\documentclass{PM}

\address[afrouzi@umz.ac.ir]{Ghasem A. Afrouzi,
Department of Mathematics, Faculty of Mathematical Sciences,
University of Mazandaran, Babolsar, Iran}

\address[a.hadjian@umz.ac.ir]{Armin Hadjian,
Department of Mathematics, Faculty of Mathematical Sciences,
University of Mazandaran, Babolsar, Iran}

\address[vicentiu.radulescu@imar.ro]{Vicen\c tiu D. R\u adulescu,
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}



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\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\newtheorem{remark}[theorem]{Remark}


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\title[Variational analysis for Dirichlet impulsive differential equations]
{Variational analysis for Dirichlet impulsive differential equations
with oscillatory nonlinearity}

\author[G.A. Afrouzi, A. Hadjian, V.D. R\u adulescu]
{Ghasem A. Afrouzi, Armin Hadjian, Vicen\c tiu D. R\u adulescu\thanks{V.~R\u adulescu acknowledges the support through  Grant CNCS PCE--47/2011.}}

\begin{document}

\maketitle

\begin{abstract}
By using variational methods and critical point theory, we establish
the existence of infinitely many solutions for second-order
impulsive differential equations with Dirichlet boundary conditions,
depending on two real parameters.
\end{abstract}


\begin{classification}
Primary 34B37; Secondary 34B15, 58E05.
\end{classification}

\begin{keywords}
Impulsive differential equations, Dirichlet condition,
variational methods, infinitely many solutions.
\end{keywords}



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

\section{Introduction}
Many dynamical systems describing models in applied sciences have an {\it impulsive} dynamical behaviour due to abrupt
changes at certain instants during the evolution process. The rigorous
mathematical
description of these phenomena leads to {\it impulsive differential equations}; they characterize
various processes of the real world described by models
 that are subject to {\it sudden changes} in
their states.
Essentially, impulsive differential equations correspond to a {\it smooth
evolution}
that may {\it change instantaneously} or even {\it abruptly},
as happens in various applications that describe mechanical or natural phenomena.
These changes correspond to impulses in the smooth system, such as for
example
in the model of a
mechanical clock.
Impulsive differential equations also study models in physics, population
dynamics, ecology, industrial robotics, biotechnology, economics,
optimal control, chaos theory. Associated with this development, a
theory of impulsive differential equations has been given extensive
attention. For an introduction of the basic theory of impulsive
differential equations in $\mathbb{R}^n$ we refer to \cite{BenHenNto},
\cite{LakBaiSim}, and \cite{sape}. Some classical tools have been used to study such
problems in the literature, such as the coincidence degree theory of
Mawhin, the method of upper and lower solutions with the monotone
iterative technique, and some fixed point theorems in cones (see
\cite{ChenTisYuan, Mawhin, QuiLi}). Recently, the existence and
multiplicity of solutions for impulsive boundary value problems by
using variational methods and critical point theory has been
considered in \cite{NieOre, TianGe,
WangShen, WangYang, XiaoNieto}.

For a general second order differential equation ${\mathcal
F}(t,u,u')=0$, one can consider impulses in the {\it position} $u$
and the {\it velocity} $u '$. However, as argued in \cite{NieOre},
it is natural to consider in the motion of spacecraft only
instantaneous impulses depending on the position that result in jump
discontinuities, but with no change in position. The impulses only
on the velocity occurs also in impulsive mechanics, see \cite{pasq}.

The purpose of this paper is to show the variational structure underlying of a class of nonlinear
impulsive differential equations. We take
as a model a Dirichlet problem with impulses. For an excellent overview of the most significant
mathematical methods employed in this paper we refer to Ciarlet \cite{ciarlet}.

\section{Statement of the problem}
The aim of this paper is to study the following nonlinear Dirichlet
boundary-value problem
\begin{equation}\label{e1.1}
\left\{\begin{array}{ll}
-(p(t)u'(t))'+q(t)u(t)=\lambda f(t,u(t))+\mu g(t,u(t)), \quad t\in [0,T],\; t\not= t_j,\\
u(0)=u(T)=0,\\
\Delta u'(t_j)=I_j(u(t_j)),\quad j=1,2,\dots,m,
\end{array}\right.
\end{equation}
where $T>0$, $p \in C^1([0,T],[0,+\infty[)$, $q \in
L^{\infty}([0,T])$ with $\operatorname{ess\,inf}_{t \in[0,T]} q(t)
\geq 0$, $\lambda \in ]0,+\infty[$, $\mu \in [0,+\infty[$,
$f,g:[0,T]\times \mathbb{R} \to \mathbb{R}$ are
$L^1$-Carath\'{e}odory functions,
$0=t_0<t_1<t_2<\dots<t_m<t_{m+1}=T$, $\Delta
u'(t_j)=u'(t^+_j)-u'(t^-_j)= \lim_{t \to t^+_j}u'(t) -\lim_{t \to
t^-_j}u'(t)$, and $I_j: \mathbb{R} \to \mathbb{R}$ are continuous
for every $j=1,2,\dots,m$.

We are motivated by the  paper of Bonanno et al.
\cite{BonaDiHen} in which, using two critical point theorems, the
authors ensured the existence of at least three classical solutions
for the nonlinear Dirichlet boundary-value problem
\begin{equation*}
\left\{\begin{array}{ll}
-u''(t)+a(t)u'(t)+b(t)u(t)=\lambda g(t,u(t)), \quad t\in [0,T],\; t\not= t_j,\\
u(0)=u(T)=0,\\
\Delta u'(t_j)=u'(t^+_j)-u'(t^-_j)=\mu I_j(u(t_j)),\quad
j=1,2,\dots,n,
\end{array}\right.
\end{equation*}
where $\lambda \in ]0,+\infty[$, $\mu \in ]0,+\infty[$,
$g:[0,T]\times \mathbb{R} \to \mathbb{R}$, $a,b \in
L^{\infty}([0,T])$ satisfy the conditions
$\operatorname{ess\,inf}_{t \in[0,T]} a(t)\geq 0$,
$\operatorname{ess\,inf}_{t \in[0,T]}b(t) \geq 0$,
$0=t_0<t_1<t_2<\dots<t_n<t_{n+1}=T$, $\Delta
u'(t_j)=u'(t^+_j)-u'(t^-_j)= \lim_{t \to t^+_j}u'(t) -\lim_{t \to
t^-_j}u'(t)$, and $I_j: \mathbb{R} \to \mathbb{R}$ are continuous
for every $j=1,2,\dots,n$, by choosing $\mu$ in a suitable way and
for every $\lambda$ lying in a precise interval.

Our goal in this paper is to obtain some sufficient conditions to
guarantee that problem \eqref{e1.1} has infinitely many classical
solutions. To this end, we require that the primitive $F$ of $f$
satisfies a suitable oscillatory behavior either at infinity (for
obtaining unbounded solutions) or at the origin (for finding arbitrarily
small solutions), while $G$, the primitive of $g$, has an
appropriate growth (see Theorems \ref{the3.1} and \ref{the3.6}). Our
analysis is mainly based on a general critical point theorem (see
Lemma \ref{lem2.1} below) contained in \cite{BonaMolica}; see also
\cite{Ricceri1}.

We also refer the interested reader to the papers \cite{AfrHad,
AfrHadHei, BonaMolica1, BonaMolicaRad, BonaMolicaRad1, CanMoli,
DagSci, GraHeiKo} and references therein, in which Ricceri's
variational principle and its variants have been successfully used
to obtain the existence of infinitely many solutions for boundary value
problems.

We end this preliminary section with the following theorem, which is a direct
consequence of our main result.

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

\begin{theorem}\label{the1.1}

Let $h:\mathbb{R}\rightarrow\mathbb{R}$ be a nonnegative continuous
and non-zero function. Define $H(t)=\int_0^t h(\xi)d\xi$ for all
$t\in\mathbb{R}$ and assume that
$$
\liminf\limits_{\xi\rightarrow +\infty}\frac{H(\xi)}{\xi^2} <
\frac{6(e^{{3T}/4}-e^{T/4})}{e^T(e^T-1)}\limsup\limits_{\xi\rightarrow
+\infty} \frac{H(\xi)}{\xi^2}.
$$
Then, for each
$$
\lambda\in\Bigg]\frac{e^T(12+T^2)}{3T(e^{{3T}/4}-e^{T/4})\limsup\limits_{\xi\rightarrow
+\infty}\frac{H(\xi)}{\xi^2}},\frac{2}{T(e^T-1)\liminf\limits_{\xi\rightarrow
+\infty}\frac{H(\xi)}{\xi^2}}\Bigg[,
$$
for every arbitrary nonnegative continuous function
$p:\mathbb{R}\rightarrow\mathbb{R}$, whose potential $P(t):=\int_0^t
p(\xi)d\xi$ for all $t\in\mathbb{R}$, satisfies the condition
$$
\limsup_{\xi\rightarrow+\infty}\frac{P(\xi)}{\xi^2}<+\infty,
$$
and for every
$$
\mu\in\Bigg[0,\frac{2}{T(e^T-1)\limsup_{\xi\rightarrow+\infty}
\frac{P(\xi)}{\xi^2}}\bigg(1-\lambda\frac{T(e^T-1)\liminf_{\xi\rightarrow
+\infty}\frac{H(\xi)}{\xi^2}}{2}\bigg)\Bigg[,
$$
the nonlinear problem
\begin{equation*}
\left\{\begin{array}{ll}
-u''(t)+u'(t)+u(t)=\lambda h(u(t))+\mu p(u(t)), \quad t\in [0,T],\; t\not= t_j,\\
u(0)=u(T)=0,\\
\Delta u'(t_j)=I_j(u(t_j)),\quad j=1,2,\dots,m,
\end{array}\right.
\end{equation*}
has a sequence of classical solutions which is unbounded in
$H_0^1(0,T)$.
\end{theorem}

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

\section{Auxiliary results}

We shall prove our results applying the following smooth version of
Theorem 2.1 of \cite{BonaMolica}, which is a more precise version of
Ricceri's variational principle \cite[Theorem 2.5]{Ricceri1}. We point out that Ricceri's variational
principle generalizes the celebrated three critical point theorem of Pucci and Serrin \cite{pucser1, pucser2}
and is an useful result that gives alternatives for the multiplicity of critical points of certain functions
depending on a parameter.

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

\begin{lemma}\label{lem2.1}
Let $X$ be a reflexive real Banach space, let
$\Phi,\Psi:X\rightarrow \mathbb{R}$ be two G\^{a}teaux
differentiable functionals such that $\Phi$ is sequentially weakly
lower semicontinuous, strongly continuous and coercive, and $\Psi$
is sequentially weakly upper semicontinuous. For every
$r>\inf_X\Phi,$ let
$$
\varphi(r):=\inf_{u\in\Phi^{-1}(-\infty,r)}
\frac{\Big{(}\sup_{v\in\Phi^{-1}(-\infty,r)}\Psi(v)\Big{)}-\Psi(u)}{r-\Phi(u)},
$$
$$
\gamma:=\liminf\limits_{r\rightarrow
+\infty}\varphi(r),\qquad\mbox{and}\qquad
\delta:=\liminf\limits_{r\rightarrow(\inf_X\Phi)^+}\varphi(r).
$$
Then the following properties hold:
\begin{enumerate}
\item[$\rm(a)$] For every $r>\inf_X\Phi$ and every $\lambda\in
(0,1/\varphi(r)),$ the restriction of the functional
$$
I_\lambda:=\Phi-\lambda\Psi
$$
to $\Phi^{-1}(-\infty,r)$ admits a global minimum, which is a
critical point (local minimum) of $I_\lambda$ in $X$.
\item[$\rm(b)$] If $\gamma<+\infty,$ then for each
$\lambda\in (0,1/\gamma)$, the following alternative holds: either
\begin{enumerate}
\item[$\rm(b_1)$] $I_\lambda$ possesses a global
minimum, or
\item[$\rm(b_2)$] there is a sequence $\{u_n\}$ of critical points
(local minima) of $I_\lambda$ such that
$$
\lim_{n\rightarrow+\infty}\Phi(u_n)=+\infty.
$$
\end{enumerate}
\item[$\rm(c)$] If $\delta<+\infty,$ then for each
$\lambda\in (0,1/\delta)$, the following alternative holds: either
\begin{enumerate}
\item[$\rm(c_1)$] there is a global minimum of $\Phi$ which is a local minimum of $I_\lambda$, or
\item[$\rm(c_2)$] there is a sequence $\{u_n\}$ of pairwise distinct critical points (local minima)
of $I_\lambda$ that converges weakly to a global minimum of $\Phi$.
\end{enumerate}
\end{enumerate}
\end{lemma}

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

In the Sobolev space $H^1_0(0,T)$, consider the inner product
$$
(u,v):=\int_0^T p(t)u'(t)v'(t)\,dt+\int_0^T q(t)u(t)v(t)\,dt \,,
$$
which induces the norm
$$
\|u\|:=\Bigg(\int_0^T p(t)(u'(t))^2\,dt +\int_0^T
q(t)(u(t))^2\,dt\Bigg)^{1/2}.
$$
Then the following Poincar\'{e}-type inequality holds:
\begin{equation}\label{e2.1}
\bigg[\int_0^Tu^2(t)\,dt \bigg]^{1/2}\leq \frac{T}{\pi}
\bigg[\int_0^T(u')^2(t)\,dt \bigg]^{1/2}.
\end{equation}

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

\begin{proposition}[{\cite[Proposition 2.1]{BonaDiHen}}]\label{prop2.2}
Let $u \in H^1_0(0,T)$. Then
\begin{equation}\label{e2.2}
\|u\|_{\infty }\leq \frac{1}{2}\sqrt{\frac{T}{p^*}}\,\|u\|,
\end{equation}
where $p^*:=\min_{t \in [0,T]}p(t)$.
\end{proposition}

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

Let $f,g:[0,T]\times\mathbb{R}\to\mathbb{R}$ be two
$L^1$-Carath\'eodory functions. We recall that
$f:[0,T]\times\mathbb{R}\to\mathbb{R}$ is an $L^1$-Carath\'eodory
function if
\begin{itemize}
\item[(a)] the mapping $t\longmapsto f(t,x)$ is measurable for every $x\in\mathbb{R}$;
\item[(b)] the mapping $x\longmapsto f(t,x)$ is continuous for almost every $t\in [0,T]$;
\item[(c)] for every $\rho>0$ there exists a function $l_\rho\in L^1([0,T])$
such that
$$
\sup_{|x|\leq \rho}|f(t,x)|\leq l_{\rho}(t)
$$
for almost every $t\in [0,T]$;
\end{itemize}

Corresponding to $f,g$ we introduce the functions
$F,G:[0,T]\times\mathbb{R}\to\mathbb{R}$ as follows
$$
F(t,x):=\int_0^x f(t,\xi)\,d\xi,\qquad\quad G(t,x):=\int_0^x
g(t,\xi)\,d\xi,
$$
for all $(t,x)\in [0,T]\times\mathbb{R}$.

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

By a \textit{classical solution} of problem \eqref{e1.1}, we mean a function
$$
u \in \big\{w \in C([0,T]: w|_{{[t_j,t_{j+1}]}} \in
H^2([t_j,t_{j+1}])\big\}
$$
that satisfies the equation in \eqref{e1.1} a.e. on $[0,T]\setminus
\{t_1,\dots,t_m\}$, the limits $u'(t^+_j)$, $u'(t^-_j)$,
$j=1,\dots,m$, exist, satisfy the impulsive conditions $\Delta
u'(t_j)=I_j(u(t_j))$ and the boundary condition $u(0)=u(T)=0$.

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

We say that a function $u \in H^1_0(0,T)$ is a \textit{weak
solution} of problem \eqref{e1.1}, if $u$ satisfies
\begin{equation*}
\begin{aligned}
&\int_0^T p(t)u'(t)v'(t) \, dt +\int_0^Tq(t)u(t)v(t) dt \\
&-\lambda\int_0^T f(t,u(t))v(t)\,dt-\mu\int_0^T
g(t,u(t))v(t)\,dt+\sum_{j=1}^m p(t_j)I_j(u(t_j))v(t_j)=0,
\end{aligned}
\end{equation*}
for any $v\in H^1_0(0,T)$.

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

\begin{lemma}[{\cite[Lemma 2.3]{BonaDiHen}}]\label{lem2.3}
The function $u \in H^1_0(0,T)$ is a weak solution of problem \eqref{e1.1} if and only if
$u$ is a classical solution of \eqref{e1.1}.
\end{lemma}

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

\begin{lemma}[{\cite[Lemma 3.1]{BonaDiHen}}]\label{lem2.4}
Assume that
\begin{enumerate}
\item [(A1)] there exist constants $\alpha,\beta>0$ and $\sigma\in [0,1[$
such that
$$
|I_j(x)|\leq \alpha+\beta|x|^{\sigma} \quad \text{for all }
 x \in \mathbb{R}, \; j=1,2,\dots,m\,.
$$
\end{enumerate} Then, for any $u \in H^1_0(0,T)$, we have
\begin{equation} \label{e2.3}
\Big| \sum_{j=1}^m p(t_j)\int_0^{u(t_j)}I_j(x)\,dx \Big| \le
\sum_{j=1}^m p(t_j) \Big( \alpha \|u\|_{\infty} +
\frac{\beta}{\sigma+1} \|u\|_{\infty}^{\sigma+1}\Big).
\end{equation}
\end{lemma}

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

Finally, put
\[
\tilde p:=\sum_{j=1}^m p(t_j),\qquad
k:=\frac{6p^*}{12\|p\|_{\infty}+T^2\|q\|_{\infty}},\qquad
\Gamma_c:=\frac{\alpha}{c}+
\big(\frac{\beta}{\sigma+1}\big)c^{\sigma-1},
\]
where $\alpha$, $\beta$, $\sigma$ are given by $\rm(A1)$ and $c$ is
a positive constant.

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

\section{Existence of infinitely many solutions}

In this section we establish the main abstract result of this paper. Let
\begin{eqnarray*}
A&:=&\liminf\limits_{\xi\rightarrow+\infty}\frac{\displaystyle\int_0^T
\max_{|x|\leq\xi}F(t,x)\,dt}{\xi^2},\\
B&:=&\limsup\limits_{\xi\rightarrow+\infty}\frac{\displaystyle\int_{T/4}^{{3T}/4}
F(t,\xi)\,dt}{\xi^2},
\end{eqnarray*}
and
$$
\lambda_1:=\frac{2p^*}{kTB},\qquad\quad\lambda_2:=\frac{2p^*}{TA}.
$$
With the above notations we establish the following multiplicity property.

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

\begin{theorem}\label{the3.1}
Let $f:[0,T]\times\mathbb{R}\to\mathbb{R}$ be an
$L^1$-Carath\'{e}odory function. Assume that $\rm(A1)$ holds and,
moreover,
\begin{enumerate}
\item[(A2)] $F(t,\xi)\geq 0$ for all $(t,\xi)\in([0,\frac{T}{4}]\cup[\frac{3T}{4},T])\times\mathbb{R};$
\item[(A3)] $A<kB$.
\end{enumerate}
Then, for every $\lambda\in(\lambda_1,\lambda_2)$ and for every
arbitrary $L^1$-Carath\'{e}odory function
$g:[0,T]\times\mathbb{R}\rightarrow\mathbb{R}$, whose potential
$G(t,x):=\int_0^x g(x,\xi)\,d\xi$ for all
$(t,x)\in[0,T]\times\mathbb{R}$, is a nonnegative function
satisfying the condition
\begin{equation}\label{e3.1}
G_\infty:=\limsup\limits_{\xi\rightarrow+\infty}\frac{\displaystyle\int_0^T
\max_{|x|\leq\xi}G(t,x)\,dt}{\xi^2}<+\infty,
\end{equation}
if we put
$$
\mu_{G,\lambda}:=\frac{2p^*}{T
G_\infty}\left(1-\lambda\frac{TA}{2p^*}\right),
$$
where $\mu_{G,\lambda}=+\infty$ when $G_\infty=0$, problem
\eqref{e1.1} has an unbounded sequence of classical solutions for
every $\mu\in[0,\mu_{G,\lambda})$ in $H_0^1(0,T)$.
\end{theorem}

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

\begin{proof}
Our aim is to apply Lemma \ref{lem2.1}(b) to problem \eqref{e1.1}. To this
end, fix $\overline{\lambda}\in (\lambda_1,\lambda_2)$ and $g$
satisfying our assumptions. Since $\overline{\lambda}<\lambda_2$, we
have
$$
\mu_{G,\overline{\lambda}}:=\frac{2p^*}{T
G_\infty}\left(1-\overline{\lambda}\frac{TA}{2p^*}\right)>0.
$$
Now fix $\overline{\mu}\in (0,\mu_{G,\overline{\lambda}})$ and set
$$
J(t,x):=F(t,x)+\frac{\overline{\mu}}{\overline{\lambda}}G(t,x)
$$
for all $(t,x)\in [0,T]\times\mathbb{R}$. Take $X=H_0^1(0,T)$ and
for each $u\in X$, let the functionals
$\Phi,\Psi:X\rightarrow\mathbb{R}$ be defined by
\begin{eqnarray*}
&\Phi(u):=\displaystyle\frac{1}{2}\|u\|^2,&\\
&\Psi(u):=\displaystyle\int_0^T
J(t,u(t))\,dt-\frac{1}{\overline{\lambda}}\sum_{j=1}^m
p(t_j)\int_0^{u(t_j)}I_j(x)\,dx,&
\end{eqnarray*}
and put
$$
E_{\overline{\lambda},\overline{\mu}}(u):=\Phi(u)-\overline{\lambda}\Psi(u),\qquad
u\in X.
$$
Using the property of $f,g$ and the continuity of $I_j$,
$j=1,2,\dots,m$, we obtain that $\Phi,\Psi\in C^1(X,\mathbb{R})$ and
for any $v\in X$, we have
$$
\Phi'(u)(v)=\int_0^T p(t)u'(t)v'(t)\,dt + \int_0^T q(t)u(t)v(t)\,dt
$$
and
$$
\Psi'(u)(v)=\int_0^Tf(t,u(t))v(t)\,dt+\frac{\overline{\mu}}{\overline{\lambda}}\int_0^Tg(t,u(t))v(t)\,dt
-\frac{1}{\overline{\lambda}}\sum_{j=1}^m p(t_j)I_j(u(t_j))v(t_j).
$$
So, with standard arguments, we deduce that the critical points of
the functional $E_{\overline{\lambda},\overline{\mu}}$ are the weak
solutions of problem \eqref{e1.1} and so they are classical. We
first observe that the functionals $\Phi$ and $\Psi$ satisfy the
regularity assumptions of Lemma \ref{lem2.1}.

First of all, we  show that $\overline{\lambda}<1/\gamma$.
Hence, let $\{\xi_n\}$ be a sequence of positive numbers such that
$\lim_{n\rightarrow +\infty}\xi_n=+\infty$ and
$$
\lim\limits_{n\rightarrow+\infty}\frac{\displaystyle\int_0^T\max_{|x|\leq\xi_n}F(t,x)\,dx}{\xi_n^2}=A.
$$
Put $r_n:=\frac{2p^*}{T}\xi_n^2$ for all $n\in\mathbb{N}$. Then, for
all $v\in X$ with $\Phi(v)<r_n$, taking \eqref{e2.2} into account,
one has $\|v\|_\infty<\xi_n$. Note that $\Phi(0)=\Psi(0)=0$. Then,
for all $n\in\mathbb{N}$,
\begin{eqnarray*}
\varphi(r_n)&=&\inf_{u\in\Phi^{-1}(-\infty,r_n)}
\frac{\Big{(}\sup_{v\in\Phi^{-1}(-\infty,r_n)}\Psi(v)\Big{)}-\Psi(u)}{r_n-\Phi(u)}\\
&\leq&\frac{\sup_{v\in\Phi^{-1}(-\infty,r_n)}\Psi(v)}{r_n}\\
&\leq&\frac{\displaystyle\int_0^T\max_{|x|\leq\xi_n}J(t,x)\,dt+\frac{1}{\overline{\lambda}}\tilde
p\Big(\alpha\xi_n+\frac{\beta}{\sigma+1}\xi_n^{\sigma+1}\Big)}
{\frac{2p^*}{T}\xi_n^2}\\
&\leq&\frac{T}{2p^*}\Bigg[\frac{\displaystyle\int_0^T\max_{|x|\leq\xi_n}F(t,x)\,dt}{\xi_n^2}
+\frac{\overline{\mu}}{\overline{\lambda}}
\frac{\displaystyle\int_0^T\max_{|x|\leq\xi_n}G(t,x)\,dt}{\xi_n^2}
+\frac{1}{\overline{\lambda}}\tilde p\, \Gamma_{\xi_n}\Bigg].
\end{eqnarray*}
Since $\lim_{n\rightarrow+\infty}\Gamma_{\xi_n}=0$, from the
assumption $\rm(A3)$ and the condition \eqref{e3.1}, we have
\begin{equation}\label{e3.2}
\gamma\leq\liminf\limits_{n\rightarrow +\infty}\varphi(r_n)\leq
\frac{T}{2p^*}\Big(A+\frac{\overline{\mu}}{\overline{\lambda}}G_\infty\Big)<+\infty.
\end{equation}
The assumption $\overline{\mu}\in (0,\mu_{G,\overline{\lambda}})$
immediately yields
$$
\gamma\leq
\frac{T}{2p^*}\Big(A+\frac{\overline{\mu}}{\overline{\lambda}}G_\infty\Big)
<\frac{T}{2p^*}A+\frac{1-\frac{T}{2p^*}\overline{\lambda}A}{\overline{\lambda}}.
$$
Hence,
$$
\overline{\lambda}=\frac{1}{\frac{T}{2p^*}A
+\big(1-\frac{T}{2p^*}\overline{\lambda}A\big)/{\overline{\lambda}}}
<\frac{1}{\gamma}.
$$
Let $\overline{\lambda}$ be fixed. We claim that the functional
$E_{\overline{\lambda},\overline{\mu}}$ is unbounded from below.
Since
$$
\frac{1}{\overline{\lambda}}<\frac{kT}{2p^*}B,
$$
there exist a sequence $\{\eta_n\}$ of positive numbers and
$\tau>0$ such that $\lim_{n\rightarrow +\infty}\eta_n=+\infty$ and
\begin{equation}\label{e3.3}
\frac{1}{\overline{\lambda}}<\tau<\frac{kT}{2p^*}\frac{\displaystyle\int_{T/4}^{{3T}/4}
F(t,\eta_n)\,dt}{\eta_n^2}
\end{equation}
for each $n\in\mathbb{N}$ large enough. For all $n\in\mathbb{N}$
define $w_n\in X$ by
\begin{equation}\label{e3.4}
w_n(t):=\begin{cases}
\displaystyle\frac{4\eta_n}{T}t, & t\in[0,T/4],\\
\displaystyle\eta_n, & t\in ]T/4,3T/4], \\
\displaystyle\frac{4\eta_n}{T}(T-t), & t\in ]3T/4,T].
\end{cases}
\end{equation}
For any fixed $n\in\mathbb{N}$, one has
\begin{equation}\label{e3.5}
\Phi(w_n)\leq\frac{(12\|p\|_{\infty}+T^2\|q\|_{\infty})}{3T}\eta_n^2
=\frac{2p^*}{kT}\eta_n^2.
\end{equation}
On the other hand, by $\rm(A2)$ and since $G$ is nonnegative, from
the definition of $\Psi$, we infer
\begin{equation}\label{e3.6}
\Psi(w_n)\geq\int_{T/4}^{{3T}/4}F(t,\eta_n)\,dt-\frac{1}{\overline{\lambda}}\frac{\tilde
p}{k}\eta_n^2\,\Gamma_{({\eta_n}/\sqrt{k})}.
\end{equation}
By \eqref{e3.3}, \eqref{e3.5} and \eqref{e3.6}, we see that
\begin{eqnarray}\label{e3.7}
E_{\overline{\lambda},\overline{\mu}}(w_n)&\leq&
\frac{2p^*}{kT}\eta_n^2-\overline{\lambda}\int_{T/4}^{{3T}/4}F(t,\eta_n)\,dt
+\frac{\tilde p}{k}\eta_n^2\,\Gamma_{({\eta_n}/\sqrt{k})}\nonumber\\
&<&\frac{2p^*}{kT}\eta_n^2(1-\overline{\lambda}\tau)+\frac{\tilde
p}{k}\eta_n^2\,\Gamma_{({\eta_n}/\sqrt{k})}
\end{eqnarray}
for every $n\in\mathbb{N}$ large enough. Since $\sigma<1$,
$\overline{\lambda}\tau>1$ and $\lim_{n\rightarrow
+\infty}\eta_n=+\infty$, we have
$$
\lim\limits_{n\rightarrow
+\infty}E_{\overline{\lambda},\overline{\mu}}(w_n)=-\infty.
$$
Then, the functional $E_{\overline{\lambda},\overline{\mu}}$ is
unbounded from below, and it follows that
$E_{\overline{\lambda},\overline{\mu}}$ has no global minimum.
Therefore, by Lemma \ref{lem2.1}(b), there exists a sequence
$\{u_n\}$ of critical points of
$E_{\overline{\lambda},\overline{\mu}}$ such that
$$
\lim\limits_{n\rightarrow +\infty}\|u_n\|=+\infty,
$$
and the conclusion is achieved.
\end{proof}

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

\begin{remark}\label{rem3.2}
Under the conditions $A=0$ and $B=+\infty$, from Theorem
\ref{the3.1} we see that for every $\lambda>0$ and for each $\mu\in
\big{[}0,\frac{2p^*}{T G_\infty}\big{)}$, problem \eqref{e1.1}
admits a sequence of classical solutions which is unbounded in $X$.
Moreover, if $G_\infty=0$, the result holds for every $\lambda>0$
and $\mu\geq 0$.
\end{remark}

%----------------------------------------------------------------------------------------------------------
\section{Further results and particular cases}
In this section we establish several useful consequences and particular cases of Theorem \ref{the3.1}.

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

\begin{corollary}\label{cor3.3}
Let $f:[0,T]\times\mathbb{R}\to\mathbb{R}$ be an
$L^1$-Carath\'{e}odory function. Suppose that hypotheses $\rm(A1)$, $\rm(A2)$ are fulfilled
and
$$
A<\frac{2p^*}{T},\qquad\quad B>\frac{2p^*}{kT}\,.
$$
 Then, for every arbitrary $L^1$-Carath\'{e}odory
function $g:[0,T]\times\mathbb{R}\rightarrow\mathbb{R}$, whose
potential $G(t,x):=\int_0^x g(t,\xi)\,d\xi$ for all
$(t,x)\in[0,T]\times\mathbb{R}$, is a nonnegative function
satisfying the condition \eqref{e3.1}, if we put
$$
\mu_{G,\lambda}:=\frac{2p^*}{T
G_\infty}\left(1-\frac{TA}{2p^*}\right),
$$
where $\mu_{G,\lambda}=+\infty$ when $G_\infty=0$, the problem
\begin{equation*}
\left\{\begin{array}{ll}
-(p(t)u'(t))'+q(t)u(t)=f(t,u(t))+\mu g(t,u(t)), \quad t\in [0,T],\; t\not= t_j,\\
u(0)=u(T)=0,\\
\Delta u'(t_j)=I_j(u(t_j)),\quad j=1,2,\dots,m,
\end{array}\right.
\end{equation*}
has an unbounded sequence of classical solutions for every
$\mu\in[0,\mu_{G,\lambda})$ in $H_0^1(0,T)$.
\end{corollary}

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

The following result is a special case of Theorem \ref{the3.1}.

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

\begin{corollary}\label{cor3.4}
Let $\rm(A1)$ holds and let $f:\mathbb{R}\to\mathbb{R}$ be a
nonnegative continuous function. Put $F(\xi):=\int_0^\xi f(t)dt$ for
all $\xi\in\mathbb{R}$ and assume that
$$
\liminf\limits_{\xi\rightarrow+\infty}\frac{F(\xi)}{\xi^2}=0,\qquad\quad
\limsup\limits_{\xi\rightarrow+\infty}\frac{F(\xi)}{\xi^2}=+\infty.
$$
Then, for every nonnegative continuous function
$g:\mathbb{R}\rightarrow\mathbb{R}$ satisfying the condition
$$
g_\infty:=\lim\limits_{\xi\rightarrow+\infty}\frac{\displaystyle\int_{0}^{\xi}g(x)\,dx}{\xi^2}<+\infty,
$$
and for every $\mu\in[0,\frac{2p^*}{T^2g_\infty}[$, the problem
\begin{equation*}
\left\{\begin{array}{ll}
-(p(t)u'(t))'+q(t)u(t)=f(u(t))+\mu g(u(t)), \quad t\in [0,T],\; t\not= t_j,\\
u(0)=u(T)=0,\\
\Delta u'(t_j)=I_j(u(t_j)),\quad j=1,2,\dots,m,
\end{array}\right.
\end{equation*}
admits infinitely many distinct pairwise classical solutions.
\end{corollary}

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

Now, we point out a special situation of our main result when
$\mu=0$ and the nonlinear term has separated variables. To be
precise, let $k\in L^1([0,T])$ such that $k(t)\geq 0$ a.e. $t\in
[0,T]$, $k\not\equiv 0,$ and let $l:\mathbb{R}\rightarrow\mathbb{R}$
be a nonnegative continuous function.

Consider the following
nonlinear Dirichlet boundary-value problem
\begin{equation}\label{e3.8}
\left\{\begin{array}{ll}
-u''(t)=\lambda k(t)l(u(t)), \quad t\in [0,T],\; t\not= t_j,\\
u(0)=u(T)=0,\\
\Delta u'(t_j)=I_j(u(t_j)),\quad j=1,2,\dots,m.
\end{array}\right.
\end{equation}
Put $L(\xi):=\displaystyle\int_0^\xi l(x)\,dx$ for all
$\xi\in\mathbb{R}$, and set $\|k\|_1:=\displaystyle\int_0^T
k(t)\,dt$ and $k_0:=\displaystyle\int_{T/4}^{{3T}/4} k(t)\,dt$.

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

\begin{corollary}\label{cor3.5}
Let $(A1)$ holds. Moreover, suppose that
$$
\liminf\limits_{\xi\rightarrow+\infty}\frac{L(\xi)}{\xi^2}
<\frac{k_0}{2\|k\|_1}\limsup\limits_{\xi\rightarrow+\infty}\frac{L(\xi)}{\xi^2}.
$$
Then, for each
$$
\lambda\in\Bigg]\frac{4}{(Tk_0)\limsup\limits_{\xi\rightarrow+\infty}
\frac{L(\xi)}{\xi^2}},\frac{2}{(T\|k\|_1)
\liminf\limits_{\xi\rightarrow+\infty}\frac{L(\xi)}{\xi^2}}\Bigg[,
$$
problem \eqref{e3.8} has an unbounded sequence of classical
solutions.
\end{corollary}

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

Now, put
\begin{eqnarray*}
\Im_c&:=&\sum_{j=1}^m\min_{|\xi| \leq c}\int_0^{\xi}I_j(x)\,dx,
\qquad \text{for all} \;c>0,\\
A'&:=&\liminf_{\xi\rightarrow
0^+}\frac{\displaystyle\int_0^T
\max_{|x|\leq\xi}F(t,x)\,dt}{\xi^2},\\
B'&:=&\limsup_{\xi\rightarrow
0^+}\frac{\displaystyle\int_{T/4}^{{3T}/4} F(t,\xi)\,dt}{\xi^2},
\end{eqnarray*}
and
$$
\lambda'_1:=\frac{2p^*}{kTB'},\qquad\quad\lambda'_2:=\frac{2p^*}{TA'}.
$$
Using Lemma \ref{lem2.1}(c) and arguing as in the proof of Theorem
\ref{the3.1}, we can obtain the following multiplicity result.

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

\begin{theorem}\label{the3.6}
Let $f:[0,T]\times\mathbb{R}\to\mathbb{R}$ be an
$L^1$-Carath\'{e}odory function and $I_j(x)\leq 0$ for all
$x\in\mathbb{R}$, $j=1,\ldots,m$. Moreover, assume that $\rm(A2)$
and
\begin{enumerate}
\item[(A4)] $A'<kB'$.
\end{enumerate}
are satisfied. Then, for every $\lambda\in(\lambda'_1,\lambda'_2)$
and for every arbitrary $L^1$-Carath\'{e}odory function
$g:[0,T]\times\mathbb{R}\rightarrow\mathbb{R}$, whose potential
$G(t,x):=\int_0^x g(x,\xi)\,d\xi$ for all
$(t,x)\in[0,T]\times\mathbb{R}$, is a nonnegative function
satisfying the condition
\begin{equation}\label{e3.9}
G_0:=\limsup\limits_{\xi\rightarrow 0^+}\frac{\displaystyle\int_0^T
\max_{|x|\leq\xi}G(t,x)\,dt}{\xi^2}<+\infty,
\end{equation}
if we put
$$
\mu'_{G,\lambda}:=\frac{2p^*}{T G_0}\left(1-\lambda\frac{TA'}{2p^*}\right),
$$
where $\mu'_{G,\lambda}=+\infty$ when $G_0=0$, for every
$\mu\in[0,\mu'_{G,\lambda})$ problem \eqref{e1.1} has a sequence of
classical solutions, which strongly converges to zero in
$H_0^1(0,T)$.
\end{theorem}

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

\begin{proof}
Fix $\overline{\lambda}\in (\lambda'_1,\lambda'_2)$ and let $g$ be a
function that satisfies the condition \eqref{e3.9}. Since
$\overline{\lambda}<\lambda'_2$, we obtain
$$
\mu'_{G,\overline{\lambda}}:=\frac{2p^*}{T
G_0}\left(1-\overline{\lambda}\frac{TA'}{2p^*}\right)>0.
$$
Now fix $\overline{\mu}\in (0,\mu'_{G,\overline{\lambda}})$ and set
$$
J(t,x):=F(t,x)+\frac{\overline{\mu}}{\overline{\lambda}}G(t,x),
$$
for all $(t,x)\in[0,T]\times\mathbb{R}$. We take $\Phi, \Psi$ and
$E_{\overline{\lambda},\overline{\mu}}$ as in the proof of Theorem
\ref{the3.1}. Now, as it has been pointed out before, the functionals
$\Phi$ and $\Psi$ satisfy the regularity assumptions required in
Lemma \ref{lem2.1}. As first step, we will prove that
$\overline{\lambda}<1/\delta$. Then, let $\{\xi_n\}$ be a sequence
of positive numbers such that $\lim_{n\rightarrow +\infty}\xi_n=0$
and
$$
\lim\limits_{n\rightarrow+\infty}\frac{\displaystyle\int_0^T\max_{|x|\leq\xi_n}F(t,x)\,dx}{\xi_n^2}=A'.
$$
By the fact that $\inf_X\Phi=0$ and the definition of $\delta$, we
have $\delta=\liminf_{r\rightarrow{0^+}}\varphi(r)$. Put
$r_n:=\frac{2p^*}{T}\xi_n^2$ for all $n\in\mathbb{N}$. Then, for all
$v\in X$ with $\Phi(v)<r_n$, taking \eqref{e2.2} into account, one
has $\|v\|_\infty<\xi_n$. Thus, for all $n\in\mathbb{N}$,
\begin{eqnarray*}
\varphi(r_n)&\leq&\frac{\sup_{v\in\Phi^{-1}(-\infty,r_n)}\Psi(v)}{r_n}\\
&\leq&\frac{\displaystyle\int_0^T\max_{|x|\leq\xi_n}J(t,x)\,dt
-\frac{1}{\overline{\lambda}}\|p\|_\infty\Im_{\xi_n}}
{\frac{2p^*}{T}\xi_n^2}\\
&\leq&\frac{T}{2p^*}\Bigg[\frac{\displaystyle\int_0^T\max_{|x|\leq\xi_n}F(t,x)\,dt}{\xi_n^2}
+\frac{\overline{\mu}}{\overline{\lambda}}
\frac{\displaystyle\int_0^T\max_{|x|\leq\xi_n}G(t,x)\,dt}{\xi_n^2}
-\frac{1}{\overline{\lambda}}\|p\|_\infty\frac{\Im_{\xi_n}}{\xi_n^2}\Bigg].
\end{eqnarray*}
Since $\lim_{n\rightarrow+\infty}\frac{\Im_{\xi_n}}{\xi_n^2}=0$,
from the assumption $\rm(A3)$ and the condition \eqref{e3.9}, we
have
\begin{equation*}
\delta\leq\liminf\limits_{n\rightarrow +\infty}\varphi(r_n)\leq
\frac{T}{2p^*}\Big(A'+\frac{\overline{\mu}}{\overline{\lambda}}G_0\Big)<+\infty.
\end{equation*}
From $\overline{\mu}\in (0,\mu'_{G,\overline{\lambda}})$, the
following inequalities hold
$$
\delta\leq
\frac{T}{2p^*}\Big(A'+\frac{\overline{\mu}}{\overline{\lambda}}G_0\Big)
<\frac{T}{2p^*}A'+\frac{1-\frac{T}{2p^*}\overline{\lambda}A'}{\overline{\lambda}}.
$$
Therefore
$$
\overline{\lambda}=\frac{1}{\frac{T}{2p^*}A'
+\big(1-\frac{T}{2p^*}\overline{\lambda}A'\big)/{\overline{\lambda}}}
<\frac{1}{\delta}\,.
$$
Let $\overline{\lambda}$ be fixed. We claim that the functional
$E_{\overline{\lambda},\overline{\mu}}$ does not have a local
minimum at zero. Since
$$
\frac{1}{\overline{\lambda}}<\frac{kT}{2p^*}B',
$$
there exists a sequence $\{\eta_n\}$ of positive numbers and
$\tau>0$ such that $\lim_{n\rightarrow +\infty}\eta_n=0$ and
\begin{equation*}
\frac{1}{\overline{\lambda}}<\tau<\frac{kT}{2p^*}\frac{\displaystyle\int_{T/4}^{{3T}/4}
F(t,\eta_n)\,dt}{\eta_n^2}
\end{equation*}
for each $n\in\mathbb{N}$ large enough. For all $n\in\mathbb{N}$,
let $w_n\in X$ defined by \eqref{e3.4} with the above $\eta_n$. Note
that $\overline{\lambda}\tau>1$. Then, since $I_j(x)\leq 0$ for all
$x\in\mathbb{R}$, $j=1,\ldots,m$, we  obtain
\begin{eqnarray*}
E_{\overline{\lambda},\overline{\mu}}(w_n)&\leq&\frac{2p^*}{kT}\eta_n^2
-\overline{\lambda}\int_{T/4}^{{3T}/4}F(t,\eta_n)\,dt
+\sum_{j=1}^m p(t_j)\int_0^{w_n(t_j)}I_j(x)\,dx\\
&<&\frac{2p^*}{kT}\eta_n^2(1-\overline{\lambda}\tau)<0,
\end{eqnarray*}
for every $n\in\mathbb{N}$ large enough. Then, since
$$
\lim_{n\rightarrow
+\infty}E_{\overline{\lambda},\overline{\mu}}(w_n)=E_{\overline{\lambda},\overline{\mu}}(0)=0,
$$
we see that zero is not a local minimum of
$E_{\overline{\lambda},\overline{\mu}}$. This, together with the
fact that zero is the only global minimum of $\Phi$, we deduce that the energy
functional $E_{\overline{\lambda},\overline{\mu}}$ does not have a
local minimum at the unique global minimum of $\Phi$. Therefore, by
Lemma \ref{lem2.1}(c), there exists a sequence $\{u_n\}$ of critical
points of $E_{\overline{\lambda},\overline{\mu}}$ which converges
weakly to zero. In view of the fact that the embedding
$H_0^1(0,T)\hookrightarrow C^0([0,T])$ is compact, we know that the
critical points converge strongly to zero, and the proof is
complete.
\end{proof}

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

\begin{remark}\label{rem3.7}
Applying Theorem \ref{the3.6}, results similar to Corollaries
\ref{cor3.3}, \ref{cor3.4} and \ref{cor3.5} can be obtained. We omit
the discussions here.
\end{remark}

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

Now, consider the nonlinear Dirichlet boundary value problem
\begin{equation}\label{e3.10}
\left\{\begin{array}{ll}
-u''(t)+a(t)u'(t)+b(t)u(t)=\lambda h(t,u(t))+\mu p(t,u(t)), \ t\in [0,T],\; t\not= t_j,\\
u(0)=u(T)=0,\\
\Delta u'(t_j)=I_j(u(t_j)),\quad j=1,2,\dots,m,
\end{array}\right.
\end{equation}
where $h,p:[0,T]\times \mathbb{R} \to \mathbb{R}$ are
$L^1$-Carath\'{e}odory functions, $a,b \in L^{\infty}([0,T])$
satisfy the conditions $\operatorname{ess\,inf}_{t \in[0,T]}
a(t)\geq 0$, $\operatorname{ess\,inf}_{t \in[0,T]}b(t) \geq 0$, and
$I_j: \mathbb{R} \to \mathbb{R}$ are continuous for every
$j=1,2,\dots,m$.

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

It is easy to see that the solutions of problem \eqref{e1.1} are solutions
of \eqref{e3.10} if
\begin{eqnarray*}
&p(t)=e^{-\int_0^t a(\tau)\,d\tau},\qquad q(t)=b(t)e^{-\int_0^t
a(\tau)\,d\tau},&\\
&f(t,u)=h(t,u)e^{-\int_0^t a(\tau)\,d\tau},\qquad
g(t,u)=p(t,u)e^{-\int_0^t a(\tau)\,d\tau}.&
\end{eqnarray*}

Let $A(t)$ be a primitive of $a(t)$ and put
\begin{eqnarray*}
&H(t,\xi):=\displaystyle\int_0^\xi h(t,x)\,dx,\qquad P(t,\xi):=\int_0^\xi p(t,x)\,dx,&\\
&\tilde{k}:=\displaystyle\frac{6e^{-A(T)}}{12+T^2\|be^{-A}\|_{\infty}}.&
\end{eqnarray*}
By Theorem \ref{the3.1}, we obtain the following multiplicity property for problem
\eqref{e3.10}.

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

\begin{theorem}\label{the3.8}
Let $h:[0,T]\times\mathbb{R}\to\mathbb{R}$ be an
$L^1$-Carath\'{e}odory function. Assume that $\rm(A1)$ holds and,
moreover,
\begin{enumerate}
\item[(A5)] $H(t,\xi)\geq 0$ for all $(t,\xi)\in([0,\frac{T}{4}]\cup[\frac{3T}{4},T])\times\mathbb{R};$
\item[(A6)] $A''<\tilde{k}B''$, where
\begin{eqnarray*}
A''&:=&\liminf_{\xi\rightarrow 0^+}\frac{\displaystyle\int_0^T
e^{-A(t)}\max_{|x|\leq\xi}H(t,x)\,dt}{\xi^2},\\
B''&:=&\limsup_{\xi\rightarrow
0^+}\frac{\displaystyle\int_{T/4}^{{3T}/4}e^{-A(t)}
H(t,\xi)\,dt}{\xi^2},
\end{eqnarray*}
\end{enumerate}
Then, for every
$$
\lambda\in\bigg]\frac{2}{\tilde{k}T e^{\|a\|_1}B''},\frac{2}{T
e^{\|a\|_1}A''}\bigg[
$$
and for every arbitrary $L^1$-Carath\'{e}odory function
$p:[0,T]\times\mathbb{R}\rightarrow\mathbb{R}$, whose potential
$P(t,x):=\int_0^x p(x,\xi)\,d\xi$ for all
$(t,x)\in[0,T]\times\mathbb{R}$, is a nonnegative function
satisfying the condition
$$
P_\infty:=\limsup\limits_{\xi\rightarrow+\infty}\frac{\displaystyle\int_0^T
e^{-A(t)}\max_{|x|\leq\xi}P(t,x)\,dt}{\xi^2}<+\infty,
$$
if we put
$$
\mu_{P,\lambda}:=\frac{2}{T e^{\|a\|_1}P_\infty}\left(1-\lambda\frac{Te^{\|a\|_1}A''}{2}\right),
$$
where $\mu_{P,\lambda}=+\infty$ when $P_\infty=0$, problem
\eqref{e3.10} has an unbounded sequence of classical solutions for
every $\mu\in[0,\mu_{P,\lambda})$ in $H_0^1(0,T)$.
\end{theorem}

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

\begin{remark}\label{rem3.9}
Theorem \ref{the1.1} follows immediately from
Theorem \ref{the3.8}, setting $a(t)=b(t)\equiv 1$ for all
$t\in\mathbb{R}$.
\end{remark}

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

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\end{document}