lane-Emden equation on unbounded domains of $\mathbb{R}\sp N$,}
J. Math. Pures Appl. \textbf{87} (2007), 537--561.

\bibitem{F2} A. Farina;
 \emph{Stable solutions of $-\Delta u=e\sp u$ on
$\mathbb{R}\sp N$,} C. R. Math. Acad. Sci. Paris \textbf{345} (2007), 63--66.

\bibitem{F} W. H. Fleming;
 \emph{On the oriented Plateau problem}, Rend. Circolo
Mat. Palermo \textbf{9} (1962), 69--89.

\bibitem{GR} M. Ghergu, V. Radulescu;
 \emph{Singular Elliptic Problems. Bifurcation and
Asymptotic Analysis}, Oxford Lecture Series in Mathematics and Its
Applications, \textbf{37}, Oxford University Press, 2008.

\bibitem{GG} N. Ghoussoub, C. Gui;
 \emph{On a conjecture of De Giorgi and some related problems,} 
Math. Ann. \textbf{311} (1998), 481--491.

\bibitem{G} E. Giusti;
 \emph{Minimal Surfaces and Functions of Bounded
Variation,} Birkh\"auser Verlag, Basel-Boston, 1984.

\bibitem{JM} D. Jerison, R. Monneau;
\emph{Towards a counter-example to a conjecture of De Giorgi in high dimensions,}
Ann. Mat. Pura Appl., \textbf{183} (2004), no 4, 439--467.

\bibitem{JL} D. D. Joseph, T. S. Lundgren;
 \emph{Quasilinear Dirichlet problems driven
by positive sources}, Arch. Ration. Mech. Anal., \textbf{49} (1973), 241--268.

\bibitem{KPZ} M. Kardar, G. Parisi, Y. C. Zhang;
 \emph{Dynamic scaling of growing  interfaces}, Phys. Rev. Lett., 
\textbf{56} (1986), 889--892.

\bibitem{LU} O. A. Ladyzhenskaya, N. N. Ural'ceva;
 \emph{Linear and quasi-linear  elliptic equations,} Academic Press, 
New York--London, 1968.

\bibitem{LL} J. Leray and J.-L. Lions;
 \emph{Quelques r\'esultats de Vi\u{s}ik sur les
 probl\`emes elliptiques non lin\'eaires par les m\'ethodes de
 Minty-Browder}, Bull. Soc. Math. France,\textbf{93} (1965), 97--107.

\bibitem{L} P. L. Lions;
 \emph{Generalized solutions of Hamilton-Jacobi equations,}
Pitman Research Notes in Math., vol. 62, 1982.

\bibitem{M1} L. Modica;
\emph{A gradient bound and a Liouville theorem for
nonlinear Poisson equations,} Comm. Pure Appl. Math. \textbf{38}
(1985), 679--684.

\bibitem{M2} L. Modica;
 \emph{Monotonicity of the energy for entire solutions of
semilinear elliptic equations,} Partial differential equations and
the calculus of variations, Vol. II., Progr. Nonlinear
Differential Equations Appl. \textbf{2}, 843--850, Birkh\"auser,
Boston (1989).


\bibitem{MM1} L. Modica, S. Mortola;
\emph{Un esempio di $\Gamma\sp{-}$-convergenza,} Boll. Un. Mat. Ital. B \textbf{14} (1977),
285--299.

\bibitem{MP} F. Mignot, J.-P. Puel;
 \emph{Sur une classe de probl\`emes non
 lin\'eaires avec nonlin\'earit\'e positive, croissante, convexe},
Comm. Partial Differential Equations \textbf{5} (1980), 791--836.

\bibitem{MR0}P. Mironescu, V. Radulescu;
 \emph{A bifurcation problem associated to a convex, asymptotically linear function}, C.R. Acad. Sci. Paris, Ser. I \textbf{316} (1993), 667--672.

\bibitem{MR}P. Mironescu, V. Radulescu;
 \emph{The study of a bifurcation problem associated
to an asymptotically linear function}, Nonlinear Analysis, T.M.A. \textbf{26} 4
(1996), 857--875.

\bibitem{Ne} G. Nedev;
 \emph{Regularity of the extremal solution of semilinear
 elliptic equations}, C. R. Acad. Sci. Paris, S\'er. I Math. \textbf{
 330} (2000), 997--1002.

\bibitem{OS} O. Savin;
\emph{Regularity of flat level sets in phase transitions},
Ann. of Math \textbf{169} 1 (2009), 41--78.

\bibitem{Sc} M. Schatzman;
\emph{On the stability of the saddle solution of
Allen-Cahn's equation,} Proc. Roy. Soc. Edinburgh Sect. A \textbf{
125} (1995), no. 6, 1241--1275.

\bibitem{S} J. Simons;
\emph{Minimal varieties in riemannian manifolds,} Ann.
Math. \textbf{88} (1968), 62--105.

\bibitem{Sh} J. Shi;
\emph{Saddle solutions of the balanced bistable diffusion equation,} 
Comm. Pure. Appl. Math \textbf{55} (2002), no. 7,
815--830.

\bibitem{V} S. Villegas;
 \emph{Boundedness of extremal solutions in dimension $4$}, 
Adv. Math. \textbf{235} (2013), 126--133.

\bibitem{Vi} S. Villegas;
\emph{Asymptotic behavior of stable radial solutions of semilinear elliptic 
equations in $\mathbb{R}^N$,}  J. Math. Pures Appl, \textbf{88} 3 (2007), 241--250.

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