\documentclass[reqno]{amsart}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2018 (2018), No. 70, pp. 1--12.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2018 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2018/70\hfil Existence and multiplicity of solutions]
{Existence and multiplicity of solutions to superlinear periodic
 parabolic problems}

\author[T. Godoy, U. Kaufmann \hfil EJDE-2018/70\hfilneg]
{Tomas Godoy, Uriel Kaufmann}

\address{Tomas Godoy \newline
FaMAF, Universidad Nacional de C\'ordoba,
 (5000)  C\'ordoba, Argentina}
\email{godoy@mate.uncor.edu}

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

\thanks{Submitted  November 3, 2017. Published March 14, 2018.}
\subjclass[2010]{35K20, 35K60, 35B10}
\keywords{Periodic parabolic problems; superlinear; sub and supersolutions;
\hfill\break\indent  elliptic problems}

\begin{abstract}
 Let $\Omega\subset\mathbb{R}^N$ be a smooth bounded domain and let
 $a,b,c$ be three (possibly discontinuous and unbounded) $T$-periodic functions
 with $c\geq0$. We study existence and nonexistence of positive solutions for
 periodic parabolic problems
 $Lu=\lambda(a(x,t)u^p-b(x,t)  u^q+c(x,t)  )  $ in
 $\Omega\times\mathbb{R}$ with Dirichlet boundary condition, where
 $\lambda>0$ is a real parameter and $p>q\geq1$.
 If $a$ and $b$ satisfy some additional conditions and $p<(N+2) /(N+1)$
 multiplicity results are also given. Qualitative properties of the
 solutions are discussed as well. Our approach relies on the sub and
 supersolution method (both to find
 the stable solution as well as the unstable one) combined with some facts
 about linear problems with indefinite weight.
 All results remain true for the corresponding elliptic problems. Moreover, in
 this case the growth restriction becomes $p<N/(N-1)$.
\end{abstract}

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

\section{Introduction}

Let $\Omega$ be a $C^{2+\theta}$ bounded domain in $\mathbb{R}^N$,
$\theta\in(0,1)  $, $N\geq2$. For $T>0$ and $1\leq p\leq\infty$,
let $L_{T}^p$ be the Banach space of $T$-periodic functions $h$ on
$\Omega\times\mathbb{R}$ (i.e. satisfying 
$h(x,t)  =h(x,t+T)  $ a.e. $(x,t)  \in\Omega\times\mathbb{R}$) such
that $h_{\mid\Omega\times(0,T)  }\in L^p(\Omega
\times(0,T)  )  $, equipped with the norm $\|
h\| _{L_{T}^p}:=\| h_{\mid\Omega\times(0,T)
}\| _{L^p(\Omega\times(0,T)  )  }$. Let
$C_{T}^{1+\theta,(1+\theta)  /2}$, $C_{T}^{1,0}$ be the spaces of
$T$-periodic functions on $\overline{\Omega}\times\mathbb{R}$ belonging to
$C^{1+\theta,(1+\theta)  /2}(\overline{\Omega}
\times\mathbb{R})  $ and $C^{1,0}(\overline{\Omega}
\times\mathbb{R})  $ respectively, and denote by
\[
P^{\circ}:=\text{the interior of the positive cone of }C_{T}^{1+\theta,(
1+\theta)  /2}.
\]


Let $\{a_{ij}\}$, $\{b_{j}\}  $, $1\leq i,j\leq N$, be two families of 
$T$-periodic functions satisfying 
$a_{ij}\in C^{0,1}(\overline{\Omega}\times\mathbb{R})$, 
$b_{j}\in L_{T}^{\infty}$, $a_{ij}=a_{ji}$ and
\[
\sum a_{ij}(x,t)  \xi_{i}\xi_{j}\geq\alpha|\xi| ^2
\]
for some $\alpha>0$ and all $(x,t)  \in\Omega\times\mathbb{R}$,
$\xi\in\mathbb{R}^N$. Let $A$ be the $N\times N$ matrix whose $i,j$ entry is
$a_{ij}$, let $\overline{b}=(b_1,\dots ,b_{N})  $, let $0\leq
c_0\in L_{T}^{\infty}$ and let $L$ be the parabolic operator given by
\[
Lu=u_{t}-div(A\nabla u)  +\langle \overline{b},\nabla u\rangle +c_0u.
\]


For $1\leq r\leq\infty$ let $W_{r}^{2,1}(\Omega\times(
t_0,t_1)  )  $ be the Sobolev space of the functions $u\in
L^{r}(\Omega\times(t_0,t_1)  )  $, $u=u(x,t)  $, $x=(x_1,\dots ,x_{N})  $
such that $u_{t}$, $u_{x_{j}}$ and $u_{x_{i}x_{j}}$ belong to 
$L^{r}(\Omega\times(t_0,t_1)  )  $ for $1\leq i,j\leq N$, and let
$W_{r,T}^{2,1}$ be the space of $T$-periodic functions such that
 $u_{\mid\Omega\times(0,T)  }\in W_{r}^{2,1}(\Omega\times(0,T)  )$. 
For $g:\Omega\times\mathbb{R\to}\mathbb{R}$ and $r>1$ we say that
$u\in W_{r,T}^{2,1}$ is a (strong) solution of the periodic problem
\begin{equation}
\begin{gathered}
Lu=g \quad \text{in }\Omega\times\mathbb{R}\\
u=0 \quad \text{on }\partial\Omega\times\mathbb{R}\\
u\quad T\text{-periodic } 
\end{gathered}  \label{Luequalf}
\end{equation}
if the equation holds a.e. in the pointwise sense. 
It is known that for $g\in L_{T}^{r}$ with $1<r<\infty$ there exists 
a unique solution $u\in W_{r,T}^{2,1}$ of \eqref{Luequalf} and that 
the associated solution operator
$L^{-1}:L_{T}^{r}\to W_{r,T}^{2,1}$ is continuous (see e.g.
\cite[Section 4]{lieber}). Moreover, if $r>N+2$ then $W_{r,T}^{2,1}\subset
C_{T}^{1+\theta,(1+\theta)  /2}$ for some $\theta\in(0,1)  $ and so
 $u\in C_{T}^{1+\theta,(1+\theta)  /2}$ (e.g.
\cite[Lemma 3.3, p. 80]{lady}), and in particular the boundary and periodicity
conditions are satisfied pointwise.

Our aim in this paper is to study existence, nonexistence and multiplicity of
(strictly) positive solutions for periodic parabolic problems of the form
\begin{equation}
\begin{gathered}
Lu=\lambda(a(x,t)  u^p-b(x,t) u^q+c(x,t)  )  \quad  \text{in }
 \Omega\times\mathbb{R}\\
u=0 \quad \text{on }\partial\Omega\times\mathbb{R}\\
u\quad T\text{-periodic} 
\end{gathered} \label{super}
\end{equation}
where $a,b,c\in L_{T}^{r}$ for some $r>N+2$, $c\geq0$, $\lambda>0$ is a real
parameter and $p>q\geq1$.

To avoid unnecessary complexity we restrict ourselves to \eqref{super}, but
one can see that most of the results are still valid for increasing
nonlinearities that behave like $u^p$ and $u^q$ near the origin and
infinity. Let us also mention that as a consequence of our proofs all results
remain true for the corresponding elliptic problems.
Let
\begin{equation}
\Lambda:=\sup\{\lambda>0:\text{there exists a solution }u_{\lambda
}>0\text{ of \eqref{super}}\}. \label{lambdagrande}
\end{equation}


If $c\not \equiv 0$, constructing (well ordered) sub and supersolutions we
shall prove that there exists some $\overline{\Lambda}>0$ such that for all
$\lambda\in(0,\overline{\Lambda}]  $ there exists $u_{\lambda}\in
P^{\circ}$ solution of \eqref{super}. Moreover, we shall see that there exist
$k_1,k_{2}>0$ not depending on $\lambda$ such that $k_1\lambda
\leq\| u_{\lambda}\| _{\infty}\leq k_{2}\lambda$ for such
$\lambda$'s. Also, if in addition $a\geq0$ and $b^{+}/c\in L_{T}^{\infty}$, 
by means of the implicit function theorem we shall show that
$u_{\lambda}$ can be chosen such that $\lambda\to u_{\lambda}$ is
differentiable and increasing for all $\lambda\in(0,\beta)  $ for
some $\beta>0$ (see Theorem \ref{thm3.1} (i) and (ii) respectively). Under an
additional condition (which is fulfilled if for instance 
$b\leq\min\{ a,c\}  $) we shall see that \eqref{super} has a solution for every
$\lambda\in(0,\Lambda)  $. Furthermore, when $a\not \equiv 0$ we
will prove that $\Lambda<\infty$ and we will provide some upper estimates for
$\Lambda$ (see Theorem \ref{thm3.1} (iii)). Let us note that if $a\equiv0\leq b$ then
\eqref{super} becomes ``sublinear'' and it is known in this case that
$\Lambda=\infty$ (see e.g. \cite{aaa}).

On the other hand, suppose $a,b,c\in L_{T}^{\infty}$ with 
$0\leq a\not \equiv 0$\ with 
$\inf_{\Omega\times\mathbb{R}}( a/b^{+})  >0$. Then for $p<(N+2)  /(N+1)  $ we
shall prove employing (non-well-ordered) sub and supersolutions that there
exists a solution $v_{\lambda}\in P^{\circ}$ for all 
$\lambda\in( 0,\alpha)  $ for some $\alpha>0$, and that $v_{\lambda}$ 
satisfies that $\| v_{\lambda}\| _{\infty}\geq k\lambda^{-1/(p-1)  }$ for all 
$\lambda>0$ small enough and $k>0$ not depending on
$\lambda$. If additionally either the aforementioned condition in 
Theorem \ref{thm3.1} (iii) holds or $c\equiv0$, then we shall prove existence 
of a positive solution for all $\lambda\in(0,\Lambda) $ 
(see Theorem \ref{thm3.3} (i) and (ii) respectively). Moreover, in many situations 
in which $c\equiv0$ we shall show that $\Lambda=\infty$ (see Theorem \ref{thm3.3} (iii)). 
Also, as a consequence of the above results we shall obtain the existence 
of at least two positive solutions of \eqref{super}, and in the case 
$c\equiv0$ and $q=1 $ we shall prove similar results even without any 
relation between $b$ and $a $ or $c$ (see Corollaries 3.4 and 3.5).
 Let us point out that for the analogous elliptic problem, 
Theorem \ref{thm3.3} and Corollaries 3.4 and 3.5 are still valid for
$p<N/(N-1)  $ (see Remark \ref{rmk2.5} (ii) below).

Problems of the form \eqref{super} have been studied by several authors. If
$b=c\equiv0$, Esteban \cite[Theorem 4]{esteban1} proved existence of a
positive solution assuming that $L$ has $\theta$-H\"{o}lder continuous
coefficients, $p<(N+2)  /N$ and that $a=a(t)  \in
C_{T}^{\theta/2}(\mathbb{R})  $ with $\min_{\mathbb{R}}a>0$. If
in addition $a\in W_{T}^{1,\infty}(\mathbb{R})  $ and satisfies
some technical conditions, she gave the same result in \cite[Theorem 7]{esteban1} 
for $L=\partial/\partial_{t}-\Delta$ and $p<(3N+8)  /(3N-4)  $, and later 
on in \cite{esteban2} she improved this last theorem
to the case $p<N/(N-2)  $. Also, Quittner in \cite{quittner}
obtained a positive solution (also for the heat operator and 
$a\in W_{T}^{1,\infty}$ as above) for $p<(N+2)  /(N-2)  $, and an extension 
of this result under some additional
hypothesis for $a=a_1(x)  a_{2}(t)  $ with
$a_1\in C^{1}(\overline{\Omega})  $,
$a_{2}\in W_{T}^{1,\infty }$, $\inf_{\Omega\times\mathbb{R}}\{a_1,a_{2}\}  >0$
and $\Omega$ convex can be found in \cite{huska}. In all these works the main
tools used are topological degree arguments together with several a priori
estimates. We would like to point out that while our approach poses a stronger
restriction on $p$, the assumptions on $a(x,t)  $ and $L$ are
considerably weakened and the proofs given here are completely different and
(in our opinion) quite more simple. We mention also that in the elliptic case
existence of a positive solution of \eqref{super} with $b=c\equiv0$ is well
known (even if $a$ changes sign) but to our knowledge it is always asked that
either $a\in C(\overline{\Omega})  $ or $a\in L^{\infty}(
\Omega)  $ but with several additional assumptions (see e.g.
\cite{beres,amann} and the references therein).

On the other hand, when $b\equiv0\not \equiv c$ Esteban
\cite[Section V]{esteban1} showed the
 existence of at least two positive solutions for
all $0<\lambda<\Lambda$ under the aforementioned hypothesis in
\cite[Theorem 7]{esteban1} and
assuming that $0\leq c\in C(\overline{\Omega}\times\mathbb{R})$. 
If $a\equiv 1$, $0\leq c\in L_{T}^{\infty}$ and $p<(N+2)/(N-2)  $, 
Hirano and Mizoguchi found also in the case of the
heat operator two positive solutions for $\lambda>0$ small enough and studied
their stability/instability (see \cite{hirano}), and an extension for a
similar problem and sign changing $c'$s with 
$c\in C(\mathbb{R},L^{\infty}(\Omega)  )  $ was later established
in \cite{dancer}. We observe that again all these results mainly rely on
topological degree arguments and a priori bounds which require restrictions on
$p$, while we do not impose any condition on $p$ in order to prove the
existence of one of the solutions (namely, the stable one). Furthermore, we
allow in this case $a$, $b$ and $c$ to be unbounded and $a,b$ may have
indefinite sign.

Finally, as far as we know no results are available specifically for
\eqref{super} neither when $b\not \equiv 0\equiv c$ nor if $b\not \equiv
0\not \equiv c$. There are, however, some bifurcation results available for
convex nonlinearities (e.g. \cite[Chapter 3]{hess1}) or increasing
nonlinearities (e.g. \cite[Section V]{esteban1}), but under strong regularity
conditions on the coefficients of $L$ and the nonlinearity. Let us note that
for example when $a\geq0$, the right member of \eqref{super} is convex either
if $q=1$ or $b\leq0$, and if $q>1$ and $b\geq0$ then it becomes
``concave-convex". As far as the elliptic problem is concerned, \eqref{super}
with $a,b,c$ positive constants and $p\leq N/(N-2)  $ is included
in some of the many types of nonlinearities covered in the nice paper
\cite[Theorem 6.21]{ouyang}. When $\Omega$ is a ball and
$L=-\Delta$, it is proved there that $\Lambda<\infty$ and that there exist
exactly two positive solutions for $\lambda\in(0,\Lambda)  $ and
exactly one for $\lambda=\Lambda$. Let us also mention that the nonlinearities
that arise in Corollary \ref{coro3.5} are included (for $\Omega$, $L$ and $p$ as above,
and $a,\pm b$ positive constants) in \cite[Theorems 6.5 and 6.11]{ouyang},
and it is also proved there that in this cases the solution is unique for every
$\lambda\in(0,\Lambda)  $. We remark that all these last results
are obtained applying variational and symmetry arguments which of course are
not eligible in our case.

\section{Preliminaries}

We start by collecting some necessary facts about periodic parabolic problems
with indefinite weight.


\begin{remark} \label{rmk2.1} \rm
(i) Let $m\in L_{T}^{r}$ with $r>(N+2)/2$,
and let
\begin{equation}
P_{\Omega}(m)  :=\int_0^{T}\operatorname{esssup}_{x\in\Omega}m(x,t)  dt.
\label{peomega}
\end{equation}
Then $P_{\Omega}(m)  >0$ is necessary and sufficient for the
existence of a (unique and simple) positive principal eigenvalue 
$\lambda _1(L,m)  $ (or $\lambda_1(m)  $ if no confusion
arises) for the problem
\begin{equation}
\begin{gathered}
Lu=\lambda mu \quad \text{in }\Omega\times\mathbb{R}\\
u=0 \quad \text{on }\partial\Omega\times\mathbb{R}\\
u\quad \text{-periodic} 
\end{gathered}  \label{lineal}
\end{equation}
(cf. \cite[Theorem 3.6]{gk}). We note that $P_{\Omega}(m)=+\infty$ 
is allowed (cf. \cite[p. 218]{gk}) and that no regularity on
$\partial\Omega$ is needed. It also holds that $m\to\lambda_1(
m)  $ is continuous (cf. \cite[Theorem 3.9]{gk}). 
If $\lambda_1( m) $ exists, we will denote (from now on) by $\Phi$ the positive
principal eigenfunction normalized by $\| \Phi\| _{\infty}=1$. 
If in addition $\Omega$ has $C^{2+\theta}$ boundary and $r>N+2$, then
$\Phi\in P^{\circ}$.

(ii) The following comparison principle holds: if $m_1,m_{2}\in L_{T}^{r}$
with $r>(N+2)  /2$, $P_{\Omega}(m_1)  >0$ and
$m_1\leq m_{2}$ in $\Omega\times\mathbb{R}$, then
$\lambda_1(m_1)  \geq\lambda_1(m_{2})  $ and, if in addition
$m_1<m_{2}$ in a set of positive measure, then
$\lambda_1( m_1)  >\lambda_1(m_{2})  $ (cf. \cite[Remark 3.7]{gk}).
\end{remark}


\begin{remark} \label{rmk2.2} \rm
(i) Let $m\in L_{T}^{r}$ with $r>(N+2)  /2$. 
For $\lambda\in\mathbb{R}$, let $\mu_{L,m}(\lambda)  $ (or simply
$\mu_{m}(\lambda)  $ if no confusion arises) be defined as the
unique $\mu\in\mathbb{R}$ such that the Dirichlet periodic problem 
$Lu=\lambda mu+\mu_{m}(\lambda)  u$ in $\Omega\times\mathbb{R}$ has a
positive solution $u$. Then $\mu_{m}(\lambda)  $ is well defined,
$\mu_{m}(0)  >0$, $\mu_{m}$ is concave and continuous, and a
given $\lambda\in\mathbb{R}$ is a principal eigenvalue for \eqref{lineal} if
and only if $\mu_{m}(\lambda)  =0$ 
(cf. \cite[Lemmas 3.2 and 3.5]{gk}). In particular, for $\lambda>0$, 
if $P_{\Omega}(m)  >0$ then
$\mu_{m}(\lambda)  >0$ if and only if $\lambda<\lambda_1(m)  $, 
and $\mu_{m}(\lambda)  >0$ for all $\lambda>0$ if
$P_{\Omega}(m)  \leq0$.

(ii) Let $m,h\in L_{T}^{r}$ with $r>(N+2)  $. Then, if $\mu_{m}(\lambda)  >0$, 
the problem
\begin{equation}
\begin{gathered}
Lu=\lambda mu+h \quad \text{in }\Omega\times\mathbb{R}\\
u=0 \quad \text{on }\partial\Omega\times\mathbb{R}\\
u\quad T\text{-periodic } 
\end{gathered}  \label{maximoconpeso}
\end{equation}
has a unique solution $u\in W_{r,T}^{2,1}$ which is positive if $h\geq0$, and
the solution operator $h\to u$ is continuous 
(cf. \cite[Lemma 2.9]{tomas}). Conversely, if $\lambda_1(m)  $ exists 
and $Lu\gneqq \lambda mu$ (respectively $\lneqq$) for some $\lambda>0$ and $u>0$ in
$\Omega\times\mathbb{R}$ with $u$ $=0$ on $\partial\Omega\times\mathbb{R}$,
then $\lambda<\lambda_1(m)  $ (respectively $\lambda
>\lambda_1(m)  $) (cf. \cite[Remark 2.1 (e)]{sofia}).
\end{remark}

We will need the following elementary lemma to provide one of the upper
estimates for $\Lambda$.

\begin{lemma} \label{lem2.3} 
Let $p>q\geq1$  and let $h(\xi)  :=\xi^p-\xi^q-c_{p,q}\xi+1$, where 
$c_{p,q}>0$ is defined by 
\begin{equation}
c_{p,q}:=\begin{cases}
\frac{p}{(p-1)  ^{(p-1)  /p}}-1 & \text{if }q=1\\
(\frac{q}{p-1})  ^{(p-1)  /(p-q)  } &
\text{if }q>1\text{ and }p-q\geq1\\
p-q & \text{if }q>1\text{ and }p-q\leq1
\end{cases}  \label{cpq}
\end{equation}
Then $h(\xi)  \geq0$ for all $\xi\geq 0$.
\end{lemma}

\begin{proof}
 Suppose $q=1$. Then $h$ attains its unique minimum at 
$\xi_0:=((1+c_{p,q})  /p)  ^{1/(p-1)  }$.
Moreover, after some computations we get
\[
h(\xi_0)  =\Big(\frac{1+c_{p,q}}{p}\Big)  ^{1/(p-1)}
\Big(\frac{1+c_{p,q}}{p}-(1+c_{p,q}) \Big)+1=0.
\]
Suppose now $q>1$ and $p-q\geq1$. Define 
$\xi_0:=(q/(p-1)  )  ^{1/(p-q)  }$. Then 
$p\xi_0^{p-1}-q\xi _0^{q-1}=c_{p,q}$ and hence $h'(\xi_0)  =0$.
Furthermore, taking into account this we find that
\begin{align*}
h(\xi_0)   
&  =(1-p)  \xi_0^p+(q-1)  \xi_0^q+1\\
&  =\Big(\frac{q}{p-1}\Big)  ^{\frac{q}{p-q}}
\Big(\frac{(1-p)  q}{(p-1)  }+(q-1)  \Big)  +1\geq0
\end{align*}
because $p-q\geq1$.

Finally, suppose $q>1$ and $p-q\leq1$. Since in this case $c_{p,q}=p-q$ it
follows that $h'(1)  =0$. Moreover, $h(1)\geq0$ because $p-q\leq1$ and 
this concludes the proof. 
\end{proof}

We say that $f:\Omega\times\mathbb{R}\times\mathbb{R}\to\mathbb{R}$
is an $L_{T}^{r}$-Carath\'{e}odory function if $f(x,t,\xi)  $ is
 $T$-periodic in $T$, $(x,t)  \to f(x,t,\xi)  $ is measurable for all 
$\xi\in\mathbb{R}$, $\xi\to f(x,t,\xi) $ is continuous on $\mathbb{R}$ a.e.
$(x,t)  \in\Omega\times\mathbb{R}$; and, for each $\rho>0$, there
exists $h\in L_{T}^{r}$ such that $| f(x,t,\xi)| \leq h(x,t)  $ for a.e.
 $(x,t) \in\Omega\times\mathbb{R}$ and every $\xi\in[-\rho,\rho]  $.
Also, if $r>N+2$, we will say that $v\in W_{r,T}^{2,1}$ is a subsolution
(respectively a supersolution) of
\begin{equation}
\begin{gathered}
Lu=f(x,t,u)  \quad \text{in }\Omega\times\mathbb{R}\\
u=0 \quad \text{on }\partial\Omega\times\mathbb{R}\\
u\quad T\text{-periodic } 
\end{gathered} \label{problema}
\end{equation}
if $Lv\leq f(x,t,v)$ (resp. $Lv\geq f(x,t,v)  $)
in $\Omega\times\mathbb{R}$ and $v\leq0$ (resp. $v\geq0$) on $\partial
\Omega\times\mathbb{R}$. Finally, we say that a subsolution $v$ of
\eqref{problema} is strict if for every solution $u$ of \eqref{problema} with
$v\leq u$ one has $v<u$ in $\Omega\times\mathbb{R}$ and either $v<u$ or $v=u $
and $\partial_{\nu}u>\partial_{\nu}v$ on $\partial\Omega\times\mathbb{R}$,
$\nu$ being the unit outer normal to $\partial\Omega$. A strict supersolution
is defined analogously.

We state for the reader's convenience the following existence result in 
the presence of non-well-ordered sub and supersolutions 
(for the proof, see \cite[Lemma 2.3]{houston}). Let us mention that for 
$m\equiv1$ this lemma can be found in \cite[Theorem 3.2]{omari}.

\begin{lemma} \label{lem2.4} 
Let $m\in L_{T}^{\infty}$ such that $P_{\Omega}(m)  >0$ and let 
$f:\Omega\times \mathbb{R}\times\mathbb{R}\to\mathbb{R}$ satisfying 
\begin{itemize}
\item[(H1)] $f$ is an $L_{T}^{r}$-Carath\'{e}odory
function for some $r>N+2$.

\item[(H2)] There exist $\gamma\in(0,1)$, $\delta\in(1,(N+2-\gamma)  /(N+1)  )$
 and $\sigma_0>0$  such that 
\[
\frac{f(x,t,\xi)  -\lambda_1(m)  m\xi}{|
\xi| ^{\gamma}}\geq-1\quad\text{and}\quad 
\frac{f(x,t,\xi)  -\lambda_1(m)  m\xi}{| \xi|^{\delta}}\leq1
\]
a.e. $(x,t)  \in\Omega\times\mathbb{R}$ for all
$\xi$ such that $| \xi| >\sigma_0$.
\end{itemize}
Suppose that there exist $v,w$ sub and supersolutions
respectively of \eqref{problema}  such that $v\nleq w$. Then
\eqref{problema} has a solution $u\in\overline{\mathcal{O}}$
where
\[
\mathcal{O}:=\{u\in C_{T}^{1,0}:v\nleq u\text{ and }u\nleq w\}.
\]
\end{lemma}

\begin{remark} \label{rmk2.5}  \rm
(i) In the same way as in \cite[Theorem 3.2 and Remark 2.2]{omari}, 
if $v$ and $w$ are strict sub and supersolutions, every solution
$u\in\overline{\mathcal{O}}$ actually satisfies $u\in\mathcal{O}$.

(ii) The restrictions $r>N+2$ and $\delta\in(1,(N+2-\gamma)  /(N+1)  )  $ 
come from the use of the strong maximum principle in the proof of 
Lemma \ref{lem2.4}. Thus, in the elliptic case one can take $r>N$ and 
$\delta\in(1,(N-\gamma)  /(N-1)  )  $ and obtain exactly the same conclusions 
(in fact, for $m\equiv1$ this is done for instance in \cite[Section 4]{eli}). 
\end{remark}

\section{Main results}

As usual, we write $f=f^{+}-f^{-}$ with $f^{+}=\max(f,0)  $ and
$f^{-}=\max(-f,0)  $. For $a,b,c\in L_{T}^{r}$ with $r>N+2$\ and
$p>q\geq1$ we set
\begin{gather}
\overline{\Lambda}:=1/\| L^{-1}(a^{+}+b^{-}+c)
\| _{L_{T}^{\infty}},\quad
\underline{\Lambda}:=\| (L+\overline{\Lambda}(a^{-}+b^{+}) )  ^{-1}c\|
_{L_{T}^{\infty}},\label{defi}\\
\beta_0:=\min\{\overline{\Lambda},(\overline{\Lambda}
^{q-1}\lambda_1(pa+qb^{-})  )  ^{1/q}\}  .
\label{defi2}
\end{gather}
Recall that $\Lambda$ is given by \eqref{lambdagrande}.


\begin{theorem} \label{thm3.1}
 Let $a,b,c\in L_{T}^{r}$ for some $r>N+2$ such that 
$0\leq c\not \equiv 0$, and let $p>q\geq1$.  Then
\begin{itemize}
\item[(i)] Equation \eqref{super} has a solution 
$u_{\lambda}\in P^{\circ}$  for all $\lambda\in(0,\overline{\Lambda}]$  and 
\begin{equation}
\underline{\Lambda}\lambda
\leq\| u_{\lambda}\| _{L_{T} ^{\infty}}
\leq\overline{\Lambda}^{-1}\lambda \label{cotiyas}
\end{equation}
for such $\lambda$ (in particular, $\Lambda\geq\overline{\Lambda}$).

\item[(ii)] Assume in addition that $a\geq0$  and 
$b^{+}/c\in L_{T}^{\infty}$. Then there exists 
$\beta>\beta_0$ such that $\lambda\to u_{\lambda}$  is a 
$C^{1}$  increasing map from $(0,\beta)$ into $P^{\circ}$.

\item[(iii)] Assume in addition that 
\begin{equation}
K_1:=\| b^{+}/c\| _{L_{T}^{\infty}}^{1/q}\leq\inf
_{\Omega\times\mathbb{R}}(a/b^{+})  ^{1/(p-q)}:=K_{2}. \label{condi}
\end{equation}
\end{itemize}
Let $m(x,t)  :=\min\{a(x,t),c(x,t)  \}$, let $c_{p,q}$ be given by
\eqref{cpq}, and let $w\in P^{\circ}$ be the solution of
\eqref{Luequalf} with $c$ in place of $g$. Then
\eqref{super} has a solution $u_{\lambda}\in P^{\circ}$
 for all $\lambda\in(0,\Lambda)$ and 
\begin{equation}
\Lambda\leq\begin{cases}
\lambda_1(m)  /c_{p,q} & \text{if }m\not \equiv 0\\
\max\{1,\lambda_1(aw^{p-1})  \}  & \text{if }m\equiv0\text{ and }a\not \equiv 0
\end{cases}  \label{cota}
\end{equation}
\end{theorem}

\begin{proof} 
Let $\lambda>0$, and let $\Phi_{\lambda}\in P^{\circ}$ be the
unique positive principal eigenfunction of \eqref{lineal} with $c$ and
$L+\lambda(a^{-}+b^{+})  $ in place of $m$ and $L$ respectively,
normalized by $\| \Phi_{\lambda}\| _{\infty}=1$ (since
$c\not \equiv 0$ such
$\Phi_{\lambda}$ exists). Let 
$\lambda^{\ast}:=\lambda_1(L+\lambda(a^{-}+b^{+})  ,c)  $ and let
$k_0=k_0(\lambda)  :=\lambda/\lambda^{\ast}$. We first claim
that $k\Phi_{\lambda}$ is a subsolution of \eqref{super} for every 
$0<k\leq k_0$. Indeed, taking into account that $p,q\geq1$ we find that
\[
\lambda(a^{+}(k\Phi_{\lambda})  ^p+b^{-}(k\Phi_{\lambda})  ^q+c)   
  \geq\lambda c\geq\lambda^{\ast}ck\Phi_{\lambda}
  =(L+\lambda(a^{-}+b^{+})  )  k\Phi_{\lambda}
\]
and since $k\Phi_{\lambda}\leq1$ the claim follows.

On the other hand, let $\overline{\Lambda}$ be given by \eqref{defi}, let
$0<\lambda\leq\overline{\Lambda}$ and let $z_{\lambda}\in P^{\circ}$ be the
unique solution of the periodic problem $Lz_{\lambda}=\lambda(
a^{+}+b^{-}+c)  $ in $\Omega\times\mathbb{R}$, $z_{\lambda}=0$ on
$\partial\Omega\times\mathbb{R}$. It is easy to check that $z_{\lambda}$ is a
supersolution for \eqref{super}. Indeed, clearly $\| z_{\lambda
}\| _{\infty}=\overline{\Lambda}^{-1}\lambda\leq1$ and then
\[
Lz_{\lambda}\geq\lambda(a^{+}\| z_{\lambda}\|
_{\infty}^p+b^{-}\| z_{\lambda}\| _{\infty}^q+c)
\geq\lambda(az_{\lambda}^p-bz_{\lambda}^q+c)  .
\]
Hence, if $k=k(\lambda)  $ is small enough, \cite[Theorem 1]{deuel} 
gives some $u_{\lambda}\in L_{T}^{\infty}$ solution of
\eqref{super} satisfying $k\Phi_{\lambda}\leq u_{\lambda}\leq z_{\lambda}$ for
all $\lambda\in(0,\overline{\Lambda}]  $. Moreover, $u_{\lambda
}\in P^{\circ}$ and $\| u_{\lambda}\| _{\infty}\leq
\overline{\Lambda}^{-1}\lambda$ for such $\lambda'$s (i.e., the second
inequality in \eqref{cotiyas} holds). Also, taking into account this last
fact, from \eqref{super} we obtain
\begin{equation}
Lu_{\lambda}\geq\lambda(-a^{-}u_{\lambda}-b^{+}u_{\lambda}+c)
\label{mayor}
\end{equation}
and thus $(L+\overline{\Lambda}(a^{-}+b^{+})  )
u_{\lambda}\geq\lambda c$ (because $\lambda\leq\overline{\Lambda}$) and the
first inequality in \eqref{cotiyas} follows. So, (i) is proved.

To prove (ii) we first note that we may assume without loss of
generality that $b\leq c$. Indeed, if $\| b^{+}/c\| _{\infty}=0$ 
then $b\leq0$ and so $b\leq c$. If not, take 
$k:=\|b^{+}/c\| _{\infty}^{1/q}$ and define $a_{k}:=ak^{1-p}$,
$b_{k}:=bk^{1-q}$ and $c_{k}:=ck$. It follows that $b_{k}\leq c_{k}$.
Furthermore, $u$ is a solution of \eqref{super} if and only if $ku$ is a
solution of \eqref{super} with $a_{k}$, $b_{k}$ and $c_{k}$ in place of $a$,
$b$ and $c$ respectively. Henceforth we assume that $b\leq c$.

Let $\lambda>0$, $u_{\lambda}>0$ be the solution of \eqref{super} found in
(i), and let $m_{\lambda}:=pau_{\lambda}^{p-1}-qbu_{\lambda}^{q-1}$.
We claim that the implicit function theorem can be applied in a point 
$(\lambda,u_{\lambda})  $ for any $\lambda>0$ sufficiently small. Indeed,
a direct computation shows that in order to see this it suffices to prove that
for a given $h\in L_{T}^{r}$ there is a unique solution $u\in W_{r,T}^{2,1}$
of problem \eqref{maximoconpeso} with $m_{\lambda}$ in place of $m$ and that
the solution operator for this problem is continuous. 
Thus, recalling Remark \ref{rmk2.2} (ii) the claim will follow if
$\lambda_1(m_{\lambda})>\lambda$ (if such $\lambda_1(m_{\lambda})  $ exists; 
if $\lambda_1(m_{\lambda})  $ does not exist we have nothing to prove). 
Now, let $\beta_0$ be given by \eqref{defi2} and let $0<\lambda \leq\beta_0$. 
Since $p>q$ and $\lambda\leq\overline{\Lambda}$, by the second
inequality in \eqref{cotiyas} we have
\[
m_{\lambda}\leq u_{\lambda}^{q-1}(pau_{\lambda}^{p-q}+qb^{-})
<(\overline{\Lambda}^{-1}\lambda)  ^{q-1}(pa+qb^{-})
\]
and therefore the comparison principle in Remark \ref{rmk2.1} (ii) yields
\[
\lambda_1(m_{\lambda})  >(\overline{\Lambda}\lambda
^{-1})  ^{q-1}\lambda_1(pa+qb^{-})  \geq\lambda
\]
(if $\lambda_1(pa+qb^{-}) $ does not exist then $m_{\lambda}\leq0
$ and we are done). Hence, the claim is proved.

Let $I:=(\alpha_1,\alpha_{2})  $ be a maximal interval centered
at $\beta_0$ provided by the implicit function theorem in which
$\lambda\to u_{\lambda}$ is a $C^{1}$ map into $P^{\circ}$.
Differentiating \eqref{super} with respect to $\lambda$ and taking into
account that $a\geq0$ and $b\leq c$ we obtain
\begin{equation}
(L-\lambda m_{\lambda})  \frac{\partial u_{\lambda}}
{\partial\lambda}=au_{\lambda}^p-bu_{\lambda}^q+c\geq c(
1-u_{\lambda}^q)  \label{deri}
\end{equation}
for all $\lambda\in I$. So, since $u_{\beta_0}$ satisfies \eqref{cotiyas},
it follows from \eqref{deri} and Remark \ref{rmk2.2} (ii) that
$\partial u_{\lambda }/\partial\lambda>0$ for some $(\alpha,\beta)  \subset I$ with
$\beta_0\in(\alpha,\beta)  $. We next observe that
$\alpha=\alpha_1=0$. Indeed, suppose first $\alpha>\alpha_1$. In this case
$\partial u_{\lambda}/\partial\lambda_{\mid\lambda=\alpha}=0$, but since
$\lambda\to u_{\lambda}$ is increasing in $(\alpha,\beta)  $ and 
$\| u_{\beta_0}\| _{\infty}\leq1$,
again \eqref{deri} and Remark \ref{rmk2.2} (ii) yield
$\partial u_{\lambda}/\partial\lambda_{\mid\lambda=\alpha}>0$.
 Assume now $\alpha>0$, and let $u_{j}\in P^{\circ}$ be the solutions 
of \eqref{super} corresponding to some sequence $\lambda_{j}\searrow\alpha$. 
Then $u_{j}$ is decreasing and so the continuity of the solution operator 
$L^{-1}$ supplies some $u_{\alpha}\geq0$ solution of \eqref{super} for 
$\lambda=\alpha$. Furthermore, $\alpha>0$ implies $\| u_{\alpha}\| _{\infty}>0$ 
(because $c\not \equiv 0$) and hence we can apply the implicit function theorem 
in the point $(\alpha,u_{\alpha})  $, contradicting the maximality of 
$( \alpha,\beta)  $. Consequently, $\alpha=0$ and (ii) is proved.

Let us prove (iii). By \eqref{condi}, as in the beginning of the proof of 
(ii) we may now assume that $b\leq\min\{a,c\}  $. Indeed, take
$0<k\in[  K_1,K_{2}]  $ (where $K_1$ and $K_{2}$ are given by
\eqref{condi}) and define $a_{k}$, $b_{k}$ and $c_{k}$ as in (ii). Then
$b_{k}\leq\min\{a_{k},c_{k}\}  $ and as before $u$ is a solution
of \eqref{super} if and only if $ku$ is a solution of \eqref{super} with
$a_{k}$, $b_{k}$ and $c_{k}$ in place of $a$, $b$ and $c$.

Now, let $\lambda\in(0,\Lambda)  $ and let $\overline{\lambda}
\in(\lambda,\Lambda)  $ such that there exists $u_{\overline
{\lambda}}>0$ solution of \eqref{super} with $\overline{\lambda}$ in place of
$\lambda$. Since $b\leq\min\{a,c\}  $ and $a,c\geq0$ it is easy
to check that $a\xi^p-b\xi^q+c\geq0$ for all $\xi\geq0$ a.e. 
$(x,t)  \in\Omega\times\mathbb{R}$. Thus $u_{\overline{\lambda}}$ is a
supersolution for \eqref{super}. Therefore, since for all $\lambda>0 $ the
first paragraph of the proof provides subsolutions for \eqref{super} of the
form $k\Phi_{\lambda}$, making $k>0$ sufficiently small we can again apply
\cite[cite 1]{deuel}, and obtain a solution of \eqref{super}.

We prove \eqref{cota}. Suppose first 
$0\not \equiv m(x,t):=\min\{a(x,t)  ,c(x,t)  \}$. We
observe that $\lambda_1(m)  $ exists because $m\geq0$. Let
$u>0$ be a solution of \eqref{super}. Taking into account Lemma \ref{lem2.3} we get
\[
Lu=\lambda(au^p-bu^q+c)  \geq\lambda m(u^p -u^q+1)  \geq\lambda mc_{p,q}u
\]
and then Remark \ref{rmk2.2} (ii) says that
 $\lambda\leq\lambda_1(mc_{p,q})  =\lambda_1(m)  /c_{p,q}$.

On the other hand, if $m\equiv0$, from $b\leq\min\{a,c\}  $ and
$a,c\geq0$ we have that $b\leq0$. Suppose now the last inequality in
\eqref{cota} is not valid. Let $w\in P^{\circ}$\ be the unique solution of
\eqref{Luequalf} with $c$\ in place of $g$. Since $c\not \equiv 0$ it holds
that $w>0$. Moreover, $0\leq aw^{p-1}\not \equiv 0$ because $a\not \equiv 0$.
Choose $\lambda>\max\{1,\lambda_1(aw^{p-1})  \}  $
such that there exists $u>0$ solution of \eqref{super}. We observe that since
$\lambda>1$ the maximum principle yields $u\geq w$. Also,
\[
Lu=\lambda(au^p-bu^q+c)  \geq\lambda au^p\geq\lambda aw^p
\]
and so again employing Remark \ref{rmk2.2} (ii) we obtain $\lambda\leq\lambda
_1(aw^{p-1})  $. Contradiction.
\end{proof}


Let us note that if in Theorem \ref{thm3.1} (iii) it holds that $m=a\equiv0$ then
\eqref{super} becomes $Lu=\lambda(b^{-}u^q+c)  $ and hence
upper bounds for $\Lambda$ can be obtained in the same way as there.


\begin{lemma} \label{lem3.2}
Let $a,b,c\in L_{T}^{\infty}$ such that
$a,c\geq0$  and $\inf_{\Omega\times\mathbb{R}}(a/b^{+}
)  >0$, and let $1\leq q<p<(N+2)  /(N+1)  $. 
Assume there exist $v,w\geq0$  sub and supersolutions respectively of 
\eqref{super} such that neither of them is a solution and 
$v\nleq w$. Then there exists $u\in P^{\circ}$ solution of \eqref{super}
satisfying $v\not \leq u\not \leq w$.
\end{lemma}

\begin{proof} 
 We note first that $v$ is a strict subsolution. Indeed, if $z$ is a solution 
of \eqref{super} with $v\leq z$ then
\[
L(z-v)  >-b^{+}(z^q-v^q)  \geq-q\|z\| _{\infty}^{q-1}b^{+}(z-v)
\]
and hence the assertion follows from the strong maximum principle (as stated
e.g. in \cite[Theorem 13.5]{dakoch}). In the same way $w$ is a strict
 supersolution.

Let $\widetilde{f}$ be defined by 
$\widetilde{f}(x,t,\xi)=\lambda(a(x,t)  \xi^p-b(x,t)  \xi^q+c(x,t)  )  $ 
for $\xi\geq0$ and $\widetilde{f}(x,t,\xi)  =\lambda c(x,t)  $ for
 $\xi<0$. Let $\mu:=(N+2)  /(N+1)  -p>0$, and choose $\alpha,\gamma>0$ small
enough such that $\alpha+\gamma/(N+1)  <\mu$. Since 
$\inf_{\Omega\times\mathbb{R}}(a/b^{+})  >0$, reasoning as in 
Theorem \ref{thm3.1} (iii) we may assume that $b\leq a$. Taking into account this,
 it is easy to check that the function $\widetilde{f}$ satisfies the 
assumptions of Lemma \ref{lem2.4} with $\gamma$ as above, $\delta:=p+\alpha$, $m:=a$ 
and any $r>N+2$.
Therefore, Lemma \ref{lem2.4} provides a solution $u\in C_{T}^{1,0}$ of \eqref{problema}
 with $\widetilde{f}$ in place of $f$ satisfying $v\not <  u\not <  w$.
Moreover, since $v$ and $w$ are strict sub and supersolutions, from 
Remark \ref{rmk2.5} (i) we get $v\nleq u\nleq w$. In particular $u\not \equiv 0$ because
$w\not \equiv 0$ (observe that if $w\equiv0$ then $c\equiv0$ and
therefore $w$ is a solution of \eqref{super}, contradicting the hypothesis).
Let $U:=\{(x,t)  \in\Omega\times\mathbb{R}:u(x,t)  <0\}  $. 
If $U\neq\varnothing$ we have $Lu=\lambda c\geq0$
in $U$ and $u=0$ on $\partial U$ and so the maximum principle (as stated e.g.
in \cite[Lemma 2.3]{publi}) implies $u\geq0$ in $U$ which is not possible.
Thus $u\geq0$ in $\Omega\times\mathbb{R}$ and hence by the aforementioned
strong maximum principle in \cite{dakoch} $u\in P^{\circ}$. It follows that
$u$ is a solution of \eqref{super} and this ends the lemma. 
\end{proof}

We focus now on what happens when $a\not \equiv 0$. The special case
$c\equiv0$ and $q=1$ will be considered separately in Corollary \ref{coro3.5} below. 
For $\overline{\Lambda}$\ as in \eqref{defi} and $\varepsilon>0$ we set
\begin{equation}
\begin{gathered}
\delta_0:=\min\{\lambda_1(a)  ,\overline{\Lambda}\} \\
\Omega_{\varepsilon}:=\{x\in\Omega:\operatorname{dist}(x,\partial\Omega)
<\varepsilon\} ,\quad \Omega_{\varepsilon}^{c}:=\Omega-\Omega_{\varepsilon}.
\end{gathered} \label{epsi}
\end{equation}


\begin{theorem} \label{thm3.3}
 Let $a,b,c\in L_{T}^{\infty}$ with $a,c\geq0$, 
$a\not \equiv 0$ and $\inf_{\Omega\times\mathbb{R}}(a/b^{+})  >0$.
 Let $1\leq q<p<(N+2)  /(N+1)  $. Then
\begin{itemize}
\item[(i)] Equation \eqref{super}  has a solution 
 $v_{\lambda}\in P^{\circ}$ for all $\lambda\in(0,\delta_0)  $ and
there exists $k>0$ not depending on $\lambda$ such that
for all $\lambda>0$  small enough
\begin{equation}
\| v_{\lambda}\| _{L_{T}^{\infty}}\geq k\lambda^{\frac{-1}{p-1}}. \label{infi}
\end{equation}


\item[(ii)] Assume that either $c\equiv0$ or \eqref{condi}
 holds.  Then \eqref{super}  has a positive
solution for all $\lambda\in(0,\Lambda)  $.

\item[(iii)]  If $c\equiv0$, $q>1$ and $b\leq0$ in
$\Omega_{\sigma}\times\mathbb{R}$ for some $\sigma>0$, then
$\Lambda=\infty$.
\end{itemize}
\end{theorem}

\begin{proof} 
As in the above lemma we assume that $b\leq a$. Let
$\lambda\in(0,\lambda_1(a) )$ ($a\not \equiv 0$
and so such $\lambda_1(a)  $ exists) and 
$m_{\varepsilon}:=a\chi_{\Omega_{\varepsilon}^{c}\times\mathbb{R}}-b^{+}\chi_{\Omega
_{\varepsilon}\times\mathbb{R}}$. 
Since $0\leq a\not \equiv 0$ and the
positive principal eigenvalue is continuous with respect to the weight (cf.
Remark \ref{rmk2.1}), we can choose $\varepsilon>0$ small enough such that $\lambda
_1(m_{\varepsilon})  $ exists and 
$\lambda_1(m_{\varepsilon})  \geq\lambda$. Let $\Phi$ be the unique positive
principal eigenfunction of \eqref{lineal} with $m_{\varepsilon}$ in place of
$m$, normalized by $\| \Phi\| _{\infty}=1$. Let
$0<\delta:=\min_{\Omega_{\varepsilon}^{c}\times\mathbb{R}}\Phi$ and let
$K_0=K_0(\lambda)  :=((1+\lambda_1(m_{\varepsilon})/\lambda)  ^{1/(p-q)  })/\delta$.
 We claim that $k\Phi$ is a subsolution of \eqref{super} for every
$k\geq K_0$. Indeed, let $k\geq K_0$ and let us first write 
$f(x,t,\xi)  :=\lambda(a\xi^p-b\xi^q+c)  $ and
\begin{equation}
\begin{gathered}
A_{k}:=\{(x,t)  \in\Omega\times\mathbb{R}:k\Phi\leq1\}  , \quad
A_{k}^{c}:=(\Omega\times\mathbb{R})  -A_{k},\\
B_{k}:=(\Omega_{\varepsilon}\times\mathbb{R})  \cap A_{k}, \quad
B_{k}^{c}:=(\Omega_{\varepsilon}\times\mathbb{R})  \cap A_{k} ^{c},\\
C_{k}:=(\Omega_{\varepsilon}^{c}\times\mathbb{R})  \cap A_{k}, \quad
C_{k}^{c}:=(\Omega_{\varepsilon}^{c}\times\mathbb{R})  \cap A_{k}^{c}.
\end{gathered} \label{conjuntos}
\end{equation}
We observe that $C_{k}=\varnothing$ because 
$k\Phi\geq K_0\Phi>\delta^{-1}\Phi\geq1$ in 
$\Omega_{\varepsilon}^{c}\times\mathbb{R}$. Now, taking
into account that $b\leq a$ and $1\leq q<p$ we get that
\begin{equation} \label{cuenta}
\begin{aligned}
f(x,t,k\Phi)   &  \geq-\lambda b^{+}(k\Phi)
^q\chi_{B_{k}}+\lambda a((k\Phi)  ^p-(
k\Phi)  ^q)  \chi_{B_{k}^{c}\cup C_{k}^{c}} \\
&  \geq-\lambda b^{+}k\Phi\chi_{B_{k}}+\lambda a(k\Phi)
^q((k\delta)  ^{p-q}-1)  \chi_{C_{k}^{c}}\\
&  \geq-\lambda_1(m_{\varepsilon})  b^{+}k\Phi\chi_{B_{k}
}+\lambda_1(m_{\varepsilon})  ak\Phi\chi_{C_{k}^{c}} \\
&  \geq\lambda_1(m_{\varepsilon})  m_{\varepsilon} k\Phi=L(k\Phi)
\end{aligned}
\end{equation}
and this proves the claim.

On the other side, let $0<\lambda<\overline{\Lambda}$ and let 
$z_{\lambda} \geq0$ be defined as in the second paragraph of the 
proof of Theorem \ref{thm3.1}.
Since $\| z_{\lambda}\| _{\infty}=\overline{\Lambda}
^{-1}\lambda$, there exists $\alpha_{\lambda}>0$ such that if 
$\overline {z}_{\lambda}:=\alpha_{\lambda}+z_{\lambda}$ then 
$\| \overline {z}_{\lambda}\| _{\infty}\leq1$. Furthermore, in a similar way as
there one can see that $\overline{z}_{\lambda}$ is a supersolution of
\eqref{super} for all $\lambda\in(0,\overline{\Lambda})  $.
Therefore, by Lemma \ref{lem3.2} there exists some $v_{\lambda}\in P^{\circ}$ solution
of \eqref{super} for every $\lambda\in(0,\min\{\lambda
_1(a)  ,\overline{\Lambda}\}  )  $ and satisfying
$k\Phi\not \leq v_{\lambda}\not \leq \overline{z}_{\lambda}$. In particular,
$\| v_{\lambda}\| _{\infty}\geq\alpha_{\lambda}$ for all such
$\lambda'$s.

To prove \eqref{infi} we proceed by contradiction.
 Let $\lambda_{j}$ be a sequence with $\lambda_{j}\searrow0$, 
let $v_{j}$ be the corresponding solutions of \eqref{super} found above, 
and suppose that $\lambda _{j}\| v_{j}\| _{\infty}^{p-1}\to0$. Without loss of
generality we can assume that $\| v_{j}\| _{\infty}\geq
\alpha$ for all $j$ large enough and $\alpha>0$ not depending on $\lambda$.
Let $w_{j}:=v_{j}/\| v_{j}\| _{\infty}$. Dividing
\eqref{super} by $\| v_{j}\| _{\infty}$ we get
\begin{equation}
Lw_{j}=\lambda_{j}\| v_{j}\| _{\infty}^{p-1}\big(
aw_{j}^p-bw_{j}^q/\| v_{j}\| _{\infty}^{p-q}+c/\|
v_{j}\| _{\infty}\big)  . \label{jota}
\end{equation}
Now, going to the limit in \eqref{jota}, the continuity of the solution
operator $L^{-1}$ yields that $w_{j}\to0$ when $j\to0$, which
is not possible.

Let us prove (ii). Assume first that \eqref{condi} holds. In this case we
start arguing as in the first part of the proof but defining now
$m_{\varepsilon}:=a\chi_{\Omega_{\varepsilon}^{c}\times\mathbb{R}}$. Then
$m_{\varepsilon}\geq0$ in $\Omega\times\mathbb{R}$ and $m_{\varepsilon}=0$ in
the sets $B_{k}$ and $B_{k}^{c}$ given by \eqref{conjuntos}. Moreover, (since
by \eqref{condi} we may suppose that $b\leq c$) we have
\[
f(x,t,k\Phi)  \geq\lambda(-b(k\Phi)^q+c)  \geq0\quad\text{in }B_{k}
\]
and hence we do not need to impose the restriction 
$\lambda\leq\lambda_1(m_{\varepsilon})  $ in \eqref{cuenta}. Furthermore, a quick
look at \eqref{cuenta} shows that the other bounds remain the same, and thus
as there we obtain a positive subsolution of the form $k\Phi$ but now for all
$\lambda>0$ (with $k\geq K_0(\lambda)  $ as in \eqref{cuenta}).

Next, we claim that reasoning as in the proof of Theorem \ref{thm3.1} (iii) 
we obtain a solution to \eqref{super} for all $\lambda\in(0,\Lambda)  $.
Indeed, let $\lambda\in(0,\Lambda)  $ and take $\overline
{\lambda}\in(\lambda,\Lambda)  $ such that there exists
$u_{\overline{\lambda}}>0$ solution of \eqref{super} with $\overline{\lambda}$
in place of $\lambda$. As before, since we are assuming that $b\leq
\min\{a,c\}  $, $f(.,\xi)  \geq0$ for all $\xi>0$.
So, $u_{\overline{\lambda}}$ is a supersolution for \eqref{super} and
therefore the claim follows from the above paragraph and Lemma \ref{lem3.2}.

Suppose now that $c\equiv0$ and $q>1$ (the case $q=1$ is included in
 Corollary \ref{coro3.5} below). In this case multiplying \eqref{super} by 
$\lambda^{1/(q-1)  }$ and writing $v:=\lambda^{1/(q-1)  }u$ we transform
\eqref{super} into the equivalent problem
\begin{equation}
Lv=\lambda^{-(p-q)  /(q-1)  }av^p-bv^q
\quad\text{in }\Omega\times\mathbb{R}. \label{aux}
\end{equation}
From (i) \eqref{aux} has a positive solution for all $\lambda>0$ small enough.
Moreover, readily \eqref{aux} has a positive solution for every $\lambda
\in(0,\Lambda)  $. Indeed, let $\lambda\in(0,\Lambda)  $ and take 
$\underline{\lambda}>0$ small enough and $\overline
{\lambda}\in(\lambda,\Lambda)  $ such that there exist
$u_{\underline{\lambda}},u_{\overline{\lambda}}>0$ solutions of \eqref{aux}
with $\underline{\lambda}$ and $\overline{\lambda}$ in place of $\lambda$
respectively. Then $u_{\underline{\lambda}}$ and $u_{\overline{\lambda}}$ are
super and subsolutions respectively of \eqref{aux} and therefore (either if
they are well-ordered or not) we obtain a solution for \eqref{aux}, and hence
for \eqref{super}.

To prove (iii) we shall supply a solution of \eqref{super} for every
$\lambda>0$. We note first that since $b\leq0$ in $\Omega_{\sigma}
\times\mathbb{R}$ for some $\sigma>0$ ($\Omega_{\sigma}$ given by
\eqref{epsi}), for any $\lambda>0$, the subsolution constructed in the first
paragraph of the proof of (ii) can still be used in this situation choosing
there $\varepsilon\leq\sigma$. Indeed, as in (ii) $m_{\varepsilon}=0$ in the
sets $B_{k}$ and $B_{k}^{c}$, $f(.,k\Phi)  \geq c\geq0$ in
$B_{k}$ and the rest also stays the same. On the other hand, let
$\underline{\lambda}<\lambda$ small enough such that there exists
$\underline{v}$ solution of \eqref{aux} with $\underline{\lambda}$ in place of
$\lambda$. Clearly $\lambda^{-1/(q-1)  }\underline{v}$ is a
supersolution of \eqref{super} and then again Lemma \ref{lem3.2} gives a solution of
\eqref{super} and this concludes the proof.
\end{proof}

\begin{corollary} \label{coro3.4}
 Let $a,b,c\in L_{T}^{\infty}$ such that 
$a,c\geq0$, $a\not \equiv 0\not \equiv c$, and let $1\leq q<p<(N+2)  /(N+1)  $.
\begin{itemize}
\item[(i)]  If $\inf_{\Omega\times\mathbb{R}}(a/b^{+})>0$, 
 then for all $\lambda>0$ small enough there exist two
positive solutions of \eqref{super}.

\item[(ii)] If in addition \eqref{condi} holds, then 
(i) is true for all $\lambda\in(0,\Lambda)  $.
\end{itemize}
\end{corollary}

\begin{proof} 
(i) is an immediate consequence of \eqref{cotiyas} and
\eqref{infi}. Let us prove (ii). We assume $b\leq\min\{a,c\}  $
and argue as before. Let $\lambda\in(0,\Lambda)  $, let
$\overline{\lambda}\in(\lambda,\Lambda)  $ and let 
$v_{\overline {\lambda}}$ be the solution of \eqref{super} with 
$\overline{\lambda}$ in place of $\lambda$ given by Theorem \ref{thm3.3} (ii). 
We have that $v_{\overline {\lambda}}$ is a supersolution for \eqref{super}. 
Also, the first paragraph of the proof of Theorem \ref{thm3.1} provides some positive 
subsolution $u_{\lambda}$ of \eqref{super} such that 
$u_{\lambda}\leq v_{\overline{\lambda}}$ and hence
\cite[Theorem 1]{deuel} gives some $\underline{w}_{\lambda}$ solution of
\eqref{super} satisfying 
$u_{\lambda}\leq\underline{w}_{\lambda}\leq v_{\overline{\lambda}}$. 
On the other hand, as in the first part of the proof of Theorem \ref{thm3.3} (ii) 
we can construct another subsolution $\widetilde {u}_{\lambda}$ such that 
$\widetilde{u}_{\lambda}\nleq v_{\overline{\lambda}}$ and thus recalling 
Lemma \ref{lem3.2} we obtain a solution $\overline{w}_{\lambda}\in
P^{\circ}$ of \eqref{super} satisfying 
$\widetilde{u}_{\lambda}\not \leq \overline{w}_{\lambda}\not \leq 
v_{\overline{\lambda}}$. In particular
$\underline{w}_{\lambda}\neq\overline{w}_{\lambda}$ and this proves (ii).
\end{proof}


For the case $c\equiv0$ and $q=1$ no relation between $b$ and $a$ or $c$ is
needed. Let us rewrite \eqref{super} as
\begin{equation}
\begin{gathered}
Lu=\lambda(a(x,t)  u^p+b(x,t)  u)  \quad \text{in }\Omega\times\mathbb{R}\\
u=0 \quad \text{on }\partial\Omega\times\mathbb{R}\\
u\quad T\text{-periodic} 
\end{gathered}  \label{super2}
\end{equation}
We recall that $P_{\Omega}$ and $\Lambda$ are given by \eqref{peomega} and
\eqref{lambdagrande} respectively. We have

\begin{corollary} \label{coro3.5}
 Let $a,b\in L_{T}^{\infty}\ $ such that $0\leq a\not \equiv 0$, and let 
$1<p<(N+2)/(N+1)  $. Then \eqref{super2} has a solution
$v_{\lambda}\in P^{\circ}$  for all $\lambda\in(0,\Lambda)  $. Moreover, 
$\Lambda=\lambda_1(b)$  if $P_{\Omega}(b)  >0$ and $\Lambda=\infty$
 if $P_{\Omega}(b)  \leq0$.
\end{corollary}

\begin{proof} 
Let us note first that since \eqref{super2} can be written as
$(L+\lambda b^{-})  u=\lambda(au^p+b^{+}u)  $,
arguing as in the first part of the proof of Theorem \ref{thm3.3} (ii) we get some
positive subsolution $k\Phi$ of \eqref{super2} for any $\lambda>0$.

On the other hand, let $\mu_{b}$ be defined as in Remark \ref{rmk2.2}.
Then there exists some $u\in P^{\circ}$ satisfying 
$Lu=\lambda bu+\mu_{b}(\lambda)  u$ in $\Omega\times\mathbb{R}$, $u=0$ 
on $\partial\Omega \times\mathbb{R}$. Furthermore, by the results listed 
in Remarks \ref{rmk2.1} and \ref{rmk2.2} it holds that $\mu_{b}(\lambda)  >0$ for all
$\lambda\in(0,\lambda_1(b)  )  $ if $P_{\Omega}(b)  >0$
and $\mu_{b}(\lambda)  >0$ for every $\lambda>0$ if $P_{\Omega}(b)  \leq0$. 
Taking this into account, it is easy to check that
for such $\lambda'$s $u$ is a supersolution of \eqref{super2} if one
takes $\| u\| _{\infty}$ sufficiently small (in fact, it suffices 
$\| u\| \leq(\mu_{b}(\lambda) /(\lambda\| a\| )  )  ^{1/(p-1)  }$). 
Hence Lemma \ref{lem3.2} applies and gives a solution for all
$\lambda<\lambda_1(b)  $ if $P_{\Omega}(b)  >0$
and for every $\lambda>0$ if $P_{\Omega}(b)  \leq0$. That is,
$\Lambda\geq\lambda_1(b)  $\ in the first case and
$\Lambda=\infty$\ in the second one. To end the proof we observe that
\eqref{super2} implies $Lu\geq\lambda bu$ and so Remark \ref{rmk2.2} (ii) says that
$\Lambda\leq\lambda_1(b)  $ when $P_{\Omega}(b)>0$. 
\end{proof}

\subsection*{Acknowledgments}
This research was partially supported by Secyt-UNC.


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