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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2014 (2014), No. 126, pp. 1--13.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2014 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2014/126\hfil Strictly positive solutions]
{Strictly positive solutions for one-dimensional nonlinear elliptic problems}

\author[U. Kaufmann, I. Medri \hfil EJDE-2014/126\hfilneg]
{Uriel Kaufmann, Iv\'an Medri}  % in alphabetical order

\address{Uriel Kaufmann \newline
 FaMAF, Universidad Nacional de C\'ordoba, (5000)
C\'ordoba, Argentina}
\email{kaufmann@mate.uncor.edu}

\address{Iv\'an Medri \newline
 FaMAF, Universidad Nacional de C\'ordoba, (5000)
C\'ordoba, Argentina}
\email{medri@mate.uncor.edu}

\thanks{Submitted June 26, 2013. Published May 14, 2014.}
\subjclass[2000]{34B15, 34B18, 35J25, 35J61}
\keywords{Elliptic one-dimensional problems; indefinite nonlinearities;
 \hfill\break\indent sub and supersolutions; positive solutions}

\begin{abstract}
 We study the existence and nonexistence of strictly positive solutions
 for the elliptic problems $Lu=m(x) u^p$ in a bounded open interval,
 with zero boundary conditions, where $L$ is a strongly uniformly
 elliptic differential operator, $p\in(0,1)$, and $m$ is a function 
 that changes sign.  We also characterize the set of values  $p$  
 for which the problem admits  a solution, and in addition an existence
 result for other nonlinearities is presented.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks


\section{Introduction}

For $\alpha<\beta$, let $\Omega:=(\alpha,\beta)$ and let
$m\in L^2 (\Omega)  $ be a function that changes sign in $\Omega$.
Let $p\in( 0,1)  $ and let $L$ be a one-dimensional strongly uniformly elliptic
differential operator given by
\begin{equation}
Lu:=-a(x)u''+b(x)u'+c(x)u, \label{L}
\end{equation}
where $a,b\in C(  \overline{\Omega})  $,
$0\leq c\in L^{\infty }(\Omega)$ and $a(  x)  \geq\lambda>0$ for all
$x\in\Omega$. Our aim in this article is to consider the  existence and
nonexistence of solutions for the problem
\begin{equation}
\begin{gathered}
Lu=mu^p \quad \text{in }\Omega\\
u>0 \quad  \text{in }\Omega\\
u=0 \quad  \text{on }\partial\Omega.
\end{gathered}  \label{prob}
\end{equation}


The question of existence of strictly positive solutions for semilinear
Dirichlet problems with indefinite nonlinearities as \eqref{prob} is
challenging and intriguing, and to our knowledge there are few results
concerning this issue. In contrast to superlinear problems where any
nonnegative (and nontrivial) solution is automatically positive (and in fact
is in the interior of the positive cone under standard assumptions), for the
analogous sublinear equations the situation is far less clear, even in the
one-dimensional case. For instance, it is known that if $m$ is smooth and
$m^{+}\not \equiv 0$ then for any $p\in(  0,1)  $ there exist
nontrivial nonnegative solutions that actually vanish in a subset of $\Omega$
(see e.g. \cite{bandle,publi}), and when $L=-u''$ one may
also construct examples of strictly positive solutions that do not belong to
the interior of the positive cone (see \cite{ultimo}).

The problem \eqref{prob} was considered recently in \cite{ultimo} for the
laplacian operator, where several \textit{non-comparable }sufficient
conditions for the existence of solutions where proved under some evenness
assumptions on $m$. In the present paper we shall adapt and extend the
approach in \cite{ultimo} in order to derive our main results for a general
operator. More precisely, in Section 3 we shall give two non-comparable
sufficient conditions on $m$ in the case $b\equiv0$ (see Theorem \ref{aa} and
Remark \ref{sepo}), and when $b\not \equiv 0$ we shall also exhibit sufficient
conditions in Theorem \ref{bien} and Corollary \ref{puf}. Let us mention that
these last conditions are non-comparable between each other nor between the
ones in Theorem \ref{aa}. Moreover, one of them substantially improves the
results known for $L=-u''$ (see Remarks \ref{ojito1},
\ref{ojito2} and \ref{lm}). Also, as a consequence of the aforementioned
results we shall characterize the set of $p'$s such that \eqref{prob}
admits a solution and we shall deduce an existence theorem for other
nonlinearities (see Corollaries \ref{ppp} and \ref{fff} respectively). Let us
finally say that necessary conditions on $m$ for the existence of solutions
are stated in Theorem \ref{necee}.

To relate our results to others already existing let us mention that
to our knowledge no necessary condition on $m$ is known in the case of a
general operator (other than the obvious one derived from the maximum
principle, i.e. $m^{+}\not \equiv 0$), and the only sufficient condition we
found in the literature is that the solution $\varphi$ of $L\varphi=m$ in
$\Omega$, $\varphi=0$ on $\partial\Omega$, satisfies $\varphi>0$ in $\Omega$
(see \cite[Theorem 4.4]{jesusultimo}, \cite[Theorem 10.6]{hand}). Let us
note that although the above condition is even true for the $n$-dimensional
problem, it is far from being necessary in the sense that there are examples
of \eqref{prob} having a solution but with the corresponding $\varphi$
satisfying $\varphi<0$ in $\Omega$ (cf. \cite{ultimo}). Concerning the
laplacian operator, \eqref{prob} was treated in  \cite[Theorem 2.1]{ultimo},
and as we said before there are also further results there under different
evenness assumptions on $m$. Let us finally mention that existence of
solutions for problem \eqref{prob} has also been studied when $L=-u^{\prime
\prime}$ and $m\geq0$ but assuming that $m\in C(  \Omega)  $ (see
e.g. \cite{zhang}, \cite{chapa} and the references therein), and  some
similar results to the ones that appear here
have been obtained recently by the authors in \cite{aust} for
some related problems involving quasilinear operators.

We would like to conclude this introduction with some few words on the
corresponding $n$-dimensional problem. As we noticed in the above paragraph
the condition in \cite{jesusultimo} is still valid in this case, and some of
the techniques in \cite{ultimo} can be applied if $L=-\Delta$ (see Section 3
in \cite{ultimo} for the radial case, and also \cite{junows}). We are strongly
convinced that some of the theorems presented here should still have some
counterpart in $n$ dimensions but we are not able to provide a proof.

\section{Preliminaries}

Since $a(x)  \geq\lambda>0$ for all $x\in\Omega$ and
$a\in C(  \overline{\Omega})  $, from now on we consider without loss of
generality that $L$ is given by
\begin{equation}
Lu:=-u''+b(x)u'+c(x)u, \label{L2}
\end{equation}
with $b$ and $c$ as in \eqref{L}. For $f\in L^{r}(\Omega)$ with $r>1$ we say
that $u$ is a (strong) solution of the problem $Lu=f$ in $\Omega$, $u=0$ in
$\partial\Omega$, if $u\in W^{2,r}(\Omega)\cap W_0^{1,r}(\Omega)  $
and the equation is satisfied $a.e.$ $x\in\Omega$. Given
$g:\Omega\times\mathbb{R}\to \mathbb{R}$ a Carathe\'odory function
such that $g( .,\xi)  \in L^2 (\Omega)$ for all $\xi$, we say
that $u$ is a (weak) subsolution of
\begin{equation}
\begin{gathered}
Lu=g(  x,u)  \quad \text{in }\Omega\\
u=0 \quad \text{on }\partial\Omega
\end{gathered} \label{g}
\end{equation}
if $u\in W^{1,2}(  \Omega)  $, $u\leq0$ on $\partial\Omega$ and
\[
\int_{\Omega}u'\phi'+bu'\phi+cu\phi\leq\int_{\Omega
}g(  x,u)  \phi\quad\text{for all }0\leq\phi\in W_0^{1,2}
(\Omega).
\]
(Weak) supersolutions are defined analogously.

The following lemma is a direct consequence of the integration by parts
formula (e.g. \cite[Corollary 8.10]{brezislibro}).

\begin{lemma}\label{lem1}
For $i:1,\dots,n$, let $u_i\in W^{2,2}(x_i,x_{i+1})$ or
$u_i\in C^2 (x_i,x_{i+1})\cap C^{1}(  [  x_i,x_{i+1}])  $ such that
$u_i(x_{i+1})=u_{i+1}(x_{i+1})$, $u_i' (x_{i+1})\leq u_{i+1}'(x_{i+1})$ and
\[
-u_i''+bu_i'+cu_i\leq g(  x,u_i)
\quad \text{ a.e. }x\in(x_i,x_{i+1})\text{ for all }i:1,\dots,n.
\]
Let $\Omega:=(  x_1,x_{n+1})  $ and set $u(x):=u_i(x)$ for all
$x\in\Omega$. Then $u\in W^{1,2}(\Omega)$ and
\[
\int_{\Omega}u'\phi'+bu'\phi+cu\phi\leq\int_{\Omega
}g(  x,u)  \phi\quad \text{for all }0\leq\phi\in W_0^{1,2}(\Omega).
\]
In particular, if also $u\leq0$ on $\partial\Omega$, then $u$ is a subsolution
of \eqref{g}.
\end{lemma}

The next remark compiles some necessary facts about problem \eqref{prob}.

\begin{remark}\label{homsup} \rm
(i) It is immediate to check that \eqref{prob} possesses a
solution if and only if it has a solution with $\tau m$ in place of $m$, for
any $\tau>0$.

(ii) Let us write as usual $m=m^{+}-m^{-}$ with
$m^{+}=\max(  m,0)  $ and $m^{-}=\max(  -m,0)  $. It is
also easy to verify that \eqref{prob} admits arbitrarily large supersolutions
(if $m^{+}\not \equiv 0$; if $m^{+}\equiv0$ there is no solution by the
maximum principle). Indeed, let $\varphi>0$ be the solution of $L\varphi
=m^{+}$ in $\Omega$, $\varphi=0$ on $\partial\Omega$.
Let $k\geq(\|\varphi\| _{\infty}+1)^{p/(1-p)}$. Then $k(\varphi+1)$ is a
supersolution since
\begin{equation}
L(k(\varphi+1))\geq kL\varphi\geq(k(\| \varphi\| _{\infty
}+1))^pm^{+}\geq(k(\varphi+1))^pm\text{\quad in }\Omega\label{ju}
\end{equation}
and $\varphi=k>0$ on $\partial\Omega$.
\end{remark}

The two following lemmas provide some useful upper bounds for the
$L^{\infty}$-norm of the nonnegative subsolutions of \eqref{prob}.
To avoid overloading the notation we write from now on
\[
\overline{B}_{\alpha}(  x)  :=e^{\int_{\alpha}^{x}b(
r)  dr},\quad\underline{B}_{\alpha}(  x)  :=e^{-\int
_{\alpha}^{x}b(  r)  dr}.
\]


\begin{lemma}\label{inerte}
Let $0\leq u\in W^{2,2}(  \Omega)  $ be such that
$Lu\leq mu^p$ in $\Omega$. Then
\begin{equation}
\| u\| _{L^{\infty}(\Omega)}\leq[  \int_{\alpha}^{\beta
}\overline{B}_{\alpha}(  x)  \| m^{+}\underline{B}_{\alpha
}\| _{L^{1}(\alpha,x)}dx]  ^{1/(  1-p)  }.
\label{iner}
\end{equation}
\end{lemma}

\begin{proof}
Since $\underline{B}_{\alpha},u'\in W^{1,2}(\Omega)$,
we may apply the product differentiation rule and hence
\begin{align*}
-(\underline{B}_{\alpha}u')'
&\leq-(\underline{B}_{\alpha}u')'+\underline{B}_{\alpha}cu \\
&=\underline{B}_{\alpha}(-u''+bu'+cu)\\
&\leq \underline{B}_{\alpha}mu^p\leq\underline{B}_{\alpha}m^{+}\|
u\| _{L^{\infty}(\Omega)}^p.
\end{align*}
Integrating on $(  \alpha,x)  $ for $x\in(\alpha,\beta)$ (see e.g.
\cite[Theorem 8.2]{brezislibro}) and noting that $\underline{B}_{\alpha
}(\alpha)u'(  \alpha)  =u'(  \alpha)
\geq0$ we obtain
\[
-\underline{B}_{\alpha}(x)u'(x)\leq\| u\|
_{L^{\infty}(\Omega)}^p\int_{\alpha}^{x}\underline{B}_{\alpha}
(t)m^{+}(t)dt.
\]
Dividing by $\underline{B}_{\alpha}(x)>0$ and integrating now on $(
y,\beta)  $ for $y\in(\alpha,\beta)$, since $u(\beta)=0$ we get
\[
0\leq\frac{u(y)}{\| u\| _{L^{\infty}(\Omega)}^p}\leq
\int_{y}^{\beta}[  \overline{B}_{\alpha}(x)\int_{\alpha}^{x}\underline
{B}_{\alpha}(t)m^{+}(t)dt]  dx\quad\text{for all }y\in(\alpha,\beta),
\]
and the lemma follows.
\end{proof}

Let
\begin{equation}
M^{+}:=\{  x\in\Omega:m\geq0\},\quad
M^{-}:=\{  x\in \Omega:m<0\}  . \label{MM}
\end{equation}


\begin{lemma}\label{dudu}
Let $0\leq u\in W^{2,2}(  \Omega)  $ be such that
$Lu\leq mu^p$ in $\Omega$, and let $M^{+}$ be given by \eqref{MM}. If $c>0$
in $M^{+}$, then
\[
\| u\| _{L^{\infty}(\Omega)}\leq[  \sup_{x\in M^{+}
}\frac{m^{+}(  x)  }{c(  x)  }]  ^{1/(1-p)  }.
\]
\end{lemma}

\begin{proof}
Without loss of generality we assume that $u\not \equiv 0$.
Furthermore, let us suppose first that $\| u\| _{L^{\infty
}(\Omega)}>1$. Let $x_0\in\Omega$ be a point where $u$ attains its absolute
maximum. There exists $\delta>0$ such that $u\geq1$ in $I_{\delta}(
x_0)  :=(  x_0-\delta,x_0+\delta)  $. There also exist
$x_1,x_2\in I_{\delta}(  x_0)  $ satisfying $x_1
<x_0<x_2$ and $u'(  x_2)  \leq0\leq u'(x_1)  $. We have that
\[
-(  \underline{B}_{\alpha}u')  '+\underline
{B}_{\alpha}cu\leq\underline{B}_{\alpha}mu^p\leq\underline{B}_{\alpha}
m^{+}u^p\quad\text{in }\Omega
\]
and so in $I_{\delta}(  x_0)  $ we get that (because $u\geq1$ in
$I_{\delta}(x_0)  $)
$-(\underline{B}_{\alpha}u')  '\leq\underline{B}_{\alpha}(  m^{+}-c)  u$.
Integrating on $(  x_1,x_2)  $ we obtain
\begin{equation}
0\leq\underline{B}_{\alpha}(  x_1)  u'(
x_1)  -\underline{B}_{\alpha}(  x_2)  u'(
x_2)  =\int_{x_1}^{x_2}-(  \underline{B}_{\alpha}u^{\prime
})  '\leq\int_{x_1}^{x_2}\underline{B}_{\alpha}(
m^{+}-c)  u. \label{cero}
\end{equation}
Since $u\geq1$ in $(  x_1,x_2)  $ and $\underline{B}_{\alpha
}\geq e^{-\| b^{+}\| _{\infty}(  x_2-\alpha)  }$
in $(  x_1,x_2)  $, from \eqref{cero} it follows that there
exists $E\subset(  x_1,x_2)  $ with $| E|>0$
 (where $| E| $ denotes the Lebesgue measure of $E$)
such that $m^{+}(  x)  \geq c(  x)  $ a.e. $x\in E$.
Moreover, due to the fact that $c>0$ a.e. $x\in M^{+}$ it must hold that
$m^{+}>0$ a.e. $x\in E$. In particular, $E\subset M^{+}$ and therefore
\begin{equation}
1\leq\sup_{x\in E}\frac{m^{+}(  x)  }{c(  x)  }\leq
\sup_{x\in M^{+}}\frac{m^{+}(  x)  }{c(  x)  }.
\label{casi}
\end{equation}


Let $u$ now be as in the statement of the lemma, and let $\varepsilon>0$. Then
\[
L\frac{u}{\| u\| _{\infty}-\varepsilon}\leq\frac{m}{(
\| u\| _{\infty}-\varepsilon)  ^{1-p}}\Big(  \frac
{u}{\| u\| _{\infty}-\varepsilon}\Big)  ^p.
\]
Applying the first part of the proof with
$m/(  \| u\| _{\infty}-\varepsilon)  ^{1-p}$ and
$u/(  \| u\| _{\infty}-\varepsilon)  $ in place of $m$ and $u$ respectively, from
\eqref{casi} we deduce that
\[
\big(  \| u\| _{L^{\infty}(\Omega)}-\varepsilon\big)^{1-p}
\leq\sup_{x\in M^{+}}\frac{m^{+}(  x)  }{c(  x)  }
\]
and since $\varepsilon$ is arbitrary this completes the proof of the lemma.
\end{proof}

We shall need the next result when we characterize the set of $p'$s
such that \eqref{prob} admits a solution.

\begin{lemma}\label{qp}
Suppose \eqref{prob} has a solution $u\in W^{2,2}(\Omega)  $, and
let $q\in(  p,1)  $. Then there exists $v\in W^{2,2}(  \Omega)  $
solution of \eqref{prob} with $q$ in place of $p$.
\end{lemma}

\begin{proof}
Let $\gamma:=(  1-p)  /(  1-q)  $. Let
$0\leq\phi\in C_{c}^{\infty}(  \Omega)  $, and let
$\Omega'$ be an open set such that
$\operatorname{supp} \phi\subset\Omega' \Subset\Omega$. One can check that
$u^{\gamma}\in W_0^{1,2}(\Omega)  \cap W^{2,2}(  \Omega')  $. Furthermore,
noticing that $\gamma>1$ and $\gamma-1+p=\gamma q$ we find that
\begin{align*}
L(  u^{\gamma})
&=-\gamma(  u''u^{\gamma -1}+(  \gamma-1)  u^{\gamma-2}(  u')
^2 )  +b\gamma u^{\gamma-1}u'+cu^{\gamma}\\
&\leq \gamma u^{\gamma-1}(  -u''+bu'+cu)  \leq\gamma
u^{\gamma-1}mu^p\\
&=\gamma m(  u^{\gamma})  ^{q}\quad\text{in }\Omega'.
\end{align*}
Multiplying the above inequality by $\phi$, integrating over $\Omega'$
and using the integration by parts formula we obtain that
\begin{align*}
\int_{\Omega}(  u^{\gamma})  '\phi'+b(
u^{\gamma})  '\phi+cu^{\gamma}\phi
&=\int_{\Omega'}[ -(  u^{\gamma})  ''+b(  u^{\gamma})
'+cu^{\gamma}]  \phi\\
&\leq \gamma\int_{\Omega}m(  u^{\gamma})  ^{q}\phi.
\end{align*}
Now, let $0\leq v\in W_0^{1,2}(  \Omega)  $. There exists
$\{  \phi_n\}  _{n\in\mathbb{N}}\subset C_{c}^{\infty}(\Omega)  $ with
$\phi_n\geq0$ in $\Omega$ and such that $\phi_n\to  v$ in $W^{1,2}(  \Omega)  $
(e.g. \cite[p. 50]{chipot}). Employing the above inequality with $\phi_n$
in place of $\phi$ and going to the limit we see that $u^{\gamma}$ is a
 subsolution of \eqref{prob}
with $\gamma m$ in place of $m$. Thus, taking into account Remark \ref{homsup}
(i) and (ii) we get a solution $v\in W_0^{1,2}(  \Omega)  $ of
\eqref{prob}, and by standard regularity arguments $v\in W^{2,2}(
\Omega)  $.
\end{proof}

\section{Main results}

We set
\begin{equation}
C_{p}:=\frac{2(  1+p)  }{(  1-p)  ^2 },\label{cp}
\end{equation}
and for any interval $I$,
\[
\lambda_1(m,I) :=\text{the positive principal eigenvalue for $m$ in }I.
\]


\begin{theorem} \label{aa}
Assume $b\equiv0$. Let $m\in L^2 (\Omega)$ with $m^{-}\in
L^{\infty}(\Omega)$ and suppose there exist $\alpha\leq x_0<x_1\leq\beta$
such that $0\not \equiv m\geq0$ in $I:=(x_0,x_1)$. Let $\gamma
:=\max\{  (\beta-x_0),(x_1-\alpha)\}  $ and let $C_{p}$ be
given by \eqref{cp}.

(i) If it holds that
\begin{equation}
\frac{\| m^{-}\| _{L^{\infty}(\Omega)}}{\|
c\| _{L^{\infty}(\Omega)}}\sinh^2 \Big[  \gamma\sqrt{\frac
{\| c\| _{\infty}}{C_{p}}}\Big]  \leq\frac{1}{\lambda
_1(m,I)} \label{seno}
\end{equation}
then there  exists a solution $u\in W^{2,2}(\Omega)$ to problem \eqref{prob}.

(ii) If it holds that
\begin{equation}
\frac{\| m^{-}\| _{L^{\infty}(\Omega)}}{\|
c\| _{L^{\infty}(\Omega)}}[  \cosh(  \gamma\sqrt{(
1-p)  \| c\| _{L^{\infty}(\Omega)}})  -1]
\leq\frac{1}{\lambda_1(m,I)} \label{expo}
\end{equation}
then there exists a solution $u\in W^{2,2}(\Omega)$ to problem
\eqref{prob}.
\end{theorem}

\begin{proof}
Recalling Remark \ref{homsup} it suffices to construct a
strictly positive (in $\Omega$) subsolution $u$ for \eqref{prob} with $\tau m$
in place of $m$, for some $\tau>0$. Moreover, without loss of generality we
may assume that $\alpha<x_0<x_1<\beta$ (in fact, it shall be clear from
the proof how to proceed if either $x_0=\alpha$ or $x_1=\beta$).
To provide such $u$ we shall employ Lemma \ref{lem1} with $n=3$ and
$g(x,\xi)  =\tau m(  x)  \xi^p$.

We shall take $u_2>0$ with $\| u_2\| _{L^{\infty}(I)  }=1$ as the
positive principal eigenfunction associated to the
weight $m$ in $I$, that is satisfying
\begin{gather*}
Lu_2=\lambda_1(m,I)mu_2 \quad \text{in }I\\
u_2=0 \quad \text{on }\partial I.
\end{gather*}
Since $m\geq0$ in $I$, for $\tau>0$ we have that
$Lu_2=\lambda _1(m,I)mu_2\leq\tau mu_2^p$ whenever
\begin{equation}
\lambda_1(m,I)\leq\tau. \label{tri}
\end{equation}
On the other hand, suppose now that \eqref{seno} holds and pick $\tau$
satisfying
\begin{equation}
\frac{\| m^{-}\| _{L^{\infty}(\Omega)}}{\|
c\| _{L^{\infty}(\Omega)}}\sinh^2 \Big[  \gamma\sqrt{\frac
{\| c\| _{\infty}}{C_{p}}}\Big]  \leq\frac{1}{\tau}
\leq\frac{1}{\lambda_1(m,I)} \label{tau}
\end{equation}
(in particular, \eqref{tri} holds). Let $x\in[  \alpha,x_1]  $
and define
\[
f(x)=\sqrt{\frac{\tau\| m^{-}\| _{\infty}}{\|
c\| _{\infty}}}\sinh[  \sqrt{\frac{\| c\|
_{\infty}}{C_{p}}}(  x-\alpha)  ]  .
\]
A few computations show that
$C_{p}(  f')  ^2 -\| c\| _{\infty}f^2 =\tau\| m^{-}\| _{\infty}$ in
$(\alpha,x_1)$. Moreover, $f(\alpha)=0$, $f(  x)  >0$ for
$x\in(  \alpha,x_1)  $ and $f',f''\geq0$ for
such $x$. Let us now fix $k:=2/(  1-p)  $. Then we have
\begin{equation}
kp=k-2,\quad k(  k-1)  =C_{p}. \label{kk}
\end{equation}
We set $u_1:=f^{k}$. Taking into account \eqref{kk} and the above mentioned
facts we find that
\begin{equation} \label{ups}
\begin{aligned}
Lu_1&=-k[  (  k-1)  f^{k-2}(  f')
^2 +f^{k-1}f'']  +cf^{k}\\
&\leq -C_{p}f^{k-2}(  f')  ^2 +\| c\| _{\infty}f^{k}\\
&=-f^{k-2}\tau\| m^{-}\| _{\infty}\\
&\leq\tau mu_1 ^p\quad\text{in }(  \alpha,x_1)  .
\end{aligned}
\end{equation}
Furthermore, since $f$ is increasing we get that
$\| u_1\|_{\infty}=[  f(x_1)]  ^{k}$ and therefore using the first
inequality in \eqref{tau} and the fact that $x_1-\alpha\leq\gamma$ one can
verify that $\| u_1\| _{\infty}\leq1$.

In a similar way, if for $x\in[  x_0,\beta]  $ we define
$u_3:=g^{k}$ where $g$ is given by
\[
g(x):=\sqrt{\frac{\tau\| m^{-}\| _{\infty}}{\|
c\| _{\infty}}}\sinh\Big[  \sqrt{\frac{\| c\|
_{\infty}}{C_{p}}}(  \beta-x)  \Big]  ,
\]
then $Lu_3\leq\tau mu_3^p$ in $(  x_0,\beta)  $,
$\| u_3\| _{\infty}\leq1$, $u_3(\beta)=0$ and
$u_3(  x)  >0$ for $x\in(  x_0,\beta)  $.

We choose now
\begin{gather*}
\underline{x}_0:=\sup\{  x\in I:u_1(  y)  >u_2(
y)  \text{ for all }y\in(  x_0,x] \}  ,\\
\overline{y}:=\max\{  x\in I:u_2(  x)  =1\},\\
\underline{y}:=\min\{  x\in I:u_2(  x)  =1\}  .
\end{gather*}
We observe that $\underline{x}_0\in I$ exists because
$u_1( \alpha)  =u_2(  x_0)  =0$ and
$u_1(x_1)\leq 1=\| u_2\| _{\infty}$. Moreover, since $u_1$ and $u_2$
are $C^{1}$, by the definition of $\underline{x}_0$ we have that
$u_1(\underline{x}_0)=u_2(\underline{x}_0)$ and $u_1^{\prime
}(\underline{x}_0)\leq u_2'(\underline{x}_0)$ (for the last
inequality it is enough to note that 
\[
\frac{u_1(x)-u_1(\underline{x}_0)}{x-\underline{x}_0}
<\frac{u_2(x)-u_2(\underline{x}_0)} {x-\underline{x}_0}
\] 
for every $x\in(  x_0,\underline{x}_0)$), and also clearly 
$\underline{x}_0<\underline{y}$. Analogously, there
exists $\overline{x}_1\in I$ such that 
$u_2(  \overline{x}_1)  =u_3(  \overline{x}_1)  $ and 
$u_2'(\overline{x}_1)\leq u_3'(\overline{x}_1)$, and satisfying
$\overline{x}_1>\overline{y}$. In particular, $\underline{x}_0
<\overline{x}_1$. Hence, defining $u$ by $u:=u_1$ in 
$[\alpha,\underline{x}_0]  $, $u:=u_2$ in 
$[  \underline{x}_0,\overline{x}_1]  $ and 
$u:=u_3$ in $[  \overline{x}_1,\beta]  $, 
we have that $u=0$ on $\partial\Omega$ and $u$ fulfills
the hypothesis of Lemma \ref{lem1} and as we said before this proves (i) 
(let us mention that if $x_0=\alpha$ then in order to build $u$ we only use
$u_2$ and $u_3$, and if $x_1=\beta$ then we do not need $u_3$).

Let us prove (ii). We shall take $u_2$ as above. We now fix $\tau$ such
that
\begin{equation}
\frac{\| m^{-}\| _{L^{\infty}(\Omega)}}{\|
c\| _{L^{\infty}(\Omega)}}[  \cosh(  \gamma\sqrt{(
1-p)  \| c\| _{L^{\infty}(\Omega)}})  -1]
\leq\frac{1}{\tau}\leq\frac{1}{\lambda_1(m,I)}. \label{tauu}
\end{equation}
We set $k:=1/(  1-p)  $, and for $x\in[  \alpha,x_1]
$ we define
\[
f(  x)  :=\frac{\tau\| m^{-}\| _{\infty}
}{\| c\| _{\infty}}\Big[  \cosh\Big(  \sqrt{\frac
{\| c\| _{\infty}}{k}}(x-\alpha)\Big)  -1\Big]  .
\]
Then $f(\alpha)=0$, $f>0$ in $(\alpha,x_1)$ and $f'\geq0$.
Furthermore, by the first inequality in \eqref{tauu}
$\|u_1\| _{\infty}\leq1$, and it can be seen that
$kf''-\| c\| _{\infty}f=\tau\| m^{-}\| _{\infty}$.
Define now $u_1:=f^{k}$. Observing that $kp=k-1$ we derive that
\begin{align*}
Lu_1&=-k[  (  k-1)  f^{k-2}(  f')
^2 +f^{k-1}f'']  +cf^{k}\\
&\leq -kf^{k-1}f''+\| c\| _{\infty}f^{k}=-f^{k-1}\tau\| m^{-}\| _{\infty}\\
&\leq\tau mu_1^p\quad\text{in }(  \alpha,x_1)  .
\end{align*}
In the same way, if for $x\in[  x_0,\beta]  $ we set
$u_3:=g^{k}$ where $g$ is given by
\[
g(x):=\frac{\tau\| m^{-}\| _{\infty}}{\| c\|
_{\infty}}\Big[  \cosh\Big(  \sqrt{\frac{\| c\| _{\infty}
}{k}}(\beta-x)\Big)  -1\Big]  ,
\]
then $Lu_3\leq\tau mu_3^p$ in $(x_0,\beta)$,
$\| u_3\| _{\infty}\leq1$, $u_3(\beta)=0$ and $u_3>0$
in $(x_0,\beta)$. Now the proof of (ii) can be finished as in
(i).
\end{proof}

\begin{remark}\label{sepo} \rm
Let us mention that the inequalities in (i) and (ii) are not
comparable. Indeed, we first check that for $p\approx1$ \eqref{seno} is better
than \eqref{expo}. Let $\kappa:=\gamma\sqrt{\| c\| _{\infty}}$.
Since $\frac{1}{\sqrt{C_{p}}}=(  1-p)  \sqrt{\frac{1}{2(1+p)  }}$,
it is sufficient to observe that
\[
0\leq\lim_{p\to 1^{-}}\frac{\sinh^2 [  \kappa(  1-p)
\sqrt{\frac{1}{2(  1+p)  }}]  }{\cosh(  \kappa\sqrt
{1-p})  -1}\leq\lim_{p\to 1^{-}}\frac{\sinh^2 (
\kappa(  1-p)  )  }{\cosh(  \kappa\sqrt{1-p})
-1}=0.
\]
We now show that for $0<p\approx0$ \eqref{expo} is better than \eqref{seno}.
It suffices to prove this for $p=0$ because the dependence on $p$ in both
inequalities is continuous. For $p=0$ \eqref{seno} and \eqref{expo} become
\begin{gather*}
\frac{\| m^{-}\| _{\infty}}{\| c\| _{\infty
}}\sinh^2 (  \kappa/\sqrt{2})  \leq\frac{1}{\lambda_1(m,I)}\\
\frac{\| m^{-}\| _{\infty}}{\| c\| _{\infty
}}(  \cosh\kappa-1)  \leq\frac{1}{\lambda_1(m,I)}
\end{gather*}
and so we only have to check that for every $x>0$ it holds that $\sinh
^2 (  x/\sqrt{2})  >\cosh x-1$ which is easy to verify.
\end{remark}

\begin{remark} \label{lapla} \rm
If in \eqref{seno} we take limit as $\| c\|
_{L^{\infty}(\Omega)}\to 0$ we arrive to the condition
\begin{equation}
\frac{\gamma^2 }{C_{p}}\| m^{-}\| _{L^{\infty}(\Omega)}
\leq\frac{1}{\lambda_1(m,I)} \label{lap}
\end{equation}
which is the one that appears for $L=-u''$ in \cite[Theorem 2.1]{ultimo}.
\end{remark}

\begin{remark} \rm
In the statement of Theorem \ref{aa} one can replace the condition
\eqref{seno} by
\begin{gather}
\frac{\| m^{-}\| _{L^{\infty}(\Omega)}}{\|
c\| _{L^{\infty}(M^{-})}}\sinh^2 \Big[  \gamma\sqrt{\frac
{\| c\| _{L^{\infty}(M^{-})}}{C_{p}}}\Big]  \leq\frac
{1}{\lambda_1(m,I)}, \label{rem}\\
c\leq m^{+}\text{ in }M^{+}, \label{remm}
\end{gather}
where $M^{+}$ and $M^{-}$ are given by \eqref{MM}. Indeed, we first observe
that if \eqref{rem} holds then one can reason as in \eqref{ups} and prove that
$Lu_1\leq\tau mu_1^p$ in $(  x_0,\beta)  \cap M^{-}$. On
the other side, if \eqref{remm} is true then since in the proof of the theorem
$f$ is chosen satisfying $f''\geq0$ and
$\|f^{k}\| _{\infty}\leq1$, then we also have
\begin{align*}
Lu_1&=-k[  (  k-1)  f^{k-2}(  f') ^2 +f^{k-1}f'']  +cf^{k}\\
&\leq cf^{k}\leq m^{+}f^{k}\\
&\leq m^{+}f^{kp}=mu_1^p\quad\text{in }(x_0,\beta)  \cap M^{+}.
\end{align*}
The same reasoning can be done for $u_3$ and hence the proof can be
continued as in the theorem. A similar observation is valid for \eqref{expo}.
\end{remark}

\begin{theorem}\label{bien}
Let $m\in L^2 (\Omega)$ and suppose there exist
$\alpha\leq x_0<x_1\leq\beta$ such that $0\not \equiv m\geq0$ in $I:=(x_0,x_1)$.
Let $C_{p}$ be given by \eqref{cp}.

(i) If $m^{-}\in L^{\infty} (\Omega)$ and it holds that
\begin{equation}
0<\frac{(  \gamma_{b}\| \underline{B}_{\alpha}\|
_{L^{\infty}(\Omega)})  ^2 }{C_{p}-\| c\| _{L^{\infty
}(\Omega)}(  \gamma_{b}\| \underline{B}_{\alpha}\|
_{L^{\infty}(\Omega)})  ^2 }\| m^{-}\| _{L^{\infty
}(\Omega)}\leq\frac{1}{\lambda_1(m,I)},\label{i1}
\end{equation}
where
\[
\gamma_{b}:=\max\{  \| \overline{B}_{\alpha
}\| _{L^{1}(\alpha,x_1)},\| \overline{B}_{\alpha
}\| _{L^{1}(x_0,\beta)}\},
\]
then there exists a solution $u\in W^{2,2}(\Omega)$  of
\eqref{prob}.

(ii) If $c\equiv0$ and it holds that
\begin{equation} \label{i2}
(1-p)  \mathcal{M}<\frac{1}{\lambda_1(m,I)}
\end{equation}
where
\[
\mathcal{M}:=\max\big\{  \int_{x_0}^{\beta}\overline{B}_{\alpha}(
x)  \| m^{-}\underline{B}_{\alpha}\| _{L^{1}(
x,\beta)  }dx,\int_{\alpha}^{x_1}\overline{B}_{\alpha}(
x)  \| m^{-}\underline{B}_{\alpha}\| _{L^{1}(
\alpha,x)  }dx\big\},
\]
then there exists a solution $u\in W^{2,2}(\Omega)$  of
\eqref{prob}.
\end{theorem}

\begin{proof}
The proof follows the lines of the proof of Theorem \ref{aa}
and hence we omit the details. Let us prove (i). We take $u_2$ as in the
aforementioned theorem, and we choose $\tau$ such that
\[
\frac{(  \gamma_{b}\| \underline{B}_{\alpha}\|
_{L^{\infty}(\Omega)})  ^2 }{C_{p}-\| c\| _{L^{\infty
}(\Omega)}(  \gamma_{b}\| \underline{B}_{\alpha}\|
_{L^{\infty}(\Omega)})  ^2 }\| m^{-}\| _{L^{\infty
}(\Omega)}\leq\frac{1}{\tau}\leq\frac{1}{\lambda_1(m,I)}.
\]
Let $x\in[  \alpha,x_1]  $ and define
\[
u_1(  x)  :=\Big(  \sigma\int_{\alpha}^{x}\overline{B}_{\alpha
}(  y)  dy\Big)  ^{k},
\]
where
\[
\sigma:=\Big[  \frac{\| \underline{B}_{\alpha}\|
_{L^{\infty}(\Omega)}^2 (  \tau\| m^{-}\| _{L^{\infty
}(\Omega)}+\| c\| _{L^{\infty}(\Omega)})  }{C_{p}
}\Big]  ^{1/2},\quad k:=\frac{2}{1-p}.
\]
We have that $u_1(\alpha)=0$, $u_1>0$ in $(\alpha,x_1)$ and that $u_1$
is increasing. Moreover, after some computations one can check that
$\| u_1\| _{\infty}\leq1$ and
\begin{align*}
-(  \underline{B}_{\alpha}(  x)  u_1'(x)  )  '
&=-k(  k-1)  \sigma^2 (  \sigma
\int_{\alpha}^{x}\overline{B}_{\alpha}(  y)  dy)
^{k-2}\overline{B}_{\alpha}(  x)  \\
&\leq -\| \underline{B}_{\alpha}\| _{L^{\infty}(\Omega)}(
\tau\| m^{-}\| _{L^{\infty}(\Omega)}+\| c\|
_{L^{\infty}(\Omega)})  (  \sigma\int_{\alpha}^{x}\overline
{B}_{\alpha}(  y)  dy)  ^{kp}\\
&\leq \underline{B}_{\alpha}(  \tau m-c)  u_1^p\leq\underline
{B}_{\alpha}(  \tau mu_1^p-cu_1)  ;
\end{align*}
that is, $Lu_1\leq\tau mu_1^p$ in $(\alpha,x_1)$. The existence of
$u_3$ follows similarly. Let us prove (ii). We pick $\tau$ satisfying
\begin{equation}
(  1-p)  \mathcal{M}<\frac{1}{\tau}<\frac{1}{\lambda_1(m,I)}.
\label{uso}
\end{equation}
For $x\in[  \alpha,x_1]  $ we define
\[
u_1(  x)  :=\Big(  \sigma\int_{\alpha}^{x}\overline{B}_{\alpha
}(  y)  \| m^{-}\underline{B}_{\alpha}+\varepsilon
\| _{L^{1}(  \alpha,y)  }dy\Big)  ^{k}
\]
where
\[
\sigma:=\tau(  1-p)  ,\quad k:=\frac{1}{1-p},\quad\varepsilon>0.
\]
Taking $\varepsilon$ small enough and employing \eqref{uso} one can see that
$\| u_1\| _{\infty}\leq1$. Also, a few computations yield
\begin{align*}
-(  \underline{B}_{\alpha}(  x)  u_1'( x)  )  '
&\leq -k\sigma^{k}\Big( \int_{\alpha}^{x}\overline{B}_{\alpha}(  y)
\| m^{-}\underline{B}_{\alpha}+\varepsilon\| _{L^{1}(
\alpha,y)  }dy\Big)  ^{k-1}(  m^{-}(  x)
\underline{B}_{\alpha}(  x)  +\varepsilon) \\
& \leq -\tau m^{-}(  x)  \underline{B}_{\alpha}(  x)  \Big(
\sigma\int_{\alpha}^{x}\overline{B}_{\alpha}(  y)  \|
m^{-}\underline{B}_{\alpha}+\varepsilon\| _{L^{1}(
\alpha,y)  }dy\Big)  ^{kp}\\
&\leq\tau\underline{B}_{\alpha}mu_1^p.
\end{align*}
Since $u_3$ can be defined analogously, this concludes the proof of (ii).
\end{proof}

\begin{remark} \label{ojito1} \rm
Let us note that the inequalities in (i) and (ii) are not
comparable because one involves the $L^{\infty}$-norm of $m^{-}$ and 
the constant $C_p$, and the other one does not.
\end{remark}

\begin{remark} \label{ojito2} \rm
(i) It can be verified that \eqref{seno} is better than
\eqref{i1} when $b\equiv0$ (noting that in this case $\underline{B}_{\alpha
}=\overline{B}_{\alpha}=1$ and $\gamma_{b}=\gamma$ ($\gamma$ as in the
statement of Theorem \ref{aa})). If also $c\equiv0$, \eqref{i1} becomes
exactly \eqref{lap}, that is, the condition deduced from the aforementioned
theorem for the laplacian operator. \newline(ii) In the case $b\equiv0$,
\eqref{i2} reads as
\begin{equation}
(  1-p)  \max\Big\{  \int_{x_0}^{\beta}\| m^{-}
\| _{L^{1}(  t,\beta)  }dt,\int_{\alpha}^{x_1}\|
m^{-}\| _{L^{1}(  \alpha,t)  }dt\Big\}  <\frac
{1}{\lambda_1(m,I)} \label{b0}
\end{equation}
which is substantially better than the condition stated in
\cite[Theorem 2.1]{ultimo}, for $L=-u''$. Also, \eqref{b0} is clearly not
comparable (for the same reason as in the above remark) with the inequalities
that can deduced from Theorem \ref{aa} in the case $c\equiv0$ (i.e., as the
one included in Remark \ref{lapla}).
\end{remark}

\begin{corollary} \label{puf}
Let
\[
K_{b}:=\int_{\alpha}^{\beta}\overline{B}_{\alpha}(x)  \|
\underline{B}_{\alpha}\| _{L^2 (\alpha,x)}dx.
\]
If \eqref{i2} holds with $m/(  K_{b}\| m^{+}\|
_{L^2 (\alpha,\beta)})  -c$ instead of $m$, then there exists
a solution $u\in W^{2,2}(\Omega)$ of \eqref{prob}.
\end{corollary}

\begin{proof} Applying H\"older's
inequality in \eqref{iner} we see that $\| u\| _{L^{\infty
}(\Omega)}^{1-p}\leq\| m^{+}\| _{L^2 (\alpha,\beta)}K_{b}$
for any nonnegative subsolution of \eqref{prob}. Now, let
$\tau:=1/( K_{b}\| m^{+}\| _{L^2 (\alpha,\beta)})  $, and let $u$
be the solution of \eqref{prob} with $\tau m-c$ in place of $m$ provided by
Theorem \ref{bien} (ii). It follows that $\| u\| _{\infty}\leq1$ and thus
\[
-u''+bu'=(  \tau m-c)  u^p\leq\tau mu^p-cu
\]
and recalling once again Remark \ref{homsup} the corollary follows.
\end{proof}

\begin{remark} \label{lm}\rm
(i) Given \emph{any operator $L$ and any $m\in
L^2 (  \Omega)  $ with $0\not \equiv m\geq0$ in
some $I\subset\Omega$}, let us note that the above corollary implies that
\eqref{prob} has a solution if $p$ is sufficiently close to $1$.

(ii) Given any operator $L$ and any $m\in L^2 (\Omega)  $ with
$m^{-}\in L^{\infty}(  \Omega)$ and $0\not \equiv m\geq0$ in some
$I\subset\Omega$, let us observe that \eqref{i1} says that \eqref{prob}
possesses a solution for $\overline{m}:=m\chi_{\Omega-I}+km\chi_{I}$ if
$k>0$ is large enough.
\end{remark}

The next result provides the structure of the set of $p'$s such that
\eqref{prob} has a solution.

\begin{corollary}\label{ppp}
Let $m\in C( M^+)  \cap L^2 (  \Omega)$ with $m^{+}\not \equiv 0$\,
and let $\mathcal{P}$ be the set of $p\in(  0,1)  $ such that \eqref{prob}
admits some solution $u\in W^{2,2}(  \Omega)  $. Then $\mathcal{P}=(  0,1)  $ or
either $\mathcal{P}=(p,1)  $ or $\mathcal{P}=[  p,1)
$ for some $p>0$.
\end{corollary}

\begin{proof}
By Remark \ref{lm} (i) we have that $\mathcal{P} \neq \emptyset$.
Let $p^{\ast}:=\inf\mathcal{P}$. If $\mathcal{P}\neq (  0,1)  $,
Lemma \ref{qp} implies that $p^{\ast}>0$ and that
\eqref{prob} has a solution for every $p>p^{\ast}$. Therefore, either
$\mathcal{P}=(  p^{\ast},1)  $ or $\mathcal{P}=[  p^{\ast},1)  $.
\end{proof}

We write
\begin{equation} \label{jota}
\begin{gathered}
I_R(  x_0)  :=(  x_0-R,x_0+R) ,\\
\mathfrak{I}:=\{  I_R(  x_0)  \subset\Omega:m\leq0\text{ in }
I_R(  x_0\}  .
\end{gathered}
\end{equation}

\begin{theorem} \label{necee}
Let $C_{p}$ and $\mathfrak{I}$ be given by \eqref{cp} and
\eqref{jota} respectively. Suppose there exists
$u\in W^{2,2}( \Omega)  $ solution of \eqref{prob}. Then
\begin{equation}
\sup_{I_R(  x_0)  \in\mathfrak{I}}\Big[  \Big[  \frac
{\gamma_{b,R}}{\| \overline{B}_{\alpha}\| _{L^{\infty}
(I_R(  x_0)  )}}\Big]  ^2 \inf_{I_R(  x_0)
}m^{-}\Big]
 \leq C_{p}\int_{\alpha}^{\beta}\overline{B}_{\alpha}(x)
 \| m^{+}\underline{B}_{\alpha}\| _{L^{1}(\alpha,x)}dx,
\label{nec}
\end{equation}
where
\[
\gamma_{b,R}:=\min\big\{  \int_{x_0}^{x_0
+R}\overline{B}_{\alpha}(  y)  dy,\int_{x_0-R}^{x_0}
\overline{B}_{\alpha}(  y)  dy\big\}.
\]
Let $M^{+}$ be given by \eqref{MM}. If also $c>0$ in $M^{+}$, then \eqref{nec}
must also hold with $C_{p}\sup_{x\in M^{+}}\frac{m^{+}(  x)
}{c(  x)  }$ in the right side of the inequality.
\end{theorem}

\begin{proof}
We proceed by contradiction. Suppose \eqref{nec} is not true
and let $I_R(  x_0)  \in\mathfrak{I}$ be such that
\begin{equation}
C_{p}\int_{\alpha}^{\beta}\overline{B}_{\alpha}(  x)  \|
m^{+}\underline{B}_{\alpha}\| _{L^{1}(\alpha,x)}dx
\leq\Big[
\frac{\gamma_{b,R}}{\| \overline{B}_{\alpha}\| _{L^{\infty
}(I_R(  x_0)  )}}\Big]  ^2 \inf_{I_R(  x_0)
}m^{-}. \label{berp}
\end{equation}
For $x\in\overline{I}_R(  x_0)$, we define a function $w$ as
follows. If $x\in[  x_0,x_0+R] $ we set
\[
w(  x)  :=\Big(\sigma\int_{x_0}^{x}\overline{B}_{\alpha
}(  y)  dy\Big)  ^{k},
\]
where
\[
\sigma:=\Big[  \frac{\inf_{I_R(  x_0)  }m^{-}}{C_{p}\|
\overline{B}_{\alpha}\| _{L^{\infty}(I_R(  x_0)
)}^2 }\Big]  ^{1/2},\quad k:=\frac{2}{1-p},
\]
and if $x\in[  x_0-R,x_0]  $ we set $w(  x)
:=(  \sigma\int_{x}^{x_0}\overline{B}_{\alpha}(  y)dy)  ^{k}$ with
$\sigma$ and $k$ as above. In $(  x_0,x_0+R)  $ we find that
\begin{align*}
(  \underline{B}_{\alpha}w')  '-\underline {B}_{\alpha}cw
&\leq k(  k-1)  \sigma^2 \Big(  \sigma\int_{\alpha
}^{x}\overline{B}_{\alpha}(  y)  dy\Big)  ^{k-2}\overline {B}_{\alpha}\\
&\leq \frac{\inf_{I_R(  x_0)  }m^{-}}{\| \overline{B}
_{\alpha}\| _{L^{\infty}(I_R(  x_0)  )}}\Big(
\sigma\int_{\alpha}^{x}\overline{B}_{\alpha}(  y)  dy\Big)^{kp}\\
&\leq\underline{B}_{\alpha}m^{-}w^p;
\end{align*}
i.e., $Lw\geq-m^{-}w^p$, and the same is also valid in
$(x_0-R,x_0)$.

Let $u$ be a solution of \eqref{prob}. We claim that $u\leq w$ in
$I_R(  x_0)  $. Indeed, if not, let $\mathcal{O}:=\{  x\in
I_R(  x_0)  :w(  x)  <u(  x)\}$. Since $Lu=-m^{-}u^p$ in $I_R(  x_0)  $,
 we have $L(w-u)  \geq m^{-}(  u^p-w^p)  \geq0$ in $\mathcal{O}$. Let
$\overline{x}\in\partial\mathcal{O}$.
Then $w(  \overline{x})=u(  \overline{x})  $ or either $\overline{x}=x_0+R$ or
$\overline{x}=x_0-R$. If $\overline{x}=x_0+R$, by Lemma \ref{inerte} and
\eqref{berp} we obtain
\begin{align*}
u(  \overline{x})  ^{1-p}
&\leq\| u\| _{L^{\infty }(  \Omega)  }^{1-p}\\
&\leq\int_{\alpha}^{\beta}\overline{B}_{\alpha
}(  x)  \| m^{+}\underline{B}_{\alpha}\|
_{L^{1}(\alpha,x)}dx\\
&\leq \Big[ \frac{\int_{x_0}^{x_0+R}\overline{B}_{\alpha}(  y)
dy}{\| \overline{B}_{\alpha}\| _{L^{\infty}(I_R(
x_0)  )}}\Big]  ^2 \frac{\inf_{I_R(  x_0)  }m^{-}
}{C_{p}}=w(  \overline{x})  ^{1-p},
\end{align*}
and we arrive to the same inequality if\ $\overline{x}=x_0-R$. Therefore the
maximum principle says that $u\leq w$ in $\mathcal{O}$ which is not possible.
Thus, $u\leq w$ in $I_R(  x_0)  $; but $u>0$ in $\Omega$ and
$w(  x_0)  =0$. Contradiction.

To conclude the proof we note that the last statement of the theorem may be
derived as above applying Lemma \ref{dudu} instead of Lemma \ref{inerte}.
\end{proof}

\begin{remark} \rm
(i) It follows from the above theorem that given $b$, $m$, $p$ fixed, there
exists $0\leq c_0\in L^{\infty}(\Omega)$ such that for all $c\in L^{\infty
}(\Omega)$ with $c\geq c_0$ the problem \eqref{prob} 
\emph{does not admit a solution}. 
Note that given $L$, $m$, $p$ fixed with $0\not \equiv m\leq0$\ in
some $I\subset\Omega$, \emph{neither there is a solution for
$\underline{m} :=m\chi_{\Omega-I}+km\chi_{I}$ if $k>0$ is large enough}.

(ii) We observe that \eqref{nec} always is true if $p$ is sufficiently close to $1$.
Let us mention that this must indeed occur by Remark \ref{lm}.
\end{remark}

As a consequence of the previous theorems we derive an existence result for
problems of the form
\begin{equation}
\begin{gathered}
Lu=mf(  u)  \quad \text{in }\Omega\\
u>0 \quad \text{in }\Omega\\
u=0 \quad \text{on }\partial\Omega,
\end{gathered}  \label{f}
\end{equation}
for certain continuous functions $f:[  0,\infty)  \to [0,\infty) $.
Now we state assumption
\begin{itemize}
\item[(H1)] There exist $k_1,k_2>0$ and $p\in(  0,1)  $ such that
\[
k_1\xi^p\leq f(  \xi)  \leq k_2\xi^p\text{ for all }\xi
\in[  0,\underline{K}] ,
\]
where
\[
\underline{K}:=\Big[  k_1\int_{\alpha}^{\beta}\overline
{B}_{\alpha}(  x)  \| m^{+}\underline{B}_{\alpha
}\| _{L^{1}(\alpha,x)}dx\Big]  ^{1/(  1-p)  },
\]
and $f(  \xi)  \leq k_3\xi^{q}$  for all $\xi\in[\overline{K},\infty)$
some $\overline{K},k_3>0$ and $q\in(0,1)$.
\end{itemize}
Note that we make no monotonicity nor concavity assumptions on $f$.

\begin{corollary} \label{fff}
Let $f$ satisfy {\rm (H1)} and suppose \eqref{prob} has a solution with
$k_1m^{+}-k_2m^{-}$ instead of $m$. Then there exists a solution
$u\in W^{2,2}(  \Omega)  $ of \eqref{f}.
\end{corollary}

\begin{proof} Let $u$ be the solution of \eqref{prob} with
$k_1 m^{+}-k_2m^{-}$ in place of $m$. It follows from Lemma \ref{inerte} that
$\| u\| _{\infty}\leq\underline{K}$, and so from (H1) we
deduce that
\[
Lu=(  k_1m^{+}-k_2m^{-})  u^p\leq mf(  u)
\quad\text{in }\Omega.
\]
On the other side, let $\varphi>0$ be the solution of $L\varphi=m^{+}$ in
$\Omega$, $\varphi=0$ on $\partial\Omega$, and let
$k\geq\max\big\{\overline{K},(k_3(  \| \varphi\| _{\infty}+1)
^{q})^{1/(1-q)}\big\}$. Recalling (H1) and reasoning as in \eqref{ju} we
see that
\[
L(k(\varphi+1))\geq km^{+}\geq k_3(k(\varphi+1))^{q}m^{+}\geq mf(k(\varphi
+1))\text{\quad in }\Omega
\]
and the corollary is proved.
\end{proof}

\subsection*{Acknowledgments}
This research was partially supported by Secyt-UNC and CONICET.
The first author wants to dedicate this work to his teacher and
friend Tom\'{a}s Godoy.

\begin{thebibliography}{99}

\bibitem{bandle} C. Bandle, M. Pozio, A. Tesei;
\emph{The asymptotic behavior of the solutions of degenerate parabolic equations,}
Trans. Amer. Math. Soc. \textbf{303} (1987), 487-501.

\bibitem{brezislibro}H. Brezis;
\emph{Functional analysis, Sobolev spaces
and partial differential equations}. Universitext. Springer, New York, 2011.

\bibitem{chapa} J. Chaparova, N. Kutev;
 \emph{Positive solutions of the generalized Emden-Fowler equation in H\"{o}lder
spaces}, J. Math. Anal. Appl. \textbf{352} (2009), 65--76.

\bibitem{chipot}M. Chipot;
\emph{Elliptic equations: an introductory course}. Birkh\"auser Advanced Texts,
Birk\-h\"auser Verlag, Basel, 2009.

\bibitem{publi} T. Godoy, U. Kaufmann;
\emph{Periodic parabolic problems with nonlinearities indefinite in sign},
Publ. Mat. \textbf{51} (2007), 45-57.

\bibitem{ultimo} T. Godoy, U. Kaufmann;
\emph{On strictly positive solutions for some semilinear elliptic problems},
 NoDEA Nonlinear Differ. Equ. Appl. \textbf{20} (2013), 779-795.

\bibitem{junows} T. Godoy, U. Kaufmann;
\emph{Existence of strictly positive solutions for sublinear elliptic problems
in bounded domains}, Adv. Nonlinear Stud. \textbf{14} (2014), 353-359.

\bibitem{hand} J. Hern\'{a}ndez, F. Mancebo;
\emph{Singular elliptic and parabolic equations}, M. Chipot (ed.) et al.,
 Handbook of differential equations: Stationary partial differential equations.
Vol. III. Amsterdam: Elsevier/North Holland. Handbook of Differential Equations,
317-400 (2006).

\bibitem{jesusultimo}J. Hern\'andez, F. Mancebo, J. Vega;
\emph{On the linearization of some singular, nonlinear elliptic problems and
applications}, Ann. Inst. H. Poincar\'{e} Anal. Non Lin\'{e}aire
\textbf{19} (2002), 777--813.

\bibitem{aust} U. Kaufmann, I. Medri;
\emph{Strictly positive solutions for one-dimensional nonlinear problems
involving the $p$-Laplacian},
Bull. Austral. Math. Soc. \textbf{89} (2014), 243-251.

\bibitem{zhang} Y. Zhang;
\emph{Positive solutions of singular sublinear Emden-Fowler boundary value problems},
J. Math. Anal. Appl. \textbf{185} (1994), 215--222.

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