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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2015 (2015), No. 103, pp. 1--18.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2015 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2015/103\hfil Nonlinear periodic problems]
{Positive solutions for parametric nonlinear periodic problems
with competing nonlinearities}

\author[S. Aizicovici, N. S. Papageorgiou, V. Staicu \hfil EJDE-2015/103\hfilneg]
{Sergiu Aizicovici, Nikolaos S. Papageorgiou, Vasile Staicu}

\address{Sergiu Aizicovici \newline
Department of Mathematics, Ohio University, Athens, OH 45701, USA}
\email{aizicovs@ohio.edu}

\address{Nikolaos S. Papageorgiou \newline
 Department of Mathematics, National Technical University,
Zografou Campus, Athens 15780, Greece}
\email{npapg@math.ntua.gr}

\address{Vasile Staicu \newline
Department of Mathematics, CIDMA, University of Aveiro,
Campus Universit\'ario de Santiago, 3810-193 Aveiro, Portugal}
\email{vasile@ua.pt}

\thanks{Submitted September 29, 2014. Published April 16, 2015.}
\subjclass[2000]{34B15, 34B18, 34C25}
\keywords{Nonhomogeneous differential operator; positive solution; 
\hfill\break\indent  local minimizer;
nonlinear maximum principle; mountain pass theorem; bifurcation}

\begin{abstract}
 We consider a nonlinear periodic problem driven by a nonhomogeneous
 differential operator plus an indefinite potential and a reaction having the
 competing effects of concave and convex terms. For the superlinear (concave)
 term we do not employ the usual in such cases Ambrosetti-Rabinowitz condition.
 Using variational methods together with truncation, perturbation and
 comparison techniques, we prove a bifurcation-type theorem describing the set
 of positive solutions as the parameter varies.
\end{abstract}

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

\section{Introduction}

In this article we study the  nonlinear periodic problem $(P_\lambda)$,
\begin{equation}
\begin{gathered}
-(a(| u'(t)  |)u'(t))  '+\beta(t)  u(t)  ^{p-1}
=\lambda u(t)  ^{q-1}+f(t,u(t)) \quad\text{a.e. on }T:=[  0,b] \\
u(0)  =u(b)  ,\quad u'(0)=u'(b)  ,\quad u>0,\;1<q<p<\infty.
\end{gathered}
 \label{ePlambda} 
\end{equation}


The function $a(| x|)  x$ involved in the
definition of the differential operator is a continuous increasing function
which satisfies certain other regularity conditions listed in hypothesis
(H1) below.

These hypotheses are general enough and incorporate as special cases many
differential operators of interest such as the scalar $p$-Laplacian. The
potential $\beta\in L^{\infty}(T)  $ may change sign (indefinite
potential). Also $\lambda>0$ is a parameter, and the term $\lambda x^{q-1}$
(for $x\geq0$) is a ``concave'' (that is, $(p-1)$-sublinear)
contribution to the reaction of problem \eqref{ePlambda}. The
perturbation $f(t,x)  $ is a Carath\'{e}odory function (i.e., for
all $x\in\mathbb{R}$, $t\to f(t,x)  $ is measurable and
for almost all $t\in T$, $x\to f(t,x)  $ is continuous),
which exhibits $(p-1)$-superlinear growth near $+\infty$, but
without satisfying the usual in such cases Ambrosetti-Rabinowitz condition
(AR-condition for short).

So, in the reaction of \eqref{ePlambda} we have the competing
effects of concave and convex nonlinearities.

Our aim is to describe the dependence on the parameter $\lambda>0$ of the set
of positive solutions of problem \eqref{ePlambda}.
We prove a bifurcation-type theorem asserting the existence of a critical
parameter value $\lambda^{\ast}>0$ such that for all $\lambda\in(
0,\lambda^{\ast})  $, problem \eqref{ePlambda} admits at
least two positive solutions, for $\lambda=\lambda^{\ast}$, problem
\eqref{ePlambda} has at least one positive solution, and for
$\lambda>\lambda^{\ast}$, there are no positive solutions for problem
\eqref{ePlambda}.

Recently, such a result was proved by Aizicovici-Papageorgiou-Staicu
\cite{Aiz-Pa-St12} for equations driven by the scalar $p$-Laplacian (that is,
$a(| x|)  x=| x| ^{p-2}x$
with $1<p<\infty$) and with $\beta\equiv0$.

In the present work, the differential operator is nonhomogeneous and this is a
source of serious difficulties in the analysis of problem
\eqref{ePlambda}, and many techniques used in \cite{Aiz-Pa-St12} are not applicable
here. We mention that recently, Aizicovici-Papageorgiou-Staicu
\cite{Aiz-Pa-St19} proved a bifurcation-type theorem for a class of nonlinear
periodic problems driven by a nonhomogeneous differential operator, but with a
positive potential function $\beta\in L^{\infty}(T)
_{+}\backslash\{  0\}  $ and a reaction $\lambda f(t,x)  $ ($\lambda>0$)
which is strictly $(p-1)  $-sublinear
near $+\infty$. So, they deal with a coercive problem with no competition of
different nonlinearities in the reaction. Nonlinear, nonhomogeneous periodic
problems with a positive potential function $\beta\in L^{\infty}(
T)  _{+}\backslash\{  0\}  $ and with competing
nonlinearities in the reaction (concave-convex terms) were studied by
Aizicovici-Papageorgiou-Staicu \cite{Aiz-Pa-St16}. In that paper the emphasis
is on the existence of nodal (that is, sign changing) solutions.

Our investigation is motivated by applications of physical interest. For
instance, in the work of Br\'{e}zis-Mawhin \cite{Br-Ma} some concrete
quasilinear inertia terms arise in the context of the study of the
relativistic motion of particles. The corresponding differential operator is
different, however it seems possible to adapt our results to the framework of
\cite{Br-Ma}. Also, our problem is related to the stationary version of the
parabolic equations studied by Badii-Diaz \cite{Ba-Di} in the context of some
catalysis and chemical reaction models.

Finally, we mention that multiplicity results for positive solutions of
equations driven by the scalar $p$-Laplacian with Dirichlet and
Sturm-Liouville boundary conditions, were proved by Ben Naoum-De Coster
\cite{BeN-DCo}, De Coster \cite{DeC}, Manasevich-Njoku-Zanolin
\cite{Ma-NJ-Za}, Njoku-Zanolin \cite{NJ-Za}. For periodic problems driven
by the scalar
$p$-Laplacian, we mention the works of Hu-Papageorgiou \cite{Hu-Pa7} and Wang
\cite{Wan2}.

Our approach is variational, based on the critical point theory, combined with
suitable truncations and comparison techniques. In the next section, for easy
reference, we recall the main mathematical tools that we will use in the
sequel. We also introduce the hypotheses on the map $x\to a(| x|)  x$
and state some useful consequences of
these conditions. Our main result is stated and proved in Section 3.

\section{Preliminaries}

Let $(X,\|\cdot\|  $ be a Banach space and
$(X^{\ast},\|\cdot\| _{\ast}) $ its topological
dual. By $\langle .,.\rangle $ we denote the duality brackets for
the pair $(X^{\ast},X)  $ and by $\overset{w}{\to}$
the weak convergence in $X$. A map $A:X\to X^{\ast}$ is said to
be \textit{of type }$(S)  _{+}$, if for every sequence
$\{x_{n}\}  _{n\geq1}\subseteq X$\ such that $x_{n}\overset{w}{\to}x$ in $X$
 and
\[
\limsup_{n\to\infty}\langle A(x_{n}),x_{n}-x\rangle \leq0,
\]
one has
\[
x_{n}\to x\quad \text{in $X$ as }n\to\infty.
\]
Let $\varphi\in C^{1}(X)  $. We say that $x^{\ast}\in X$ is a
critical point of $\varphi$ if $\varphi'(x^{\ast})  =0$.
If $x^{\ast}\in X$ is a critical point of $\varphi$, then
$c=\varphi(x^{\ast})  $ is called a critical value of $\varphi$. The set of all
critical points of $\varphi$ will be denoted by $K_{\varphi}$.

Given $\varphi\in C^{1}(X)  $, we say that $\varphi$ satisfies
the \textit{Cerami condition} (the $C$-condition for short), if the
following is true:
\begin{quote}
 every sequence $\{  u_{n}\}  _{n\geq1}\subset X$
such that $\{  \varphi(u_{n})  \}  _{n\geq1}$
is bounded in $\mathbb{R}$ and
\[
(1+\| u_{n}\|  \varphi'(u_{n})  \to0\quad \text{in $X^{\ast}$  as }n\to\infty
\]
admits a strongly convergent subsequence.
\end{quote}
This is a compactness-type condition on the functional $\varphi$, which
compensates for the fact that the ambient space $X$ needs not be locally
compact (in general, $X$ is infinite dimensional). It leads to a deformation
theorem from which we can derive the minimax theory for critical values of
$\varphi$. Prominent in that theory, is the so-called \textit{mountain pass
theorem, }due to Ambrosetti-Rabinowitz \cite{Am-Ra}. Here we state the result
in a slightly more general form (see \cite{Gas-Pa2}).

\begin{theorem}\label{Thm:1}
If $(X,\|\cdot\|  $ is a Banach
space, $\varphi\in C^{1}(X)  $ satisfies the $C-$condition,
$u_0$, $u_1\in X$, $\| u_1-u_0\| >r>0$,
\begin{gather*}
\max\{  \varphi(u_0)  ,\varphi(u_1)
\}  <\inf\{  \varphi(u)  :\| u-u_0 \| =r\}  =:m_{r}, \\
c:=\inf_{\gamma\in\Gamma}\max_{t\in[  0,1]  }\varphi(
\gamma(t))\quad\text{with} \\
\Gamma:=\{  \gamma\in C([  0,1]  ,X)
:\gamma(0)  =u_0,\; \gamma(1)  =u_1\}
\end{gather*}
then $c\geq m_{r}$ and $c$ is a critical value of $\varphi$.
\end{theorem}

In the study of problem \eqref{ePlambda} we will use the
following spaces:
\begin{gather*}
W_{p}:=W_{per}^{1,p}(0,b)  =\{  u\in W^{1,p}(0,b)  :u(0)  =u(b)  \}  ,\\
\widehat{C^{1}}(T)  :=C^{1}(T)  \cap W_{p}.
\end{gather*}
Recall that the Sobolev $W^{1,p}(0,b)  $ is embedded continuously
(in fact compactly) in $C(T)  $. Hence the evaluations at $t=0$
and $t=b$ in the definition of $W_{p}$ make sense. The Banach space
$\widehat{C^{1}}(T)  $ is an ordered Banach space with positive
cone
\[
\widehat{C_{+}}=\{  u\in C^{1}(T)  :u(t)
\geq0\text{ for all }t\in T\}  .
\]
This cone has nonempty interior given by
\[
\operatorname{int} \widehat{C_{+}}=\{  u\in C^{1}(T)  :u(
t)  >0\text{ for all }t\in T\}  .
\]

Now we introduce the following hypotheses on the map $x\to a(| x|)  x$:
\begin{itemize}
\item[(H1)] $a:(0,\infty)  \to(0,\infty)  $ is a $C^{1}$-function such that:
\begin{itemize}
\item[(i)]] $x\to a(x)  x$ is strictly
increasing on $(0,\infty)  $, with
\[
a(x)  x\to0\text{ and }\frac{a'(x)
x}{a(x)  }\to c>-1\text{ as }x\to0^{+};
\]


\item[(ii)] there exists $\widehat{c}>0$ such that
\[
| a(| x|)  x| \leq
\widehat{c}\big(1+| x| ^{p-1}\big)  \quad \text{for all }x\in\mathbb{R};
\]


\item[(iii)] there exists $C_0>0$ such that
$a(| x|)  x^2\geq C_0|x| ^{p}$ for all $x\in\mathbb{R}$;

\item[(iv)] if $G_0(t)  :=\int_0^{t}a(
s)  sds\ $for all $t\geq0$, then there exists $\xi_0>0$ such that
\[
pG_0(t)  -a(| t|)  t^2
\geq-\xi_0\quad \text{for all }t\geq0;
\]


\item[(v)] there exists $\tau\in(q,p)  $ such
that $t\to G(| x| ^{\frac{1}{\tau}})
$ is convex on $(0,\infty)  $ and
\[
\underset{t\to0^{+}}{\lim}\frac{G_0(t)  }{t^{\tau}}=0.
\]
\end{itemize}
\end{itemize}

\noindent\textbf{Remarks:} From the above hypotheses, it is clear that
$G_0(\cdot)  $ is strictly convex and strictly increasing on
$(0,\infty)  $. We set $G(x)  =G_0(
| x|)  $ for all $x\in R$. Then $G(\cdot)$ is convex, too, and for
all $x\neq0$, we have
\[
G'(x)  =G_0'(| x|)  =a(| x|)  | x|\frac{x}{| x| }=a(| x|)x.
\]
So, $G(\cdot)  $ is the primitive of the function $x\to
a(| x|)  x$ involved in the definition of the
differential operator. The convexity of $G(\cdot)  $ and
$G(0)  =0$ imply
\begin{equation}
G(x)  \leq a(| x|)  x^2\quad \text{for all }x\in\mathbb{R}. \label{1}
\end{equation}
Then, using \eqref{1} and hypotheses (H1)(ii), (H1)(iii), we have
the following growth estimates for the
primitive $G(\cdot)$:
\begin{equation}
\frac{C_0}{p}| x| ^{p}\leq G(x)  \leq
C_1(1+| x| ^{p}) \quad \text{for some }
C_1>0,\text{ all }x\in\mathbb{R}. \label{2}
\end{equation}


\noindent\textbf{Examples:}
The following functions satisfy hypotheses (H1):
\begin{gather*}
a_1(x)  =| x| ^{p-2}x\quad \text{with } 1<p<\infty;\\
a_2(x)  =| x| ^{p-2}x+\mu|x| ^{q-2}x\quad \text{with
$1<q<p<\infty$  and }\mu>0;\\
a_3(x)  =(1+| x| ^2)^{\frac{p-2}{2}}x\quad \text{with }1<p<\infty;\\
a_{4}(x)  =| x| ^{p-2}x(1+\frac{1}{1+| x| ^{p}}) \quad \text{with }1<p<\infty.
\end{gather*}
The first function corresponds to the scalar $p$-Laplacian, the second
corresponds to the scalar $(p,q)$-differential operator (that
is, the sum of a $p$-Laplacian and of a $q$-Laplacian) and the third function
is the generalized scalar $p$-mean curvature differential operator.

Let $A:W_{p}\to W_{p}^{\ast}$ be the nonlinear map defined by
\begin{equation}
\langle A(u)  ,y\rangle =\int_0^ba(
| u'|)  u'y'dt\quad \text{for all }u,y\in W_{p}. \label{2'}
\end{equation}


The following result is well known; see, e.g., \cite{Aiz-Pa-St16}.

\begin{proposition}\label{Prop:2}
The nonlinear map $A:W_{p}\to W_{p}^{\ast}$ defined by
\eqref{2'}   is bounded (that is, it maps bounded sets to
bounded sets), demicontinuous, monotone (hence maximal monotone, too) and of
type $(S)  _{+}.$
\end{proposition}

Let $f_0:T\times\mathbb{R\to}\mathbb{R}$ be a Carath\'{e}odory
function such that
\[
| f_0(t,x)  | \leq\alpha_0(t)  \big(1+| x| ^{r-1}\big) \quad \text{for a.a. }
t\in T,\text{ all }x\in\mathbb{R}
\]
with $\alpha_0\in L^{1}(T)  _{+}$, $1<r<\infty$. We set
\[
F_0(t,x)  =\int_0^x f_0(t,s) \, ds
\]
and consider the $C^{1}$-functional $\varphi_0:W_{p}\to\mathbb{R}$
defined by
\[
\varphi_0(u)  =\int_0^b
G(u'(t))  dt-\int_0^b
F_0(t,u(t))  dt\quad \text{for all }u\in W_{p}.
\]
From Aizicovici-Papageorgiou-Staicu \cite{Aiz-Pa-St16}, we have:

\begin{proposition} \label{Prop:3}
If hypotheses {\rm(H1) (i)-(iii)} hold and $u_0\in W_{p}$ is a
local $\widehat{C^{1}}(T)  $-minimizer of $\varphi_0$; that is,
there exists $\rho_0>0$ such that
\[
\varphi_0(u_0)  \leq\varphi_0(u_0+h)  \quad
\text{for all }h\in\widehat{C^{1}}(T)  \text{ with }\|
h\| _{\widehat{C^{1}}(T)  }\leq\rho_0,
\]
then $u_0\in\widehat{C^{1}}(T)  $ and $u_0$ is also a local
$W_{p}$-minimizer of $\varphi_0$; that is, there exists $\rho_1>0$ such
that
\[
\varphi_0(u_0)  \leq\varphi_0(u_0+h)  \quad \text{for all }h\in W_{p}
\text{ with }\| h\| \leq\rho_1.
\]
\end{proposition}

In the above result and in the sequel, by $\| \cdot\| $ we
denote the norm of $W_{p}$ defined by
\[
\| u\| =(\| u\| _{p}^{p}+\|
u'\| _{p}^{p})  ^{1/p}\quad \text{for all }u\in W_{p},
\]
where $\|\cdot\| _{p}$ stands for the norm in $L^{p}(T)  $. Also,
if $x\in\mathbb{R}$, then $x^{\pm}=\max\{  \pm x,0\}  $. Then given
$u\in W_{p}$, we set $u^{\pm}(.)=u(.)  ^{\pm}$. We have
\[
u^{+}, u^{-}\in W_{p},\quad
| u| =u^{+}+u^{-},\quad u=u^{+}-u^{-}.
\]
Finally, for any Carath\'{e}odory function $g:T\times\mathbb{R\to R}$,
we denote by $N_{g}$ the Nemytskii operator corresponding to $g$, defined by
\[
N_{g}(u)  (\cdot)  =g(\cdot,u(\cdot)
)  \quad \text{for all }u\in W_{p}.
\]
Note that $t\to N_{g}(u)  (t)  =g(t,u(t))  $ is measurable.

\section{A bifurcation-type theorem}

In this section, we prove a bifurcation-type theorem describing the set of
positive solutions of problem \eqref{ePlambda}, as the parameter
$\lambda>0$ varies. The following hypotheses will be needed:

\begin{itemize}
\item[(H2)] $\beta\in L^{\infty}(T)_{+}$.

\item[(H3)] $f:T\times\mathbb{R}\to\mathbb{R}$ is
a Carath\'{e}odory function such that $f(t,0)  =0$ for a.a.
$t\in T$ and
\begin{itemize}
\item[(i)] $| f(t,x)  |
\leq\alpha(t)  (1+| x| ^{r-1})  $
for a.a. $t\in T$, all $x\geq0$, with $\alpha\in L^{\infty}(T)_{+}$, $p<r<\infty$;

\item[(ii)] if $F(t,x)  =\int_0^x f(t,s)  ds$, then 
$\lim_{x\to+\infty} \frac{F(t,x)  }{x^{p}}=+\infty$ uniformly for a.a. $t\in T$;

\item[(iii)] there exist $\mu>\max\{  r-p,q\}  $
and $\eta_0>0$ such that
\[
0<\eta_0\leq\liminf_{x\to+\infty} \frac{f(t,x)  x-pF(t,x)}{x^{\mu}}
\quad \text{uniformly for a.a. }t\in T;
\]


\item[(iv)] $\lim_{x\to0^{+}} \frac{f(t,x)  }{x^{p-1}}=0$ 
uniformly for a.a. $t\in T$;

\item[(v)] for every $\rho>0$, there exists $\xi_{\rho}>0$
such that for a.a. $t\in T$ the function
$x\to f(t,x)  +\xi_{\rho}x^{p-1}$ is nondecreasing on
$[  0,\rho]$.
\end{itemize}
\end{itemize}

\noindent\textbf{Remarks:}
Since we are looking for positive solutions and all
the above hypotheses concern the positive half-axis, the values of 
$f(t,\cdot)  $ on $(-\infty,0)  $ are irrelevant and so, without
any loss of generality, we may assume that $f(t,x)  =0$ for a.a.
$t\in T$, all $x<0$. Note that hypotheses
(H3) (ii), (iii)  imply that
\[
\lim_{x\to+\infty} \frac{f(t,x)  }{x^{p-1}}=+\infty\quad 
\text{ uniformly for a. a. }t\in T.
\]
So, the perturbation $f(t,\cdot)  $ is $(p-1)-$superlinear
 (``convex'' nonlinearity$)$. Hence in the reaction of problem
\eqref{ePlambda} we have the competing effects of concave and
convex terms.

However, for the $(p-1) $-superlinear (convex) term, we do not
assume the usual in such cases AR-condition, unilateral version. This
condition says that there exist $r_0>p$ and $M>0$ such that
\begin{gather}
0<r_0F(t,x)  \leq f(t,x)  x\quad \text{for a.a. }t\in T,
\text{ all }x\geq M, \label{3a} \\
\operatorname{essinf} F(.,M)  >0. \label{3b}
\end{gather}
Integrating \eqref{3a}  and using \eqref{3b}
we obtain the following weaker condition
\begin{equation}
C_2x^{r_0}\leq F(t,x)  \quad\text{for a.a. }t\in T,\text{ all
}x\geq M\text{ with }C_2>0. \label{4'}
\end{equation}
Evidently \eqref{4'} implies the much weaker hypothesis in
(H3) (ii). Here we use this superlinearity
condition together with (H3) (iii), and the
two together are weaker than the AR-condition and incorporate in our framework
$(p-1)$-superlinear perturbations with slower growth near
$+\infty$.

\noindent\textbf{Examples:} 
The following functions satisfy 
(H3) (For the sake of simplicity we drop the $t$-dependence):
\begin{gather*}
f_1(x)    =x^{r-1}\quad \text{for all }x\geq0\text{ with
}p<r<\infty,\\
f_2(x)   =x^{p-1}(\ln(x+1)  +\frac{1}
{p}) \quad \text{for all }x\geq0.
\end{gather*}
Note that $f_2$ does not satisfy the AR-condition.

Our main result reads as follows.

\begin{theorem}\label{Thm:14} 
If hypotheses {\rm (H1)--(H3)} hold, then there exists
$\lambda^{\ast}>0$ such that:
\begin{itemize}
\item[(i)] for $\lambda\in(0,\lambda^{\ast})  $,
problem \eqref{ePlambda} admits at least two positive solutions
$u_0,\widehat{u}\in \operatorname{int} \widehat{C}_{+}$
 with $\widehat{u}-u_0\in\operatorname{int}\widehat{C}_{+}$;

\item[(ii)]  for $\lambda=\lambda^{\ast}$, problem 
\eqref{ePlambda}  has at least one positive solution $u_{\ast}\in \operatorname{int}\widehat{C}_{+}$;

\item[(iii)] for $\lambda>\lambda^{\ast}$, problem 
\eqref{ePlambda}  has no positive solution.
\end{itemize}
Moreover, for every $\lambda\in(0,\lambda^{\ast}]  $ problem
\eqref{ePlambda} has a smallest positive solution 
$u_{\lambda }^{\ast}$ and the curve $\lambda\to u_{\lambda}^{\ast}$ is
nondecreasing from $(0,\lambda^{\ast}]  $ into $\widehat{C^{1}}(T)  $.
\end{theorem}

The proof of Theorem \ref{Thm:14} is based on several propositions of
independent interest.
Let
\[
\mathcal{P}=\{  \lambda>0:\text{problem \eqref{ePlambda}
 admits a positive solution}\}
\]
and for every $\lambda\in\mathcal{P}$, let 
$\mathcal{S}(\lambda)$ be the set of positive solutions of problem \eqref{ePlambda}.

First we establish the nonemptiness of the set $\mathcal{P}$ of admissible
parameters. To this end, let $\gamma>\| \beta\| _{\infty}$
(see hypothesis (H2)) and consider the following
truncation-perturbation of the reaction of \eqref{ePlambda}:
\begin{equation}
h_{\lambda}(t,x)  =\begin{cases}
0 & \text{if }  x\leq0,\\
\lambda x^{q-1}+f(t,x)  +\gamma x^{p-1} & \text{if }  x>0.
\end{cases}  \label{4}
\end{equation}
This is a Carath\'{e}odory function. We set 
$H_{\lambda}(t,x)  :=\int_0^xh_{\lambda}(t,s)  ds$ and introduce 
the $C^{1}$-functional
$\widehat{\varphi}_{\lambda}:W_{p}\to\mathbb{R}$ defined by
\[
\widehat{\varphi}_{\lambda}(u)  =
\int_0^b G(u'(t))  dt+\frac{1}{p}
\int_0^b(\beta(t)  +\gamma)  | u(t)| ^{p}dt
-\int_0^bH_{\lambda}(t,u(t))  dt
\]
 for all $u\in W_{p}$.
Next we show that $\widehat{\varphi}_{\lambda}$ satisfies the $C$-condition.

\begin{proposition}\label{Prop:4} 
If hypotheses {\rm (H1)--(H3)}  hold and $\lambda>0$, then the
functional $\widehat{\varphi}_{\lambda}$ satisfies the $C$-condition.
\end{proposition}

\begin{proof}
Let $\{  u_{n}\}  _{n\geq1}$ be a sequence in $W_{p}$ such that
\begin{equation}
| \widehat{\varphi}_{\lambda}(u_{n})  | \leq
M_1\quad\text{for some $M_1>0$ and all }n\geq1, \label{5}
\end{equation}
and
\begin{equation}
(1+\| u_{n}\|  \widehat{\varphi}_{\lambda
}'(u_{n})  \to0 \quad\text{in }W_{p}^{\ast}\text{ as
}n\to\infty. \label{6}
\end{equation}
From \eqref{6} we have
\[
| \langle \widehat{\varphi}_{\lambda}'(
u_{n})  ,v\rangle | \leq\frac{\varepsilon_{n}\|
v\| }{1+\| u_{n}\| }\quad \text{for all }v\in
W_{p},\text{ all }n\geq1,\text{ with }\varepsilon_{n}\to0^{+},
\]
hence
\begin{equation}
\big| \langle A(u_{n})  ,v\rangle +
\int_0^b(\beta(t)  +\gamma)  | u_{n}|
^{p-2}u_{n}vdt
-\int_0^bh_{\lambda}(t,u_{n})  vdt\big|
\leq\frac{\varepsilon _{n}\| v\| }{1+\| u_{n}\| }
\label{7}
\end{equation}
for all $n\geq1$. 
In \eqref{7} we choose $v=-u_{n}^{-}\in W_{p}$. Then we have
\[
\int_0^b
a(| (u_{n}^{-})  '|)
[  (u_{n}^{-})  ']  ^2dt
+\int_0^b (\beta(t)  +\gamma)  (u_{n}^{-})
^{p}dt\leq\varepsilon_{n}\text{ for all }n\geq1
\]
(see \eqref{4}), hence
\[
C_0\| (u_{n}^{-})  '\| _{p}
^{p}+(\gamma-\| \beta\| _{\infty})  \|
u_{n}^{-}\| _{p}^{p}\leq\varepsilon_{n}\quad \text{for all }n\geq1
\]
(with $\gamma>\| \beta\| _{\infty}$), therefore
\begin{equation}
u_{n}^{-}\to0\quad \text{in }W_{p}\text{ as }n\to\infty. \label{8}
\end{equation}
Next, in \eqref{7}  we choose $v=u_{n}^{+}\in W_{p}$. Then
\begin{equation}
\begin{aligned}
&\int_0^b a(| (u_{n}^{+})  '|)
[  (u_{n}^{+})  ']  ^2dt-
\int_0^b (\beta(t))  (u_{n}^{+})
^{p}dt+\lambda\| u_{n}^{+}\| _{q}^{q}+
\int_0^b f(t,u_{n}^{+})  u_{n}^{+}dt\\
&\leq\varepsilon_{n} \quad \text{for all }n\geq1 (\text{see }\eqref{4}).
\end{aligned}\label{9}
\end{equation}
From \eqref{5}  and \eqref{8}, we have
\begin{equation}
\int_0^b pG((u_{n}^{+})  ')  dt
+ \int_0^b\beta(t)  (u_{n}^{+})  ^{p}dt
-\frac{\lambda p} {q}\| u_{n}^{+}\| _{q}^{q}
-\int_0^b pF(t,u_{n}^{+})  dt
\leq M_2 \label{10}
\end{equation}
for some $M_2>0$, all $n\geq1$. We add \eqref{9}  and
\eqref{10} and obtain
\begin{align*}
&\int_0^b[  pG\big((u_{n}^{+})  '\big)  -a(
| u_{n}^{+}| ')  \big((u_{n}
^{+})  '\big)  ^2]  dt
+
\int_0^b[  f(t,u_{n}^{+})  u_{n}^{+}-pF(t,u_{n}^{+})
]  dt\\
&  \leq M_3+\lambda(\frac{p}{q}-1)  \| u_{n}
^{+}\| _{q}^{q}\quad \text{for some }M_3>0\text{ all }n\geq1,
\end{align*}
hence
\begin{equation}
\int_0^b[  f(t,u_{n}^{+})  u_{n}^{+}-pF(t,u_{n}^{+})
]  dt\leq M_{4}+\lambda(\frac{p}{q}-1)  \|
u_{n}^{+}\| _{q}^{q} \label{11}
\end{equation}
for some $M_{4}>0$ and all $n\geq1$
(see (H1) (iv)). 
Hypotheses (H3) (i), (iii)  imply that we can find
$\eta_1\in(0,\eta_0)  $ and $\alpha_1\in L^{1}(T)  _{+}$ such that
\begin{equation}
\beta_1| x| ^{\mu}-\alpha_1(t)  \leq
f(t,x)  x-pF(t,x)  \quad \text{ for a.a. $t\in T$ and all }x\geq0. \label{12}
\end{equation}
Returning to \eqref{11} and using \eqref{12}, we obtain
\[
\eta_1\| u_{n}^{+}\| _{\mu}^{\mu}\leq C_{4}(
1+\| u_{n}^{+}\| _{\mu}^{q}) \quad \text{for some }
C_{4}>0,\text{ all }n\geq1
\]
(since $q<\mu$ and $q<p)$, therefore,
\begin{equation}
\{  u_{n}^{+}\}  _{n\geq1}\text{ is bounded in }L^{\mu}(
T)  . \label{13}
\end{equation}
It is clear from hypothesis (H3) (iii), that
we may assume that $\mu<r$. Then we can find $t\in(0,1)  $ such
that
\begin{equation}
\frac{1}{r}=\frac{t}{\mu}. \label{14}
\end{equation}
Invoking the interpolation inequality (see, for example, Gasinski-Papageorgiou
\cite[ p. 905]{Gas-Pa2}), we have
\[
\| u_{n}^{+}\| _{r}
\leq\| u_{n}^{+}\| _{\mu}^{t}\| u_{n}^{+}\| _{\infty}^{1-t}
\]
hence
\begin{equation}
\| u_{n}^{+}\| _{r}^{r}\leq C_{5}\| u_{n}^{+}\| ^{(1-t)  r}\quad 
\text{for some }C_{5}>0,\text{ all }n\geq1 \label{15}
\end{equation}
(see \eqref{13} and use the Sobolev embedding theorem).

Hypotheses (H3) (i), (iv) 
imply that we can find $C_{6}=C_{6}(\lambda)  >0$ such that
\begin{equation}
\lambda x^{q}+f(t,x)  x\leq C_{6}(1+x^{r}) \quad
 \text{for a.a. }t\in T,\text{ all }x\geq0. \label{16}
\end{equation}
In \eqref{7} we choose $v=u_{n}^{+}\in W_{p}$. Then
\begin{align*}
& \int_0^b a(| u_{n}^{+}| ') \big((
u_{n}^{+})  '\big)  ^2dt+
\int_0^b\beta(t)  (u_{n}^{+})  ^{p}dt-\lambda\|
u_{n}^{+}\| _{q}^{q}-
\int_0^b f(t,u_{n}^{+})  u_{n}^{+}dt\\
&  \leq\varepsilon_{n}\quad \text{for all }n\geq1.
\end{align*}
(see \eqref{4}), hence
\begin{equation}
C_0\| (u_{n}^{+})  '\| _{p}^{p}\leq
C_{7}(1+\| u_{n}^{+}\| _{r}^{r})  \quad \text{for some }C_{7}>0,
\text{ all }n\geq1 \label{17}
\end{equation}
(see (H1) (iii) , (H2), \eqref{16} and recall $p<r$).
 We know that
$y\to\| y'\| _{p}+\| y\|_{\mu}$ is an equivalent norm on $W_{p}$ 
(see, for example, Gasinski-Papageorgiou \cite{Gas-Pa2}, p.227). 
So, from \eqref{13}, \eqref{15} and \eqref{17} we have
\begin{equation}
\begin{aligned}
\| u_{n}^{+}\| ^{p}
&\leq C_{8}(1+\| u_{n}
^{+}\| _{r}^{r})  \quad \text{for some }C_{8}>0,\text{ all }n\geq1\\
&\leq C_{9}(1+\| u_{n}^{+}\|^{(1-t)  r})  \quad \text{for some }
C_{9}>0,\text{ all }n\geq1.
\end{aligned}
\label{18}
\end{equation}


From \eqref{14} we have $\mu=tr$. Hence $(1-t)r=r-\mu<p$;
see hypothesis (H1) (iii). So,
from \eqref{18} it follows that $\{  u_{n}^{+}\}
_{n\geq1}$ is bounded in $W_{p}$; consequently
\begin{equation}
\{  u_{n}\}  _{n\geq1}\quad \text{is bounded in }W_{p}
\text{ (see \ref{8}).} \label{19}
\end{equation}
By \eqref{19} and passing to a subsequence if
necessary, we may assume that
\begin{equation}
u_{n}\overset{w}{\to}u\text{ in }W_{p}\text{ and }u_{n}\to
u\text{ in }C(T)  \text{ as }n\to\infty. \label{20}
\end{equation}
In \eqref{7} we choose $v=u_{n}-u\in W_{p}$, pass to the
limit as $n\to\infty$ and use \eqref{20}. Then
\[
\lim_{n\to\infty}\langle A(u_{n})  ,u_{n}
-u\rangle =0,
\]
hence $u_{n}\to u$ in $W_{p}$ (see Proposition \ref{Prop:2}).
Therefore $\widehat{\varphi}_{\lambda}$ satisfies the C-condition.
\end{proof}

To establish the nonemptiness of the set $\mathcal{P}$, we  use Theorem
\ref{Thm:1}. To this end, we need to satisfy the mountain pass geometry for
the functional $\widehat{\varphi}_{\lambda}$. The next proposition is a
crucial step in this direction.

\begin{proposition} \label{Prop:5} 
If hypotheses {\rm (H1)--(H3)} hold, then there exists
$\lambda_{+}>0$ such that for all $\lambda\in(0,\lambda_{+})  $
we can find $\rho_{\lambda}>0$ for which we have
\[
\inf\{  \widehat{\varphi}_{\lambda}(u)  :\|
u\| =\rho_{\lambda}\}  =\widehat{m}_{\lambda}>0.
\]
\end{proposition}

\begin{proof}
Hypotheses (H3) (i), (iv) 
imply that given $\varepsilon>0$, there exists 
$C_{10}=C_{10}(\varepsilon)  >0$ such that
\begin{equation}
\lambda x^{q-1}+f(t,x)  \leq(\lambda+\varepsilon)
x^{q-1}+C_{10}x^{r-1}\quad \text{for a.a. }t\in T,\text{ all }x\geq0. \label{21}
\end{equation}
Then, for every $u\in W_{p}$, we have
\begin{equation}
\begin{aligned}
\widehat{\varphi}_{\lambda}(u)
&  =\int_0^b G(u'(t))  dt+\frac{1}{p}
\int_0^b(\beta(t)  +\gamma)  | u(t)
| ^{p}dt
-\int_0^b H_{\lambda}(t,u(t))  dt\\
&\geq \frac{C_0}{p}\| u'\| _{p}^{p}+\frac{1}{p}
\int_0^b (\beta(t)  +\gamma)  | u(t)
| ^{p}dt-\frac{\lambda+\varepsilon}{q}\| u^{+}\|
_{q}^{q}\\
&\quad -\frac{C_{10}}{r}\| u^{+}\| _{r}^{r}
-\frac{\gamma}{p}\| u^{+}\| _{p}^{p}\quad \text{(see \eqref{2}, \eqref{4} and
 \eqref{21})}\\
&\geq C_{11}\| u\| ^{p}-\frac{\gamma}{p}\|
u\| ^{p}-C_{12}(\frac{\lambda+\varepsilon}{q}\|
u\| ^{q}+\| u\| ^{r}) 
\end{aligned} \label{22}
\end{equation}
for some $C_{11},C_{12}>0$
(recall that $\gamma>\| \beta\| _{\infty}$). Since $q<p<r$,
given $\varepsilon>0$, one can find $C_{\varepsilon}>0$ such that
\[
\| u\| ^{p}\leq\frac{\varepsilon p}{\gamma q}\|
u\| ^{q}+C_{\varepsilon}\| u\| ^{r}\quad \text{for all }u\in W_{p}.
\]
So, from \eqref{22} we have
\begin{equation}
\begin{aligned}
\widehat{\varphi}_{\lambda}(u)  
&\geq C_{11}\| u\| ^{p}-C_{13}(\frac{\lambda+2\varepsilon}{q}
 \|u\| ^{q}+\| u\| ^{r}) \quad \text{for some } C_{13}>0\\
&=[  C_{11}-C_{13}(\frac{\lambda+2\varepsilon}{q}\|
u\| ^{q-p}+\| u\| ^{r-p})  ]  \|u\| ^{p}.
\end{aligned} \label{23}
\end{equation}
Consider the function
\[
\theta_{\lambda}(t)  =\frac{\lambda+2\varepsilon}{q}
t^{q-p}+t^{r-p}\quad \text{for all }t>0.
\]
Evidently, $\theta_{\lambda}\in C^{1}(0,\infty)  $, and because
$q<p<r$, we have
\[
\theta_{\lambda}(t)  \to+\infty\quad \text{as }t\to
0^{+}\text{ and }t\to+\infty.
\]
So, we can find $t_0\in(0,\infty)  $ such that
\[
\theta_{\lambda}(t_0)  =\inf\{  \theta_{\lambda}(
t)  :t>0\}  ,
\]
hence
$\theta_{\lambda}'(t_0)  =0$;
therefore
\[
\frac{\lambda+2\varepsilon}{q}(p-q)  t_0^{q-p-1}=(r-p)  t_0^{r-p-1}.
\]
We obtain
\[
t_0=t_0(\lambda,\varepsilon)  =\big[  \frac{(
\lambda+2\varepsilon)  (p-q)  }{q(r-p)
}\big]  ^{\frac{1}{r-q}}.
\]
Note that
$\theta_{\lambda}(t_0(\lambda,\varepsilon))
\to0^{+}$ as $\lambda,\varepsilon\to 0^{+}$.
Therefore, we can find $\lambda_{+}$, $\varepsilon_{+}>0$ small such that
\begin{equation}
\theta_{\lambda}(t_0(\lambda,\varepsilon))
<\frac{C_{11}}{C_{13}}\quad \text{for all }\lambda\in(0,\lambda_{+})
,\; \varepsilon\in(0,\varepsilon_{+})  . \label{24}
\end{equation}
So, fixing $\varepsilon\in(0,\varepsilon_{+})  $, from 
\eqref{23} and \eqref{24} we have
$\widehat{\varphi}_{\lambda}(u)  \geq\widehat{m}_{\lambda}>0$
for all $u\in W_{p}$ with $\| u\| =\rho_{\lambda}
=t_0(\lambda,\varepsilon)$   and all $\lambda\in(0,\lambda_{+})$.
\end{proof}

By adapting the proof of Proposition 6 in \cite{Aiz-Pa-St16}, 
we arrive at the following result, which completes the mountain
 pass geometry for the functional $\widehat{\varphi}_{\lambda}$.

\begin{proposition}\label{Prop:6} 
If hypotheses {\rm (H1)--(H3)} hold, $\lambda>0$ and $u\in \operatorname{int}\widehat{C}_{+}$, then
$\widehat{\varphi}_{\lambda}(tu)  \to-\infty$  as
$t\to+\infty$.
\end{proposition}

Now we establish the nonemptiness of the set $\mathcal{P}$ and also determine
the nature of the solution set $\mathcal{S}(\lambda)  $ when
$\lambda\in\mathcal{P}$.

\begin{proposition} \label{Prop:7}
 If hypotheses {\rm (H1)--(H3)} hold, then 
$\mathcal{P} \neq \varnothing$ and for all $\lambda\in\mathcal{P}$
 we have $\mathcal{S}
(\lambda)  \subseteq \operatorname{int}\widehat{C}_{+}$.
\end{proposition}

\begin{proof}
Propositions \ref{Prop:4}, \ref{Prop:5} and \ref{Prop:6} permit the use of
Theorem \ref{Thm:1} (the mountain pass theorem) on 
$\widehat{\varphi} _{\lambda}$ when $\lambda\in(0,\lambda_{+})  $. 
So, we can find $u_{\lambda}\in W_{p}$ such that
\begin{equation}
u_{\lambda}\in K_{\widehat{\varphi}_{\lambda}}\text{ and }
0=\widehat{\varphi}_{\lambda}(0)  <\widehat{m}_{\lambda}\leq\widehat{\varphi
}_{\lambda}(u_{\lambda})  . \label{25}
\end{equation}
Evidently $u_{\lambda}\neq0$. Also, since 
$u_{\lambda}\in K_{\widehat{\varphi}_{\lambda}}$ we have
$\widehat{\varphi}_{\lambda}'(u_{\lambda})  =0$,
hence
\begin{equation}
A(u_{\lambda})  +(\beta(t)  +\gamma)
| u_{\lambda}| ^{p-2}u_{\lambda}=N_{h_{\lambda}}(
u_{\lambda})  . \label{26}
\end{equation}
On \eqref{26} we act with $-u_{\lambda}^{-}\in W_{p}$. Then
\[
\int_0^b a(| (u_{\lambda}^{-})  '|)  ((u_{\lambda}^{-})  ')  ^2dt+
\int_0^b(\beta(t)  +\gamma)  (u_{\lambda}^{-})
^{p}dt=0\quad \text{(see \eqref{4})},
\]
hence
\[
C_0\| (u_{\lambda}^{-})  '\| _{p}
^{p}+C_{14}\| u_{\lambda}^{-}\| _{p}^{p}\leq0\quad 
\text{for some }C_{14}>0,
\]
(see (H1) (iii) and recall that
$\gamma>\| \beta\| _{\infty}$), therefore
$u_{\lambda}\geq0$, $u_{\lambda}\neq0$.
Then, because of \eqref{4}, equation \eqref{26} becomes
\[
A(u_{\lambda})  +\beta(t)  | u_{\lambda
}| ^{p-1}=\lambda u_{\lambda}^{q-1}+N_{f}(u_{\lambda}),
\]
therefore $u_{\lambda}\in\mathcal{S}(\lambda)  $ and $u_{\lambda
}\in\widehat{C}_{+}\backslash\{  0\}  $ (see \eqref{4}).

Let $\rho=\| u_{\lambda}\| _{\infty}$ and let 
$\xi_{\rho}>0$ be as postulated by hypothesis (h3) (v).
Then
\begin{align*}
& -(a(| u_{\lambda}'(t)
|)  u_{\lambda}'(t))  '+(\beta(t)  +\xi_{\rho})  [  u_{\lambda
}(t)  ]  ^{p-1}\\
& =\lambda[  u_{\lambda}(t)  ]  ^{q-1}+f(
t,u_{\lambda}(t))  +\xi_{\rho}[  u_{\lambda}(
t)  ]  ^{p-1}\geq0\quad \text{a.e. on }T,
\end{align*}
hence
\[
-(a(| u_{\lambda}'(t)  |
)  u_{\lambda}'(t))  '\leq(
\| \beta\| _{\infty}+\xi_{\rho})  [  u_{\lambda
}(t)  ]  ^{p-1}\quad \text{a.e. on }T,
\]
and we infer that $u_{\lambda}\in \operatorname{int}\widehat{C}_{+}$ 
(see Pucci-Serrin \cite[pp. 111, 120]{Pu-Se}). Therefore we conclude that
\[
(0,\lambda_{+})  \subseteq\mathcal{P}\text{ and }\mathcal{S}
(\lambda)  \subseteq \operatorname{int}\widehat{C}_{+}\text{ for all
}\lambda\in\mathcal{P}.
\]
\end{proof}

\begin{proposition}\label{Prop:8} 
If hypotheses {\rm (H1)--(H3)} hold and 
$\lambda\in\mathcal{P}$, then $(0,\lambda]  \subseteq\mathcal{P}$.
\end{proposition}

\begin{proof}
Let $\mu\in(0,\lambda)  $ and let 
$u_{\lambda}\in\mathcal{S}(\lambda) $. We introduce the following 
truncation-perturbation of the reaction of \eqref{ePlambda}
with $\mu$ instead of $\lambda$, 
$(P_\mu)$:

\begin{equation}
e_{\mu}(t,x)  =\begin{cases}
0 & \text{if }  x<0\\
\mu x^{q-1}+f(t,x)  +\gamma x^{p-1} & \text{if } 
0\leq x\leq u_{\lambda}(t) \\
\mu u_{\lambda}(t)  ^{q-1}+f(t,u_{\lambda}(t))  
+\gamma u_{\lambda}(t)  ^{p-1} & \text{if } u_{\lambda}(t)  <x.
\end{cases} \label{27}
\end{equation}

This is a Carath\'{e}odory function. We set 
$E_{\mu}(t,x) =\int_0^{x}e_{\mu}(t,s)  ds$ and consider the 
$C^{1}$-functional 
$\psi_{\mu}:W_{p}\to\mathbb{R}$ defined by
\[
\psi_{\mu}(u)  =
\int_0^b
G(u'(t))  dt+\frac{1}{p}
\int_0^b
(\beta(t)  +\gamma)  | u(t)
| ^{p}dt-
\int_0^b
E_{\mu}(t,u(t))  dt
\]
 for all $u\in W_{p}$.
By \eqref{27} and since $\gamma>\| \beta\|_{\infty}$, it is clear that 
$\psi_{\mu}$ is coercive. Also, it is
sequentially weakly lower semicontinuous (just use the Sobolev embedding
theorem and the fact that since $G(\cdot)  $ is convex, the
integral functional $y\to
\int_0^bG(y'(t))  dt$ is sequentially weakly lower
semicontinuous). So, by the Weierstrass theorem, we can find 
$u_{\mu}\in W_{p}$ such that
\begin{equation}
\psi_{\mu}(u_{\mu})  =\inf\{  \psi_{\mu}(u)
:u\in W_{p}\}  . \label{28}
\end{equation}
By  hypothesis (H3) (iv) , given
$\varepsilon>0$, there exists $\delta=\delta(\varepsilon)\in(0,1)$ such that
\begin{equation}
F(t,x)  \geq-\frac{\varepsilon}{p}x^{p}\text{ for a.a. }t\in
T,\text{ all }x\in[  0,\delta]  . \label{29}
\end{equation}
Let $\xi\in(0,\min\{  \delta,\min_{T}u_{\lambda}\})  $ 
(recall that $u_{\lambda}\in \operatorname{int}\widehat{C}_{+}$). Then
\begin{align*}
\psi_{\mu}(\xi)  
 &  \leq\frac{\xi^{p}}{p}\|
\beta\| _{\infty}b-\frac{\mu\xi^{q}}{q}b-
\int_0^bF(t,\xi)  dt\text{ (see }\eqref{27}  \text{)}\\
&  \leq\frac{\xi^{p}}{p}[  \| \beta\| _{\infty
}+\varepsilon]  b-\frac{\mu\xi^{q}}{q}b.
\end{align*}
Since $q<p$, choosing $\xi\in(0,1)  $ even smaller if necessary,
we have
$\psi_{\mu}(\xi)  <0$.
Then
\[
\psi_{\mu}(u_{\mu})  <0\quad \text{ (see \eqref{28})}.
\]
hence $u_{\mu}\neq0$.
From \eqref{28}, we have
$\psi_{\mu}'(u_{\mu})  =0$, hence
\begin{equation}
A(u_{\mu})  +(\beta(t)  +\gamma)
| u_{\mu}| ^{p-2}u_{\mu}=N_{e_{\mu}}(u_{\mu})
. \label{30}
\end{equation}
On \eqref{30}, first we act with $-u_{\mu}^{-}\in W_{p}$ and
then with $(u_{\mu}-u_{\lambda})  ^{+}\in W_{p}$ and obtain
\[
u_{\mu}\in[  0,u_{\lambda}]  :=\{  u\in W_{p}:0\leq u(
t)  \leq u_{\lambda}(t)  \text{ for all }t\in T\}  .
\]
From \eqref{27} it follows that $u_{\mu}\in\mathcal{S}(
\mu)  \subseteq \operatorname{int} \widehat{C}_{+}$ and so,
 $(0,\lambda ]  \subseteq\mathcal{P}$.
\end{proof}

Let $\lambda^{\ast}:=\sup\mathcal{P}$.
\
\begin{proposition} \label{Prop:9}
 If hypotheses {\rm (H1)--(H3)} hold, then $\lambda^{\ast }<\infty$.
\end{proposition}

\begin{proof}
Fix $\gamma_0\geq\| \beta\| _{\infty}$. Hypotheses
(H3) (i)--(iv) imply that there exists $\widetilde {\lambda}>0$ such that
\begin{equation}
\widetilde{\lambda}x^{q-1}+f(t,x)  \geq\gamma_0x^{p-1}\quad 
\text{for a.a. }t\in T,\text{ all }x\geq0 \label{31}
\end{equation}
(recall that $q<p$). Let $\lambda>\widetilde{\lambda}$ and suppose that
$\lambda\in\mathcal{P}$. Then we can find $u_{\lambda}\in\mathcal{S}(
\lambda)  \subseteq \operatorname{int}\widehat{C}_{+}$ (see Proposition
\ref{Prop:7}). Let
\[
m=\min_{T}u_{\lambda}>0.
\]
For $\delta>0$, let $m_{\delta}=m+\delta\in \operatorname{int}\widehat{C}_{+}$. For
$\rho=\| u_{\lambda}\| _{\infty}$, let $\xi_{\rho}>0$ be as
postulated by hypothesis (H3) (v).
Evidently, we may assume that $\xi_{\rho}>\| \beta\|_{\infty}$. Then
\begin{align*}
&  -(a(| m_{\delta}'|)
m_{\delta}')  '+(\beta(t)
+\xi_{\rho})  m_{\delta}^{p-1}\\
&  \leq(\gamma_0+\xi_{\rho})  m^{p-1}+\chi(
\delta)  \text{ with }\chi(\delta)  \to0^{+}\quad \text{as }\delta\to0^{+}\\
&  \leq\widetilde{\lambda}m^{q-1}+f(t,m)  +\xi_{\rho}m^{p-1}
+\chi(\delta)  \quad \text{(see \eqref{31})} \\
&  \leq\widetilde{\lambda}u_{\lambda}(t)  ^{q-1}+f(
t,u_{\lambda}(t))  +\xi_{\rho}u_{\lambda}(
t)  ^{p-1}+\chi(\delta)  \quad\text{(see (H3) (v))}\\
&  =\lambda u_{\lambda}(t)  ^{q-1}+f(t,u_{\lambda}(
t))  +\xi_{\rho}u_{\lambda}(t)  ^{p-1}-(
\lambda-\widetilde{\lambda})  u_{\lambda}(t)  ^{q-1}
+\chi(\delta) \\
&  \leq\lambda u_{\lambda}(t)  ^{q-1}+f(t,u_{\lambda
}(t))  +\xi_{\rho}u_{\lambda}(t)
^{p-1}-(\lambda-\widetilde{\lambda})  m^{q-1}+\chi(
\delta) \quad \text{(since $\lambda>\widetilde{\lambda}$)}\\
&  \leq\lambda u_{\lambda}(t)  ^{q-1}+f(t,u_{\lambda
}(t))  +\xi_{\rho}u_{\lambda}(t)
^{p-1}\quad \text{for }\delta>0 \\
&  =-(a(| u_{\lambda}'(t)
|)  u_{\lambda}'(t))'+(\beta(t)  +\xi_{\rho})  | u_{\lambda
}(t)  | ^{p-1}\text{ a.e. on $T$, for $\delta>0$ small,}
\end{align*}
hence
$m_{\delta}\leq u_{\lambda}(t)$ for all $t\in T$;
therefore
$m_{\delta}\leq m$  for $\delta>0$ small, which is a contradiction.
So, $\lambda\notin\mathcal{P}$ and we have
$\lambda^{\ast}\leq\widetilde{\lambda}<\infty$.
\end{proof}

From Proposition \ref{Prop:8}, we see that
$(0,\lambda^{\ast})  \subseteq\mathcal{P}$.

\begin{proposition} \label{Prop:10} 
If hypotheses {\rm (H1)--(H3)} hold and $\lambda\in(0,\lambda^{\ast})$, 
then problem \eqref{ePlambda} admits at least two positive solutions
\[
u_0, \widehat{u}\in \operatorname{int}\widehat{C}_{+},\quad
\widehat{u}-u_0\in \operatorname{int}\widehat{C}_{+},
\]
and $u_0$ is a local minimizer of the functional $\widehat{\varphi}_{\lambda}$.
\end{proposition}

\begin{proof}
Let $\theta\in(\lambda,\lambda^{\ast})  $ and let
 $u_{\theta} \in\mathcal{S}(\theta)  \subseteq \operatorname{int}\widehat{C}_{+}$.
 As in the proof of Proposition \ref{Prop:8}, we truncate the reaction of problem
\eqref{ePlambda} at $u_{\theta}(t) $ and use the
direct method to obtain
\[
u_0\in[  0,u_{\theta}]  \cap\mathcal{S}(\lambda)  .
\]
For $\delta>0$, let $u_0^{\delta}=u_0+\delta\in \operatorname{int}\widehat{C}_{+}$.
Let $\rho=\| u_{\theta}\| _{\infty}$ and let $\xi_{\rho}>0$
be as postulated by hypothesis (H3) (v). We
can always assume that $\xi_{\rho}>\| \beta\| _{\infty}$. We
have
\begin{align*}
&  -(a(| (u_0^{\delta})  ^{\prime
}|)  (u_0^{\delta})  ')
'+(\beta(t)  +\xi_{\rho})  (
u_0^{\delta})  ^{p-1}\\
&  \leq-(a(| u_0'|)
u_0')  '+(\beta(t)  +\xi_{\rho
})  u_0^{p-1}+\chi(\delta)  \quad \text{with }\chi(
\delta)  \to0^{+}\text{ as }\delta\to0^{+}\\
&  =\lambda u_0^{q-1}+f(t,u_0)  +\xi_{\rho}u_0{}^{p-1}
+\chi(\delta)  \text{ (since $u_0\in\mathcal{S}(\lambda)$)}\\
&  =\theta u_0^{q-1}+f(t,u_0)  +\xi_{\rho}u_0{}
^{p-1}-(\theta-\lambda)  u_0^{q-1}+\chi(\delta) \\
&  \leq\theta u_{\theta}^{q-1}+f(t,u_{\theta})  +\xi_{\rho
}u_{\theta}{}^{p-1}-(\theta-\lambda)  m_0^{q-1}+\chi(
\delta) \\
&  \quad\text{(recall that }u_0\leq u_{\theta},u_0\in
 \operatorname{int}\widehat{C}_{+}\text{ see (H3) (v)})\\
&  \leq\theta u_{\theta}^{q-1}+f(t,u_{\theta})  +\xi_{\rho
}u_{\theta}^{p-1}\text{ for $\delta>0$ small}\\
&  =-(a(| u_{\theta}'(t)|)  u_{\theta}'(t)) '+(\beta(t)  
+\xi_{\rho})  u_{\theta}(t)  ^{p-1}\text{ a.e. on }T.
\end{align*}
Then
$u_0^{\delta}\leq u_{\theta}$  for $\delta>0$ small,
hence
$u_{\theta}-u_0\in \operatorname{int}\widehat{C}_{+}$.
So, we have that
\begin{equation}
u_0\in \operatorname{int}{}_{\widehat{C^{1}}(T)  }[  0,u_{\theta}]  .\label{32}
\end{equation}
Let $\psi_{\lambda}$ be the $C^{1}$-functional corresponding to the
truncation-perturbation of the reaction of \eqref{ePlambda} at
$u_{\theta}(t)  $ (see the proof of Proposition \ref{Prop:8} and
in particular \eqref{27} with $\mu$ replaced by $\lambda$ and
$\lambda$ replaced by $\theta)$. We know that $u_0$ is a minimizer of
$\psi_{\lambda}$ and
\begin{equation}
\psi_{\lambda}\mid_{[  0,u_{\theta}]  }=\widehat{\varphi}_{\lambda
}\mid_{[  0,u_{\theta}]  }. \label{33}
\end{equation}
From \eqref{32} and \eqref{33} it follows
that $u_0$ is a local $\widehat{C^{1}}(T)  $-minimizer of
$\widehat{\varphi}_{\lambda}$. Invoking Proposition \ref{Prop:3}, we infer
that $u_0$ is a local $W_{p}$-minimizer of $\widehat{\varphi}_{\lambda}$. We
consider the following Carath\'{e}odory function
\begin{equation}
\eta_{\lambda}(t,x)  =\begin{cases}
\lambda u_0(t)  ^{q-1}+f(t,u_0(t))  +\gamma u_0(t)  ^{p-1} & \text{if } x\leq
u_0(t) \\
\lambda x^{q-1}+f(t,x)  +\gamma x^{p-1} & \text{if } 
u_0(t)  <x.
\end{cases}
  \label{34}
\end{equation}
As before, $\gamma>\| \beta\| _{\infty}$. 
Let $H_{\lambda}(t,x)  =\int_0^{x}\eta_{\lambda}(t,s)  ds$ and
consider the $C^{1}-$functional $\sigma_{\lambda}:W_{p}\to\mathbb{R}$
defined by
\[
\sigma_{\lambda}(u) 
 =\int_0^bG(u'(t))  dt+\frac{1}{p}\int_0^b(\beta(t)  +\gamma)  | u(t)
| ^{p}dt
-\int_0^bH_{\lambda}(t,u(t))  dt\text{ for all }u\in W_{p}.
\]
From \eqref{34} it is clear that
$\sigma_{\lambda}=\widehat{\varphi}_{\lambda}+\widehat{\zeta}_{\lambda}$
 for some $\widehat{\zeta}_{\lambda}\in\mathbb{R}$,
hence
\begin{equation}
\sigma_{\lambda}\text{ satisfies the }C-\text{condition.} \label{35}
\end{equation}
Also, $u_0$ is a local $W_{p}$-minimizer of $\sigma_{\lambda}$ (since it is
a local $W_{p}-$minimizer of $\widehat{\varphi}_{\lambda})$. So, we can find
$\rho\in(0,1)  $ small such that
\begin{equation}
\sigma_{\lambda}(u_0)  <\inf\{  \sigma_{\lambda}(
u)  :\| u\| =\rho\}  =m_{\rho} \label{36}
\end{equation}
(see Aizicovici-Papageorgiou-Staicu \cite{Aiz-Pa-St3}, proof of Proposition 29).

Hypothesis (H3) (iii)  and \eqref{34}) imply that
\begin{equation}
\sigma_{\lambda}(\xi)  \to-\infty\quad \text{as }
\xi\to+\infty,\text{ }\xi\in\mathbb{R}. \label{37}
\end{equation}
Then \eqref{35}, \eqref{36}, \eqref{37} enable us to use
Theorem \ref{Thm:1} (the mountain pass
theorem). So, we can find $\widehat{u}\in W_{p}$ such that
\begin{equation}
\widehat{u}\in K_{\sigma_{\lambda}}\text{ and } m_{\rho}\leq\sigma_{\lambda
}(\widehat{u})  . \label{38}
\end{equation}
From \eqref{36}, \eqref{38} it follows that
$\widehat{u}\neq u_0$. Also
$\sigma_{\lambda}'(\widehat{u})  =0$,
hence
\begin{equation}
A(\widehat{u})  +(\beta(t)  +\gamma)
| \widehat{u}| ^{p-2}\widehat{u}=N_{\eta_{\lambda}}(
\widehat{u})  . \label{39}
\end{equation}
Acting on \eqref{39} with $(u_0-\widehat{u})
^{+}\in W_{p}$ and using \eqref{34}, we show that
$u_0\leq\widehat{u}$,
and so,
\[
\widehat{u}\in\mathcal{S}(\lambda)  \subseteq \operatorname{int}
\widehat{C}_{+}.
\]
Moreover, reasoning as in the first part of the proof, we conclude that
\[
\widehat{u}-u_0\in \operatorname{int}\widehat{C}_{+}
\]
\end{proof}

Next we deal with the critical case $\lambda=\lambda^{\ast}$.
 To treat this case, we first need some auxiliary results.

Hypotheses (H2) and (H3) (i), (iv) imply that given $\lambda>0$ and
$\varepsilon\in(0,\lambda)  $, there is a $C_{15}>0$ such that
\begin{equation}
\lambda x^{q-1}+f(t,x)  -\beta(t)  x^{p-1}
\geq(\lambda-\varepsilon)  x^{q-1}-C_{15}x^{r-1}\quad
\text{for a.a. $t\in T$, all }x\geq0.
\label{40}
\end{equation}
This unilateral growth condition on the reaction of \eqref{ePlambda}
 leads to the auxiliary periodic problem $(AP_\lambda$),
\begin{equation}
\begin{gathered}
-(a(| u'(t)  |)u'(t))  '=(\lambda-\varepsilon)  u(t)  ^{q-1}
-C_{15}u(t)  ^{r-1}\quad \text{a.e. on }T,\\
u(0)  =u(b)  ,\quad u'(0)=u'(b)  ,\quad u>0,\; \varepsilon\in(0,\lambda)  .
\end{gathered}  \label{APlambda}
\end{equation}


\begin{proposition} \label{Prop:11} 
If hypotheses {\rm (H1)--(H3)} hold and $\lambda>0$, then
problem \eqref{APlambda} has a unique positive solution
$\overline{u}_{\lambda}\in \operatorname{int}\widehat{C}_{+}$ and the map
 $\lambda \to\overline{u}_{\lambda}$ is nondecreasing from 
$(0,\infty)  $ into $\widehat{C^{1}}(T)  $ (that is, if
$\lambda<\mu$, then $\overline{u}_{\lambda}\leq\overline{u}_{\mu})$
\end{proposition}

\begin{proof}
First we show the existence of a positive solution for the problem 
\eqref{APlambda}. For this purpose, we introduce the $C^{1}$-functional
$\epsilon_{\lambda}:W_{p}\to\mathbb{R}$ defined by
\[
\epsilon_{\lambda}(u)
=\int_0^b G(u'(t))  dt+\frac{1}{p}\|
u\| _{p}^{p}-\frac{\lambda-\varepsilon}{q}\| u^{+}\|_{q}^{q}
+\frac{C_{15}}{r}\| u^{+}\| _{r}^{r}-\frac{1}{p}\|
u^{+}\| _{p}^{p}\]
 for all $u\in W_{p}$.
We observe that since by hypothesis $q<p<r$, $\epsilon_{\lambda}$ is coercive.
Also, it is sequentially weakly lower semicontinuous. So, we can find
$\overline{u}_{\lambda}\in W_{p}$ such that
\begin{equation}
\epsilon_{\lambda}(\overline{u}_{\lambda})  =\inf\{
\epsilon_{\lambda}(u)  :u\in W_{p}\}  . \label{42}
\end{equation}
Since $q<p<r$, for $\xi\in(0,1)  $ small, we have
$\epsilon_{\lambda}(\xi)  <0$.
Then
\[
\epsilon_{\lambda}(\overline{u}_{\lambda})  <0=\epsilon_{\lambda
}(0)  \text{,}
\]
hence
$\overline{u}_{\lambda}\neq0$.
From \eqref{42} we have
\[
\epsilon_{\lambda}'(\overline{u}_{\lambda})  =0,
\]
hence
\begin{equation}
A(\overline{u}_{\lambda})  +| \overline{u}_{\lambda
}| ^{p-2}\overline{u}_{\lambda}=(\lambda-\varepsilon)
(\overline{u}_{\lambda}^{+})  ^{q-1}-C_{15}(\overline
{u}_{\lambda}^{+})  ^{r-1}+(\overline{u}_{\lambda}^{+})
^{p-1}\text{.} \label{43}
\end{equation}
On \eqref{43} we act with $-\overline{u}_{\lambda}^{-}\in
W_{p}$ and obtain
\[
\overline{u}_{\lambda}\geq0,\text{ }\overline{u}_{\lambda}\neq0.
\]
Then \eqref{43} becomes
\[
A(\overline{u}_{\lambda})  =(\lambda-\varepsilon)
(\overline{u}_{\lambda})  ^{q-1}-C_{15}(\overline
{u}_{\lambda})  ^{r-1},
\]
hence $\overline{u}_{\lambda}\in C_{+}\backslash\{  0\}  $ is a
solution of \eqref{APlambda}. Moreover, we have
\[
(a(| \overline{u}_{\lambda}'|
)  \overline{u}_{\lambda}')  '\leq C_{15}
\| \overline{u}_{\lambda}\| _{\infty}^{r-p}\overline
{u}_{\lambda}^{p-1},
\]
hence
$\overline{u}_{\lambda}\in \operatorname{int}\widehat{C}_{+}$
(see Pucci-Serrin \cite[pp. 111, 120]{Pu-Se}, ).

Next we show the uniqueness of this solution. To do this, we introduce the
integral functional $\chi:L^{1}(T)  \to\overline
{\mathbb{R}}=\mathbb{R\cup}\{  +\infty\}  $ defined by
\[
\chi(u)  =\begin{cases}
\int_0^bG((u^{\frac{1}{\tau}})  ')  dt &
\text{if }u\geq0,u^{\frac{1}{\tau}}\in W_{p}\\
+\infty & \text{otherwise,}
\end{cases}
\]
where $\tau$ is as in (H1) (v). Let
\[
\operatorname{dom}\chi=\{  u\in L^{1}(T)  :\chi(u)<+\infty\}
\]
be the effective domain of $\chi$ and consider $u_1$, 
$u_2\in \operatorname{dom}\chi$.
Let $t\in[  0,1]  $ and set
\[
y=[  tu_1+(1-t)  u_2]  ^{1/\tau},\text{
}v_1=u_1^{1/\tau},\text{ }v_2=u_2^{1/\tau}.
\]
From Diaz-Saa \cite{Di-Sa} (Lemma 1), we have
\[
| y'(t)  | =[  t|v_1'(t)  | ^{\tau}+(1-t)
| v_2'(t)  | ^{\tau}]^{1/\tau}\quad \text{for a.a. }t\in T,
\]
hence
\begin{align*}
G_0(| y'(t)  |)   
&\leq G_0([  t| v_1'(t) | ^{\tau}+(1-t)  | v_2'(
t)  | ^{\tau}]  ^{1/\tau})  \quad (G_0(\cdot)  s\text{ is increasing})\\
&  \leq tG_0(| v_1'(t)  |)  +(1-t)  G_0(| v_2'(
t)  |)  \quad \text{for a.a. $t\in T$ (see (H1) (v))}
\end{align*}
therefore
\[
G(y'(t))  \leq tG((
u_1(t)  ^{1/\tau})  ')  +(
1-t)  G((u_2(t)  ^{^{\frac{1}{\tau}}
})  ') \quad \text{for a.a. }t\in T,
\]
hence $\chi(\cdot)$ is convex. Moreover, via Fatou's lemma we
can see that $\chi(\cdot)  $ is also lower semicontinuous.

Suppose that $u$, $v\in W_{p}$ are two solutions of 
\eqref{APlambda}. From the first part of the proof, we have that
 $u$, $v\in \operatorname{int}\widehat{C}_{+}$. 
Therefore, if $h\in\widehat{C^{1}}(T)  $ and
$| \mu| <1$ is small, then
\[
u^{\tau}+\mu h\in \operatorname{dom}\chi\text{ and }v^{\tau}
+\mu h\in \operatorname{dom}\chi.
\]
It follows that $\chi$ is G\^{a}teaux differentiable at $u^{\tau}$ and
$v^{\tau}$ in the direction $h$. Moreover, using the chain rule and the
density of $\widehat{C^{1}}(T)  $ in $W_{p}$, we have for all
$h\in W_{p}$,
\begin{gather*}
\chi'(u^{\tau})  (h)  =\frac{1}{\tau}
\int_0^b\frac{-(a(| u'|)
u')  '}{u^{\tau-1}}h\,dt, \\
\chi'(v^{\tau})  (h)  =\frac{1}{\tau}
\int_0^b\frac{-(a(| v'|)
v')  '}{v^{\tau-1}}h\,dt.
\end{gather*}
The convexity of $\chi(\cdot)  $ implies the monotonicity of
$\chi(\cdot)  $. Therefore
\begin{align*}
0 
 &  \leq\frac{1}{\tau}\int_0^b(\frac{-(a(|
u'|)  u')  '}{u^{\tau-1}}
+\frac{(a(| v'|)  v')  '}{v^{\tau-1}})  (u^{\tau}-v^{\tau})
dt\\
&  =\frac{1}{\tau}\int_0^b[  (\lambda-\varepsilon)
(\frac{1}{u^{\tau-q}}-\frac{1}{v^{\tau-q}})  -C_{15}(
u^{r-\tau}-v^{r-\tau})  ]  (u^{\tau}-v^{\tau})  dt\\
&  \leq0,
\end{align*}
and we conclude that $u=v$. This proves the uniqueness of the solution
$\overline{u}_{\lambda}\in \operatorname{int}\widehat{C}_{+}$.

Next we show that $\lambda\to\overline{u}_{\lambda}$ is nondecreasing
from $(0,\infty)  $ into $\widehat{C^{1}}(T)  $.
Indeed, let $\lambda<\mu$ and let 
$\overline{u}_{\mu}\in \operatorname{int}\widehat{C}
_{+}$ be the unique positive solution of problem $(AP_\mu)$.
We consider the following truncation-perturbation of the reaction in
problem \eqref{APlambda}:
\[
\theta_{\lambda}(t,x)  =\begin{cases}
0 & \text{if }  x<0\\
(\lambda-\varepsilon)  x^{q-1}-C_{15}x^{r-1}+x^{p-1} 
& \text{if }  0\leq x\leq\overline{u}_{\mu}(t) \\
(\lambda-\varepsilon)  \overline{u}_{\mu}(t)
^{q-1}-C_{15}\overline{u}_{\mu}(t)  ^{r-1}+\overline{u}_{\mu
}(t)  ^{p-1} & \text{if } \overline{u}_{\mu}(t)<x.
\end{cases}
\]
This is a Carath\'{e}odory function. We set 
$\Theta_{\lambda}(t,x)  =\int_0^{x}\theta_{\lambda}(t,s)  ds$ and consider
the $C^{1}$-functional $\psi_{\lambda}:W_{p}\to\mathbb{R}$ defined by
\[
\psi_{\lambda}(u)  
=\int_0^b G(u'(t))  dt+\frac{1}{p}\|u\| _{p}^{p}
-\int_0^b \Theta_{\lambda}(t,u(t))  dt\quad \text{for all }u\in
W_{p}.
\]
It is clear that $\psi_{\lambda}(\cdot)  $ is coercive. Also,
$\psi_{\lambda}(\cdot)  $ is sequentially weakly lower
semicontinuous. So, we can find $\widetilde{u}_{\lambda}\in W_{p}$ such that
\begin{equation}
\psi_{\lambda}(\widetilde{u}_{\lambda})  
=\inf\{\psi_{\lambda}(u)  :u\in W_{p}\}. \label{45}
\end{equation}
As in the proof of Proposition \ref{Prop:8}, since $q<p<r$, for
$\xi\in(0,\min\{  1,\min_{T}\overline{u}_{\mu}\})  $ small 
(recall that $\overline{u}_{\mu}\in \operatorname{int}\widehat{C}_{+}$),
we have
$\psi_{\lambda}(\xi)  <0$.
Then
\[
\psi_{\lambda}(\widetilde{u}_{\lambda})  <0=\psi_{\lambda}(
0)  \quad \text{(see \eqref{45})},
\]
hence
$\widetilde{u}_{\lambda}\neq0$.
From \eqref{45}, we have
\[
\psi_{\lambda}'(\widetilde{u}_{\lambda})  =0,
\]
which implies
\begin{equation}
A(\widetilde{u}_{\lambda})  +| \widetilde{u}_{\lambda
}| ^{p-2}\widetilde{u}_{\lambda}=N_{\theta_{\lambda}}(
\widetilde{u}_{\lambda})  . \label{46}
\end{equation}


On \eqref{46} we first act with $-\widetilde{u}_{\lambda}^{-}\in W_{p}$ and obtain
\[
\widetilde{u}_{\lambda}\geq0,\quad \widetilde{u}_{\lambda}\neq0
\]
(see hypothesis (H1) (iii). Also we act
with $(\widetilde{u}_{\lambda}-\overline{u}_{\mu})  ^{+}\in
W_{p}$. Then
\begin{align*}
&  \langle A(\widetilde{u}_{\lambda})  ,(
\widetilde{u}_{\lambda}-\overline{u}_{\mu})  ^{+}\rangle +\int
_0^b\widetilde{u}_{\lambda}^{p-1}(\widetilde{u}_{\lambda}
-\overline{u}_{\mu})  ^{+}dt\\
&  =\int_0^b\theta_{\lambda}(t,\widetilde{u}_{\lambda})
(\widetilde{u}_{\lambda}-\overline{u}_{\mu})  ^{+}dt\\
&  =\int_0^b[  (\lambda-\varepsilon)  \overline{u}_{\mu
}^{q-1}-C_{15}\overline{u}_{\mu}^{r-1}+\overline{u}_{\mu}^{p-1}]
(\widetilde{u}_{\lambda}-\overline{u}_{\mu})  ^{+}dt\\
&  =\langle A(\overline{u}_{\mu})  ,(\widetilde
{u}_{\lambda}-\overline{u}_{\mu})  ^{+}\rangle +\int_0
^b\widetilde{u}_{\mu}^{p-1}(\widetilde{u}_{\lambda}-\overline{u}
_{\mu})  ^{+}dt
\end{align*}
(since $\overline{u}_{\mu}$ is a solution of $(AP_\mu)$,
hence
$\widetilde{u}_{\lambda}\leq\overline{u}_{\mu}$,
therefore
\[
\widetilde{u}_{\lambda}\in[  0,\overline{u}_{\mu}]  \backslash
\{  0\}.
\]
Consequently \eqref{46} becomes
\[
A(\widetilde{u}_{\lambda})  =(\lambda-\varepsilon)
(\widetilde{u}_{\lambda})  ^{q-1}-C_{15}(\widetilde
{u}_{\lambda})  ^{r-1},
\]
hence $\widetilde{u}_{\lambda}$ is a positive solution of 
\eqref{APlambda}. Invoking the uniqueness of solutions to 
\eqref{APlambda} we get
$\widetilde{u}_{\lambda}=\overline{u}_{\lambda}$,
therefore
\[
\overline{u}_{\lambda}\leq\overline{u}_{\mu}.
\]
This proves that the map $\lambda\to\overline{u}_{\lambda}$ is
nondecreasing from $(0,\infty)  $ into $\widehat{C^{1}}(T)  $.
\end{proof}

\begin{proposition} \label{Prop:12} 
If hypotheses {\rm (H1)--(H3)} hold and 
$\lambda\in(0,\lambda^{\ast})$, then
$\overline{u}_{\lambda}\leq u\text{ for all }u\in\mathcal{S}(\lambda)$.
\end{proposition}

\begin{proof}
Let $u\in\mathcal{S}(\lambda)  \subseteq \operatorname{int}\widehat{C}_{+}$
and consider the following Carath\'{e}odory function
\[
\gamma_{\lambda}(t,x)  =\begin{cases}
0 & \text{if }  x<0\\
(\lambda-\varepsilon)  x^{q-1}-C_{15}x^{r-1}+x^{p-1} & \text{if }
 0\leq x\leq u(t) \\
(\lambda-\varepsilon)  u(t)  ^{q-1}-C_{15}u(
t)  ^{r-1}+u(t)  ^{p-1} & \text{if }  u(t)<x.
\end{cases}
\]
Let $\Gamma_{\lambda}(t,x)  =\int_0^{x}\gamma_{\lambda}(t,s)  ds$ 
and consider the $C^{1}-$functional $\widehat{\sigma}_{\lambda}:W_{p}\to\mathbb{R}$ 
defined by
\[
\widehat{\sigma}_{\lambda}(u)  
=\int_0^b G(u'(t))  dt+\frac{1}{p}\|
u\| _{p}^{p}-\int_0^b \Gamma_{\lambda}(t,u(t))  dt\quad
\text{for all }u\in W_{p}.
\]
As in the proof of Proposition \ref{Prop:11}, using the direct method, we can
find $\widehat{u}_{\lambda}\in W_{p}$ such that
\[
\widehat{\sigma}_{\lambda}(\widehat{u}_{\lambda})  =\inf\{
\widehat{\sigma}_{\lambda}(u)  :u\in W_{p}\}
<0=\widehat{\sigma}_{\lambda}(0)  ,
\]
hence
$\widehat{u}_{\lambda}\neq0$.
In fact, we can show that $\widehat{u}_{\lambda}\in[  0,u]
\backslash\{  0\}  $ (see the proof of Proposition \ref{Prop:11}
and \eqref{40}). Then we have
\[
A(\widehat{u}_{\lambda})  =(\lambda-\varepsilon)
\widehat{u}_{\lambda}^{q-1}-C_{15}\widehat{u}_{\lambda}^{r-1},
\]
hence
$\widehat{u}_{\lambda}=\overline{u}_{\lambda}$
(see Proposition \ref{Prop:11}), therefore
$\overline{u}_{\lambda}\leq u$  for all $u\in\mathcal{S}(\lambda)$.
\end{proof}

Now we can deal with the critical case $\lambda=\lambda^{\ast}$.

\begin{proposition} \label{Prop:13} 
If hypotheses {\rm (H1)--(H3)} hold, then 
$\lambda^{\ast} \in\mathcal{P}$ and so $\mathcal{P}=(0,\lambda^{\ast}]  $.
\end{proposition}

\begin{proof}
Let $\{\lambda_{n}\}  _{n\geq1}\subseteq\mathcal{P}$ be such that
$\lambda_{n}\to(\lambda^{\ast})  _{-}$ and let 
$u_{n}\in\mathcal{S}(\lambda_{n})  \subseteq \operatorname{int}\widehat{C}_{+}$ for
all $n\geq1$. From the proof of Proposition \ref{Prop:8}, we see that
\begin{equation}
\varphi_{\lambda_{n}}(u_{n})  <0\quad \text{for all }n\geq1.
\label{47}
\end{equation}
Also, we have
\begin{equation}
A(u_{n})  +\beta(t)  u_{n}^{p-1}
=\lambda_{n} u_{n}^{q-1}+N_{f}(u_{n})  \text{ for all }n\geq1, \label{48}
\end{equation}
hence
\begin{equation}
\int_0^ba(| u_{n}'|)  (
u_{n}')  ^2dt+\int_0^b\beta(t)  |
u_{n}| ^{p}dt=\lambda_{n}\| u_{n}\| _{q}^{q}
+\int_0^bf(t,u_{n})  u_{n}dt \label{49}
\end{equation}
 for all $n\geq1$.
From \eqref{47} it follows that
\begin{equation}
\int_0^bpG(u_{n}')  dt+\int_0^b\beta(
t)  | u_{n}| ^{p}dt-\frac{\lambda_{n}p}{q}\|
u_{n}\| _{q}^{q}-\int_0^bpF(t,u_{n})  dt<0
\label{50}
\end{equation}
for all $n\geq1$. Using \eqref{49}, \eqref{50}, hypothesis
 (H1) (iv), and recalling that
$\lambda_{n}<\lambda^{\ast}$ for all $n\geq1$, we have
\begin{equation}
\int_0^b[  f(t,u_{n})  u_{n}-pF(t,u_{n})
]  dt\leq\lambda^{\ast}(\frac{p}{q}-1)  \|
u_{n}\| _{q}^{q}+C_{16}\text{ for some }C_{16}>0. \label{51}
\end{equation}
From \eqref{51}, reasoning as in the proof of Proposition
\ref{Prop:4}, and using hypothesis (H3) (iii),
 we infer that
$\{  u_{n}\}  _{n\geq1}\subseteq W_{p}$  is bounded.
So, we may assume that
\begin{equation}
u_{n}\overset{w}{\to}u_{\ast}\text{ in }W_{p}\quad\text{and}\quad 
u_{n}\to u_{\ast}\text{ in }C(T) \quad \text{as }
n\to\infty. \label{52}
\end{equation}
On \eqref{48} we act with $u_{n}-u_{\ast}\in W_{p}$, pass to
the limit as $n\to\infty$ and use \eqref{52}. Then
\[
\lim_{n\to\infty}\langle A(u_{n})  ,u_{n}-u_{\ast
}\rangle =0,
\]
hence
$u_{n}\to u_{\ast}\text{ in }W_{p}$ (see Proposition \ref{Prop:2});
therefore
\begin{equation}
A(u_{\ast})  +\beta(t)  u_{\ast}^{p-1}=\lambda
^{\ast}u_{\ast}^{q-1}+N_{f}(u_{\ast})  \text{ for all }n\geq1.
\label{53}
\end{equation}
From Propositions \ref{Prop:11} and \ref{Prop:12}, we have
\[
\overline{u}_{\lambda_1}\leq\overline{u}_{\lambda_{n}}\leq u_{n}\quad
\text{for all }n\geq1.
\]
Then
$\overline{u}_{\lambda_1}\leq u_{\ast}$;
therefore
$u_{\ast}\in\mathcal{S}(\lambda^{\ast})$  (see \eqref{53}).
Hence
$\lambda^{\ast}\in\mathcal{P}$ and $\mathcal{P}=(0,\lambda^{\ast}]$.
\end{proof}



\begin{proof}[Proof of Theorem \ref{Thm:14} ]
We just observe that the conclusions of
Theorem \ref{Thm:14} follow directly from Propositions \ref{Prop:7},
\ref{Prop:9}, \ref{Prop:11}, \ref{Prop:13}.

The existence of the smallest positive solution follows as in
\cite{Aiz-Pa-St16} using the lower bound for the elements of 
$\mathcal{S} (\lambda)  $ established in Proposition \ref{Prop:12}. The
monotonicity of the curve $\lambda\to u_{\lambda}^{\ast}$ is
established as the corresponding result for $\lambda\to\overline
{u}_{\lambda}$ in the proof of Proposition \ref{Prop:11}, using 
hypothesis (H3) (v).
\end{proof}

\subsection*{Acknowledgements} 
The authors wish to thank the knowledgeable referee for his/her useful remarks 
and for providing additional references. 
The third author gratefully acknowledges the partial support by
FEDER funds through COMPETE - Operational Programme Factors
of Competitiveness and by Portuguese funds through the Center for
Research and Development in Mathematics and Applications,
 and by the Portuguese Foundation for Science and Technology (FCT), 
within project PEst-C/MAT/UI4106/2011 with COMPETE
 number FCOMP-01-0124-FEDER-022690.

\begin{thebibliography}{99}


\bibitem{Aiz-Pa-St3} S.~Aizicovici, N.~S. Papageorgiou,  V.~Staicu;
\newblock Degree theory for operators of monotone type and nonlinear elliptic
equations with inequality constraints. \newblock {\em Mem. Amer. Math. Soc.},
196(915), 2008.

\bibitem{Aiz-Pa-St12} S.~Aizicovici, N.~S. Papageorgiou,  V.~Staicu;
\newblock Positive solutions for nonlinear periodic problems with concave
terms. \newblock {\em J. Math. Anal. Appl.}, 381:866--883, 2011.

\bibitem{Aiz-Pa-St19} S.~Aizicovici, N.~S. Papageorgiou,  V.~Staicu;
\newblock Positive solutions for nonlinear nonhomogeneous periodic eigenvalue
problems. \newblock {\em Ann. Univ. Buch. Math., Ser. 3(LXI).}, 3:129--144, 2012.

\bibitem{Aiz-Pa-St16} S.~Aizicovici, N.~S. Papageorgiou,  V.~Staicu;
\newblock Nodal and multiple solutions for nonlinear periodic problems with
competing nonlinearities. \newblock {\em Commun. Contemp. Math.},
15:1350001--1350030, 2013.

\bibitem{Am-Ra} A.~Ambrosetti, P.~Rabinowitz;
 \newblock Dual variational
methods in critical point theory and applications.
\newblock {\em J. Functional. Anal.}, 14:349--381, 1973.

\bibitem{Ba-Di} M.~Badii, J.~I.~Diaz;
 \newblock On the time periodic free boundary associated to some nonlinear 
parabolic equations.
\newblock {\em Bound. Value Probl.}, Vol. 2010:17 pages, 2010. 
doi:10.1155/2010/147301.

\bibitem{BeN-DCo} A.~Ben~Naoum, C.~De~Coster;
 \newblock On the existence
and multiplicity of positive solutions of the {p-L}aplacian separated boundary
value problem. \newblock {\em Differential Integral Equations}, 10:1093--1112, 1997.

\bibitem{Br-Ma} H.~Brezis, J.~Mawhin;
 \newblock Periodic solutions of the
forced relativistic pendulum. \newblock {\em Differential Integral Equations},
23:801--810, 2010.

\bibitem{DeC} C.~De~Coster;
\newblock Pairs of positive solutions for the one
dimensional p-{L}aplacian. \newblock {\em Nonlinear Anal.}, 23:669--681, 1994.

\bibitem{Di-Sa} J.~I. Diaz, J.~E. Saa;
 \newblock Existence et unicit\'e de
solutions positives pour certaines equations elliptiques quasilineaires.
\newblock {\em C. R. Math. Acad. Sci. Paris}, 305:521--524, 1987.

\bibitem{Gas-Pa2} L.~Gasinski, N.~S. Papageorgiou;
\newblock {\em Nonlinear Analysis}. \newblock Chapman \&Hall/ CRC Press, Boca
Raton, 2006.

\bibitem{Hu-Pa7} S.~Hu, N.~S. Papageorgiou;
 \newblock Multiple positive
solutions for nonlinear periodic problems. \newblock {\em Nonlinear Anal.},
73:3675--3687, 2010.

\bibitem{Ma-NJ-Za} R.~Manasevich, F.~Njoku, F.~Zanolin;
 \newblock Positive solutions for the one-dimensional {p-L}aplacian.
\newblock {\em Differential Inegral Equations}, 8:1321--1338, 1995.

\bibitem{NJ-Za} F.~Njoku, F.~Zanolin;
\newblock Positive solutions for two-point {BVPs}: existence 
and multiplicity results.
\newblock {\em Nonlinear Anal.}, 13:191--202, 1989.

\bibitem{Pu-Se} P.~Pucci, J.~Serrin;
 \newblock {\em The Maximum Principle}.
\newblock Birkhauser, Basel, 2007.

\bibitem{Wan2}J.~Wang;
 \newblock The existence of positive solutions for the
one-dimensional {p-L}aplacian. \newblock {\em Proc.Amer. Math. Soc.},
125:2275--2283, 1997.

\end{thebibliography}

\end{document}