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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2014 (2014), No. 106, pp. 1--14.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2014 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2014/106\hfil Multiple solutions]
{Multiple solutions for perturbed $p$-Laplacian boundary-value problems
 with impulsive effects}

\author[M. Ferrara, S. Heidarkhani \hfil EJDE-2014/106\hfilneg]
{Massimiliano Ferrara, Shapour Heidarkhani}  % in alphabetical order

\address{Massimiliano Ferrara \newline
Department of Law and Economics,
University Mediterranea of Reggio Calabria,
 Via dei Bianchi, 2 - 89131 Reggio Calabria, Italy}
\email{massimiliano.ferrara@unirc.it}

\address{Shapour Heidarkhani \newline
Department of Mathematics, Faculty of Sciences,
Razi University,  67149 Kermanshah, Iran}
\email{s.heidarkhani@razi.ac.ir}

\thanks{Submitted August 8, 2013. Published April 15, 2014.}
\subjclass[2000]{34B15, 34B18, 34B37, 58E30}
\keywords{Multiple solutions; perturbed $p$-Laplacian; critical point theory;
\hfill\break\indent boundary-value  problem with impulsive effects; 
variational methods}

\begin{abstract}
 We establish the existence of three distinct solutions for a
 perturbed $p$-Laplacian boundary value problem with impulsive
 effects. Our approach is  based on variational methods.
\end{abstract}

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

\section{Introduction}

In this work, we show the existence of at least three solutions for
the nonlinear perturbed problem
\begin{equation}\label{e1}
\begin{gathered}
-(\rho(x)\Phi_{p}(u'(x)))'+s(x)\Phi_{p}(u'(x))
=\lambda f(x,u(x))+\mu g(x,u(x))\quad \text{a.e. }x\in (a,b),\\
\alpha_1 u'(a^{+})-\alpha_2u(a)=0,\quad \beta_1u'(b^{-})+\beta_2u(b)=0
\end{gathered}
\end{equation}
with the impulsive conditions
\begin{equation}\label{e2}
\Delta(\rho(x_j) \Phi_{p}(u'(x_j)))=I_j(u(x_j)),\quad
j=1,2,\dots ,l
\end{equation}
where $a,b\in \mathbb{R}$ with $a<b$, $p>1$,
$\Phi_{p}(t)=|t|^{p-2}t$,    $\rho,s\in L^\infty([a,b])$ with
$\rho_{0}:=\operatorname{ess\,inf}_{x\in[a,b]}\rho(x)>0$,
$s_0:=\operatorname{ess\,inf}_{x\in[a,b]}s(x)>0$,
$\rho(a^{+})=\rho(a)>0$, $\rho(b^-)=\rho(b)>0$,
$\alpha_1$, $\alpha_2$, $\beta_1$,
$\beta_2$ are positive constants, $f,g:[a,b]\times \mathbb{R}\to
\mathbb{R}$ are two $L^1$-Carath\'{e}odory functions,
$x_{0}=a<x_1<x_2<\dots <x_{l}<x_{l+1}=b$,
$$
\Delta(\rho(x_j) \Phi_{p}(u'(x_j)))
=\rho(x_j^{+})\Phi_{p}(u'(x_j^{+}))-\rho(x_j^{-})\Phi_{p}(u'(x_j^{-}))
$$
where $z(y^+)$ and $z(y^-)$ denote the right and left limits of
$z(y)$ at $y$, respectively, $I_j:\mathbb{R}\to\mathbb{R}$ for
$j=1,\dots ,l$ are continuous satisfying the condition
$\sum_{j=1}^p(I_j(t_1)-I_j(t_2))(t_1-t_2)\geq 0$ for every
$t_1,t_2\in \mathbb{R}$, $\lambda$ is a positive parameter and
$\mu$ is a non-negative parameter. 

 The theory of impulsive differential equations describes processes
which experience a sudden
change of their state at certain moments. Processes with such a
character arise naturally and often, especially in phenomena
studied in mechanical systems with impact, biological systems such
as heart beats, population dynamics, theoretical physics,
radiophysics, pharmacokinetics, mathematical economy, chemical
technology, electric technology, metallurgy, ecology, industrial
robotics, biotechnology processes, chemistry, engineering, control
theory and so on. For the background, theory and applications of
impulsive differential equations, we refer the interest readers to
\cite{BS,BHN,Ca,LAS,LBS,LW,NR1,SP}.

  Existence and multiplicity of solutions for impulsive differential
equations have been studied by several authors and,  for an
 overview on this subject, we refer the reader to the papers
 \cite{BD2,BD3,LJ,NO,TG1,TG2,ZD,ZY}.
 For instance, Tian and Ge in \cite{TG1}, using variational methods, have
studied the existence of at least two positive solutions for the
nonlinear impulsive boundary-value  problem
\begin{gather*}
-(\rho(t)\Phi_{p}(u'(t)))'+s(t)\Phi_{p}(u'(t))=
f(t,u(t))\quad \text{a.e. }t\neq t_i,\; t\in (a,b),\\
\Delta(\rho(t_i) \Phi_{p}(u'(t_j)))=I_{i}(u(t_{i})),\quad i=1,2,\dots ,l\\
\alpha u'(a)-\beta u(a)=A,\quad \gamma u'(b)+\sigma u(b)=B,
\end{gather*}
where $a,b\in \mathbb{R}$ with $a<b$, $p>1$,
$\Phi_{p}(t)=|t|^{p-2}t$,    $\rho,s\in L^\infty([a,b])$ with
$\operatorname{ess\,inf}_{t\in[a,b]}\rho(t)>0$,
$\operatorname{ess\,inf}_{t\in[a,b]}s(t)>0$, $0<\rho(a),\rho(b)<+\infty$,
$A\leq 0,\ B\geq 0$, $\alpha$, $\beta$, $\gamma$, $\sigma$ are
positive constants, $I_i\in C( [0,+\infty),\ [0,+\infty))$ for
$i=1,\dots ,l$, $f\in C([a,b]\times [0,+\infty), [0,+\infty))$,
$f(t,0)\neq 0$ for $t\in[a,b]$,
$t_{0}=a<t_1<t_2\dots <t_{l}<t_{l+1}=b$, $\Delta(\rho(t_i)
\Phi_{p}(u'(t_i)))=\rho(t_i^{+})\Phi_{p}(u'(t_{i}^{+}))
-\rho(t_i^{-})\Phi_{p}(u'(t_{i}^{-}))$
where $x(t_i^+)$ (respectively $x(t_i^-)$) denotes the right limit
(respectively left limit) of $x(t)$ at $t=t_i$ for $i=1,\dots ,l$.
Also, Tain and Ge in \cite{TG2} have studied the existence of
positive solutions to the linear and nonlinear Sturm-Liouville
impulsive problem by using variational methods. In fact they have
generalized the results of \cite{NO,TG1}. In \cite{BD2}, Bai and
Dai by using critical point theory, some criteria have obtained to
guarantee that the impulsive problem
\begin{gather*}
-(\rho(t)\Phi_{p}(u'(t)))'+s(t)\Phi_{p}(u'(t))=
\lambda f(t,u(t))\quad\text{a.e. } t\neq t_i,\; t\in (a,b),\\
\Delta(\rho(t_i) \Phi_{p}(u'(t_j)))
 =I_{i}(u(t_{i})),\quad i=1,2,\dots ,l\\
\alpha u'(a)-\beta u(a)=A,\quad \gamma u'(b)+\sigma u(b)=B,
\end{gather*}
where $a,b\in \mathbb{R}$ with $a<b$, $p>1$,
$\Phi_{p}(t)=|t|^{p-2}t$,  $\rho,s\in L^\infty([a,b])$ with
$\operatorname{ess\,inf}_{t\in[a,b]}\rho(t)>0$,
$\operatorname{ess\,inf}_{t\in[a,b]}s(t)>0$, $0<\rho(a),\rho(b)<+\infty$,
$\lambda$ is a positive parameter, $A,B$ are constant,
$\alpha$, $\beta$, $\gamma$, $\sigma$ are positive constants,
$f:[a,b]\times \mathbb{R}\to \mathbb{R}$ is a continuous function,
$I_i:\mathbb{R}\to \mathbb{R}$  for $i=1,\dots ,l$ are continuous
functions, $t_{0}=a<t_1<t_2\dots <t_{l}<t_{l+1}=b$,
$\Delta(\rho(t_i) \Phi_{p}(u'(t_i)))
=\rho(t_i^{+})\Phi_{p}(u'(t_{i}^{+}))-\rho(t_i^{-})\Phi_{p}(u'(t_{i}^{-}))$
where $x(t_i^+)$ (respectively $x(t_i^-)$) denotes the right limit
(respectively left limit) of $x(t)$ at $t=t_i$ for $i=1,\dots ,l$,
has at least one solution, two solutions and infinitely many
solutions when the parameter lies in different intervals. In
particular, in \cite{BD3}, Bai and Dai, employing a three critical
points theorem due to Ricceri \cite{R3} have ensured the existence
of at least three solutions for \eqref{e1}-\eqref{e2} in the case
$\mu=0$.

 In this article, motivated by \cite{BD3}, employing a three critical
points theorem obtained in \cite{BM} which we recall in the next section
(Theorem \ref{t1}), we ensure the existence of at least three weak
solutions for the problem \eqref{e1}-\eqref{e2}. We explicitly observe that in
\cite{BD3}, $\mu=0$ and no exact estimate of $\lambda$ for which
the problem \eqref{e1}-\eqref{e2} admits multiple solutions is ensured. The aim of
this work is to establish precise values of $\lambda$ and $\mu$
for which the problem \eqref{e1}-\eqref{e2} admits at least three weak solutions.

 Theorem \ref{t1} has been used for establishing the
existence of at least three solutions for eigenvalue problems in
the papers \cite{BM1,BMR1,BMR2,HH1}.
Fora review on the subject, we refer the reader to \cite{FKH}.

\section{Preliminaries}

Our main tool is the following three critical points theorem.

\begin{theorem}[{\cite[Theorem 2.6]{BM}}] \label{t1}
 Let $X$ be a reflexive real Banach space, $ \Phi:X \to \mathbb{R}$ be a coercive
continuously G\^{a}teaux differentiable and sequentially weakly
lower semicontinuous functional whose G\^{a}teaux derivative admits
a continuous inverse on $X^{*}$, and $\Psi:X\to \mathbb{R}$
be a continuously  G\^{a}teaux differentiable functional whose
 G\^{a}teaux derivative is compact, such that $ \Phi(0)=\Psi(0)=0$.
 Assume that there exist $r>0$ and $\overline{x}\in X$, with
$r< \Phi(\overline{x})$ such  that
\begin{itemize}
\item[(a1)]  $\frac{1}{r} \sup_{ \Phi(x)\leq r} \Psi(x)
< \frac{\Psi(\overline{x})}{\Phi(\overline{x})}$,

\item[(a2)] for each $\lambda\in \Lambda_{r}:=
]\frac{\Phi(\overline{x})}{\Psi(\overline{x})},
\frac{r}{\sup_{\Phi(x)\leq r}\Psi(x)}[$ the functional $
\Phi-\lambda \Psi$ is coercive.
\end{itemize}
Then, for each $\lambda\in\Lambda_{r}$ the functional
$\Phi-\lambda \Psi$ has at least three distinct critical points in
$X$.
\end{theorem}

Let $ X:=W^{1,p}([a,b])$ equipped with the norm
 $$
\|u\| := \Big(\int_a^b\rho(x)|u'(x)|^pdx + \int_a^b
s(x)|u(x)|^pdx\Big)^{1/p}
$$
which is equivalent to the usual one.
The following lemma is useful for proving our main result.

 \begin{lemma}[{\cite[Lemma 2.6]{TG1}}]\label{lem1}
 Let $u\in X$. Then
 \begin{equation}\label{e3}
\| u\|_{\infty}=\max_{x\in[a,b]}| u(x)|\leq M\|u\|
\end{equation}
where
$$
M=2^{1/q}\max\Big\{\frac{1}{(b-a)^{1/p}s_0^{1/p}},\;
 \frac{(b-a)^{1/p}}{\rho_{0}^{1/p}}\Big\}, \quad
 \frac{1}{p}+\frac{1}{q}=1.
$$
\end{lemma}
 By a classical solution of the problem \eqref{e1}-\eqref{e2}, we mean a function
$u\in\{u(x)\in X: \rho(x)\Phi_{p}(u')(.)\in
  W^{1,\infty}(x_j,x_{j+1}),\,
j=0,1,\dots ,l\}$ such that   $u$
satisfies \eqref{e1}-\eqref{e2}. We say that a function $u\in X$ is
a weak solution of the problem \eqref{e1}-\eqref{e2} if
\begin{align*}
&\int_a^b \rho(x)\Phi_{p}(u'(x))v'(x)dx +\int_a^b s(x)\Phi_{p}(u(x))v(x)dx \\
&+\rho(a)\Phi_{p}\Big(\frac{\alpha_2u(a)}{\alpha_1}\Big)v(a)
 +\rho(b)\Phi_{p}\Big(\frac{\beta_2u(b)}{\beta1}\Big)v(b)
 +\sum_{j=1}^{l}I_j(u(x_j))v(x_j)\\
&-\lambda\int_{a}^{b}f(x,u(x))v(x)dx-\mu\int_{a}^{b}g(x,u(x))v(x) dx=0
\end{align*}
for every $v\in X$.

 For the sake of convenience, in the sequel, we define
\begin{gather*}
F(x,t)=\int_{0}^{t}f(x,\xi)d\xi\quad \text{for all }(x,t)\in[a,b]\times\mathbb{R},\\
G(x,t)=\int_{0}^{t}g(x,\xi)d\xi\quad \text{for all }(x,t)\in[a,b]\times\mathbb{R},\\
C_1=\frac{M^p}{p}\Big(\frac{\rho(a)\alpha_2^{p-1}}{\alpha_1^{p-1}}
 +\frac{\rho(b)\beta_2^{p-1}}{\beta_1^{p-1}}\Big)\\
C_2=\frac{1}{p}-\sum_{j=1}^{l}\frac{b_j}{\gamma_j+1}M^{\gamma_j+1},\\
C_3=\frac{1}{p}+\sum_{j=1}^{l}\frac{b_j}{\gamma_j+1}M^{\gamma_j+1},\\
C_4=\sum_{j=1}^{l}\Big(a_jM+\frac{b_j}{\gamma_j+1}
M^{\gamma_j+1}\Big).
\end{gather*}

 For given constants $\delta_1,\ \delta_2,\ \eta_1$ and $\eta_2$
 put
\begin{gather*}
K_1:=\Big((b-a)\Big(\frac{\delta_1}{\eta_1}+\frac{\delta_2}{\eta_2}\Big)
+\frac{\alpha_1}{\alpha_2}\delta_1+  \frac{\beta_1}{\beta_2}\delta_2\Big)/
\Big((b-a)\Big(\frac{1}{\eta_1}+\frac{1}{\eta_2}-1\Big)\Big),\\
K_2:=|\delta_1|^p\int_{a}^{a+\frac{b-a}{\eta_1}}\rho(x)dx+|K_1|^p
 \int_{a+\frac{b-a}{\eta_1}}^{b-\frac{b-a}{\eta_2}}\rho(x)dx
+|\delta_2|^p\int_{b-\frac{b-a}{\eta_1}}^{b}\rho(x)dx,\\
K_3=\max\Big{\{}\frac{\alpha_1}{\alpha_2}|\delta_1|,\;
 \Big(\frac{b-a}{\eta_1}+\frac{\alpha_1}{\alpha_2} \Big)|\delta_1|,\;
 \Big(\frac{b-a}{\eta_2}+\frac{\beta_1}{\beta_2}
 \Big)|\delta_2|,\ \frac{\beta_1}{\beta_2}|\delta_2|\Big{\}},\\
K_4:=(C_1+C_3)\Big(K_2+K_3^p\int_{a}^{b}s(x)dx\Big)
  +C_4\Big(K_2+K_3^p\int_{a}^{b}s(x)dx\Big)^{1/p},\\
h_1(x)=\delta_1\Big(x+\frac{\alpha_1}{\alpha_2}-a\Big),\
 h_2(x)=K_1\Big(x-a-\frac{b-a}{\eta_1}\Big)
+\delta_1\Big(\frac{b-a}{\eta_1}+\frac{\alpha_1}{\alpha_2}\Big),\\
h_3(x)=\delta_2\Big(x-\frac{\beta_1}{\beta_2}-b\Big),
\end{gather*}
 and
  $$
K^F:=\int_{a}^{a+\frac{b-a}{\eta_1}}F(x,h_1(x))dx
 +\int_{a+\frac{b-a}{\eta_1}}^{b-\frac{b-a}{\eta_2}}F(x,h_2(x))dx
+\int_{b-\frac{b-a}{\eta_2}}^{b}F(x,h_3(x))dx.
$$

  In this article, we assume throughout, and without further mention,
that the following condition holds:
\begin{itemize}
\item[(A1)] The impulsive functions $I_j$ have sublinear growth, i.e.,
there exist constants $a_j>0$, $b_j>0$, and
$\gamma_j\in[0,p-1)$ for $j=1,2,\dots ,l$ such that
$$
| I_j(t)| \leq a_j+b_j|t|^{\gamma_j}\quad \text{for very }
 t\in \mathbb{R},\; j=1,2,\dots ,l.
$$
\end{itemize}
Moreover, set
$G^\theta:=\int_{\Omega}\max_{|t|\leq \theta}G(x,t)dt$ for all $\theta>0$, and
$G_\eta:=\inf_{\Omega\times [0,\eta]}G$ for all $\eta>0$. If $g$
is sign-changing, then clearly, $G^\theta\geq 0$ and $G_\eta\leq 0$.

 A special case of our main results is the following theorem, whose proof we delay
  until the end of the paper.

 \begin{theorem}\label{t1.1}
Assume that $C'_2:=\frac{1}{p}-\sum_{j=1}^{l}\frac{b_j}{\gamma_j+1}
2^{\frac{\gamma_j+1}{q}}>0$. Let $f:\mathbb{R}\to \mathbb{R}$
 be a non-negative continuous function. Put $F(t)=\int_0^tf(\xi)d\xi$ for each
$t\in \mathbb{R}$.
 Suppose that
 $$
\liminf_{\xi\to 0}\frac{F(\xi)}{\frac{C'_2}{2^{p/q}}\xi^p
-\frac{C'_4}{2^{1/q}}\xi}= \limsup_{\xi\to
  +\infty}\frac{F(\xi)}{\frac{C'_2}{2^{p/q}}\xi^p
-\frac{C'_4}{2^{1/q}}\xi}=0
$$
where
$$
C'_4:=\sum_{j=1}^{l}\Big(a_j2^{1/q}+\frac{b_j}{\gamma_j+1}
2^{\frac{\gamma_j+1}{q}}\Big).
$$
 Then, there is $\lambda^*>0$ such that for each
 $\lambda>\lambda^*$ and for every $L^1$-Carath\'eodory
function $g:[0,1]\times \mathbb{R}\to \mathbb{R}$
satisfying the condition
$$
\limsup_{|t|\to\infty}\frac{\sup_{x\in [0,1]}
\int_{0}^{t}g(x,s)ds}{\frac{C'_2}{2^{p/q}}t^p
-\frac{C'_4}{2^{1/q}}t}<+\infty,
$$
there exists
$\delta^{*}_{\lambda, g}>0$ such that, for each
$\mu\in[0,\delta^{*}_{\lambda, g}[$, the problem
\begin{gather*}
-(\Phi_{p}(u'(x)))'+\Phi_{p}(u'(x))=\lambda f(u(x))+\mu g(x,u(x))\quad
\text{a.e. }x\in (0,1),\\
 u'(0^{+})-u(0)=0,\quad u'(1^{-})+u(1)=0
\end{gather*}
with the impulsive conditions
$$
\Delta(\rho(x_j) \Phi_{p}(u'(x_j)))=I_j(u(x_j)),\quad j=1,2,\dots ,l
$$
admits at least three weak solutions.
\end{theorem}

  We need the following proposition in the proof our main result.

\begin{proposition}\label{p1}
Let $T:X\to X^{*}$ be the operator defined by
\begin{align*}
T(u)v&=\int_a^b \rho(x)\Phi_{p}(u'(x))h'(x)dx
+\int_a^b s(x)\Phi_{p}(u(x))h(x)dx \\
&\quad +\rho(a)\Phi_{p}\Big(\frac{\alpha_2u(a)}{\alpha_1}\Big)h(a)
 +\rho(b)\Phi_{p}\Big(\frac{\beta_2u(b)}{\beta1}\Big)h(b)\\
&\quad +\sum_{j=1}^{l}I_j(u(x_j))v(x_j)
\end{align*}
for every $u,h\in X$. Then $T$ admits a continuous inverse on
$X^{*}$.
\end{proposition}

\begin{proof}
For any $u\in X\setminus\{0\}$,
\begin{align*}
&\lim_{\|u\|\to\infty}\frac{\langle T(u), u\rangle}{\|u\|} \\
&= \lim_{\|u\|\to\infty}\Big(\frac{\int_a^b
\rho(x)\Phi_{p}(u'(x))u'(x)dx +\int_a^b
s(x)\Phi_{p}(u(x))u(x)dx}{\|u\|}\\
&\quad + \frac{\rho(a)\Phi_{p}
\Big(\frac{\alpha_2u(a)}{\alpha_1}\Big)u(a)
+\rho(b)\Phi_{p}\Big(\frac{\beta_2u(b)}{\beta1}\Big)u(b)
+\sum_{j=1}^{l}I_j(u(x_j))u(x_j)}{\|u\|}\Big)
\\
&= \lim_{\|u\|\to\infty}\Big(\frac{\int_a^b
 \rho(x)|u'(x)|^pdx +\int_a^b s(x)|u(x)|^pdx}{\|u\|}\\
&\quad + \frac{\rho(a)\Phi_{p}\Big(\frac{\alpha_2u(a)}{\alpha_1}\Big)u(a)
 +\rho(b)\Phi_{p}\Big(\frac{\beta_2u(b)}{\beta1}\Big)u(b)
 +\sum_{j=1}^{l}I_j(u(x_j))u(x_j)}{\|u\|}\Big)\\
&= \lim_{\|u\|\to\infty}\frac{\|u\|^p+\rho(a)\Phi_{p}
 \Big(\frac{\alpha_2u(a)}{\alpha_1}\Big)u(a)
+\rho(b)\Phi_{p}\Big(\frac{\beta_2u(b)}{\beta1}\Big)u(b)}{\|u\|}\\
&\quad + \frac{\sum_{j=1}^{l}I_j(u(x_j))u(x_j)}{\|u\|}
=\infty.
\end{align*}
Thus, the map $T$ is coercive.

For any $u\in X$ and $v\in X$, we have
\begin{align*}
&\langle T(u)-T(v), u-v\rangle \\
&=\int_{a}^{b}\Big(\rho(x)(\Phi_{p}(u'(x))-\Phi_{p}(v'(x)))(u'(x)-v'(x))\\
&\quad +s(x)(\Phi_{p}(u(x))-\Phi_{p}(u(x))) (u(x)-v(x))\Big)dx \\
&\quad +\rho(a)(\Phi_{p}\Big(\frac{\alpha_2u(a)}{\alpha_1}\Big)-
\Phi_{p}\Big(\frac{\alpha_2v(a)}{\alpha_1}\Big))(u(a)-v(a))
 +\rho(b)(\Phi_{p}\Big(\frac{\beta_2u(b)}{\beta_1}\Big)\\
&\quad - \Phi_{p}\Big(\frac{\beta_2v(b)}{\beta_1}\Big))(u(b)-v(b))
 +\sum_{j=1}^{l}(I_j(u(x_j))-I_j(v(x_j)))(u(x_j)-v(x_j)).
\end{align*}
Hence, from our assumptions on the data, we have
\begin{align*}
\langle T(u)-T(v), u-v\rangle
&\geq\int_{a}^{b}\Big(\rho(x)(\Phi_{p}(u'(x))-\Phi_{p}(v'(x)))(u'(x)-v'(x))\\
&\quad +s(x)(\Phi_{p}(u(x))-\Phi_{p}(u(x))) (u(x)-v(x))\Big)dx.
\end{align*}
Now, taking into account
\cite[(2.)]{S}, there exist $c_p,\, d_p>0$ such that
\begin{equation}\label{e4}
\begin{aligned}
&\langle T(u)-T(v), u-v\rangle \\
&\geq \begin{cases}
c_p\int_{a}^{b}\Big(\rho(x)|u'(x)-v'(x)|^p+s(x)|u(x)-v(x)|^p\Big)
dx &\text{if } p\geq 2,\\[4pt]
d_p\int_{a}^{b}\Big(\frac{\rho(x)|u'(x)-v'(x))|^2}{(|u'(x)|+|v'(x)|)^{2-p}}+
\frac{s(x)|u(x)-v(x))|^2}{(|u(x)|+|v(x)|)^{2-p}}\Big)dx &\text{if } 1<p<2.
\end{cases}
\end{aligned}
\end{equation}
At this point, if $p\geq 2$, then it follows that
$$
\langle T(u)-T(v), u-v\rangle\geq c_p\|u-v\|^p,
$$
so $T$ is uniformly monotone. By \cite[Theorem 26.A (d)]{Z},
$T^{-1}$ exists and is continuous on $X^*$. On the other hand, if
$1<p<2$, by H\"older's inequality, we obtain
\begin{equation}\label{e5}
\begin{aligned}
&\int_a^bs(x)|u(x)-v(x)|^pdx \\
&\leq \Big(\int_a^b\frac{s(x)|u(x)-v(x)|^2}{(|u(x)|+|v(x)|)^{2-p}}dx\Big)^{p/2}
\Big(\int_a^bs(x)(|u(x)|+|v(x)|)^pdx\Big)^{(2-p)/2}\\
&\leq \Big(\int_a^b\frac{s(x)|u(x)-v(x)|^2}{(|u(x)|+|v(x)|)^{2-p}}dx\Big)^{p/2}
 2^{\frac{(p-1)(2-p)}{2}}\Big(\int_a^bs(x)(|u(x)|^p+|v(x)|^p)dx
 \Big)^{\frac{2-p}{2}}\\
&\leq 2^{\frac{(p-1)(2-p)}{2}}
\Big(\int_a^b\frac{s(x)|u(x)-v(x)|^2}{(|u(x)|+|v(x)|)^{2-p}}dx\Big)^{p/2}
\big(\|u\|+\|v\|\big)^{(2-p)p/2}.
\end{aligned}
\end{equation}
Similarly, one has
\begin{equation}\label{e6}
\begin{aligned}
&\int_a^b\rho(x)|u'(x)-v'(x)|^pdx \\
&\leq 2^{\frac{(p-1)(2-p)}{2}}
\Big(\int_a^b\frac{\rho(x)|u'(x)-v'(x)|^2}{(|u'(x)|+|v'(x)|)^{2-p}}dx\Big)^{p/2}
\big(\|u\|+\|v\|\big)^{(2-p)p/2}.
\end{aligned}
\end{equation}
Then, relation
\eqref{e4} together with \eqref{e5} and \eqref{e6}, yields
\begin{align*}
&\langle T(u)-T(v), u-v\rangle\\
&\geq \frac{2^{\frac{(p-1)(2-p)}{2}}d_p}{(\|u\|+\|v\|)^{2-p}}
\Big(\Big(\int_a^b\rho(x)|u'(x)-v'(x)|^pdx\Big)^{2/p}\\
&\quad + \Big(\int_a^bs(x)|u(x)-v(x)|^pdx\Big)^{2/p}\Big) \\
&\geq \frac{2^{p-2}d_p}{(\|u\|+\|v\|)^{2-p}}
\Big(\int_a^b\rho(x)|u'(x)-v'(x)|^pdx
+\int_a^bs(x)|u(x)-v(x)|^pdx\Big)^{2/p}\\
&= 2^{p-2}d_p\frac{\|u-v\|^2}{(\|u\|+\|v\|)^{2-p}}.
\end{align*}
Thus, $T$ is strictly monotone. By
\cite[Theorem 26.A (d)]{Z}, $T^{-1}$ exists and is bounded.
Moreover, given $g_1, g_2\in X^{*}$, by the inequality
$$
\langle T(u)-T(v), u-v\rangle\geq
2^{p-2}d_p\frac{\|u-v\|^2}{\left(\|u\|+\|v\|\right)^{2-p}},
$$
choosing $u=T^{-1}(g_1)$ and $v=T^{-1}(g_2)$ we have
$$
\|T^{-1}(g_1)-T^{-1}(g_2)\|
\leq\frac{1}{2^{p-2}d_p}(\|T^{-1}(g_1)\|+\|T^{-1}(g_2)\|)^{2-p}\|g_1-g_2\|_{X^*}.
$$
So $T^{-1}$ is locally Lipschitz continuous and hence continuous.
This completes the proof.
\end{proof}

\section{Main results}

To introduce our result, we fix
three constants $\theta>0$, $\delta_1$ and $\delta_2$ such that
 $$
\frac{K_4}{K^F}<
\frac{\frac{C_2}{M^p}\theta^p-\frac{C_4}{M}\theta}{\int_{a}^{b}\sup_{|
t|\leq \theta}F(x,t)dx}
$$
and taking
$$
\lambda\in\Lambda:=\big]\frac{K_4}{K^F},\
\frac{\frac{C_2}{M^p}\theta^p-\frac{C_4}{M}\theta}{\int_{a}^{b}\sup_{|
t|\leq \theta}F(x,t)dx}\big[,
$$
we set
\begin{equation}\label{e7}
\delta_{\lambda,
g}:=\min\Big\{\frac{\frac{C_2}{M^p}\theta^p-\frac{C_4}{M}\theta-\lambda
\int_{a}^{b}\sup_{|t|\leq \theta}F(x,t)dx}{G^\theta},\
\frac{K_4-\lambda K^F}{(b-a)G_\eta}\Big\}
\end{equation}
and
\begin{equation}\label{e8}
\overline{\delta}_{\lambda,
g}:=\min\Big\{\delta_{\lambda, g},\
\frac{1}{\max\{0,(b-a)\limsup_{|t|\to\infty}\frac{\sup_{x\in
[a,b]}G(x,t)}{\frac{C_2}{M^p}t^p-\frac{C_4}{M}t}\}}\Big\},
\end{equation}
where we define $\frac{r}{0}=+\infty$, so that, for instance,
$\overline{\delta}_{\lambda, g}=+\infty$ when
\[
\limsup_{|t|\to\infty}\frac{\sup_{x\in
[a,b]}G(x,t)}{\frac{C_2}{M^p}t^p-\frac{C_4}{M}t}\leq 0,
\]
and $G_\eta=G^\theta=0$.

Now, we formulate our main result.
\begin{theorem}\label{thm3}
Assume that $C_2>0$ and there exist constants $\delta_1$ and $\delta_2$,
and positive constants $\theta$, $\eta_1$ and
$\eta_2$ with $\delta_1^{2}+\delta_2^{2}\neq 0$,
$\eta_1+\eta_2<\eta_1\eta_2$ and
\[
K_2^{1/p}>\frac{\theta}{M}>(\frac{C_4}{C_1})^{1/(p-1)}
\]
 such that
\begin{itemize}
\item[(A2)] $\frac{\int_{a}^{b}\sup_{|t|\leq \theta}F(x,t)dx}
{\frac{C_2}{M^p}\theta^p-\frac{C_4}{M}\theta}<\frac{K^F}{K_4}$;

\item[(A3)] $\limsup_{|t|\to +\infty}\frac{\sup_{x\in[a,b]}
F(x,t)}{\frac{C_2}{M^p}t^p-\frac{C_4}{M}t}\leq0$.
\end{itemize}
Then, for each
$$
\lambda\in\Lambda:=\big]\frac{K_4}{K^F},\,
\frac{\frac{C_2}{M^p}\theta^p-\frac{C_4}{M}\theta}{\int_{a}^{b}\sup_{|
t|\leq \theta}F(x,t)dx}\big[
$$
and for every $L^1$-Carat\'eodory function $g:[a,b]\times \mathbb{R}\to
\mathbb{R}$ satisfying the condition
$$
\limsup_{|t|\to\infty}\frac{\sup_{x\in [a,b]}G(x,t)}{\frac{C_2}{M^p}t^p
-\frac{C_4}{M}t}<+\infty,
$$
there exists $\overline{\delta}_{\lambda,
g}>0$ given by \eqref{e8} such that, for each
$\mu\in[0,\overline{\delta}_{\lambda, g}[$, the problem \eqref{e1}-\eqref{e2}
admits at least three distinct weak solutions in $X$.
\end{theorem}

\begin{proof}
To apply Theorem \ref{t1} to our problem, we introduce the functionals
 $\Phi,  \Psi:X \to \mathbb{R} $ for each $u\in X$, as follows
\begin{gather*}
\Phi(u)=\frac{1}{p}\|u\|^p
+\sum_{j=1}^{l}\int_{0}^{u(x_j)}I_j(t)dt
+\frac{\rho(a)\alpha_2^{p-1}}{p\alpha_1^{p-1}}|u(a)|^p
+\frac{\rho(b)\beta_2^{p-1}}{p\beta_1^{p-1}}|u(b)|^p,,
\\
\Psi(u)=\int_{a}^{b}[F(x,u(x))+\frac{\mu}{\lambda}G(x,u(x))]dx.
\end{gather*}
Now we show that the functionals $\Phi$ and $\Psi$ satisfy the
required conditions. It is well known that $\Psi$ is a
differentiable functional whose differential at the point $u\in X$
is
$$
\Psi'(u)(v)=\int_{a}^{b}[f(x,u(x))+\frac{\mu}{\lambda}g(x,u(x))]v(x)dx,
$$
for every $v\in X$, as well as, is sequentially weakly upper
semicontinuous. Furthermore, $\Psi':X \to X^{*}$ is a compact
operator. Indeed, it is enough to show that $\Psi'$ is strongly
continuous on $X$. For this, for fixed $u\in X$, let $u_{n}\to u$
weakly in $X$ as $n\to +\infty$.  Then we have $u_{n}$ converges
uniformly to $u$ on $[a,b]$ as $n\to +\infty$ (see \cite{Z}).
Since $f$ and $g$ are $L^1$-Carath\'eodory functions, $f$
and $g$ are continuous in $\mathbb{R}$ for every $x\in [a,b]$. So
$f(x,u_{n})+\frac{\mu}{\lambda}g(x,u_{n})\to f(x,u)+\frac{\mu}{\lambda}g(x,u)$
strongly as $n\to +\infty$, from
which follows $\Psi'(u_{n})\to \Psi'(u)$ strongly as $n\to +\infty$.
Thus we have established that $\Psi'$ is strongly
continuous on $X$, which implies that $\Psi'$ is a compact
operator by Proposition 26.2 of \cite{Z}. Moreover, $\Phi$ is
continuously differentiable and whose differential at the point
$u\in X$ is
\begin{align*}
\Phi'(u)v
&=\int_a^b \rho(x)\Phi_{p}(u'(x))v'(x)dx
+\int_a^b s(x)\Phi_{p}(u(x))v(x)dx\\
&\quad +\rho(a)\Phi_{p}\Big(\frac{\alpha_2u(a)}{\alpha_1}\Big)v(a)
 +\rho(b)\Phi_{p}\Big(\frac{\beta_2u(b)}{\beta1}\Big)v(b)
 +\sum_{j=1}^{l}I_j(u(x_j))v(x_j)
\end{align*}
for every $v\in X$, while Proposition \ref{p1} gives that
$\Phi'$ admits a continuous inverse on $X^{*}$. Furthermore, $\Phi$ is
sequentially weakly lower semicontinuous. Indeed, let for fixed
$u\in X$, assume $u_{n}\to u$ weakly in $X$ as $n\to +\infty$. The
continuity and convexity of $\|u\|^p$ imply $\|u\|^p$ is
sequentially weakly lower semicontinuous, which combining the
continuity of $I_j$ for $j=1,\dots ,l$ yields that
\begin{align*}
&\lim_{n\to +\infty}\Big(\frac{1}{p}\|u_{n}\|^p
+\sum_{j=1}^{l}\int_{0}^{u_{n}(x_j)}I_j(t)dt
+\frac{\rho(a)\alpha_2^{p-1}}{p\alpha_1^{p-1}}|u_{n}(a)|^p
+\frac{\rho(b)\beta_2^{p-1}}{p\beta_1^{p-1}}|u_{n}(b)|^p\Big)\\
&\geq \frac{1}{p}\|u\|^p +\sum_{j=1}^{l}\int_{0}^{u(x_j)}I_j(t)dt
+\frac{\rho(a)\alpha_2^{p-1}}{p\alpha_1^{p-1}}|u(a)|^p
+\frac{\rho(b)\beta_2^{p-1}}{p\beta_1^{p-1}}|u(b)|^p,
\end{align*}
namely
$$
\liminf_{n\to +\infty}\Phi(u_n)\geq\Phi(u)
$$
which means $\Phi$ is sequentially weakly lower semicontinuous.
Clearly, the weak solutions of the problem \eqref{e1} are exactly
the solutions of the equation $\Phi'(u)-\lambda\Psi'(u)=0$.
Put $r=\frac{C_2}{M^p}\theta^p-\frac{C_4}{M}\theta$ and
\begin{equation}\label{e9}
w(x)= \begin{cases}
h_1(x), &x\in[a,a+\frac{b-a}{\eta_1}),\\
h_2(x), &x\in[a+\frac{b-a}{\eta_1},b-\frac{b-a}{\eta_1}],\\
h_3(x), &x\in(a+\frac{b-a}{\eta_1},b].
\end{cases}
\end{equation}
It is easy to see that $w\in X$ and, in particular,
in view of
$$
\int_{a}^{b}\rho(x)|w'(x)|^pdx=K_2\quad \text{and}\quad
0\leq\int_{a}^{b}s(x)|w(x)|^pdx\leq K_3^p\int_{a}^{b}s(x)dx,
$$
we have
$$
\|w\|\leq \Big(K_2 +K_3^p\int_{a}^{b}s(x)dx\Big)^{1/p},
$$
which in conjunction with the inequality
\begin{equation}\label{e10}
\Phi(u)\leq (C_1+C_3)\|u\|^p+C_4\|u\|
\end{equation}
for all $u\in X$ (see\cite{BD3}), yields
\begin{equation}\label{e11}
 \Phi(w)\leq K_4.
\end{equation}
Moreover, by the same reasoning as given given in the proof
\cite[Lemma 5]{BD3}, using \eqref{e11},
 from the condition
 $$
K_2^{1/p}>\frac{\theta}{M}>\big(\frac{C_4}{C_1}\big)^{1/(p-1)}
$$
one  has  $0<r<\Phi(w)$.
Taking \eqref{e3} into account, by the same arguing as
 given in the proof \cite[Lemma 5]{BD3} we have
$$
 \Phi^{-1}(]-\infty,r])\subseteq \left\{ u\in X; \|u\|_\infty\leq
 \theta\right\},
$$
and it follows that
\begin{align*}
\sup_{u\in\Phi^{-1}(]-\infty,r])}\Psi(u)
&= \sup_{u\in\Phi^{-1}(]-\infty,r])}\int_{a}^{b}[F(x,u(x))
 +\frac{\mu}{\lambda}G(x,u(x))]dx\\
&\leq \int_{a}^{b}
 \sup_{|t|\leq \theta}F(x,t)dx+\frac{\mu}{\lambda}G^{\theta}.
\end{align*}
On the other hand, from the definition of $\Psi$, we infer
\begin{align*}
\Psi(w)&= \int_{a}^{b}F(x,w(x))dx+\frac{\mu}{\lambda}\int_{a}^{b}G(x,w(x))dx\\
&= K^F+\frac{\mu}{\lambda}\int_{a}^{b}G(x,w(x))dx\\
&\geq K^F+(b-a)\frac{\mu}{\lambda}\inf_{[a,b]]\times[0,\eta]}G\\
&= K^F +(b-a)\frac{\mu}{\lambda}G_{\eta}.
\end{align*}
 Therefore, owing to Assumption (A2) and \eqref{e11}, we have
\begin{equation} \label{e12}
\begin{aligned}
\frac{\sup_{u\in\Phi^{-1}(]-\infty,r])}\Psi(u)}{r}
&= \frac{\sup_{u\in\Phi^{-1}(]-\infty,r])}\int_{a}^{b}[F(x,u(x))
+\frac{\mu}{\lambda}G(x,u(x))]dx}{r}\\
&\leq   \frac{\int_{a}^{b}
 \sup_{|t|\leq \theta}F(x,t)dx+\frac{\mu}{\lambda}G^{\theta}}{\frac{C_2}{M^p}
\theta^p-\frac{C_4}{M}\theta}
 \end{aligned}
\end{equation}
 and
\begin{equation} \label{e13}
\begin{aligned}
\frac{\Psi(w)} {\Phi(w)}
&\geq \frac{K^F+\frac{\mu}{\lambda}\int_{a}^{b}G(x,w(x))dx}{K_4}\\
&\geq \frac{\int_{a}^{b}F(x,w(x))dx+(b-a)\frac{\mu}{\lambda}G_{\eta}}
{K_4}.
\end{aligned}
\end{equation}
 Since $\mu<\delta_{\lambda, g}$, one has
$$
\mu<\frac{\frac{C_2}{M^p}\theta^p-\frac{C_4}{M}\theta-\lambda
\int_{a}^{b}\sup_{|t|\leq \theta}F(x,t)dx}{G^\theta},
$$
which means
$$
\frac{\int_{a}^{b}
 \sup_{|t|\leq \theta}F(x,t)dx+\frac{\mu}{\lambda}G^{\theta}}
 {\frac{C_2}{M^p}\theta^p-\frac{C_4}{M}\theta}<\frac{1}{\lambda}.
$$
Furthermore,
 $$
\mu<\frac{K_4-\lambda K^F}{(b-a)G_\eta},
$$
and this means
$$
\frac{K^F+(b-a)\frac{\mu}{\lambda} G_{\eta}}{K_4}>\frac{1}{\lambda}.
$$
Then
\begin{equation}\label{e14}
\frac{\int_{a}^{b}
 \sup_{|t|\leq \theta}F(x,t)dx
+\frac{\mu}{\lambda}G^{\theta}}{\frac{C_2}{M^p}\theta^p-\frac{C_4}{M}\theta}
 <\frac{1}{\lambda}<\frac{K^F+(b-a)\frac{\mu}{\lambda}
G_{\eta}}{K_4}.
\end{equation}
Hence from \eqref{e12}-\eqref{e14},
the condition (a1) of Theorem \ref{t1} is verified.

Finally, since $\mu<\overline{\delta}_{\lambda, g}$, we
can fix $l>0$ such that
$$
\limsup_{|t|\to\infty}\frac{\sup_{x\in
[a,b]}G(x,t)}{\frac{C_2}{M^p}t^p-\frac{C_4}{M}t}<l
$$
and $\mu l<M^p$. Therefore, there exists a function $h\in L^1([a,b])$
such that
$$
G(x,t)\leq l (\frac{C_2}{M^p}t^p-\frac{C_4}{M}t)+h(x) \quad
 \text{for all $x\in [a,b]$ and for all } t\in \mathbb{R}.
$$
Now, fix $0<\epsilon<\frac{M^p}{\lambda}-\frac{\mu l}{\lambda}$. From
(A3) there is a function  $h_{\epsilon}\in L^1([a,b])$ such that
$$
F(x,t)\leq \epsilon (\frac{C_2}{M^p}t^p-\frac{C_4}{M}t)+h_{\epsilon}(x) \quad
 \text{for all} \ x\in [a,b]
\text{and for all } t\in \mathbb{R}.
$$
Using \eqref{e10}, it follows that, for each $u\in X$,
\begin{align*}
&\Phi(u)-\lambda \Psi(u)\\
&=  \frac{1}{p}\|u\|^p +\sum_{j=1}^{l}\int_{0}^{u(x_j)}I_j(t)dt
+\frac{\rho(a)\alpha_2^{p-1}}{p\alpha_1^{p-1}}|u(a)|^p
+\frac{\rho(b)\beta_2^{p-1}}{p\beta_1^{p-1}}|u(b)|^p\\
&\quad -\lambda\int_{\Omega}[F(x,u(x))+\frac{\mu}{\lambda}G(x,u(x))]dx\\
&\geq (C_2-\lambda\epsilon\frac{C_2}{M^p}-\mu
l\frac{C_2}{M^p})\|u\|^p-(C_4+\lambda\epsilon\frac{C_4}{M}+\mu
l\frac{C_4}{M})\|u\|
-\lambda\|h_{\epsilon}\|_1-\mu\|h\|_1,
\end{align*}
and thus
$$
\lim_{\| u\|\to+\infty} (\Phi(u)-\lambda \Psi(u))=+\infty,
$$
which means the functional $\Phi-\lambda \Psi$ is coercive, and the
condition (a2) of Theorem \ref{t1} is satisfied. Since, from \eqref{e12} and
\eqref{e14},
$$
\lambda\in\big]\frac{\Phi(w)}{\Psi(w)},
\frac{r}{\sup_{\Phi(x)\leq r}\Psi(x)}\big[\,,
$$
Theorem \ref{t1}, with $\overline{x}=w$, assures the existence of three critical
points for the functional $\Phi-\lambda\Psi$, and the proof is complete.
\end{proof}

 Here, we exhibit an example whose construction is motivated
by \cite[Example 1]{BD3}, in which the hypotheses of Theorem \ref{thm3}
 are satisfied.

\begin{example}\label{examp1} \rm
Consider the problem
 \begin{equation}\label{e15}
\begin{gathered}
-((x+3)|u'(x)|u'(x))'+(2x+2)|u(x)|u(x)=\lambda f(x,u(x))+\mu
g(x,u(x)) \\
\text{a.e. }x\in (1,2),\\
u'(1^{+})-u(1)=0,\quad u'(2^{-})+u(2)=0,\\
\Delta((x_1+3)|u'(x_1)|u'(x_1)=-(\frac{1}{12}+\frac{5}{24}|u(x_1)|^{3/2}),\quad
 x_1\in(1,2)
\end{gathered}
\end{equation}
 where
$$
f(x,t)=\begin{cases}
x(3t^2-2t) &\text{if } (x,t)\in[1,2]\times(-\infty,1],\\
xt &\text{if }  (x,t)\in[1,2]\times[1,+\infty).
\end{cases}
$$
  $g(x,t)=e^{x-t}t^3$ for all $x\in[1,2]$ and $t\in\mathbb{R}$, and
$I_1(u(x_1))=-(\frac{1}{12}+\frac{5}{24}|u(x_1)|^{3/2})$
satisfying the condition
$(|v(x_1)|^{3/2}-|u(x_1)|^{3/2})(u(x_1)-v(x_1))\geq 0$ for all
$u,v\in W^{1,3}([1,2])$. A direct calculation shows
$$
F(x,t)=\begin{cases}
x(t^3-t^2) &\text{if }(x,t)\in[1,2]\times(-\infty,1],\\
\frac{x}{2}(t^2-1) &\text{if }  (x,t)\in[1,2]\times[1,+\infty).
\end{cases}
$$
In view of Lemma \ref{lem1}, $M=1$. Choose $\eta_1=\eta_2=4$,
$\delta_1=1$, $\delta_2=-1$ and $\theta=1$. We observe that
$C_1=3$, $C_2=\frac{1}{4}$, $C_3=5/12$, $C_4=1/6$,
$K_1=0$, $K_2=9/4$, $K_3=5/4$,
$K_4\approx\frac{1}{12\times 2.011\times 10^{-3}}$,
$K^F\approx 3.125\times 10^{-1}$ and
$\int_1^{2}\sup_{| t|\leq \theta}F(x,t)dx\leq 0$. So, since
$$
\limsup_{|t|\to +\infty }\frac{\sup_{x\in[1,2]}F(x,t)}{\frac{t^{3}}{4}-\frac{t}{6}}=0,$$
we see that all assumptions of Theorem \ref{thm3} are satisfied. Hence,
for each $\lambda>\frac{\frac{1}{12\times 2.011\times
10^{-3}}}{3.125\times 10^{-1}}$ and every $\mu\geq 0$
(since $g_\infty=0$), the problem \eqref{e15} has at least three
solutions in $W^{1,3}([1,2])$.
\end{example}

The following example illustrates the
result in Theorem \ref{t1.1}.

\begin{example}\label{examp2} \rm
Consider the problem
 \begin{equation}\label{e16}
\begin{gathered}
-(|u'(x)|u'(x))'+|u(x)|u(x)=\lambda e^{-u(x)}u^{2}(x)(3-u(x))+\mu
e^{x-u(x)^{+}}(u(x)^{+})^\gamma,\\ 
\text{a.e. } x\in(0,1)\\
u'(0^{+})-u(0)=0,\quad u'(1^{-})+u(1)=0,\\
\Delta((x_1+3)|u'(x_1)|u'(x_1)=-(\frac{1}{12}+\frac{5}{24}|u(x_1)|^{3/2}),\quad
x_1\in(0,1)
\end{gathered}
\end{equation}
where $u^{+}=\max\{u,0\}$,
$I_1(u(x_1))=-(\frac{1}{12}+\frac{5}{24}|u(x_1)|^{3/2})$
satisfying the condition
$(|v(x_1)|^{3/2}-|u(x_1)|^{3/2})(u(x_1)-v(x_1))\geq 0$ for all
$u,v\in W^{1,3}([1,2])$ and $\gamma$ is a positive real number. It
is obvious that $C'_2=1/4$ and $C'_4=1/6$. Also a
direct calculation shows $F(t)=e^{-t}t^3$ for all
$t\in\mathbb{R}$. So, one has
$$
\liminf_{\xi\to 0}\frac{F(\xi)}{\frac{1}{16}\xi^{3}-\frac{1}{6\sqrt[3]{4}}\xi}=
\limsup_{\xi\to  +\infty}\frac{F(\xi)}{\frac{1}{16}\xi^{3}
-\frac{1}{6\sqrt[3]{4}}\xi}=0.
$$
Hence, using Theorem \ref{t1.1}, there is $\lambda^*>0$ such that, since
$g_\infty=0$, for each
 $\lambda>\lambda^*$ and $\mu\geq 0$, the problem
\eqref{e16}  admits at least three solutions.
\end{example}

\begin{proof}[Proof of Theorem \ref{t1.1}]
 Fix $\lambda>\lambda^*:=\frac{K'_4}{K'^{F}}$ for some
constants $\delta_1$ and $\delta_2$, and positive constants
$\eta_1$ and $\eta_2$ with $\delta_1^{2}+\delta_2^{2}\neq 0$,
$\eta_1+\eta_2<\eta_1\eta_2$ where
\begin{align*}
K_4'&:=(C'_1+C'_3)\Big(\frac{|\delta_1|^p}{4}
+\frac{5^p}{2^{p+1}}(|\delta_1|+|\delta_2|)^p+\frac{|\delta_2|^p}{4}
+(\frac{5}{4}\max\{|\delta_1|,|\delta_2|\})^p\Big) \\
&\quad +C'_4\Big(\frac{|\delta_1|^p}{4}
 +\frac{5^p}{2^{p+1}}(|\delta_1|+|\delta_2|)^p+\frac{|\delta_2|^p}{4}
+(\frac{5}{4}\max\{|\delta_1|,|\delta_2|\})^p\Big)^{1/p}
\end{align*}
where $C'_1:=\frac{2^p}{p}$ and
$C'_3=\frac{1}{p}+\sum_{j=1}^{l}\frac{b_j}{\gamma_j+1}
2^{\frac{\gamma_j+1}{q}}$,
and
\begin{align*}
K'^F&:=\int_{0}^{1/4}F(|\delta_1|(x+1))dx
 +\int_{1/4}^{3/4}F\Big(-\frac{5}{2}(|\delta_1|+|\delta_2|)(x-\frac{1}{4})
+\frac{5|\delta_1|}{4}\Big)dx\\
&\quad +\int_{3/4}^1F(|\delta_2|(x-2))dx.
\end{align*}
 Recalling that
$$
\liminf_{\xi\to 0}\frac{F(\xi)}{\frac{C'_2}{2^{p/q}}\xi^p
-\frac{C'_4}{2^{1/q}}\xi}=0,
$$
there is a sequence $\{\theta_n\}\subset ]0,+\infty[$ such that
$\lim_{n\to \infty} \theta_{n}=0$ and
$$
\lim _{n\to \infty}\frac{\sup_{|\xi| \leq
\theta_{n}}F(\xi)}{\frac{C'_2}{2^{p/q}}\theta_n^p
-\frac{C'_4}{2^{1/q}}\theta_n}=0.
$$
 Indeed, one has
$$
\lim _{n\to \infty}\frac{\sup_{|\xi| \leq
\theta_n}F(\xi)}{\frac{C'_2}{2^{p/q}}\theta_n^p
-\frac{C'_4}{2^{1/q}}\theta_n}
=\lim _{n\to\infty}\frac{F(\xi_{\theta_n})}{\frac{C_2}{2^{p/q}}\xi_{\theta_n}^p
 -\frac{C'_4}{2^{1/q}}\xi_{\theta_n}}
\frac{\frac{C'_2}{2^{p/q}}\xi_{\theta_n}^p-\frac{C'_4}{2^{1/q}}
\xi_{\theta_n}}{\frac{C'_2}{2^{p/q}}\theta_n^p
-\frac{C'_4}{2^{1/q}}\theta_n}=0,
$$
where $F(\xi_{\theta_n})=\sup_{|\xi| \leq \theta_{n}}F(\xi)$.
 Hence, there exists $\overline{\theta}>0$ such that
$$
\frac{\sup_{|\xi| \leq\overline{\theta}}F(\xi)}{\frac{C'_2}{2^{p/q}}
\overline{\theta}^p-\frac{C'_4}{2^{1/q}}
\overline{\theta}}< \min\big\{\frac{K'^{F}}{ (b-a)K'_4};\
\frac{1}{ (b-a)\lambda}\big\}
$$
and
\[
\Big(\frac{|\delta_1|^p}{4}+\frac{5^p}{2^{p+1}}
(|\delta_1|+|\delta_2|)^p+\frac{|\delta_2|^p}{4}\Big)^{1/p}
>\frac{\overline{\theta}}{2^{1/q}}>(\frac{C'_4}{C'_1})^{1/(p-1)}.
\]
The conclusion follows by using Theorem \ref{thm3} with
$\eta_1=\eta_2=4$.
\end{proof}

 \begin{remark} \rm
The methods used here can be applied studying discrete boundary
value problems as in \cite{CMo}, and also non-smooth variational
problems as in \cite{MMM}.
 \end{remark}

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\end{document}