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\begin{document}

\title[Nonlinear Sturm-Liouville problems]
{A variational approach of Sturm-Liouville problems with the nonlinearity
depending on the derivative}

\centerline{}
%-----------------------------------------------------------------------------------------------------------
\author[G.A. Afrouzi]{Ghasem A. Afrouzi}
\address{Ghasem A. Afrouzi, \newline
Department of Mathematics, Faculty of Mathematical Sciences,
University of Mazandaran, Babolsar, Iran} \email{\href{mailto: G.A.
Afrouzi <afrouzi@umz.ac.ir>}{afrouzi@umz.ac.ir}}
%-----------------------------------------------------------------------------------------------------------
\author[A. Hadjian]{Armin Hadjian}
\address{Armin Hadjian \newline
Department of Mathematics, Faculty of Basic Sciences, University of
Bojnord, P.O. Box 1339, Bojnord 94531, Iran} \email{\href{mailto: A.
Hadjian <hadjian83@gmail.com>}{hadjian83@gmail.com}}
%------------------------------------------------------------------------------------------------------------
\author[V.D. R\u adulescu]{Vicen\c tiu D. R\u adulescu$^\ast$}
\address{Vicen\c tiu D. R\u adulescu \newline
Institute of Mathematics ``Simion Stoilow" of the Romanian Academy,
P.O. Box 1-764, 014700 Bucharest, Romania \& Department of
Mathematics, University of Craiova,  Street A.I. Cuza No. 13, 200585
Craiova, Romania} \email{\href{mailto: V.D. R\u adulescu
<vicentiu.radulescu@imar.ro>}{vicentiu.radulescu@imar.ro}}
%-----------------------------------------------------------------------------------------------------------
\keywords{Dirichlet problem; Critical points; Variational methods.\\
\hspace*{.3cm} {\it 2010 Mathematics Subject Classifications}:
34B15, 35B38, 58E05\\
$^\ast$ Corresponding author.}

\begin{abstract}

In this paper, we are concerned with the existence  of positive classical solutions for
a class of second-order differential equations with the nonlinearity
dependent. We also provide a range of the parameter in order to obtain the existence of multiple solutions. The approach is based on variational methods. An example illustrates the abstract results of this paper.

\end{abstract}

\maketitle

\numberwithin{equation}{section}
%---------------------------------------------------------------------------------------------------------------

\section{Introduction}

In 1836-1837, the French mathematicians Jacques Charles Fran\c cois Sturm (1803-1855) and Joseph Liouville (1809-1882) published several papers that initiated a new subtopic of mathematical analysis: the Sturm-Liouville theory. Sturm and Liouville were concerned with the general {\it linear}, homogeneous second-order differential equation of the form
\begin{equation}\label{sturm}(p(x)u')'+q(x)u=\lambda w(x)u\quad\mbox{if $x\in [a,b]$},\end{equation}
where the potentials are given functions. Under various boundary conditions, Sturm and Liouville established that solutions of problem \eqref{sturm} can exist only for particular values
of the real parameter $\lambda$, which is called an {\it eigenvalue}. Relevant examples of linear Sturm-Liouville problems are the Bessel equation and the Legendre equation.

The classical Sturm-Liouville theory does not depend upon the calculus of variations, but stems from the theory of ordinary linear or nonlinear differential equations.
Linear Sturm-Liouville equations can be also studied in the context of functional analysis by means of self-adjoint operators or integral operators with a continuous symmetric kernel (the Green's function of the problem). Certain applications involving linear partial differential equations can be treated with the help of the Sturm-Liouville theory, for instance the normal modes of vibration of a thin membrane. We also refer to \cite{basel}, where it is studied a perturbed {\it nonlinear} Sturm-Liouville problem with superlinear convex nonlinearity. In the recent paper \cite{molrep}, the authors study a class of discrete anisotropic Sturm-Liouville problems. We also refer to \cite{aga1}, \cite{aga2}, \cite{aga3} for related properties of solutions of Sturm-Liouville problems.

In the present paper, we are concerned with a class of {\it nonlinear} Sturm-Liouville problems and we establish some qualitative properties of the eigenvalues by using variational principles. A feature of our work is the presence of the derivative in the nonlinear term, which creates further technical constraints.

Our main purpose in  this paper is to establish a range of eigenvalues in a suitable interval in order to create at least one
eigenfunction. As a consequence, we establish sufficient conditions for the existence of two or three
solutions.

Consider the following quasilinear Sturm-Liouville problem with Dirichlet boundary condition
on a bounded interval $[a,b]$ in $\mathbb{R}:$
\begin{equation}\label{e1.1}
\left\{\begin{array}{ll}
-(p-1)|u'(x)|^{p-2}u''(x)=\lambda f(x,u)h(x,u'),\qquad x\in(a,b),\\
u(a)=u(b)=0.
\end{array}\right.
\end{equation}
We assume that $p>1$, $\lambda$ is a positive parameter,
$h:[a,b]\times\mathbb{R}\rightarrow [0,+\infty)$ is a bounded and
continuous function with
$m:=\inf_{(x,t)\in[a,b]\times\mathbb{R}}h(x,t)>0$ and
$f:[a,b]\times\mathbb{R}\to\mathbb{R}$ is an $L^1$-Carath\'{e}odory
function.

Dirichlet boundary value problems have been widely studied because
of their applications to various fields of applied sciences, as
mechanical engineering, control systems, computer science,
economics, artificial or biological neural networks and many others.

In this connection, several existence and multiplicity results for
solutions to second-order ordinary differential nonlinear equations,
with the nonlinearity dependent on the derivative and Dirichlet
conditions at the ends, have been investigated making use of
variational methods.

For instance, Graef \textit{et al.} in \cite{GraHeiKong}, by using a
three critical point theorem of Ricceri \cite{Ricceri1}, proved the
existence of at least three classical solutions for the Dirichlet
quasilinear elliptic system
\begin{equation}\label{ghk}
\left\{\begin{array}{ll}
-(p_i-1)|u'_i(x)|^{p_i-2}u''_i(x)\\
\hspace{.5cm}=\left[\lambda\, F_{u_i}(x,u_1,\ldots,u_n)
+\mu\, G_{u_i}(x,u_1,\ldots,u_n)\right]h_i(x,u'_i),\qquad x\in(a,b),\\
u_i(a) = u_i(b) = 0,\qquad \mbox{for } 1\leq i\leq n,
\end{array}\right.
\end{equation}
where $p_i>1$ for $1\leq i\leq n$, $\lambda>0$ and $\mu\geq 0$ are
parameters, $a, b\in\mathbb{R}$ with $a<b$,
$h_i:[a,b]\times\mathbb{R}\rightarrow [0,+\infty)$ is a bounded and
continuous function with
$m_i:=\inf_{(x,t)\in[a,b]\times\mathbb{R}}h_i(x,t) > 0$ for $1\leq
i\leq n,$ $F:[a,b]\times \mathbb{R}^{n}\to \mathbb{R}$ is a function
such that the mapping $(t_1,t_2,\ldots,t_n)\rightarrow
F(x,t_1,t_2,\ldots,t_n)$ is in $C^1$ in $\mathbb{R}^n$ for all
$x\in[a,b],$ $F_{t_i}$ is continuous in $[a,b]\times\mathbb{R}^n$
for $1\leq i\leq n$, and $F(x,0,\ldots,0)=0$ for all $x\in[a,b]$.
Here, $F_{t_i}$ denotes the partial derivative of $F$ with respect
to $t_i$. Similarly, $G:[a,b]\times \mathbb{R}^{n}\to \mathbb{R}$ is
a function such that $(t_1,t_2,\ldots,t_n)\rightarrow
G(x,t_1,t_2,\ldots,t_n)$ is in $C^1$ in $\mathbb{R}^n$ for all
$x\in[a,b],$ and $G_{t_i}$ is continuous in
$[a,b]\times\mathbb{R}^n$ for $1\leq i\leq n$.

Successively, in \cite{AfrHad1}, the authors considered the system
\eqref{ghk} in the case $\mu=0$, finding the existence of infinitely
many classical solutions for certain values of the parameter
$\lambda$ by using variational methods.

Further, the authors in \cite{AfHadHei}, applying a recent critical
point theorem of Ricceri \cite{Ricceri1}, established the existence
of at least one nontrivial solution for the problem
\begin{equation*}
\left\{\begin{array}{ll}
-u''=\big{[}\lambda f(x,u)+g(u)\big{]}h(x,u')\qquad \textrm{ in } (0,1),\\
u(0)=u(1)=0,
\end{array}\right.
\end{equation*}
where $\lambda$ is a positive parameter,
$f:[0,1]\times\mathbb{R}\rightarrow\mathbb{R}$ is an
$L^1$-Carath\'{e}odory function, $g:\mathbb{R}\rightarrow\mathbb{R}$
is a Lipschitz continuous function with $g(0)=0$, and
$h:[0,1]\times\mathbb{R}\rightarrow [0,+\infty)$ is a bounded and
continuous function with
$m:=\inf_{(x,t)\in[0,1]\times\mathbb{R}}h(x,t)>0$.

%----------------------------------------------------------------------------------------------------------

In the present paper, first we obtain the existence of at least one
solution for problem \eqref{e1.1}. It is worth noticing that,
usually, to obtain the existence of one solution, asymptotic
conditions both at zero and at infinity on the nonlinear term are
requested, while, here, it is assumed only a unique algebraic
condition (see $\rm(A_4)$ in Theorem \ref{the3.3}). As a
consequence, by combining with the classical Ambrosetti-Rabinowitz
condition (see \cite{AmbRab}), the existence of two solutions is
obtained (see Theorem \ref{the4.1}). Subsequently, an existence
result of three nonnegative solutions is obtained combining two
algebraic conditions which guarantee the existence of two local
minima for the Euler-Lagrange functional and applying the mountain
pass theorem as given by Pucci and Serrin (see \cite{PuSe1}) to
ensure the existence of the third critical point (see Theorem
\ref{the4.5}).

%----------------------------------------------------------------------------------------------------------

As an example, we state here the following special case of Theorem
\ref{the4.5}.

%----------------------------------------------------------------------------------------------------------

\begin{theorem}\label{the1.1}
Let $h:\mathbb{R}\rightarrow [0,+\infty)$ be a bounded and
continuous function with $\overline{m}:=\inf_{t\in\mathbb{R}}h(t)>0$
and $\overline{M}:=\sup_{t\in\mathbb{R}}h(t)$, and
$g:\mathbb{R}\rightarrow\mathbb{R}$ a nonnegative continuous
function with $g(0)\neq 0$, such that
$$
\lim_{t\rightarrow
0^+}\frac{g(t)}{t}=+\infty,\qquad\lim_{t\rightarrow+\infty}\frac{g(t)}{t}=0,
$$
and
$$
\displaystyle\int_0^1
g(x)dx<\frac{\overline{m}}{16\overline{M}}\int_0^2 g(x)dx.
$$
Then, for each
$$
\lambda\in\left]\frac{32}{\displaystyle\overline{m}\int_0^2
g(x)dx},\frac{2}{\displaystyle\overline{M}\int_0^1 g(x)dx}\right[,
$$
the problem
\begin{equation}\label{especial}
\left\{\begin{array}{ll}
-u''(x)=\lambda g(u)h(u'),\qquad x\in(0,1),\\
u(0)=u(1)=0
\end{array}\right.
\end{equation}
admits at least three positive classical solutions in
$W_0^{1,2}([0,1])$.
\end{theorem}

%---------------------------------------------------------------------------------------------------------------

Finally, we point out that Theorem \ref{the1.1} ensures a
precise conclusion, in the sense that a location of the parameter $\lambda$ in order to obtain at least three distinct solutions is also provided.


%---------------------------------------------------------------------------------------------------------------

\section{Preliminaries}


For a given non-empty set $X$, and two functionals
$\Phi,\Psi:X\to\mathbb{R}$, we define the following functions
$$
\beta(r_1,r_2):=\inf_{v\in \Phi^{-1}(]r_1,r_2[)}\frac{\sup_{u\in
\Phi^{-1}(]r_1,r_2[)}\Psi(u)-\Psi(v)}{r_2-\Phi(v)},
$$
$$
\rho_2(r_1,r_2):=\sup_{v\in \Phi^{-1}(]r_1,r_2[)}
\frac{\Psi(v)-\sup_{u\in
\Phi^{-1}(]-\infty,r_1])}\Psi(u)}{\Phi(v)-r_1},
$$
for all $r_1,r_2\in\mathbb{R},$ with $r_1<r_2$, and
$$
\rho(r):=\sup_{v\in \Phi^{-1}(]r,+\infty[)} \frac{\Psi(v)-\sup_{u\in
\Phi^{-1}(]-\infty,r])}\Psi(u)}{\Phi(v)-r},
$$
for all $r\in\mathbb{R}$.

%---------------------------------------------------------------------------------------------------------------

\begin{theorem}[{\cite{Ricceri1}}]\label{the2.1}
Let $X$ be a real Banach space; $\Phi:X\to\mathbb{R}$ be a
sequentially weakly lower semicontinuous, coercive and continuously
G\^ateaux differentiable function whose G\^ateaux derivative admits
a continuous inverse on $X^\ast$; $\Psi:X\to\mathbb{R}$ be a
continuously G\^ateaux differentiable function whose G\^ateaux
derivative is compact. Put $I_{\lambda}:=\Phi-\lambda \Psi$ and
assume that there are $r_1,r_2\in\mathbb{R}$, with $r_1<r_2$, such
that
\begin{equation}\label{e2.1}
\beta(r_1,r_2)<\rho_2(r_1,r_2).
\end{equation}

Then, for each
$\lambda\in\Big]\frac{1}{\rho_2(r_1,r_2)},\frac{1}{\beta(r_1,r_2)}\Big[$
there is $u_{0,\lambda}\in\Phi^{-1}(]r_1,r_2[)$ such that
$I_{\lambda}(u_{0,\lambda})\leq I_\lambda(u)$ for all $u\in
\Phi^{-1}(]r_1,r_2[)$ and $I'_\lambda(u_{0,\lambda})=0.$
\end{theorem}

%---------------------------------------------------------------------------------------------------------------

\begin{theorem}[{\cite{Ricceri1}}]\label{the2.2}
Let $X$ be a real Banach space; $\Phi:X\to\mathbb{R}$ be a
continuously G\^ateaux differentiable function whose G\^ateaux
derivative admits a continuous inverse on $X^*$;
$\Psi:X\to\mathbb{R}$ be a continuously G\^ateaux differentiable
function whose G\^ateaux derivative is compact. Fix
$\inf_X\Phi<r<\sup_X\Phi$ and assume that
\begin{equation}\label{e2.2}
\rho(r)>0,
\end{equation}
and for each $\lambda>\frac{1}{\rho(r)}$ the function
$I_{\lambda}=\Phi-\lambda \Psi$ is coercive.

Then, for each $\lambda>\frac{1}{\rho(r)}$ there is
$u_{0,\lambda}\in\Phi^{-1}(]r,+\infty[)$ such that
$I_{\lambda}(u_{0,\lambda})\leq I_\lambda(u)$ for all $u\in
\Phi^{-1}(]r,+\infty[)$ and $I'_\lambda(u_{0,\lambda})=0.$
\end{theorem}

%---------------------------------------------------------------------------------------------------------------

Let $h:[a,b]\times\mathbb{R}\rightarrow [0,+\infty)$ be a bounded
and continuous function with
$$
m:=\inf_{(x,t)\in[a,b]\times\mathbb{R}}h(x,t)>0,
$$
and
$f:[a,b]\times\mathbb{R}\to\mathbb{R}$ be an $L^1$-Carath\'{e}odory
function.

We recall that $f:[a,b]\times\mathbb{R}\rightarrow\mathbb{R}$ is an
$L^1$-Carath\'{e}odory function if
\begin{enumerate}
\item[$\rm(a)$] $x\mapsto f(x,\xi)$ is measurable for every $\xi\in\mathbb{R}$;
\item[$\rm(b)$] $\xi\mapsto f(x,\xi)$ is continuous for almost every
$x\in[a,b]$;
\item[$\rm(c)$] for every $\rho>0$ there is a function $l_\rho\in
L^1([a,b])$ such that
$$
\sup_{|\xi|\leq\rho}|f(x,\xi)|\leq l_\rho(x)
$$
for almost every $x\in[a,b]$.
\end{enumerate}

Corresponding to $f$ and $h$ we introduce the functions
$F:[a,b]\times\mathbb{R}\rightarrow\mathbb{R}$ and
$H:[a,b]\times\mathbb{R}\rightarrow[0,+\infty)$, respectively, as
follows
$$
F(x,t):=\int_0^t f(x,\xi)d\xi
$$
and
$$
H(x,t):=\int_0^t\bigg{(}\int_0^{\tau}\frac{(p-1)|\delta|^{p-2}}{h(x,\delta)}\,d\delta\bigg{)}d\tau
$$
for all $x\in [a,b]$ and $t\in\mathbb{R}$. Also, we use the
following notation:
$$
M:=\sup\limits_{(x,t)\in [a,b]\times\mathbb{R}}h(x,t).
$$

Here and in the following, let $X:=W_0^{1,p}([a,b])$, equipped with
the norm
$$
\|u\|:=\bigg(\int_a^b |u'(x)|^{p}dx\bigg)^{1/p}.
$$
Then, $X$ is a reflexive real Banach space. Since $p>1$, $X$ is
compactly embedded in $C^0([a,b])$ and
\begin{equation}\label{e2.3}
\|u\|_\infty\leq\frac{(b-a)^{(p-1)/p}}{2}\|u\|,
\end{equation}
for all $u\in X$ (see, e.g., \cite{Talenti}).

By a \textit{classical solution} of problem (\ref{e1.1}), we mean a
function $u$ such that $u\in C^1([a,b]),$ $u'\in AC([a,b]),$ and
$u(x)$ satisfies (\ref{e1.1}) a.e. on $[a,b]$. We say that a
function $u\in X$ is a \textit{weak solution} of problem
(\ref{e1.1}) if
\begin{eqnarray*}
\int_a^b\bigg{(}\int_0^{u'(x)}\frac{(p-1)|\tau|^{p-2}}{h(x,\tau)}\,d\tau\bigg{)}
v'(x)dx-\lambda\int_a^b f(x,u(x))v(x)dx=0
\end{eqnarray*}
for all $v\in X$.

The following lemma is taken from \cite[Lemma 2.2]{GraHeiKong}.

%---------------------------------------------------------------------------------------------------------

\begin{lemma}\label{lem2.3}
A weak solution to (\ref{e1.1}) in $X$ coincides with a classical
solution to (\ref{e1.1}).
\end{lemma}

%---------------------------------------------------------------------------------------------------------

We cite recent monographs \cite{ciarlet,GherRad, KrisRadVar} as general
references for the basic notions used in the paper.

%----------------------------------------------------------------------------------------------------------

\section{Main results}

In this section we present our main results. To be precise, we
establish an existence result of at least one solution, Theorem
\ref{the3.1}, which is based on Theorem \ref{the2.1}, and we point
out some consequences, Theorems \ref{the3.2}, \ref{the3.3} and
\ref{the3.4}. Finally, we present an other existence result of at
least one solution, Theorem \ref{the3.7}, which is based in turn on
Theorem \ref{the2.2}.

%----------------------------------------------------------------------------------------------------------

Throughout the sequel, $\alpha, \beta$ are two positive constants
such that $\alpha+\beta<b-a$. Now, put
$$
D:=\frac{(p-1)^{p-2}}{p}\left(\alpha^{-p+1}+\beta^{-p+1}\right).
$$
Given two nonnegative constants $c, d$, with
$$
m(2c)^p\neq Dd^p(b-a)^{p-1}pM,
$$
put
$$
a_d(c):=\frac{\displaystyle\int_a^b\max_{|t|\leq
c}F(x,t)dx-\int_{a+\alpha}^{b-\beta}F(x,d)dx}{\displaystyle
m(2c)^p-Dd^p(b-a)^{p-1}pM}.
$$

%----------------------------------------------------------------------------------------------------------

\begin{theorem}\label{the3.1}
Assume that there exist a nonnegative constant $c_1$ and two
positive constants $c_2,d$, with
\begin{equation}\label{e3.1}
c_1<\frac{(b-a)^{(p-1)/p}(pD)^{1/p}}{2}d<\left(\frac{m}{M}\right)^{1/p}c_2,
\end{equation}
such that
\begin{enumerate}
\item[$\rm(A_1)$] $F(x,t) \geq 0$ for all $(x,t)\in ([a,a+\alpha]\cup[b-\beta,b])\times[0,d]$;
\item[$\rm(A_2)$] $a_d(c_2)<a_d(c_1)$.
\end{enumerate}
Then, for each
$$
\lambda\in\frac{1}{(b-a)^{p-1}pmM}\left]\frac{1}{a_d(c_1)},\frac{1}{a_d(c_2)}\right[,
$$
problem \eqref{e1.1} admits at least one nontrivial classical
solution $\bar{u}\in X$, such that
$$
\frac{2m^{1/p}}{(b-a)^{(p-1)/p}M^{1/p}}c_1<\|\bar{u}\|<\frac{2}{(b-a)^{(p-1)/p}}c_2.
$$
\end{theorem}

\begin{proof}
Our aim is to apply Theorem \ref{the2.1} to our problem. To this
end, for each $u\in X$, let the functionals
$\Phi,\Psi:X\rightarrow\mathbb{R}$ be defined by
$$
\Phi(u):=\int_a^b H(x,u'(x))dx
$$
and
$$
\Psi(u):=\int_a^b F(x,u(x))dx,
$$
and put
$$
I_{\lambda}(u):=\Phi(u)-\lambda\Psi(u)\qquad \forall\ u\in X.
$$
Note that the weak solutions of \eqref{e1.1} are exactly the
critical points of $I_{\lambda}$. The functionals $\Phi$ and $\Psi$
satisfy the regularity assumptions of Theorem \ref{the2.1}. Indeed,
by standard arguments, we have that $\Phi$ is continuously
G\^{a}teaux differentiable and its G\^{a}teaux derivative is the
functional $\Phi'(u)\in X^\ast$, given by
$$
\Phi'(u)(v)=\int_a^b\bigg{(}\int_0^{u'(x)}\frac{(p-1)|\tau|^{p-2}}{h(x,\tau)}d\tau\bigg{)}
v'(x)dx
$$
for every $v\in X$. Since $\Phi'$ is monotone (see proof of
\cite[Lemma 2.1]{GraHeiKong}), by applying \cite[Proposition
25.20]{Zeidler}, $\Phi$ is sequentially weakly lower semicontinuous.
Again by \cite[Lemma 2.1]{GraHeiKong}, $\Phi':X\rightarrow X^\ast$
admits a continuous inverse. On the other hand, the fact that $X$ is
compactly embedded into $C^0([a,b])$ implies that the functional
$\Psi$ is well defined, continuously G\^{a}teaux differentiable and
with compact derivative, whose G\^{a}teaux derivative is given by
$$
\Psi'(u)(v)=\int_a^b f(x,u(x))v(x)dx
$$
for every $v\in X$.

Also, since $m\leq h(x,t)\leq M$ for all
$(x,t)\in[a,b]\times\mathbb{R}$, we see that
\begin{equation}\label{e3.2}
\frac{1}{pM}\|u\|^p\leq\Phi(u)\leq \frac{1}{pm}\|u\|^p \qquad \mbox{
for all } u\in X.
\end{equation}
Now, put
$$
r_1:=\frac{2^p}{(b-a)^{p-1}pM}c_1^p,\qquad
r_2:=\frac{2^p}{(b-a)^{p-1}pM}c_2^p,
$$
and
\begin{equation*}
w(x):=\left\{\begin{array}{ll}
\displaystyle\frac{1}{\alpha^{p-1}}\,d(x-a)^{p-1}, & \mbox{if } a\leq x<a+\alpha,\\
d, & \mbox{if } a+\alpha\leq x\leq b-\beta,\\
\displaystyle\frac{1}{\beta^{p-1}}\,d(b-x)^{p-1}, & \mbox{if }
b-\beta<x\leq b.
\end{array}\right.
\end{equation*}
It is easy to verify that $w\in X$ and, in particular, one has
$$
\|w\|^p=d^p(p-1)^{p-2}\left(\alpha^{-p+1}+\beta^{-p+1}\right)=pDd^p.
$$
So, from \eqref{e3.2}, we have
$$
\frac{Dd^p}{M}\leq\Phi(w)\leq\frac{Dd^p}{m}.
$$
From the condition \eqref{e3.1}, we obtain $r_1<\Phi(w)<r_2$. For
all $u\in X$ such that $\Phi(u)<r_2$, taking \eqref{e2.3} into
account, one has $|u(x)|<c_2$ for all $x\in [a,b]$, from which it
follows
\begin{eqnarray*}
\sup_{u\in \Phi^{-1}(]-\infty,r_2[)}\Psi(u) &=& \sup_{u\in
\Phi^{-1}(]-\infty,r_2[)}\int_a^b F(x,u(x))dx\\
&\leq& \int_a^b \max_{|t|\leq c_2}F(x,t)dx.
\end{eqnarray*}
Arguing as before, we obtain
$$
\sup_{u\in \Phi^{-1}(]-\infty,r_1])}\Psi(u) \leq \int_a^b
\max_{|t|\leq c_1}F(x,t)dx.
$$
Since $0\leq w(x)\leq d$ for each $x\in [a,b]$, the assumption
$\rm(A_1)$ ensures that
$$
\Psi(w)\geq\int_{a+\alpha}^{b-\beta}F(x,d)dx.
$$
Then, we get
$$
\int_a^b\max_{|t|\leq
c_2}F(x,t)dx\geq\int_{a+\alpha}^{b-\beta}F(x,d)dx,
$$
and thus $a_d(c_2)\geq 0$. At this point, one has
\begin{eqnarray*}
\beta(r_1,r_2)&\leq&\frac{\displaystyle\sup_{u\in
\Phi^{-1}(]-\infty,r_2[)}\Psi(u)-\Psi(w)}{r_2-\Phi(w)}\\
&\leq&\frac{\displaystyle\int_a^b\max_{|t|\leq
c_2}F(x,t)dx-\int_{a+\alpha}^{b-\beta}F(x,d)dx}
{\displaystyle\frac{2^p}{(b-a)^{p-1}pM}c_2^p-\frac{Dd^p}{m}}\\
&=&(b-a)^{p-1}pmM\frac{\displaystyle\int_a^b\max_{|t|\leq
c_2}F(x,t)dx-\int_{a+\alpha}^{b-\beta}F(x,d)dx}
{\displaystyle m(2c_2)^p-Dd^p(b-a)^{p-1}pM}\\
&=&\left[(b-a)^{p-1}pmM\right]a_d(c_2).
\end{eqnarray*}
Since $a_d(c_2)\geq 0$, the hypothesis $\rm(A_2)$ implies that
$$
\int_a^b\max_{|t|\leq
c_1}F(x,t)dx<\int_{a+\alpha}^{b-\beta}F(x,d)dx.
$$
So, one has
\begin{eqnarray*}
\rho_2(r_1,r_2)&\geq&\frac{\displaystyle\Psi(w)-\sup_{u\in
\Phi^{-1}(]-\infty,r_1])}\Psi(u)}{\Phi(w)-r_1}\\
&\geq&\frac{\displaystyle\int_{a+\alpha}^{b-\beta}F(x,d)dx-\int_a^b\max_{|t|\leq
c_1}F(x,t)dx}{\displaystyle\frac{Dd^p}{m}-\frac{2^p}{(b-a)^{p-1}pM}c_1^p}\\
&=&(b-a)^{p-1}pmM\frac{\displaystyle\int_{a+\alpha}^{b-\beta}F(x,d)dx-\int_a^b\max_{|t|\leq
c_1}F(x,t)dx}
{\displaystyle Dd^p(b-a)^{p-1}pM-m(2c_1)^p}\\
&=&\left[(b-a)^{p-1}pmM\right]a_d(c_1).
\end{eqnarray*}
Hence, from the assumption $\rm(A_2)$, one has
$\beta(r_1,r_2)<\rho_2(r_1,r_2)$. Therefore, from Theorem
\ref{the2.1}, for each
$\lambda\in\frac{1}{(b-a)^{p-1}pmM}\left]\frac{1}{a_d(c_1)},\frac{1}{a_d(c_2)}\right[$,
the functional $I_\lambda$ admits at least one critical point
$\bar{u}$ such that
$$
r_1<\Phi(\bar{u})<r_2,
$$
that is
$$
\frac{2m^{1/p}}{(b-a)^{(p-1)/p}M^{1/p}}c_1<\|\bar{u}\|<\frac{2}{(b-a)^{(p-1)/p}}c_2.
$$
So, applying Lemma \ref{lem2.3}, the conclusion is achieved.
\end{proof}

%----------------------------------------------------------------------------------------------------------

Now, we point out an immediate consequence of Theorem \ref{the3.1}.

%----------------------------------------------------------------------------------------------------------

\begin{theorem}\label{the3.2}
Assume that there exist two positive constants $c,d$, with
$$
\frac{(b-a)^{(p-1)/p}(pD)^{1/p}}{2}d<\left(\frac{m}{M}\right)^{1/p}c,
$$
such that the assumption $\rm(A_1)$ in Theorem \ref{the3.1} holds.
Furthermore, suppose that
\begin{enumerate}
\item[$\rm(A_3)$] $\frac{\displaystyle\int_a^b\max_{|t|\leq
c}F(x,t)dx}{\displaystyle
m(2c)^p}<\frac{\displaystyle\int_{a+\alpha}^{b-\beta}F(x,d)dx}{\displaystyle
Dd^p(b-a)^{p-1}pM}$.
\end{enumerate}
Then, for each
$$
\lambda\in\left]\frac{Dd^p}{\displaystyle
m\int_{a+\alpha}^{b-\beta}F(x,d)dx},\frac{(2c)^p}
{\displaystyle(b-a)^{p-1}pM\int_a^b\max_{|t|\leq c}F(x,t)dx}\right[,
$$
problem \eqref{e1.1} admits at least one nontrivial classical
solution $\bar{u}\in X$, such that $|\bar{u}(x)|<c$ for all
$x\in[a,b]$.
\end{theorem}

\begin{proof}
The conclusion follows from Theorem \ref{the3.1}, by taking $c_1=0$
and $c_2=c$. Indeed, owing to the assumption $\rm(A_3)$, one has
\begin{eqnarray*}
a_d(c)&=&\frac{\displaystyle\int_a^b\max_{|t|\leq
c}F(x,t)dx-\int_{a+\alpha}^{b-\beta}F(x,d)dx}{m(2c)^p-Dd^p(b-a)^{p-1}pM}\\
&<&\frac{\displaystyle\left(1-\frac{Dd^p(b-a)^{p-1}pM}{m(2c)^p}\right)\int_a^b\max_{|t|\leq
c}F(x,t)dx}{m(2c)^p-Dd^p(b-a)^{p-1}pM}\\
&=&\frac{1}{m(2c)^p}\int_a^b\max_{|t|\leq c}F(x,t)dx.
\end{eqnarray*}
On the other hand, one has
$$
a_d(0)=\frac{\displaystyle\int_{a+\alpha}^{b-\beta}F(x,d)dx}{Dd^p(b-a)^{p-1}pM}.
$$
Hence, taking the assumption $\rm(A_3)$ and \eqref{e2.3} into
account, Theorem \ref{the3.1} ensures the conclusion.
\end{proof}

%----------------------------------------------------------------------------------------------------------

Now, we point out a special situation of our main result when the
nonlinear term has separable variables. To be precise, let
$\gamma\in L^1([a,b])$ such that $\gamma(x)\geq 0$ a.e. $x\in
[a,b],\; \gamma\not\equiv 0,$ and let
$g:\mathbb{R}\rightarrow\mathbb{R}$ be a nonnegative continuous
function. Consider the following Dirichlet boundary value problem
\begin{equation}\label{e3.3}
\left\{\begin{array}{ll}
-(p-1)|u'(x)|^{p-2}u''(x)=\lambda \gamma(x)g(u)h(x,u'),\qquad x\in(a,b),\\
u(a)=u(b)=0.
\end{array}\right.
\end{equation}
Put $\displaystyle G(t):=\int_0^t g(\xi)d\xi$ for all
$t\in\mathbb{R}$, and set
$\displaystyle\|\gamma\|_1:=\int_a^b\gamma(x)dx$.

%----------------------------------------------------------------------------------------------------------

\begin{theorem}\label{the3.3}
Assume that there exist two positive constants $c,d$, with
$$
\frac{(b-a)^{(p-1)/p}(pD)^{1/p}}{2}d<\left(\frac{m}{M}\right)^{1/p}c,
$$
such that
\begin{enumerate}
\item[$\rm(A_4)$] $\displaystyle\frac{G(c)}{c^p}
<\left(\frac{\displaystyle
2^pm\int_{a+\alpha}^{b-\beta}\gamma(x)dx}{\displaystyle
D(b-a)^{p-1}pM\|\gamma\|_1}\right)\frac{G(d)}{d^p}$.
\end{enumerate}
Then, for each
$$
\lambda\in\left]\frac{D}{\displaystyle
m\int_{a+\alpha}^{b-\beta}\gamma(x)dx}\frac{d^p}{G(d)},
\frac{2^p}{\displaystyle(b-a)^{p-1}pM\|\gamma\|_1}\frac{c^p}{G(c)}\right[,
$$
problem \eqref{e3.3} admits at least one positive classical solution
$\bar{u}\in X$, such that $\bar{u}(x)<c$ for all $x\in[a,b]$.
\end{theorem}

\begin{proof}
Put $f(x,\xi):=\gamma(x)g(\xi)$ for all $(x,\xi)\in
[a,b]\times\mathbb{R}$. Clearly, one has $F(x,t)=\gamma(x)G(t)$ for
all $(x,t)\in [a,b]\times\mathbb{R}$. Therefore, taking into account
that $G$ is a nondecreasing function, Theorem \ref{the3.2} ensures
the existence of a non-zero classical solution $\bar{u}$. We claim
that it is nonnegative. In fact, arguing by contradiction and
setting $A:=\{x\in[a,b] : \bar{u}(x)<0\}$, one has
$A\neq\varnothing$. Put $\bar{v}:=\min\{\bar{u},0\}$, one has
$\bar{v}\in X$. So, taking into account that $\bar{u}$ is a weak
solution and by choosing $v=\bar{v}$, from our sign assumptions on
the data, one has
$$
\int_A\bigg{(}\int_0^{\bar{u}'(x)}\frac{(p-1)|\tau|^{p-2}}{h(x,\tau)}\,d\tau\bigg{)}
\bar{u}'(x)dx=\lambda\int_A \gamma(x)g(\bar{u}(x))\bar{u}(x)dx\leq
0.
$$
On the other hand,
$$
\frac{1}{M}\|\bar{u}\|_{W_0^{1,p}(A)}^p\leq\int_A\bigg{(}\int_0^{\bar{u}'(x)}
\frac{(p-1)|\tau|^{p-2}}{h(x,\tau)}\,d\tau\bigg{)} \bar{u}'(x)dx.
$$
So, $\|\bar{u}\|_{W_0^{1,p}(A)}=0$ which is absurd. Hence, our claim
is proved. Now, owing to the strong maximum principle (see, e.g.,
\cite[Theorem 11.1]{PuSe2}) the classical solution $\bar{u}$, being
non-zero, is positive and the conclusion is achieved.
\end{proof}

%----------------------------------------------------------------------------------------------------------

We now give a special case of our main result as follows.

%----------------------------------------------------------------------------------------------------------

\begin{theorem}\label{the3.4}
Assume that
\begin{enumerate}
\item[$\rm(A_5)$] $\displaystyle\lim_{t\to 0^{+}}\frac{g(t)}{t^{p-1}}=+\infty$.
\end{enumerate}
Furthermore, for each $c>0$, set
$$
\lambda_c^\star:=\frac{2^p}{(b-a)^{p-1}pM\|\gamma\|_1}\frac{c^p}{G(c)}.
$$
Then, for every $\lambda\in]0,\lambda_c^\star[$, problem
\eqref{e3.3} admits at least one positive classical solution.
\end{theorem}

\begin{proof}
Fix $c>0$ and $\lambda\in]0,\lambda_c^\star[$. From $\rm(A_5)$ we
get $\lim_{t\to 0^{+}}\frac{t^p}{G(t)}=0$. Thus, there exists a
positive constant $d$ satisfying
$$
d<\frac{2m^{1/p}}{(b-a)^{(p-1)/p}(pMD)^{1/p}}c,
$$
such that
$$
\frac{D}{\displaystyle
m\int_{a+\alpha}^{b-\beta}\gamma(x)dx}\frac{d^p}{G(d)}
<\lambda<\frac{2^p}{(b-a)^{p-1}pM\|\gamma\|_1}\frac{c^p}{G(c)}.
$$
Hence, owing to Theorem \ref{the3.3}, for every
$\lambda\in]0,\lambda_c^\star[$ problem \eqref{e3.3} admits at least
one positive classical solution $\bar{u}\in X$. The proof is
complete.
\end{proof}

%----------------------------------------------------------------------------------------------------------

\begin{remark}\label{rem3.5}
\rm{ \noindent We claim that under the above assumptions, the
mapping $\lambda\mapsto I_{\lambda}(\bar{u})$ is negative and
strictly decreasing in $]0,\lambda_c^\star[$. Indeed,  the
restriction of the functional $I_{\lambda}$ to $\Phi^{-1}(]0,r_2[)$,
where $r_{2}:=\frac{(2c_2)^p}{(b-a)^{p-1}pM}$, admits a global
minimum, which is a critical point (local minimum) of $I_{\lambda}$
in $X$. Moreover, since $w\in\Phi^{-1}(]0,r_2[)$ and
$$
\frac{\Phi(w)}{\Psi(w)}\leq\frac{Dd^p}{\displaystyle
mG(d)\int_{a+\alpha}^{b-\beta}\gamma(x)dx}<\lambda,
$$
we have
$$
I_\lambda(\bar{u})\leq I_\lambda(w)=\Phi(w)-\lambda\Psi(w)<0.
$$
Next, we observe that
$$
I_{\lambda}(u)=\lambda\left(\frac{\Phi(u)}{\lambda}-\Psi(u)\right),
$$
for every $u\in X$ and fix $0<\lambda_1<\lambda_2<\lambda_c^\star$.
Set
$$
m_{\lambda_1}:=\left(\frac{\Phi(\bar{u}_1)}{\lambda_1}-\Psi(\bar{u}_1)\right)=\inf_{u\in
\Phi^{-1}(]0,r_2[)}\left(\frac{\Phi(u)}{\lambda_1}-\Psi(u)\right),
$$
and
$$
m_{\lambda_2}:=\left(\frac{\Phi(\bar{u}_2)}{\lambda_2}-\Psi(\bar{u}_2)\right)=\inf_{u\in
\Phi^{-1}(]0,r_2[)}\left(\frac{\Phi(u)}{\lambda_2}-\Psi(u)\right).
$$
Clearly, as claimed before, $m_{\lambda_i}<0$ (for $i=1,2$), and
$m_{\lambda_2}\leq m_{\lambda_1}$ thanks to $\lambda_1<\lambda_2$.
Then the mapping $\lambda\mapsto I_{\lambda}(\bar{u})$ is strictly
decreasing in $]0,\lambda_c^\star[$ owing to
$$
I_{\lambda_2}(\bar{u}_2)=\lambda_2m_{\lambda_2}\leq
\lambda_2m_{\lambda_1}<\lambda_1m_{\lambda_1}=I_{\lambda_1}(\bar{u}_1).
$$ This concludes the proof of our claim.}
\end{remark}

%----------------------------------------------------------------------------------------------------------

\begin{remark}\label{rem3.6}\rm{
In other words, Theorem \ref{the3.4} ensures that if the asymptotic
condition at zero $\rm(A_5)$ is verified then, for every parameter
$\lambda$ belonging to the real interval $]0,\lambda^\star[$, where
$$
\lambda^\star:=\frac{2^p}{(b-a)^{p-1}pM\|\gamma\|_1}\sup_{c>0}\frac{c^p}{G(c)},
$$
problem \eqref{e3.3} admits at least one positive classical solution
$\bar{u}\in X$.}
\end{remark}

%----------------------------------------------------------------------------------------------------------

Finally, we present an application of Theorem \ref{the2.2} which we
will use in next section to obtain multiple solutions.

%----------------------------------------------------------------------------------------------------------

\begin{theorem}\label{the3.7}
Assume that there exist two constants $\bar{c},\bar{d}$, with
$$
\bar{c}<\frac{(b-a)^{(p-1)/p}(pD)^{1/p}}{2}\bar{d},
$$
such that
\begin{enumerate}
\item[$\rm(A_6)$] $\displaystyle\int_a^b\max_{|t|\leq\bar{c}}F(x,t)dx<\int_{a+\alpha}^{b-\beta}F(x,\bar{d})dx$;
\item[$\rm(A_7)$] $\displaystyle\limsup_{|t|\rightarrow +\infty}\frac{F(x,t)}{|t|^p}\leq
0$ uniformly in $x$.
\end{enumerate}
Then, for each $\lambda>\tilde{\lambda}$, where
$$
\tilde{\lambda}:=\frac{\displaystyle
D\bar{d}^p(b-a)^{p-1}pM-m(2\bar{c})^p}{\displaystyle(b-a)^{p-1}pmM
\left(\int_{a+\alpha}^{b-\beta}
F(x,\bar{d})dx-\int_a^b\max_{|t|\leq\bar{c}}F(x,t)dx\right)},
$$
problem \eqref{e1.1} admits at least one nontrivial classical
solution $\tilde{u}\in X$ such that
$$
\|\tilde{u}\|>\frac{2m^{1/p}}{(b-a)^{(p-1)/p}M^{1/p}}\bar{c}.
$$
\end{theorem}

\begin{proof}
The functionals $\Phi$ and $\Psi$ defined in the proof of Theorem
\ref{the3.1} satisfy all regularity assumptions requested in Theorem
\ref{the2.2}. Moreover, by standard computations, the assumption
$\rm(A_7)$ implies that $I_\lambda,\,\lambda>0$, is coercive. So,
our aim is to verify condition \eqref{e2.2} of Theorem \ref{the2.2}.
To this end, put
$$
r:=\frac{2^p}{(b-a)^{p-1}pM}\bar{c}^p
$$
and
\begin{equation*}
w(x):=\left\{\begin{array}{ll}
\displaystyle\frac{1}{\alpha^{p-1}}\,\bar{d}(x-a)^{p-1}, & \mbox{if } a\leq x<a+\alpha,\\
\bar{d}, & \mbox{if } a+\alpha\leq x\leq b-\beta,\\
\displaystyle\frac{1}{\beta^{p-1}}\,\bar{d}(b-x)^{p-1}, & \mbox{if }
b-\beta<x\leq b.
\end{array}\right.
\end{equation*}
Arguing as in the proof of Theorem \ref{the3.1} we obtain that
$$
\rho(r)\geq(b-a)^{p-1}pmM\frac{\displaystyle\int_{a+\alpha}^{b-\beta}F(x,\bar{d})dx-\int_a^b\max_{|t|\leq
\bar{c}}F(x,t)dx} {\displaystyle
D\bar{d}^p(b-a)^{p-1}pM-m(2\bar{c})^p}.
$$
So, from our assumption it follows that $\rho(r)>0$.

Hence, from Theorem \ref{the2.2} for each $\lambda>\tilde{\lambda}$,
the functional $I_\lambda$ admits at least one local minimum
$\tilde{u}$ such that
$$
\|\tilde{u}\|>\frac{2m^{1/p}}{(b-a)^{(p-1)/p}M^{1/p}}\bar{c},
$$
and the conclusion is achieved.
\end{proof}

%----------------------------------------------------------------------------------------------------------

\section{Multiplicity results}

The main aim of this section is to present multiplicity results.
First, as consequence of Theorem \ref{the3.1}, taking into account
the classical theorem of Ambrosetti and Rabinowitz, we have the
following multiplicity result.

%----------------------------------------------------------------------------------------------------------

\begin{theorem}\label{the4.1}
Let the assumptions of Theorem \ref{the3.1} be satisfied, and
$f(\cdot,0)\neq 0$ in $(a,b)$. Moreover, let
\begin{enumerate}
\item[$\rm(A_8)$] there exist constants $\nu$ and $R>0$ such
that $\nu m>pM$, and for all $|t|\geq R$ and for all $x\in[a,b]$,
one has
\begin{equation*}
0<\nu F(x,t)\leq t\cdot f(x,t).
\end{equation*}
\end{enumerate}
Then, for each
$$
\lambda\in\frac{1}{(b-a)^{p-1}pmM}\left]\frac{1}{a_d(c_1)},\frac{1}{a_d(c_2)}\right[,
$$
problem \eqref{e1.1} admits at least two nontrivial classical
solutions $\bar{u}_1,\bar{u}_2$, such that
\begin{equation}\label{e4.1}
\frac{2m^{1/p}}{(b-a)^{(p-1)/p}M^{1/p}}c_1<\|\bar{u}_1\|<\frac{2}{(b-a)^{(p-1)/p}}c_2.
\end{equation}
\end{theorem}

\begin{proof}
Fix $\lambda$ as in the conclusion. So, Theorem \ref{the3.1} ensures
that problem \eqref{e1.1} admits at least one nontrivial classical
solution $\bar{u}_1$ satisfying the condition \eqref{e4.1} which is
a local minimum of the functional $I_\lambda$.

Now, we prove the existence of the second solution distinct from the
first one. To this end, we must show that the functional $I_\lambda$
satisfies the hypotheses of the mountain pass theorem.

Clearly, the functional $I_\lambda$ is of class $C^1$ and
$I_\lambda(0)=0$.

We can assume that $\bar{u}_1$ is a strict local minimum for
$I_\lambda$ in $X$. Therefore, there is $\rho>0$ such that
$\inf_{\|u-\bar{u}_1\|=\rho}I_\lambda(u)>I_\lambda(\bar{u}_1)$, so
condition \cite[$(I_1)$, Theorem 2.2]{Rab} is verified.

From $\rm(A_8)$, by standard computations, there is a positive
constant $C$ such that
\begin{equation}\label{e4.2}
F(x,t)\geq C|t|^\nu
\end{equation}
for all $x\in[a,b]$ and $|t|>R$. In fact, setting
$a(x):=\min_{|\xi|=R}F(x,\xi)$ and
\begin{equation}\label{e4.3}
\varphi_t(s):=F(x,st),\quad \forall s>0,
\end{equation}
by $\rm(A_8)$, for every $x\in[a,b]$ and $|t|>R$ one has
$$
0<\nu\varphi_t(s)=\nu F(x,st)\leq st\cdot
f(x,st)=s\varphi'_t(s),\quad \forall s>\frac{R}{|t|}.
$$
Therefore,
$$
\int_{R/|t|}^1\frac{\varphi'_t(s)}{\varphi_t(s)}ds\geq
\int_{R/|t|}^1\frac{\nu}{s}ds.
$$
Then
$$
\varphi_t(1)\geq\varphi_t\Big(\frac{R}{|t|}\Big)\frac{|t|^\nu}{R^\nu}.
$$
Taking into account of \eqref{e4.3}, we obtain
$$
F(x,t)\geq F\Big(x,\frac{R}{|t|}t\Big)\frac{|t|^\nu}{R^\nu}\geq
a(x)\frac{|t|^\nu}{R^\nu}\geq C|t|^\nu,
$$
where $C>0$ is a constant. Thus, \eqref{e4.2} is proved. Now,
choosing any $u\in X\setminus\{0\}$, one has
\begin{eqnarray*}
I_\lambda(tu)&=&(\Phi-\lambda\Psi)(tu)\\
&\leq&\frac{t^p}{pm}\|u\|^p-\lambda t^\nu C\int_a^b|u(x)|^\nu
dx\rightarrow -\infty
\end{eqnarray*}
as $t\rightarrow +\infty$ (since $\nu>p$). So, the functional
$I_\lambda$ is unbounded from below and condition \cite[$(I_2)$,
Theorem 2.2]{Rab} is verified. Therefore, $I_\lambda$ satisfies the
geometry of mountain pass.

Now, to verify the Palais-Smale condition it is sufficient to prove
that any sequence of Palais-Smale is bounded. To this end, taking
into account $\rm(A_8)$ one has
\begin{eqnarray}\label{PS}
\nu I_\lambda(u_n)-\|I_\lambda'(u_n)\|_{X^*}\|u_n\|&\geq&\nu I_\lambda(u_n)-I_\lambda'(u_n)(u_n)\nonumber\\
&=&\nu\Phi(u_n)-\lambda\nu\Psi(u_n)-\Phi'(u_n)(u_n)+\lambda\Psi'(u_n)(u_n)\nonumber\\
&\geq&\left(\frac{\nu}{pM}-\frac{1}{m}\right)\|u_n\|^p-\lambda\int_a^b
\left[\nu F(x,u_n(x))-f(x,u_n(x))u_n(x)\right]dx\nonumber\\
&\geq&\left(\frac{\nu}{pM}-\frac{1}{m}\right)\|u_n\|^p.
\end{eqnarray}
If $\{u_n\}$ is not bounded, from \eqref{PS} we have a
contradiction. Thus, $I_\lambda$ satisfies the Palais-Smale
condition.

Hence, the classical theorem of Ambrosetti and Rabinowitz ensures a
critical point $\bar{u}_2$ of $I_\lambda$ such that
$I_\lambda(\bar{u}_2)>I_\lambda(\bar{u}_1)$. So, $\bar{u}_1$ and
$\bar{u}_2$ are two distinct classical solutions of \eqref{e1.1} and
the proof is complete.
\end{proof}

%----------------------------------------------------------------------------------------------------------

\begin{corollary}\label{cor4.2}
Assume that there exist two positive constants $c,d$, with
$$
\frac{(b-a)^{(p-1)/p}(pD)^{1/p}}{2}d<\left(\frac{m}{M}\right)^{1/p}c,
$$
such that $\rm(A_4)$ holds. Assume also that
\begin{enumerate}
\item[$\rm(A_9)$] there exist constants $\nu$ and $R>0$ such
that $\nu m>pM$, and for all $|t|\geq R$, one has
$$
0<\nu G(t)\leq t\cdot g(t).
$$
\end{enumerate}
Then, for each
$$
\lambda\in\left]\frac{D}{\displaystyle
m\int_{a+\alpha}^{b-\beta}\gamma(x)dx}\frac{d^p}{G(d)},
\frac{2^p}{\displaystyle(b-a)^{p-1}pM\|\gamma\|_1}\frac{c^p}{G(c)}\right[,
$$
problem \eqref{e3.3} admits at least two nonnegative classical
solutions $\bar{u}_1,\bar{u}_2$, such that $\bar{u}_1(x)<c$ for all
$x\in[a,b]$.
\end{corollary}

%----------------------------------------------------------------------------------------------------------

\begin{corollary}\label{cor4.3}
Assume that $\rm(A_5)$ and $\rm(A_9)$ are satisfied. Then, for each
$\lambda\in]0,\lambda^\star[$, problem \eqref{e3.3} admits at least
two nonnegative classical solutions.
\end{corollary}

%----------------------------------------------------------------------------------------------------------

Next, as a consequence of Theorems \ref{the3.7} and \ref{the3.2},
the following theorem of the existence of three classical solutions
is obtained and its consequence for the nonlinearity with separable
variables is presented.

%----------------------------------------------------------------------------------------------------------

\begin{theorem}\label{the4.4}
Assume that $\rm(A_7)$ holds. Moreover, assume that there exist four
positive constants $c,d,\bar{c},\bar{d}$, with
$$
\frac{(b-a)^{(p-1)/p}(pD)^{1/p}}{2}d
<\left(\frac{m}{M}\right)^{1/p}c\leq\bar{c}
<\frac{(b-a)^{(p-1)/p}(pD)^{1/p}}{2}\bar{d},
$$
such that $\rm(A_3)$, $\rm(A_6)$ and
\begin{enumerate}
\item[$\rm(A_{10})$] $\frac{\displaystyle\int_a^b\max_{|t|\leq c} F(x,t)dx}{\displaystyle(2c)^p}
<\frac{\displaystyle m\left(\int_{a+\alpha}^{b-\beta}
F(x,\bar{d})dx-\int_a^b\max_{|t|\leq\bar{c}}F(x,t)dx\right)}{\displaystyle
D\bar{d}^p(b-a)^{p-1}pM-m(2\bar{c})^p}$
\end{enumerate}
are satisfied. Then, for each
$$
\lambda\in\Lambda:=\left]\max\left\{\tilde{\lambda},\frac{Dd^p}{\displaystyle
m\int_{a+\alpha}^{b-\beta}F(x,d)dx}\right\},
\frac{(2c)^p}{\displaystyle(b-a)^{p-1}pM\int_a^b\max_{|t|\leq
c}F(x,t)dx}\right[,
$$
problem \eqref{e1.1} admits at least three classical solutions.
\end{theorem}

\begin{proof}
First, we observe that $\Lambda\neq\varnothing$ owing to
$\rm(A_{10})$. Next, fix $\lambda\in\Lambda$. Theorem \ref{the3.2}
ensures a nontrivial classical solution $\bar{u}$ such that
$$
\|\bar{u}\|<\frac{2}{(b-a)^{(p-1)/p}}c,
$$
which is a local minimum for the associated functional $I_\lambda$,
as well as Theorem \ref{the3.7} guarantees a nontrivial classical
solution $\tilde{u}$ such that
$$
\|\tilde{u}\|>\frac{2m^{1/p}}{(b-a)^{(p-1)/p}M^{1/p}}\bar{c},
$$
which is a local minimum for $I_\lambda$. Hence, the mountain pass
theorem as given by Pucci and Serrin (see \cite{PuSe1}) ensures the
conclusion.
\end{proof}

%----------------------------------------------------------------------------------------------------------

\begin{theorem}\label{the4.5}
Assume that
\begin{enumerate}
\item[$\rm(A_{11})$]
$\displaystyle\limsup_{t\rightarrow 0^+}\frac{G(t)}{t^p}=+\infty$;
\item[$\rm(A_{12})$]
$\displaystyle\limsup_{t\rightarrow +\infty}\frac{G(t)}{t^p}=0$.
\end{enumerate}
Further, assume that there exist two positive constants
$\bar{c},\bar{d}$, with
$$
\bar{c}<\frac{(b-a)^{(p-1)/p}(pD)^{1/p}}{2}\bar{d}
$$
such that
\begin{enumerate}
\item[$\rm(A_{13})$] $\displaystyle\frac{G(\bar{c})}{\bar{c}^p}
<\left(\frac{\displaystyle2^pm\int_{a+\alpha}^{b-\beta}\gamma(x)dx}
{\displaystyle
D(b-a)^{p-1}pM\|\gamma\|_1}\right)\displaystyle\frac{G(\bar{d})}{\bar{d}^p}$.
\end{enumerate}
Then, for each
$$
\lambda\in\left]\frac{D}{\displaystyle
m\int_{a+\alpha}^{b-\beta}\gamma(x)dx}\frac{\bar{d}^p}{G(\bar{d})},
\frac{2^p}{(b-a)^{p-1}pM\|\gamma\|_1}\frac{\bar{c}^p}{G(\bar{c})}\right[,
$$
problem \eqref{e3.3} admits at least three nonnegative classical
solutions.
\end{theorem}

\begin{proof}
Clearly, $\rm(A_{12})$ implies $\rm(A_7)$. Moreover, by choosing $d$
small enough and $c=\bar{c}$, simple computations show that
$\rm(A_{11})$ implies $\rm(A_3)$. Finally, from $\rm(A_{13})$ we get
$\rm(A_6)$ and, arguing as in the proof of Theorem \ref{the3.2},
also $\rm(A_{10})$. Hence, Theorem \ref{the4.4} ensures the
conclusion.
\end{proof}

%----------------------------------------------------------------------------------------------------------

\begin{remark}\label{rem4.6}\rm{
If $g(0)\neq 0$, Corollaries \ref{cor4.2} and \ref{cor4.3} ensure
two positive classical solutions while Theorem \ref{the4.5} ensures
three positive classical solutions (see proof of Theorem
\ref{the3.3}).}
\end{remark}

%----------------------------------------------------------------------------------------------------------

Finally, we present the following example to illustrate our results.

%----------------------------------------------------------------------------------------------------------

\begin{example}\rm{
Consider the following Dirichlet problem
\begin{equation}\label{e4.4}
\left\{\begin{array}{ll}
-u''(x)=\lambda e^x\left(\frac{1}{6}+|u|^2u\right)h(u'),\qquad x\in(0,1),\\
u(0)=u(1)=0,
\end{array}\right.
\end{equation}
where
$$
h(t):=\left\{\begin{array}{ll}
1, & \textrm{ if }\, t < 0,\\
\frac{6-t}{6}, & \textrm{ if }\, 0 \leq t \leq 1,\\
\frac{5}{6}, & \textrm{ if }\, t > 1.
\end{array}\right.
$$
Then, $h:\mathbb{R}\rightarrow [0,+\infty)$ is a bounded and
continuous function with
$$
m=\inf_{t\in\mathbb{R}}h(t)=\frac{5}{6}>0 \qquad \mbox{and} \qquad
M=\sup_{t\in\mathbb{R}}h(t)=1.
$$
Let $g(t):=\frac{1}{6}+|t|^2t$ for all $t\in\mathbb{R}$. Obviously,
$g(0)\neq 0$. Since
$$
\lim_{t\to 0^{+}}\frac{g(t)}{t}=\lim_{t\to
0^{+}}\left(\frac{1}{6t}+|t|^2\right)=+\infty,
$$
condition $\rm(A_5)$ holds true. Choose $\nu=3$ and $R=2$, we have
$\nu m>pM$, and
$$
0<3 G(t)\leq t\cdot g(t),
$$
for all $|t|\geq 2$. Moreover, one has
$$
\lambda^\star=\frac{2}{\displaystyle\int_0^1e^xdx}\sup_{c>0}\frac{c^2}{G(c)}
=\left(\frac{2}{e-1}\right)\sup_{c>0}\frac{12c}{2+3c^3}\geq\left(\frac{2}{e-1}\right)\frac{12c}{2+3c^3}
\Big|_{c=\frac{1}{2}}=\frac{96}{19(e-1)}.
$$
Then, owing to Corollary \ref{cor4.3} and Remark \ref{rem4.6}, for
each $\lambda\in\left]0,\frac{96}{19(e-1)}\right[$, problem
\eqref{e4.4} admits at least two positive classical solutions.}
\end{example}

%----------------------------------------------------------------------------------------------------------

\subsection*{Acknowledgment}
V.~R\u adulescu acknowledges the support through Grant CNCS-PCE-47/2011.

%----------------------------------------------------------------------------------------------------------

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\end{document} 