\documentclass[reqno]{amsart} 

\AtBeginDocument{{\noindent\small 
{\em Electronic Journal of Differential Equations},
Vol. 2004(2004), No. 132, pp. 1--15.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2004 Texas State University - San Marcos.} 
\vspace{9mm}}

\begin{document} 

\title[\hfilneg EJDE-2004/132\hfil Existence and multiplicity of positive solutions]
{Existence and multiplicity of positive solutions
 for a singular problem associated to the p-Laplacian operator} 

\author[Carlos Aranda \& Tomas Godoy\hfil EJDE-2004/132\hfilneg]
{Carlos Aranda, Tomas Godoy}  % in alphabetical order

\address{Carlos Aranda \hfill\break
Departamento de Matematica de la Facultad de Ciencias, Universidad de Tarapaca,
Av. General Velasquez 1775
Casilla 7-D, Arica, Chile}
\email{caranda@uta.cl}

\address{Tomas Godoy \hfill\break
FaMAF, Universidad Nacional de Cordoba,
Medina Allende y Haya de la Torre,
Ciudad Universitaria, 5000 Cordoba, Argentina}
\email{godoy@mate.uncor.edu}


\date{}
\thanks{Submitted June 18, 2004. Published November 16, 2004.}
\thanks{Research partialy supported by ANPCYT, CONICET, SECYT-UNC, 
\hfill\break\indent
Fundacion Antorchas, and Agencia Cordoba Ciencia}
\subjclass[2000]{35J60, 35J65}
\keywords{Singular problems; $p$-laplacian operator; \hfill\break\indent
nonlinear eigenvalue problems}


\begin{abstract}
 Consider the problem
 $$
 -\Delta_{p}u=g(u)  +\lambda h(u)\quad\hbox{in }\Omega
 $$
 with $u=0$ on the boundary, where $\lambda\in(0,\infty)$,
 $\Omega$ is a strictly convex bounded and $C^{2}$ domain in
 $\mathbb{R}^{N}$ with $N\geq2$, and $1<p\leq2$.
 Under suitable assumptions on $g$ and $h$
 that allow a singularity of $g$ at the origin, we show that for
 $\lambda$ positive and small enough the above problem has at least
 two positive solutions in $C(\overline{\Omega})\cap C^{1}(\Omega)$
 and that $\lambda=0$ is a bifurcation point from infinity.
 The existence of positive solutions for problems of the form
 $-\Delta_{p}u=K(x)  g(u)+\lambda h(u)+f(x)$ in $\Omega$,
 $u=0$ on $\partial\Omega$ is also studied.
\end{abstract}

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

\section{Introduction}

This paper  concerns problems of the form
\begin{equation} \label{eeqq11}
\begin{gathered}
-\Delta_{p}u    =Kg(u)+\lambda h(u)  +f \quad \text{in } \Omega,\\
u  =0\quad \text{on }\partial\Omega \\
u  >0\quad \text{in }\Omega.
\end{gathered}
\end{equation}
Here $\lambda$ is a nonnegative parameter, $\Delta_{p}$ is the
$p$-laplacian operator defined by
$\Delta_{p}u:=\mathop{\rm div}(|  \nabla u|  ^{p-2}\nabla u)  $
with $1<p<\infty$. We assume that 
\begin{itemize}
\item[(H1)] $\Omega$ is a $C^{2}$ and bounded domain in $\mathbb{R}^{N}$ with
$N\geq2$

\item[(H2)] $g:(0,\infty)  \to(0,\infty)  $ is a
continuous and non increasing function (that may be singular at the origin)

\item[(H3)] $h:[ 0,\infty) \to [ 0,\infty)  $ is a
continuous and non decreasing function

\item[(H4)] $K$ and $f$ are nonnegative functions defined on $\Omega$ which 
satisfy that $K$ is non identically zero, $K\in L^{\infty}(\Omega)  $ and
$f\in C(\overline{\Omega}) $.
\end{itemize}
As usual, $g(u)  $, $h(u)  $ denote the Nemitskii operators associated with
$g$ and $h$ respectively. The solutions of (\ref{eeqq11}) will be understood 
in the following weak sense:
 $u\in W_{\rm loc}^{1,p}(\Omega)  \cap C(\overline{\Omega})  $
satisfying $u=0$ on $\partial\Omega$ and
\[
\int_{\Omega}|  \nabla u|  ^{p-2}\langle \nabla u,\nabla
\varphi\rangle =\int_{\Omega}(Kg(u)  +\lambda
h(u)  +f)  \varphi
\]
for all $\varphi\in C_{c}^{\infty}(\Omega)$.

Singular bifurcation problems of the form $-\Delta u=g(x,u)+h(x, \lambda u)$ 
in $\Omega$, $u=0$ on $\partial\Omega$, $u>0$ in $\Omega$  have been 
considered in \cite{cp} for the case where, for some $\alpha >0$ and 
$p>0$ $g(x,u)$ and $h(x,\lambda u)$ behave like $u^{-\alpha}$ and 
$(\lambda u)^p $ respectively. There, existence of positive solutions for 
$\lambda$ nonnegative and small enough is obtained via a sub and 
supersolutions method and non existence of such solutions is also shown for 
large values of $\lambda$. From these results it seems a natural question 
to ask for similar results when the laplacian is replaced by the degenerated 
operator $\Delta_{p}$. Our aim in this paper is to study existence and 
(at least for the case $K=1$, $f=0$) multiplicity of positive solutions of 
(\ref{eeqq11}). Our approach to this problem is somewhat different from the 
followed in [4] and it is more in the line of fixed point theorems for 
nonlinear eigenvalue problems. We first study in section 2, for a nonnegative 
$F\in C(\overline{\Omega})$, the problem $-\Delta_{p}u=Kg(u)+F$ in $\Omega$, 
$u=0$ on $\partial\Omega$, $u>0$ in $\Omega$. Lemma \ref{lem2.6} states that this 
problem has unique solution and Lemma \ref{lem2.10} says that the corresponding 
solution operator $S$ for this problem, defined by $S(F):=u$ is a compact, 
continuous and non decreasing map from $P\cup\{0\}$ into $P$, where $P$ is
 the positive cone in $C(\overline{\Omega})$. These results 
(Lemmas \ref{lem2.6} and \ref{lem2.10}) are suggested by the work of several authors in 
\cite{ag,cp,crandrabi,g,lazermck} where existence of positive solutions 
for this problem is obtained under different assumptions on $K$ and $f$.

In section 3 we consider problem (\ref{eeqq11}). We write it as 
$u=S(\lambda h(u)+f)$ with $S$ as above. The above stated properties of $S$ 
allow us to apply a classical fixed point theorem for nonlinear eigenvalue 
problems to obtain in Theorem \ref{thm3.1} that for $\lambda$ nonnegative and small 
enough there exists at least a (positive) solution of (\ref{eeqq11}) and 
that the solution set for this problem (i.e., the set of the pairs 
$(\lambda,u)$ that solve it) contains an unbounded subcontinuum (i.e., an 
unbounded connected subset) emanating from $(0,u^{\ast})$, where 
$u^{\ast}$ is the (unique) solution of the problem $-\Delta_{p}u=Kg(u)+f$ 
in $\Omega$, $u=0$ on $\partial\Omega$, $u>0$ in $\Omega$.

Concerning multiplicity of positive solutions of (\ref{eeqq11}), 
in section 4, Theorem \ref{thm4.6}, we prove  that, if in addition, 
\begin{itemize}

\item[(H5)] $\Omega$ is a strictly convex domain in $R^{N}$

\item[(H6)] $g$ and $h$ are locally Lipchitz on $(0,\infty)  $ and
$[  0,\infty)  $ respectively

\item[(H7)] $1<p\leq2$, $\inf_{s>0}h(s)  /s^{p-1}>0$ and
$\lim_{s\to\infty}h(s)  /s^{q}<\infty$ for some
$q\in(p-1,\frac{N(p-1)  }{N-p}]$,
\end{itemize}
then the problem
\begin{equation}\label{eeqq12}
\begin{gathered}
-\Delta_{p}u =g(u)+\lambda h(u)  \text{ in }\Omega ,\\
u   =0\quad \text{on }\partial\Omega \\
u   >0 \quad \text{in }\Omega
\end{gathered}
\end{equation}
has at least two positive solutions for $\lambda$
positive and small enough and that $\lambda=0$ is a bifurcation point 
from infinity for this problem.

To see this in section 4 we study, for $j\in\mathbb{N}$, the
 problem \begin{equation} \label{eeqq111tris}
\begin{gathered}
-\Delta_{p}u    =g(u+\frac{1}{j})+\lambda h(u)  \text{ in } \Omega,\\
u  =0\quad \text{on }\partial\Omega \\
u  >0\quad \text{in }\Omega.
\end{gathered}
\end{equation}
Lemma \ref{lem4.1} provides, for a given $\lambda_{0}>0$, an a priori bound for the  
$L^{\infty}$ norm of the solutions $u$ of (\ref{eeqq111tris}) corresponding 
to some $\lambda\geq \lambda_{0}$. On the other hand, from 
Theorem \ref{thm3.1} we have an unbounded subcontinuum $C_{j}$ of the solution set 
for (\ref{eeqq111tris}) emanating now from $(0, u_{j}^{\ast})$ where 
$u_{j}^{\ast}$ is the (unique) solution of the problem 
$-\Delta_{p}v =g(v+\frac{1}{j})+\lambda h(v)$ in $\Omega$, $v=0$ on 
$\partial\Omega$, $v>0$ in $\Omega$. Also (cf. Remark \ref{rmk3.2}, part ii)) 
$C_{j} \subset [0,c)\times P$ for some positive constant $c$. 
Since $C_{j}$ is connected and unbounded, from these facts we obtain, 
for $\lambda$ positive and small enough, two positive solutions of 
(\ref{eeqq111tris}) and then, going to the limit as $j$ tends to infinity 
(perhaps after passing to a subsequence) we obtain two positive solutions 
for (\ref{eeqq111tris}). Lemmas \ref{lem4.2}, \ref{lem4.3}, \ref{lem4.5} and
 Remark \ref{rmk4.4} provide the 
necessary intermediate statements on order to do it.

\section{Preliminaries}

For this section, we assume  that the conditions (H1), (H2), (H3) and (H4)
stated at the introduction hold. Let us start with some preliminary remarks
collecting some known facts about the $p$-Laplacian operator.

\begin{remark} \label{rmk2.1} \rm
Let us recall \cite{li,dibenedetto,vas}
 that for $v\in L^{\infty}(\Omega)$ and $1<p<\infty$ the problem
$-\Delta_{p}u=v$ in $\Omega$, $u=0$ on $\partial\Omega$ has a unique (weak)
solution $u$ which belongs to $C^{1,\alpha}(\overline{\Omega})  $
for some $\alpha\in(0,1)  $ and that the associated solution
operator $(-\Delta_{p})  ^{-1}:L^{\infty}(\Omega)
\to C^{1}(\overline{\Omega})  $ is a positive, continuous
and compact map. Moreover, if $v\geq0$ and $v\neq0$ then $u$ belongs to the
interior of the positive cone in $C^{1}(\overline{\Omega})  $ So
$\frac{\partial u}{\partial\nu}<0$ on $\partial\Omega$ and $u$ is bounded from
above and from below by positive multiples of the distance function
\[
\delta(x)  :=\mathop{\rm dist}(x,\partial\Omega)  .
\]
So $(-\Delta_{p})  ^{-1}$ is a strongly positive operator on
$C(\overline{\Omega})  $, i.e., $v\in P$ implies
$(-\Delta_{p})  ^{-1}v\in Int(P)$ where $P$ denotes the positive cone in
$C(\overline{\Omega})$.

In addition, for the $p$-laplacian operator the following comparison principle
holds: If $U$ is a bounded domain (non necessarily regular) in 
$\mathbb{R}^{N}$ and if $u,v\in W_{\rm loc}^{1,p}(U)  \cap C(\overline{U})  $ 
with $1<p<\infty$ satisfy (in weak sense)
$-\Delta_{p}u\leq-\Delta_{p}v$ on $U,$ $u\leq v$ on $\partial U$, then
$u\leq v$.
\end{remark}


\begin{remark} \label{rmk2.2} \rm
 If $U$ is a bounded domain (i.e an open and connected
set, non necessarily regular) in $\mathbb{R}^{N}$ and if $u,v\in
W_{\rm loc}^{1,p}(U)  \cap C(\overline{U})  $ satisfy
(in weak sense) $-\Delta_{p}u-Kg(u)  \leq-\Delta_{p}v-g(
v)  $ in $U$ with $u\leq v$ on $\partial U,$ then $u\leq v$ on $U$.
Indeed, suppose $u>v$ somewhere and consider the non empty open set
$V=\{  x\in U:u(x)  >v(x)\}  $. Since
$-\Delta_{p}u+\Delta_{p}v\leq K(g(u)  -g(v))  \leq0$ in $V$ and
$u=v$ on $\partial V$ the comparison principle
gives a contradiction.
\end{remark}


\begin{lemma} \label{lem2.3}
For a nonnegative $F\in L^{\infty}(\Omega) $
 and for $j\in\mathbb{N}$,  the problem
\begin{equation} \label{eeqq21}
\begin{gathered}
-\Delta_{p}u_{j}    =Kg(u+\frac{1}{j})  +F\text{ in } \Omega,\;
u_{j}   \in W_{\rm loc}^{1,p}(\Omega)  \cap C(\overline {\Omega})  , \\
u_{j}   =0\quad \text{ on }\partial\Omega, \\
u_{j}   >0\quad \text{ in }\Omega
\end{gathered}
\end{equation}
has a unique positive (weak) solution satisfying
$u_{j}\in C^{1}(\overline{\Omega})$ and
$j\to\frac{1}{j}+u_{j}$ is non increasing. Moreover,
$u_{j}\geq c\,\delta$  for some positive constant $c$
independent of $j$.
\end{lemma}

\begin{proof}
Let $g_{j}:\mathbb{R}\to\mathbb{R}$ be defined by
$g_{j}(s)  =g(s)  $ for $s\geq\frac{1}{j}$ and
$g_{j}(s)  =g(\frac{1}{j})  $ for $s<\frac{1}{j}$,
let $T_{j}:C(\overline{\Omega})  \to C(\overline{\Omega})  $
be given by $T_{j}(v)  =(-\Delta_{p})  ^{-1}(Kg(\frac{1}{j}+v)  +F)  $.
Since for $v\in C(\overline{\Omega})  $ we have
\[
\|  Kg(\frac{1}{j}+v)  +F\|  _{L^{\infty}(
\Omega)  }\|  K\|  _{L^{\infty}(\Omega)
}g(\frac{1}{j})  +\|  F\|  _{L^{\infty}(
\Omega)  },
\]
it follows that $T_{j}$ is a compact operator. Moreover,
\[
0\leq T(v)  \leq(-\Delta_{p})  ^{-1}\big(g(
\frac{1}{j})  \|  K\|  _{L^{\infty}(\Omega)
}+\|  F\|  _{L^{\infty}(\Omega)  }\big) \quad \text{on }\Omega,
\]
and so the existence assertion of the lemma follows from the Schauder fixed
point theorem (as stated in \cite[Corollary 11.2]{trudi}) applied to
$T_{j}$ on a closed ball (in $C(\overline{\Omega})  $) around $0$
with radius large enough.

If $v$ and $w$ are two different solutions of (\ref{eeqq21}) in
$W_{\rm loc}^{1,p}(\Omega)  \cap C(\overline{\Omega})  $,
consider the open set $\Omega':=\{x\in\Omega:v(x)
>w(x)  \}$. If $\Omega'\neq\emptyset$ then
\begin{equation}
-\Delta_{p}v+\Delta_{p}w=K\big(g_{j}(\frac{1}{j}+v)
-g_{j}(\frac{1}{j}+w)  \big)  \quad \text{in }\Omega '
\end{equation}
and also $v=w$ on $\partial\Omega'$, but, from our assumptions on $K$
and $g$, the comparison principle gives $v\leq w$ on $\Omega'$ which
is a contradiction. A similar contradiction is obtained if $v<w$ somewhere.
thus the uniqueness assertion of the lemma holds. From the facts in Remark
\ref{rmk2.1}, the solution of (\ref{eeqq21}) belongs to $C^{1}(\overline{\Omega})$ and
it is positive because $(-\Delta_{p})  ^{-1}$ is a positive
operator Again by the comparison principle $\frac{1}{j+1}+u_{j+1}\leq\frac
{1}{j}+u_{j}$. Indeed, consider the set $U=\{x\in\Omega:\frac{1}%
{j+1}+u_{j+1}>\frac{1}{j}+u_{j}\}$ and observe that
$-\Delta_{p}(\frac{1}{j+1}+u_{j+1})  +\Delta_{p}(\frac{1}{j}
+u_{j})  =Kg(\frac{1}{j+1}+u_{j+1})  -Kg(\frac{1}
{j}+u_{j})  \leq0$ in $U$ and $\frac{1}{j+1}+u_{j+1}\leq\frac{1}
{j}+u_{j}$ on
$\partial U$, thus the comparison principle gives $U=\emptyset$.

Finally, $-\Delta_{p}(u_{j})  =Kg(\frac{1}{j}+u_{j})  +F\geq Kg(1+u_{1})  $,
so the strong positivity of
$(-\Delta_{p})  ^{-1}$ gives the last assertion of the
lemma.
\end{proof}


\begin{remark}[Tolksdorf's estimates] \label{rmk2.4}  \rm
Let $\Omega'$ and $\Omega''$ be open subsets of $\Omega$ such that
$\Omega ''\subset\subset\Omega'\subset\subset\Omega$ and suppose
that $u\in W_{\rm loc}^{1,p}(\Omega)\cap C(\overline{\Omega})$
satisfies $-\Delta_{p}u=v$ on $\Omega$ for some
$v\in L^{\infty}(\Omega)$. Then there exist $\alpha\in(0,1)
$ such that $u\in C^{1,\alpha}(\overline{\Omega''})  $. Moreover,
an upper bound of
$\|  u\|  _{C^{1,\alpha}(\overline {\Omega''})  }$ can be found
depending only on $p$, $\Omega$, $\Omega'$ $\Omega''$,
$\|  u\|_{L^{\infty}(\Omega')  }$ and $\|  v\|_{L^{\infty}(\Omega')  }$
(cf. \cite[Theorem 1]{t}).
\end{remark}


The Tolksdorf's estimates imply the following result.


\begin{remark} \label{rmk2.5} \rm
 Assume that the sequences $\{  F_{j}\}  _{j\in\mathbb{N}}$ and
$\{  u_{j}\}  _{j\in\mathbb{N}}$  are  in
$L_{\rm loc}^{\infty}(\Omega)  $ and
$W_{\rm loc}^{1,p}(\Omega)\cap C(\overline{\Omega})  $ respectively
with $1<p<\infty$ and $u_{j}\geq0$ such that $-\Delta_{p}u_{j}=F_{j}$ on
$\Omega$ for all $j\in\mathbb{N}$. Assume also that for each open set
 $\Omega'' \subset\subset\Omega$ there exist positive constants
$c_{\Omega''}$, $\widetilde{c}_{\Omega''}$ such that
$\|  F_{j}\|  _{L^{\infty}(\Omega'') }\leq c_{\Omega''}$ and
$\|  u_{j}\|  _{L^{\infty}(\Omega'')  }\leq\widetilde{c}_{\Omega
''}$ for all $j\in\mathbb{N}$ and that $\lim_{j\to
\infty}F_{j}=F$ a.e. in $\Omega$ for some $F:\Omega\to\mathbb{R}$.
Then there exists a subsequence $\{  u_{j_{k}}\}  _{k\in\mathbb{N}}$ and
a nonnegative function $v\in C^{1}(\Omega)  $ satisfying
$-\Delta_{p}v=F$ on $\Omega$ and such that $\{u_{j_{k}}\}
_{k\in\mathbb{N}}$ converges, in the $C^{1}$ norm, to $v$ on each compact
subset of $\Omega$.

Indeed, if $\Omega'\subset\subset\Omega$, let $\Omega''$
be a domain such that $\Omega'\subset\subset\Omega'' \subset\Omega$.
We have $\|  F_{j}\|  _{L^{\infty}(
\Omega')  }\leq c_{\Omega''}$, $\|u_{j}\|  _{L^{\infty}(\Omega'') }
\leq\widetilde{c}_{\Omega''}$. Taking into account the
Tolksdorf's estimates in b), a Cantor diagonal process gives a subsequence
$\{u_{j_{k}}\}_{k\in\mathbb{N}}$ that converges to some
function $u\in C^{1}(\Omega)  $ on each compact subset of
$\Omega$ in the $C^{1}$ norm. So, we have, for all
$\varphi\in C_{c}^{\infty}(\Omega)  $
\[
\int_{\Omega}|  \nabla u|  ^{p-2}\langle \nabla u,\nabla
\varphi\rangle =\lim_{j\to\infty}\int_{\Omega}|  \nabla
u_{j}|  ^{p-2}\langle \nabla u_{j},\nabla\varphi\rangle
=\lim_{j\to0}\int_{\Omega}F_{j}\varphi=\int_{\Omega}F\varphi
\]
and then $u$ satisfies $-\Delta_{p}u=F$ on $\Omega$.
\end{remark}

\begin{lemma} \label{lem2.6}
For a nonnegative function $F\in L^{\infty}(\Omega) $\ the problem
\begin{equation}  \label{eeqq22}
\begin{gathered}
-\Delta_{p}u    =Kg(u)  +F\quad \text{in }\Omega, \\
u    =0\quad\text{on }\partial\Omega, \\
u    >0\quad \text{in }\Omega
\end{gathered}
\end{equation}
has a unique positive solution in
$W_{\rm loc}^{1,p}(\Omega)  \cap C(\overline{\Omega}) $
and this solution belongs to $C^{1}(\Omega)  \cap C(\overline{\Omega})$.
 Moreover, $u\geq c\delta$  where
$c$ is the positive constant given by Lemma \ref{lem2.3} and
$u=\lim_{j\to\infty}u_{j}$ (in the pointwise sense) with
$u_{j}$ as there.
\end{lemma}

\begin{proof}
Let $u_{j}$ be as in Lemma \ref{lem2.3} and let $u=\lim_{j\to \infty}u_{j}$.
Since $\frac{1}{j}+u_{j}\geq c\delta$ (with $c$ as there, and
so independent of $j$) we have, for each subdomain
$\Omega'\subset\subset\Omega$,
\[
\|  Kg(\frac{1}{j}+u_{j})  +F\|  _{L^{\infty}(
\Omega')  }\leq\|  K\|  _{L^{\infty}(\Omega)  }g(c\delta)  +\|  F\|  _{L^{\infty
}(\Omega)  }.
\]
Also,
\[
\|  u_{j}\|  _{L^{\infty}(\Omega')  }%
\leq\|  \frac{1}{j}+u_{j}\|  _{L^{\infty}(\Omega^{\prime
})  }\leq1+\|  u_{1}\|  _{L^{\infty}(\Omega)
}<\infty.
\]
After passing to a subsequence, from Remark \ref{rmk2.5} we can assume that $\{
u_{j}\}_{j\in\mathbb{N}}$ converges, in the $C^{1}$ norm, on each
compact subset of $\Omega$, to a solution $u\in C^{1}(\Omega)  $
of the problem $-\Delta_{p}u=Kg(u)  +F$ in $\Omega$.

Since (as shown in Lemma \ref{lem2.3}) $\frac{1}{j}+u_{j}$ is decreasing in $j$, we
have $0\leq u\leq\frac{1}{j}+u_{j}$ for all $j$. Also, $u_{j}\in C(
\overline{\Omega})  $, $u_{j}=0$ on $\partial\Omega$ and so $u=0$ on
$\partial\Omega$ and $u$ is continuous up to the boundary. Moreover,
$\frac{1}{j}+u_{j}\geq c\delta$ gives, going to the limit, that $u\geq c\delta$.

If $z\in W_{\rm loc}^{1,p}(\Omega)  \cap C(\overline{\Omega})  $ is another
solution of (\ref{eeqq22}), consider the open set
$U:=\{x\in\Omega:z(x)>u(x)\}$. From (\ref{eeqq22}) we have
$-\Delta_{p}z\leq-\Delta_{p}(u)  $ in $U$ and $z=u$ in $\partial U$,
the comparison principle leads to $U=\emptyset$. Then $z\leq u$ in
$\Omega$. Similarly we see that $u\leq z$.
\end{proof}


\begin{remark} \label{rmk2.7} \rm
 It is known \cite[section 4]{godoy} that if
$m\in L^{\infty}(\Omega)  $ and $|  \{x\in
\Omega:m(x)  >0\}|  >0$ then there exists a unique
$\lambda=\lambda_{1}(-\Delta_{p},m,\Omega)  \in(
0,\infty)  $ such that the problem $-\Delta_{p}\Phi=\lambda m|
\Phi|  ^{p-2}\Phi$ in $\Omega$, $\Phi=0$ in $\partial\Omega$, $\Phi>0$
in $\Omega$ has a solution $\Phi\in W^{1,p}(\Omega)  \cap
C(\overline{\Omega})  $. This solution is unique up to a
multiplicative constant, belongs to $C^{1,\alpha}(\overline{\Omega
})  $ for some $\alpha\in(0,1)  $, satisfies that
$\nabla\Phi(x)  \neq0$ for all $x\in\partial\Omega$ and there
exists positive constants $c_{1}$ and $c_{2}$ such that $c_{1}\delta(
x)  \leq\Phi(x)  \leq c_{2}\delta(x)  $ for
all $x\in\Omega$ (so $\Phi(x)  >0$ for all $x\in\Omega$). For the
case $m=1$ we will write $\lambda_{1}(-\Delta_{p},\Omega)  $
instead of $\lambda_{1}(-\Delta_{p},m,\Omega)  $.
\end{remark}

We recall also that if $0\leq h\in L^{\infty}(\Omega)  $,
$\lambda>0$ and if there exists a nonnegative and non identically zero
solution $w\in W_{0}^{1,p}(\Omega)  $ of the problem $-\Delta
_{p}w=\lambda mw^{p-1}+h$ in $\Omega$ then $\lambda\geq\lambda_{1}(
-\Delta_{p},m)  $  \cite[Proposition 4.1]{godoy}.
This implies the following result.


\begin{remark} \label{rmk2.8} \rm
Let $m\in L^{\infty}(\Omega)  $ and let, as
usual, $m^{+}=\max(m,0)  $. Assume $m^{+}\neq0$ and let
$\lambda\geq0$ such that there exists a nonnegative and non identically zero
function $w\in W_{\rm loc}^{1,p}(\Omega)  \cap C(
\overline{\Omega})  $ such that $-\Delta_{p}w\geq\lambda mw^{p-1}$ in
$\Omega$. Then $\lambda\leq\lambda_{1}(-\Delta_{p},m,\Omega)  $.
Indeed, Let $v\in W_{0}^{1,p}(\Omega)  $ be the (positive)
solution of the problem $-\Delta_{p}v=\lambda mw^{p-1}$ in $\Omega$. Then
$v\in C^{1}(\overline{\Omega})  $ and $-\Delta_{p}w\geq
-\Delta_{p}v$ in $\Omega$, $w=v$ on $\partial\Omega$, Thus the comparison
principle gives $w\geq v$ in $\Omega$. Since $-\Delta_{p}v=\lambda mv^{p-1}+h$
with $h=\lambda m(w^{p-1}-v^{p-1})  $, we have $0\leq h\in
L^{\infty}(\Omega)  $ and so Remark \ref{rmk2.7} applies to give that
$\lambda\leq\lambda_{1}(-\Delta_{p},m,\Omega)$.
\end{remark}

\begin{remark} \label{rmk2.9} \rm
 This remark concerns to the behavior near the boundary of
the solution of problem (\ref{eeqq22}). We will say that two functions
$v_{1}$, $v_{2}:\Omega\to(0,\infty)  $ are comparable if
there exist positive constants $c_{1}$, $c_{2}$ such that $c_{1}v_{1}\leq
v_{2}\leq c_{2}v_{1}$. Consider in Lemma \ref{lem2.6} the case $F=0$ and assume that
$K$ is comparable with $\Phi^{\gamma}$ for some $\gamma\geq0$ and that
$0<\liminf_{s\to0^{+}}s^{\alpha}g(s)  \leq\limsup
_{s\to0^{+}}s^{\alpha}g(s)  <\infty$ for some
$\alpha>\gamma+1$. Then the solution $u$ given there is comparable with
$\Phi^{\frac{\gamma+p}{\alpha+p-1}}$ (and so with $\delta^{\frac{\gamma
+p}{\alpha+p-1}}$) where $\Phi$ is a positive principal eigenfunction for
$-\Delta_{p}$ in $\Omega$ with homogeneous Dirichlet boundary condition
associated to the weight $m\equiv1$. Indeed, let $\beta=(\gamma
+p)  /(\alpha+p-1)  $ and let $v=\Phi^{\beta}$. Since
$0<\beta<1$ it follows that $v\in C^{1}(\Omega)  \cap C(
\overline{\Omega})  $. A computation shows that
$-\Delta_{p}v=\widetilde{K}v^{-\alpha}$ on $\Omega$, where
\[
\widetilde{K}=\beta^{p-1}((1-\beta)  (p-1)
|  \nabla\Phi|  ^{p}+\lambda_{1}\Phi^{p})  .
\]
Taking into account that $0<\beta<1$, the properties of $\Phi$ stated in
Remark \ref{rmk2.7} imply that $\widetilde{K}$ is comparable with $1$ and so, from our
assumptions on $g$, we can choose positive constants $c$ and $c'$ such
that $-\Delta_{p}(cv)  =c^{p-1}\widetilde{K}v^{-\alpha}\leq
g(v)  $ and $-\Delta_{p}(c'v)  =(
c')  ^{p-1}\widetilde{K}v^{-\alpha}\geq g(v)  $.
Let $U=\{x\in\Omega:u(x)  <cv(x)  \}$. Thus $U$ is open. Since $g$ is
 non increasing we have $-\Delta_{p}u\geq-\Delta_{p}(cv)  $ on $U$ on $\Omega$.
Also $u=cv$ on $\partial U$ and so the comparison principle implies
$U=\emptyset$. Then
$u\geq cv=c\Phi^{\beta}$ in $\Omega$. Similarly, we obtain also that $u\leq
c'\Phi^{\beta}$ in $\Omega$.
\end{remark}

Let $P$ be the positive cone in $C(\overline{\Omega})  $. For
$j\in\mathbb{N}$, let $S_{j}:P\cup\{0\}\to P$ be the
solution operator for problem (\ref{eeqq21}) gives by $S_{j}(f)
=u$ and let $S:P\cup\{  0\}  \to P$ be the analogous
solution map of (\ref{eeqq22}).



\begin{lemma} \label{lem2.10}
\begin{itemize}
\item[(i)] $S:P\cup\{0\}\to P$ is a continuous, non
decreasing and compact map and the same is true for each $S_{j}$.

\item[(ii)] $0<j\leq k$  implies $S_{k}(u)  \leq S_{j}(u)  $
for $u\in P\cup\{0\}$.

\item[(iii)] $S(u)  \leq S_{j}(u)  $ for
$u\in P\cup\{  0\}$, $j\in\mathbb{N}$.
\end{itemize}
\end{lemma}

\begin{proof}
To see that $S$ is non decreasing, suppose $F_{1}$,
$F_{2}\in P$ with $F_{1}\geq F_{2}\geq0$. Let $v_{1}=S(F_{1})  $,
$v_{2}=S(F_{2})  $. If $v_{1}<v_{2}$ somewhere in $\Omega$, let
$U:=\{  x\in\Omega:v_{2}(x)  >v_{1}(x)\}  $.
Thus $U$ is a non empty open set and, from our assumptions on $g$
and $K$,
\begin{gather*}
-\Delta_{p}v_{1}    =Kg(v_{1})  +F_{1}\geq Kg(
v_{2})  +F_{2}=-\Delta_{p}v_{2}\quad \text{in }U,\\
v_{1}   =v_{2}\quad \text{on }\partial U.
\end{gather*}
Then the comparison principle gives $v_{1}\geq v_{2}$ on $U$ which is a
contradiction. Then $S$ is non decreasing.

To see that $S$ is continuous, consider $F\in P\cup\{0\}$ and a
sequence $\{F_{j}\}_{j\in\mathbb{N}}$ in $P\cup\{
0\}$ that converges to $F$ in $C(\overline{\Omega})  $.
Let $M$ be an upper bound of $\{F_{j}\}_{j\in\mathbb{N}}$.
Then
\begin{equation} \label{eeqq23}
0<S(0)  \leq S(F_{j})  \leq S(M)  .
\end{equation}
Let $u_{j}=S(F_{j})  $, thus $-\Delta_{p}u_{j}=Kg(
u_{j})  +F_{j}$ in $\Omega$, $u_{j}=0$ on $\partial\Omega$. Taking into
account that by Lemma \ref{lem2.6}) $S(0)  \geq c\delta$ and that $S$ is
non decreasing, we have
\[
0\leq g(u_{j})  =g(S(F_{j})  )  \leq
g(S(0)  )  \leq g(c\delta)  \in
L_{\rm loc}^{\infty}(\Omega)  .
\]
Also $0\leq u_{j}\leq S(M)  \in C(\overline{\Omega})  $.
Then Remark \ref{rmk2.5} gives a subsequence $\{F_{j_{k}}\}_{k\in\mathbb{N}}$
such that $S(F_{j_{k}})  $ converges, in the
$C^{1}$ norm, on each compact subset of $\Omega$ to a positive solution
$z\in C^{1}(\Omega)  $ of the problem
\[
-\Delta_{p}z=Kg(z)  +F\quad \text{in }\Omega.
\]
Since $u_{j_{k}}=S(F_{j_{k}})  \geq S(0)  \geq
S(c\delta)  $, we have $z\geq c\delta$. Also, $z\leq S(
M)  \in C(\overline{\Omega})  $. Since $S(M)
=0$ on $\partial\Omega$ it follows that $z$ is continuous up to the boundary
and $z=0$ on $\partial\Omega$. Thus $z=S(F)  $.

Let $\varepsilon>0$ and let $\eta=\eta(\varepsilon)  >0$ such
that $S(M)  \leq\varepsilon$ on $\Omega-\Omega_{\eta}$ where
\begin{equation} \label{eeqq24}
\Omega_{\eta}:=\{x\in\Omega:\mathop{\rm dist}(x,\partial\Omega)
>\eta\}.
\end{equation}
We have $S(F_{j_{k}})  \leq S(M)  \leq\varepsilon$
on $\Omega-\Omega_{\eta}$ for all $k$. Also $S(F)  \leq
\varepsilon$ on $\Omega-\Omega_{\eta}$, thus
$\|  S(F_{j_{k}})  -S(F)  \|  _{L^{\infty}(\Omega
-\Omega_{\eta})  }\leq2\varepsilon$ for all $k$. On the other hand,
since $\{S(F_{j_{k}})  \}$ converges in
$C^{1}(\overline{\Omega}_{\eta})  $ to $S(F)  $, for
$k$ large enough we have $\|  S(F_{j_{k}})  -S(
F)  \|  _{L^{\infty}(\Omega_{\eta})  }\leq \varepsilon$.
Then $\{S(F_{j_{k}})  \}$ converges
in $C(\overline{\Omega})  $ to $S(F)  $. Then $S$ is continuous.

To prove that $S$ is a compact map, consider a bounded sequence $\{
F_{j}\}$ in $P\cup\{0\}$ and let $M\in(
0,\infty)  $ be an upper bound of $\{F_{j}\}$. For
$\varepsilon>0$ let $\eta=\eta(\varepsilon)  $ be chosen as
above. As before, Remark \ref{rmk2.5} gives a subsequence $\{F_{j_{k}}\}
$ that converges, in the $C^{1}$ norm, on each compact subset of $\Omega$.
Thus, for $k$ and $s$ large enough,
\[
\|  S(F_{j_{k}})  -S(F_{j_{s}})  \|
_{C(\overline{\Omega}_{\eta})  }\leq\varepsilon
\]
and
\begin{align*}
\|  S(F_{j_{k}})  -S(F_{j_{s}})  \|
_{C(\Omega-\Omega_{\eta})  }  &  \leq\|  S(F_{j_{k}
})  \|  _{C(\Omega-\Omega_{\eta})  }+\|
S(F_{j_{s}})  \|  _{C(\Omega-\Omega_{\eta})}\\
&  \leq2\|  S(F_{j_{s}})  \|  _{C(\Omega
-\Omega_{\eta})  }\leq2\varepsilon
\end{align*}
Then $\{S(F_{j_{k}})  \}_{k\in\mathbb{N}}$ is a
Cauchy's sequence in $C(\overline{\Omega})  $ and the compactness
of $S$ follows.
Since for each $j$, $g(.+\frac 1j)$ satisfies the assumptions made for on
$g$, (i) holds for each $S_{j}$.
Finally, (ii) is a direct consequence of the comparison principle and,
since $S(u)  =\lim_{j\to\in}S_{j}(u)  $ 
(by Lemma \ref{lem2.6}), (iii) follows from (ii).
\end{proof}

\section{An existence result}

Our assumptions for this section are those stated at the beginning
of the Section 2. Let us introduce some additional notations.
Consider, for $j\in\mathbb{N}$ and $\lambda\geq0$ the problem
\begin{equation} \label{eeqq31}
\begin{gathered}
-\Delta_{p}u   =Kg(u+\frac{1}{j})  +\lambda h(u)
+f\quad \text{ in }\Omega,\;
u   \in W_{\rm loc}^{1,p}(\Omega)  \cap C(\overline{\Omega })  , \\
u   =0\quad \text{on }\partial\Omega, \\
u   >0\quad \text{in }\Omega
\end{gathered}
\end{equation}
Let $\pi:[  0,\infty)  \times P\to[  0,\infty)
$ be defined by $\pi(\lambda,u)  =\lambda$ and for $j$ as above,
let
\begin{gather*}
\Sigma_{j}   =\{(\lambda,u)  \in[  0,\infty)
\times P:u\in W_{\rm loc}^{1,p}(\Omega)  \cap C(\overline
{\Omega})  \text{ and $u$ solves (\ref{eeqq31})}\},\\
\Lambda_{j}  =\pi(\Sigma_{j})  ,\quad
u_{j}^{\ast}   =S_{j}(f)
\end{gather*}
and let $\Sigma_{\infty}$, $\Lambda_{\infty}$ and $u_{\infty}^{\ast}$ be the
sets and the function analogously defined replacing (\ref{eeqq31}) by
(\ref{eeqq11}). Finally, let $C_{j}$ (respectively $C_{\infty}$) be the
connected component of $\Sigma_{j}$ containing $u_{j}^{\ast}$
(respectively of $\Sigma_{\infty}$ containing $u_{\infty}^{\ast}$).

With this notation, we have the following theorem.


\begin{theorem} \label{thm3.1}
\begin{itemize}
\item[(i)] $(\lambda,u)  \in \Sigma_{\infty}$
implies that $u\in C^{1}(\Omega)$.

\item[(ii)]  For $f\in P\cup\{0\}$ it holds that
$C_{\infty}$ is unbounded in $[0,\infty)\times P$.

\item[(iii)] $\Lambda_{\infty}$ is an interval containing $0$.

\item[(iv)] For $j\in N$, (i), (ii) and (iii) hold with
$\Sigma_{\infty}$, $\Lambda_{\infty}$ and $u_{\infty}^{\ast}$
replaced by $\Sigma_{j}$, $\Lambda_{j}$ and $u_{j}^{\ast}$
respectively and with \eqref{eeqq31} replaced by \eqref{eeqq11}.

\item[(v)] There exists $\widetilde{\lambda}>0$ such that
$[0,\widetilde{\lambda}]  \subset\Lambda_{\infty}$  and
$[0,\widetilde{\lambda}]  \subset\Lambda_{j}$ for each $j$.
\end{itemize}
\end{theorem}

\begin{proof}
(i) is given by Lemma \ref{lem2.3}. To see (ii) and (iii), observe that
(\ref{eeqq11}) is equivalent to $S(\lambda h(u)+f)=u$. Let
$T:[ 0,\infty)  \times(P\cup\{0\})  \to C(\overline{\Omega})$ be
defined by $T(\lambda,v)  =S(\lambda h(v+u_{\infty}^{\ast})  +f)
-u_{\infty}^{\ast}$ (since $S$ is non decreasing
we have $T(\lambda,v)  \geq0$ for $v\geq0$). Lemma \ref{lem2.10} implies
that $T$ is a continuous, non decreasing and compact map. Moreover,
$T(0,0)  =0$ and, since $T(0,v)  =0$ for all $v\in P\cup\{0\}$, $v=0$ is
the unique fixed point of $T(0,.)  $. For each $\sigma\geq1$ and $\rho>0$,
we have also that $T(0,u)  \neq\sigma u$ for $u\in P\cap\rho\partial B$,
where $B$ denotes the open unit ball centered at $0$ in
$C(\overline{\Omega})$. Since $u$ solves (\ref{eeqq11}) if and only if
$u=v+u_{\infty}^{\ast}$ with $v$ a fixed point for $T$, 
in \cite[Theorem 17.1]{amann}, applied to $T$, gives that $C_{\infty}$ is unbounded in
$[0,\infty)  \times P$ and that $\Lambda_{\infty}$ is an interval. Thus
(i), (ii) and (iii) hold for $S$ and, replacing in the above argument $g$ by
$g(.+\frac{1}{j})  $, we see that the same is true for each $S_{j}$.

To prove (v) one observes that, by Lemma \ref{lem2.3} the problem
\begin{gather*}
-\Delta_{p}u   =Kg(1+u)  +f\quad \text{in }\Omega,\;
u   \in W_{\rm loc}^{1,p}(\Omega)  \cap C(\overline{\Omega}) \\
u   =0\quad \text{on }\partial\Omega
\end{gather*}
has a unique solution $u=u_{1}$ that belongs to
$C^{1}(\overline{\Omega})$. Thus $0\in\Lambda_{1}$. Since, by ii), $C_{1}$ is
unbounded, it follows that there exists $\widetilde{\lambda}>0$ such that
$\widetilde{\lambda}\in\Lambda_{1}-\{0\}$. Thus, by (iii), for
$0<\lambda<\widetilde{\lambda}$ there exists a positive solution
$u_{\lambda,1}$ of
\begin{gather*}
-\Delta_{p}u_{\lambda,1}  =Kg(1+u_{\lambda,1})  +\lambda
h(u_{\lambda,1})  +f\quad \text{in }\Omega,\;
u_{\lambda,1}   \in W_{\rm loc}^{1,p}(\Omega)  \cap C(
\overline{\Omega}) \\
u_{\lambda,1}   =0\quad \text{ on }\partial\Omega
\end{gather*}
Now, by Lemma \ref{lem2.10}, $S(\lambda h(u_{\lambda,1})
+f)  \leq S_{1}(\lambda h(1+u_{\lambda,1})
+f)  $ and since the operator $u\to U(u)
:=S(\lambda h(u)  +f)  $ is a positive, non
decreasing, continuous and compact map. \cite[Theorem 17.1]{amann} implies
that the sequence $\{U^{j}(0)  \}_{j\in\mathbb{N}}$ converges to a fixed
point of $U$. Then $\lambda\in\Sigma_{\infty}$.
Similarly, by considering $S_{j}$ instead of $S$ we get that $\lambda
\in\Lambda_{j}$ for all $j$.
\end{proof}

\begin{remark} \label{rmk3.2} \rm
(i) If for some $\lambda_{0}>0$ and $j\in\mathbb{N}$ we
know that an a priori estimate $\|  u\|  _{L^{\infty}(\Omega) }\leq c$
holds for each positive solution of (\ref{eeqq31}) associated to each
$\lambda\geq\lambda_{0}$ then an upper bound for $\Lambda_{j}$ can be given.
Indeed, in this case we have
\[
-\Delta_{p}u=g(\frac{1}{j}+u)  +\lambda h(u)
+f\geq\lambda c^{1-p}h(u)  u^{p-1}\quad \text{in }\Omega.
\]
Also, $u=S_{j}(\lambda h(u)  +f)  \geq S(0)  \geq c\delta$ for
some positive constant $c$ (cf. Lemmas \ref{lem2.10} and \ref{lem2.6}), then
$h(u)  \geq h(c\delta)  $ and so, by Remark \ref{rmk2.8},
$\lambda$ does not exceed the principal eigenvalue for $-\Delta_{p}$
associated to the weight function $c^{1-p}h(c\delta)  $.

(ii) On the other hand if $\inf_{s>0}\frac{h(s)  }{s^{p-1}}>0$ a
similar result holds. Indeed, in this case from (\ref{eeqq31}) we have
$-\Delta_{p}u\geq\lambda\inf_{s>0}\frac{h(s)  }{s^{p-1}}u^{p-1}$
in $\Omega$, $u=0$ on $\partial\Omega$ and so, again by Remark \ref{rmk2.8},
$\lambda\inf_{s>0}\frac{h(s)  }{s^{p-1}}\leq\lambda_{1}(-\Delta_{p},\Omega)$.
\end{remark}

The following proposition gives some additional information about the
regularity of the solutions of (\ref{eeqq11}).

\begin{proposition} \label{prop3.3}
Assume that $\sup_{s>0}s^{\alpha}g(s)  <\infty$ for some
$\alpha\in[0,\frac{2p-1}{p-1})$.
Then $u\in W_{0}^{1,p}(\Omega)$ for all positive weak solution
$u\in W_{\rm loc}^{1,p}(\Omega)\cap C(\overline{\Omega})$ of
\eqref{eeqq11} with $\lambda>0$.
\end{proposition}

\begin{proof}  We have
\begin{equation} \label{eeqq32}
\int_{\Omega}|  \nabla u|  ^{p-2}\langle \nabla u,\nabla
\varphi\rangle =\int_{\Omega}(Kg(u)  +\lambda
h(u)  +f)  \varphi
\end{equation}
for all $\varphi\in C_{c}^{\infty}(\Omega)$ and so, since
$u\in W_{\rm loc}^{1,p}(\Omega)$ this equality holds also for all
$\varphi\in W_{0}^{1,p}(\Omega)$ such that
$\operatorname{\rm supp}\varphi\subset\Omega$. For $\varepsilon>0$, let
$u_{\varepsilon}:=\max(u,\varepsilon)  -\varepsilon$. Since $u\in
C(\overline{\Omega})  $ and $u=0$ on $\partial\Omega$, we have
$\operatorname*{supp}u_{\varepsilon}\subset\Omega$. So we can take
$\varphi=u_{\varepsilon}$ as test function in (\ref{eeqq11}) to obtain
\begin{equation}\label{eeqq33}
\begin{aligned}
\int_{\Omega}\chi_{\{u>\varepsilon\}}|  \nabla u|^{p}
&=\int_{\Omega}(\lambda h(u)  +Kg(u) +f)  u_{\varepsilon}\\
&\leq \int_{u>\varepsilon}(\lambda h(u)  +Kg(u)  +f)  u\\
&\leq\int_{\Omega}(\lambda h(u)+Kg(u)  +f)  u
\end{aligned}
\end{equation}
We claim that the last integral is finite. Indeed, it is enough to show that
$ug(u)  \in L^{1}(\Omega)$ and clearly this holds
if $\alpha\leq1$. Suppose now $\alpha>1$. We have
\begin{equation} \label{eeqq34}
\begin{aligned}
-\Delta_{p}u  &  =\lambda h(u)  +Kg(u)+f \\
&  \leq \Big (\big(\lambda h(\|  u\|  _{L^{\infty
}(\Omega)  })  +\|  f\|  _{L^{\infty}(
\Omega)  }\big)  \|  u\|  _{L^{\infty}(
\Omega)  }^{\alpha}+c_{1}\|  K\|  _{L^{\infty}(
\Omega)  }\Big)  u^{-\alpha}\\
&=cu^{-\alpha}
\end{aligned}
\end{equation}
where $c=c_{\lambda,u}=(\lambda h(\|  u\|
_{L^{\infty}(\Omega)  }+\|  f\|  _{L^{\infty}(
\Omega)  })  )  \|  u\|  _{L^{\infty}(
\Omega)  }+c_{1}\|  K\|  _{L^{\infty}(\Omega)  }$.

Let $w\in W_{\rm loc}^{1,p}(\Omega)  \cap C(\overline{\Omega
})  $ be the solution (provided by Lemma \ref{lem2.6}) of the problem
$-\Delta_{p}w=cw^{-\alpha}$ in $\Omega$, $w=0$ on $\partial\Omega$. Then, from
(\ref{eeqq34}), $-\Delta_{p}u-cu^{-\alpha}\leq-\Delta_{p}w-cw^{-\alpha}$ in
$\Omega$, also $u=w=0$ on $\partial\Omega$ and so Remark \ref{rmk2.1} gives $u\leq w$
in $\Omega$. On the other hand, Remark \ref{rmk2.8} gives
$w\leq c'\Phi^{\frac{p}{\alpha+p-1}}$ for some constant $c'$
 where $\Phi$ is a positive principal eigenfuntion for $-\Delta_{p}$ on
$\Omega$. Then
$$
0\leq ug(u)  \leq c''\Phi^{\frac{p}{\alpha+p-1}}
\Phi^{-\frac{\alpha p}{\alpha+p-1}}=c''
\Phi^{-\frac{p(\alpha-1)  }{\alpha+p-1}}.$$
Since $0\leq \alpha<\frac{2p-1}{p-1}$ implies
$\frac{p(\alpha-1)  }{\alpha+p-1}<1$ our claim holds. By Lemma \ref{lem2.6}, $u(x)  >0$
for all $x\in\Omega$ and so, from \eqref{eeqq34} and from the monotone convergence
Theorem, we get $|\nabla u|^{p}\in L^{1}(\Omega)$.
\end{proof}

\section{A multiplicity result}

In this section we assume that in addition to the conditions (H1) (H2) and (H3)
stated at the introduction, the conditions (H5), (H6) and (H7) also hold.

In \cite[Proposition 4.1]{azizieh} it is proved that if $\Omega$ is a
strictly convex and bounded domain with $C^{2}$ boundary and if $G:\mathbb{R}%
\to\mathbb{R}$ is a locally Lipchitz function, then there exists
$\rho>0$, with $\rho$ depending only on $\Omega$ and $N$, such that if
$1<p\leq2$ and $u\in C^{1}(\overline{\Omega})  $ is a positive
weak solution of the problem $-\Delta_{p}u=G(u)  $ in $\Omega$,
$u=0$ on $\partial\Omega$ then the global maximum of $u$ in $\overline{\Omega
}$ is achieved at least at some point $y\in\Omega$ satisfying $\mathop{\rm dist}(
y,\partial\Omega)  \geq\rho$. From this fact and using the Gidas Spruck blow up
technique \cite{gs}, in \cite[Lemmas 3.1 and 3.2]{azizieh}
is obtained an a priori estimate for the solutions of the above problem.
Following a similar approach, Lemma \ref{lem4.1} below adapts to our actual setting,
with a similar purpose, the arguments in \cite{azizieh}.


\begin{lemma} \label{lem4.1}
Let $\Omega$ be a strictly convex, $C^{2}$ and bounded domain in
$\mathbb{R}^{N},N\geq2$. Assume that $1<p\leq2$ and that, in addition to
the hypothesis stated at the introduction, $g$ and $h$ are locally
Lipchitz on their domains and that
$\inf_{s>0}s^{-p+1}h(s) >0$ and
$0<\lim_{s\to\infty}s^{-q}h(s)  <\infty$ for some
$q\in(p-1,\frac {N(p-1)}{N-p}]$.  Then for each $\lambda_{0}>0$
there exists a positive constant $c_{\lambda_{0}}$ such
that for all $j$ and for all positive solution $u$ of the
problem
\begin{equation} \label{eeqq35}
\begin{gathered}
-\Delta_{p}u  =g(u+\frac{1}{j})  +\lambda h(u)\quad \text{in }\Omega,\;
u  \in W_{\rm loc}^{1.p}(\Omega)  \cap C(\overline{\Omega})  , \\
u   =0\quad \text{on }\partial\Omega, \\
u   >0\quad \text{in }\Omega,
\end{gathered}
\end{equation}
with $\lambda\geq\lambda_{0}$ it holds that
$\|u\|  _{L^{\infty}(\Omega)  }<c_{\lambda_{0}}$
\end{lemma}

\begin{proof}
Let $c=\inf_{s>0}(h(s)  /s^{p-1})  $
and let $u$ be a positive solution of (\ref{eeqq35}) corresponding to some
$\lambda>0$. We have $-\Delta_{p}u=c\lambda u^{p-1}+H$ in $\Omega$ with
$H:=g(u+\varepsilon)  +\lambda(h(u)-cu^{p-1})  $. Since $H\in L^{\infty}(\Omega)$
and $H>0$ in $\Omega$, Remark \ref{rmk2.7} gives that $\lambda\leq c^{-1}\lambda_{1}(
-\Delta_{p},\Omega)  $.

To prove the Lemma we proceed by contradiction. Assume that there exists a
sequence $\{u_{n},\lambda_{n},\varepsilon_{n}\}_{n\in
\mathbb{N}}$ such that $j_{n}\in\mathbb{N}$, $\lambda_{n}\geq\lambda_{n}$,
with $u_{n}$ satisfying (\ref{eeqq35}) for $\lambda=\lambda_{n}$ and such that
$\|  u_{n}\|  _{L^{\infty}(\Omega)  }\geq n$ and let
$G_{n}:\mathbb{R}\to\mathbb{R}$ be defined by
\[
G_{n}(s)=\begin{cases}
g(s+\frac{1}{j_{n}})  +\lambda _{n}h(s) & \text{for }s>0,\\
g(\frac{1}{j_{n}})  +\lambda_{n}h(0)  &  \text{for }s\leq 0.
\end{cases}
\]
Thus each $G_{n}$ is locally Lipchitz and so, by \cite[Proposition 4.1]{azizieh},
 there exists $x_{n}\in\Omega$ such that
$\|  u_{n}\|_{L^{\infty}(\Omega)  }=u_{n}(x_{n})  $ and
$\mathop{\rm dist}(x_{n},\partial\Omega)  \geq\rho$ with $\rho$ as described
at the beginning of the section.

Let $\alpha_{n}=\|u_{n}\|  _{L^{\infty}(\Omega)  }$
and let $\Omega_{n}=\alpha_{n}^{k}(\Omega-x_{n})  $ where
$\Omega_{n}:=\{x-x_{n}:x\in\Omega\}$ and $k=\frac{q-p+1}{p}$.
Observe that, since $q>p-1$, we have $k>0$.

Let $w_{n}:\Omega_{n}\to\mathbb{R}$ be defined by
\[
w_{n}(y)  =\frac{1}{\alpha_{n}}u_{n}\big(\frac{1}{\alpha
_{n}^{k}}y+x_{n}\big)  .
\]
Lemma \ref{lem2.3} applied to $F:=\lambda_{n}h(u_{n})  \in C(
\overline{\Omega})$ gives $u_{n}\in C^{1}(\overline{\Omega
})$ and so $w_{n}C^{1}(\overline{\Omega}_{n})  $.
Let $v\in C^{1}(\overline{\Omega})  $ be the solution of the
problem
\begin{gather*}
-\Delta_{p}v   =\lambda_{0}h(v) \quad \text{in }\Omega,\\
v   =0\quad \text{on }\partial\Omega.
\end{gather*}
We have, for some positive constant $c_{1}$ that $v\geq c_{1}\delta$ in
$\Omega$. Since $-\Delta_{p}u_{n}\geq\lambda_{0}h(u_{n})
=-\Delta_{p}v$ in $\Omega$ and $u_{n}=v$ on $\partial\Omega$ we get that
$u_{n}\geq c_{1}\delta$ and so $w_{n}(y)  \neq0$ for $y\in
\Omega_{n}$.
A computation gives that
\begin{equation} \label{eeqq36}
\begin{gathered}
-\Delta_{p}w_{n}(y)  =\frac{1}{\alpha_{n}^{q}}g(\alpha_{n}
w_{n}+\frac{1}{j_{n}})+\lambda_{n}w_{n}^{q}\frac{h(\alpha_{n}
w_{n})  }{(\alpha_{n}w_{n})  ^{q}}\quad \text{in }\Omega_{n}\\
w_{n}=0\quad \text{on }\partial\Omega_{n}.
\end{gathered}
\end{equation}
For $r>0$, let $\overline{B}_{r}(0)  $ be the closed ball in
$\mathbb{R}^{n}$ centered at $0$ with radius $r$. Since (by our contradiction
hypothesis) $\lim_{n\to\infty}\alpha_{n}^{k}=\infty$, by our choice of
$x_{n}$ there exists $n_{0}=n_{0}(r)  $ such that $\overline
{B}_{r}(0)  \subset\Omega_{n}$ for all $n\geq n_{0}$.

For $c_{1}$ as above and for $n$ large enough we have
\[
u_{n}(\frac{1}{\alpha_{n}^{k}}y+x_{n})  \geq c_{1}\delta(
\frac{1}{\alpha_{n}^{k}}y+x_{n})  \geq c_{1}\delta(\frac{\rho}%
{2})
\]
for all $y\in\Omega_{n}$. Then (recalling that, by Remark \ref{rmk3.2}, $\lambda
_{n}\leq c_{2}^{-1}\lambda_{1}(-\Delta_{p},\Omega)  $ with
$c_{2}=\inf_{s>0}(h(s)  /s^{p-1})  $) we get that,
for $y\in\overline{B}_{r}(0)  ,$%
\begin{align*}
0  &  \leq\alpha_{n}^{-q}1g(\alpha_{n}w_{n}(y)  +\frac{1}{j_{n}
})+\lambda_{n}h(\alpha_{n}w_{n}(y)  ) \\
&  \leq\alpha_{n}^{-q}g(c_{1}\frac{\rho}{2})+\frac{1}{c_{2}}\lambda_{1}(
-\Delta_{p},\Omega)  \alpha_{n}^{q}u_{n}^{q}(\alpha_{n}
^{-k}1y+x_{n})  \frac{h(u_{n}(\alpha_{n}^{-k}
y+x_{n})  )  }{u_{n}^{q}(\alpha_{n}^{-k}y+x_{n})  }\\
&  \leq\alpha_{n}^{-q}g(c_{1}\frac{\rho}{2})+\frac{1}{c_{2}}\lambda_{1}(
-\Delta_{p},\Omega)  \sup_{s>c_{1}\frac{\rho}{2}}\frac{h(
s)  }{s^{q}}
\end{align*}
Thus, from (\ref{eeqq36}) and Remark \ref{rmk2.4}, there exist positive constants
$c_{2}$ and $\alpha\in(0,1)$ such that $\|  w_{n}\|  _{C^{1,\alpha
}(\overline{B}_{r/2}(0)  )}\leq c_{2}$ for all $n$ large enough.
Then we can find a subsequence $\{w_{n_{j}}\}_{j\in\mathbb{N}}$
that converges in $C^{1}(\overline{B}_{r/2}(0)  )  $
to some nonnegative $w\in C^{1}(\overline{B}_{r}(0)
)  $ with $\|  w\|  _{L^{\infty}(\overline{B}%
_{r}(0)  )  }=1$. After passing to a furthermore
subsequence we can assume that $\lambda_{n_{j}}$ converges to some
$\lambda^{\ast}\in[  \lambda_{0},\lambda_{1}(-\Delta_{p}%
,\Omega)]  $. We take test functions in $C_{c}^{\infty}(
\overline{B}_{r/2}(0)  )  $ in (\ref{eeqq36}) and taking
the limit as $n$ tends to $\infty$ we get $-\Delta_{p}w\geq\lambda^{\ast
}Bw^{q}$ on $\overline{B}_{r/2}(0)  $, where $B=\lim
_{s\to\infty}(s^{-q}h(s)  )  $. Since
$w\not \equiv0$, Remark \ref{rmk2.4} gives $w(x)  >0$ for all
$x\in\overline{B}_{r/2}(0)  $ and so, again taking test functions
in $C_{c}^{\infty}(\overline{B}_{r/2}(0)  )  $ in
(\ref{eeqq36}) and going to the limit as $n$ tends to $\infty$, we obtain now
\begin{equation}
-\Delta_{p}w=B\lambda^{\ast}w^{q}\text{ on }\overline{B}_{r/2}(
0)  . \label{eeqq37}%
\end{equation}
Taking a sequence of balls $\overline{B}_{r_{i}}(0)  $ with
radius increasing to $\infty$ and repeating the above argument on the
subsequence $w_{n_{j}}$ obtained in the previous step, we can obtain a Cantor
diagonal subsequence, still denoted by $w_{n_{j}}$, which converges in the
$C^{1}$ norm on each compact set in $\mathbb{R}^{N}$ to a function
$\widetilde{w}\in C^{1}(\mathbb{R}^{N})  $ satisfying
$-\Delta_{p}\widetilde{w}=B\lambda^{\ast}\widetilde{w}^{q}$ on $\mathbb{R}%
^{N}$. Since, under our assumptions on $p$ and $q$, this problem has no
solution  \cite{mp2} we obtain a contradiction.
\end{proof}

\begin{lemma} \label{lem4.2}
For $\sigma>\|  S(0)  \|_{L^{\infty}(\Omega)  }$ there exists
$\lambda_{\sigma}$ and $j_{\sigma}\in\mathbb{N}$ such that for
$j>j_{\sigma}$ and $0\leq\lambda<\lambda_{\sigma}$,
problem \eqref{eeqq35} has no positive solution $u$
 satisfying $\|  u\|  _{L^{\infty}(\Omega)}=\sigma$.
\end{lemma}

\begin{proof} We proceed by contradiction. Suppose that there exists a
sequence \break
$\{j_{n},u_{n},\lambda_{n}\}_{n=1}^{\infty}$ with
$\lim_{n\to\infty}j_{n}=\infty$, $\lambda_{n}>0$, $\lim_{n\to
\infty}\lambda_{n}=0$, and where $u_{n}$ is a positive solution of
(\ref{eeqq35}) for $\lambda=\lambda_{n}$ and $j=j_{n}$ satisfying $\|
u_{n}\|  _{L^{\infty}(\Omega)  }=\sigma$. Let $M>0$ be an
upper bound of $\{\lambda_{n}\}$. By Lemma \ref{lem2.10} we have
\[
0  <S(0)  \leq S(\lambda_{n}h(u_{n}))  =u_{n}
 \leq S_{1}(\lambda_{n}h(u_{n})  )  \leq
S_{1}(Mh(\sigma)  )  .
\]
Then $\{u_{n}\}_{n\in\mathbb{N}}$ is bounded in $C(
\overline{\Omega})  $. Also, for $\Omega'\subset\subset
\Omega''\subset\subset\Omega$ we have
\[
\|  g(u_{n}+\frac{1}{j_{n}})  +\lambda_{n}h(
u_{n})  \|  _{L^{\infty}(\Omega'')
}\leq\|  g(S(0)  )  \|  _{L^{\infty
}(\Omega'')  }+Mh(\sigma)
\]
thus Remark \ref{rmk2.5} gives a subsequence $\{u_{j_{k}}\}$ that
converges, in the $C^{1}$ norm, to some function $u\in C^{1}(
\Omega)  $ on each compact subset of $\Omega$.

Since $0<S(0)  <u_{n}<S_{1}(Mh(\sigma)
)  $ and also $S_{1}(Mh(\sigma)  )  \in
C(\overline{\Omega})  $, $S_{1}(Mh(\sigma)
)  =0$ on $\partial\Omega$, we get that $u\in C(\overline{\Omega
})  \cap C^{1}(\Omega)  $ and $u=0$ on $\partial\Omega$
$ $Going to the limit in the weak form of
\[
-\Delta_{p}u_{n_{k}}=g(u_{n_{k}}+\frac{1}{j_{n_{k}}})  +\lambda
h(u_{n_{k}})
\]
we find that $-\Delta_{p}u=g(u)  +\lambda h(u)  $ in
$\Omega$. So $u=S(0)$.

Observe that $\{u_{n_{k}}\}$ converges to $u$ in $C(
\overline{\Omega})  $. Indeed, given $\varepsilon>0$, let $\eta
=\eta(\varepsilon)  >0$ such that $S_{1}(Mh(\sigma)  )  <\varepsilon$
on $\Omega-\Omega_{\eta}$ (with $\Omega_{\eta}$ defined by (\ref{eeqq24})).
Proceeding as in the
proof of the continuity of $S$ in Lemma \ref{lem2.10}, we get that
$\|  u_{n_{k}}-u\|  _{L^{\infty}(\Omega_{\eta})  }<\varepsilon$ for $k$
large enough and that $\|  u_{n_{k}}-u\|  _{L^{\infty}(
\Omega-\Omega_{\eta})  }<2\varepsilon$ for all $k$.
Then $\{u_{n_{k}}\}_{k\in\mathbb{N}}$ converges to $u$ in $C(\overline{\Omega})$.

Since $\|S(0)\|_{L^{\infty}(\Omega)}<\sigma=\|  u_{n}\|  _{L^{\infty}(\Omega)  }$
for all $n$, we get a contradiction.
\end{proof}

\begin{lemma} \label{lem4.3}
Let $\widetilde{\lambda}$ be as in Theorem \ref{thm3.1} and let
$\widetilde{u}$ be a positive solution of \eqref{eeqq31} corresponding to
$j=1$ and $\lambda=\widetilde{\lambda}$ (taking there $K=1$ and $f=0$).
Then for $\sigma>\|  \widetilde{u}\|  _{L^{\infty}(\Omega)}$,
$0\leq\lambda\leq\widetilde{\lambda}$ and $j\in\mathbb{N}$
there exists a positive solution $u$ of (\ref{eeqq35}) satisfying
$u\in C^{1}(\Omega)  \cap C(\overline{\Omega})$, $u\geq S(0)$ and
$\|  u\|  _{L^{\infty}(\Omega)}\leq\sigma$.
\end{lemma}

\begin{proof}
For $0<\lambda\leq \widetilde{\lambda}$, $j\in\mathbb{N}$,
Lemma \ref{lem2.10} gives
\begin{equation} \label{eeqq38}
0<S(0)  <S_{j}(\lambda h(\widetilde{u}))
\leq S_{1}(\widetilde{\lambda}h(\widetilde{u}))
 =\widetilde{u}\leq\sigma.
\end{equation}
Let $T:P\cup\{0\}\to P$ be defined by $T(v)  =S_{j}(\lambda h(v)  )  $.
Then $T$ is a non decreasing continuous and compact map and (\ref{eeqq38})
 says that
$T(\widetilde{u})  \leq\widetilde{u}$ and \cite[Theorem 17.1]{amann}
applies to see that $\{T^{k}(0)  \}_{k\in\mathbb{N}}$ is a non decreasing
sequence that converges in $C(\overline{\Omega})$ to a fixed point
$u\in P$ for $T$, which solves (\ref{eeqq35}) and since
$T^{k}(0)  \leq T^{k}(\widetilde{u})  \leq\widetilde{u}\leq\sigma$ we get
$\|  u\|_{L^{\infty}(\Omega)  }\leq\sigma$.
Also
$u\geq T^{k}(0)  =S_{j}(\lambda T^{k-1}(0))\geq S_{j}(0)  \geq S(0)$
(the last inequality by Lemma \ref{lem2.10} applied with $F:=\lambda h(u)$)
and since $\lambda h(u)  \in C(\overline{\Omega})$, from
(\ref{eeqq35}) Lemma \ref{lem2.3} gives $u\in C^{1}(\overline{\Omega})$
\end{proof}

\begin{remark} \label{rmk4.4} \rm
The following analogous of the Lemmas \ref{lem4.2} and \ref{lem4.3} hold:
\begin{itemize}
\item[(i)] For $\sigma>\|  \widetilde{u}\|  _{L^{\infty}(
\Omega)  }$ there exists $\lambda_{\sigma}>0$ such that for
$0\leq\lambda\leq\widetilde{\lambda}$ (\ref{eeqq12}) has a positive solution
$u$ satisfying $u\in C^{1}(\Omega)  \cap C(\overline{\Omega})  $ and
$\|  u\|  _{L^{\infty}(\Omega)}=\sigma$.

\item[(ii)] For $\sigma>\|  \widetilde{u}\|  _{L^{\infty}(
\Omega) }$ and for $0\leq\lambda\leq\widetilde{\lambda}$ there exists
a positive solution $u$ of (\ref{eeqq12}) satisfying $u\in C^{1}(
\Omega)  \cap C(\overline{\Omega})  $, $u\geq S(0)  $ and
$\|  u\|  _{L^{\infty}(\Omega)  } \leq\sigma$.
\end{itemize}
Indeed, the proofs are the same, replacing there $S_{j}$ by $S$ and
$g(\frac{1}{j}+.)  $ by $g$ whenever they appear and using Lemma \ref{lem2.6}
instead of Lemma \ref{lem2.3}.
\end{remark}


\begin{lemma} \label{lem4.5}
For $\sigma\geq\|  \widetilde{u}\|_{L^{\infty}(\Omega)}
+\|  S(0)  \|_{L^{\infty}(\Omega)}$  we have
\begin{itemize}
\item[(i)] There exist $\eta>0$ and $j_{\sigma}\in N$ such
that for $0<\lambda<\eta$ and $j\geq j_{\sigma}$, problem
\eqref{eeqq35} has a positive solution $u_{j}$ satisfying
$\|u_{j}\|  _{L^{\infty}(\Omega)  }\geq\sigma$.

\item[(ii)] There exist $\eta>0$ such that for $0<\lambda<\eta$
problem (\ref{eeqq12}) has a positive solution $u$
satisfying $\|  u\|_{L^{\infty}(\Omega)}\geq\sigma$.
\end{itemize}
\end{lemma}

\begin{proof}
Let $\Sigma_{j}$, $C_{j}$, $u_{j}^{\ast}$, and
$\widetilde {\lambda}$ be as in Theorem \ref{thm3.1} and let $\widetilde{u}$ be as
in Lemma \ref{lem4.3}. Let $\sigma$, $j_{\sigma}$, $\lambda_{\sigma}$ be as in
Lemma \ref{lem4.2} and let
$\eta=\min(\lambda_{\sigma,}\widetilde{\lambda})  $. For
$0<\lambda<\eta$ and $j\geq j_{\sigma}$ let $\lambda_{0}\in(
0,\lambda)  $ and let $c_{\lambda_{0}}$ be the constant provided by
Lemma \ref{lem4.1}. Clearly we can assume that $c_{\lambda_{0}}\geq\sigma$. Let
$O_{1}=O_{11}\cup O_{12}\cup O_{13}$ with
\begin{gather*}
O_{11}    =\{(\overline{\lambda},\overline{u})  \in
\Sigma_{j}:0\leq\overline{\lambda}<\lambda\text{ and }\|  \overline
{u}\|  _{L^{\infty}(\Omega)  }<\sigma\},\\
O_{12}    =\{(\overline{\lambda},\overline{u})  \in
\Sigma_{j}:\lambda<\overline{\lambda}<\lambda_{1}(-\Delta_{p}%
,\Omega)  \text{ and }\|  \overline{u}\|  _{L^{\infty}(
\Omega)  }<c_{\lambda_{0}}\},\\
O_{13}    =\{(\overline{\lambda},\overline{u})  \in
\Sigma_{j}:\overline{\lambda}=\lambda\text{ and }\|  \overline
{u}\|  _{L^{\infty}(\Omega)  }<c_{\lambda_{0}}\},
\end{gather*}
and let
\[
O_{2}=\{(\overline{\lambda},\overline{u})  \in\Sigma
_{j}:0\leq\overline{\lambda}<\lambda\text{ and }\|  \overline{u}\|
_{L^{\infty}(\Omega)  }>\sigma\}\,.
\]
Suppose, by contradiction, that there not exists a positive solution $u_{j}$
of problem (\ref{eeqq35}) such that
$\|  u_{j}\|  _{L^{\infty}(\Omega)  }\geq\sigma$.
Clearly, this assumption implies that
$O_{1}$ and $O_{2}$ are disjoint relative open sets in $\Sigma_{j}$. Moreover,
$\Sigma_{j}\subset O_{1}\cup O_{2}$. Indeed, suppose that $(
\overline{\lambda},\overline{u})  \in\Sigma_{j}$ and consider the case
$\overline{\lambda}<\lambda$. Then $\overline{\lambda}<\lambda_{\sigma}$ and
so, by Lemma \ref{lem4.2}, $\|  \overline{u}\|  _{L^{\infty}(
\Omega)  }\neq\sigma$. Thus $(\overline{\lambda},\overline
{u})  \in O_{11}\cup O_{2}$ In the case $\overline{\lambda}=\lambda$,
taking into account that $\lambda>\lambda_{0}$ and Lemma \ref{lem4.1} we get that
$(\overline{\lambda},\overline{u})  \in O_{13}$ and in the case
$\overline{\lambda}>\lambda$, again by Lemma \ref{lem4.1} we get that $(
\overline{\lambda},\overline{u})  \in O_{12}$. Then $\Sigma_{j}\subset
O_{1}\cup O_{2}$. Let $C_{j}$ be the unbounded connected component of
$\Sigma_{j}$ containing $(0,u_{j}^{\ast})  $. Thus $C_{j}\subset
O_{1}\cup O_{2}$. Since, by Theorem \ref{thm3.1}, $C_{j}$ is unbounded and since
$O_{1}$ is bounded, we get that $C_{j}\cap O_{2}\neq\emptyset$. Since $C_{j}$
is connected this implies that $C_{j}\cap O_{1}=\emptyset$. But, since
$(0,u_{j}^{\ast})  \in C_{j}$ and
\[
\|  u_{j}^{\ast}\|  _{L^{\infty}(\Omega)  }
=\|  S_{j}(0)  \|  _{L^{\infty}(\Omega)}
\leq\|  S_{1}(0)  \|  _{L^{\infty}(\Omega)  }
\leq\|  S_{1}(\widetilde{\lambda}h(\widetilde{u})
)  \|  _{L^{\infty}(\Omega)  }
=\|  \widetilde {u}\|  _{L^{\infty}(\Omega)  }<\sigma
\]
and so we get that $u_{j}^{\ast}\in O_{1}$. Then $C_{j}\cap O_{1}\neq
\emptyset$ which is a contradiction. Thus i) holds.

To prove (ii), consider for $j\geq j_{\sigma}$ the solution $u_{j}$ given by
the part (i) and observe that
\[
u_{j}=S_{j}(\lambda h(u_{j})  )  \geq S_{j}(
0)  \geq S(0)  \geq\widetilde{c}\delta
\]
where the constant $\widetilde{c}$ is independent of $j$ (these inequalities
follow from Lemma \ref{lem2.10} part (i) and from Lemma \ref{lem2.6} applied 
with $K=1$ and $f=0$). Also, by Lemma \ref{lem4.1}, 
$u_{j}\leq c_{\frac{\lambda}{2}}$ and so
\begin{equation} \label{eeqq39}
\begin{gathered}
-\Delta_{p}u_{j}=g(\frac{1}{j}+u_{j})  +\lambda h(
u_{j})  \leq g(\widetilde{c}\delta)  +\lambda_{1}(
-\Delta_{p},m,\Omega)  h(c_{\frac{\lambda}{2}}) \quad \text{in }\Omega\\
u_{j}=0\quad \text{ on }\partial\Omega
\end{gathered}
\end{equation}
Since $0\leq u_{j}\leq c_{\frac{\lambda}{2}}$, from Remark \ref{rmk2.5}, 
after passing to some subsequence, we can assume that 
$\{u_{j}\}_{j\in\mathbb{N}}$ converges, in the $C^{1}$ norm, on each 
compact subset of $\Omega$, to some function $u\in C^{1}(\Omega)  $ satisfying
$u\geq\widetilde{c}\delta$ (and so $u(x)  >0$ for $x\in\Omega$)
which is a solution of the problem $-\Delta_{p}u=g(u)  +\lambda
h(u)  $ in $\Omega$. Let $w=(-\Delta_{p})
^{-1}(h(c_{\frac{\lambda}{2}})  )  $.
From(\ref{eeqq39}) we have $0\leq u_{j}\leq w$. Since $w\in C(
\overline{\Omega})  $ and $w=0$ on $\partial\Omega$ we obtain that $u$
is continuous up to the boundary and that $u=0$ on $\partial\Omega$.

Finally, let $\rho=\rho(\Omega,N)  $ be as at the beginning of
this section. Thus $\|  u_{j}\|  _{L^{\infty}(\Omega)
}=\|  u_{j}\|  _{L^{\infty}(\overline{\Omega_{\rho}})
}$ for all $j$ and since $\{u_{j}\}_{j\in\mathbb{N}}$ converges
in $C^{1}(\overline{\Omega_{\rho}})  $ to $u$ we get that
$\|  u\|  _{L^{\infty}(\Omega)  }\geq$ $\|
u\|  _{L^{\infty}(\overline{\Omega_{\rho}})  }%
=\lim_{j\to\infty}\|  u_{j}\|  _{L^{\infty}(
\overline{\Omega_{\rho}})  }=\lim_{j\to\infty}\|
u_{j}\|  _{L^{\infty}(\Omega)  }\geq\sigma$ and the proof
of the lemma is complete.
\end{proof}

\begin{theorem} \label{thm4.6}
Assume the conditions (H1), (H2), (H3), (H5), (H6) and (H7) are satisfied.
Then
\begin{itemize}
\item[(i)]  For $\lambda$ positive and small enough there exist at
least two positive solutions of the problem \eqref{eeqq12}.

\item[(ii)] $\lambda=0$ is a bifurcation point from infinity.
\end{itemize}
\end{theorem}

\begin{proof} To prove (i) observe that for $\lambda$ positive and small
enough, taking into account Lemma \ref{lem4.5} (ii) we have a solution
$u\in C(\overline{\Omega})  \cap C^{1}(\Omega)  $ of \eqref{eeqq12}
which satisfies $\|  u\|  _{L^{\infty}(\Omega)}
\geq\sigma+1$ and, by Remark \ref{rmk4.4} (ii), a solution
$v\in$ $C(\overline{\Omega})  \cap C^{1}(\Omega)  $ such that
$\|  v\|  _{L^{\infty}(\Omega)  }\leq\sigma$.
To prove (ii) note that, proceeding as in Remark \ref{rmk3.2}, we have
$\Lambda_{\infty}\subset[  0,c^{-1}\lambda_{1}(-\Delta_{p},\Omega)]$
with $c=1/\inf_{s>0}(h(s)  /s^{p-1})$. Since by
Theorem \ref{thm3.1} $C_{\infty}$ is unbounded, (ii) follows).
\end{proof}

\subsection*{Acknowledgements}
We wish to thank Djairo G. De Figueiredo for his useful suggestions 
and to the referee for his/her careful reading and interesting comments. 


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\end{document}