\documentclass[reqno]{amsart}
\usepackage{hyperref}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2015 (2015), No. 207, pp. 1--9.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2015 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2015/207\hfil Multiple solutions]
{Multiple solutions for p-Laplacian boundary-value problems with
impulsive effects}

\author[H. Shi, H. Chen \hfil EJDE-2015/207\hfilneg]
{Hongxia Shi, Haibo Chen}

\address{Hongxia Shi \newline
School of Mathematics and Statistics,
Central South University, Changsha, 410083 Hunan, China}
\email{shihongxia5617@163.com}

\address{Haibo Chen (corresponding author)\newline
School of Mathematics and Statistics,
Central South University, Changsha, 410083 Hunan, China}
\email{math\_chb@163.com,  math\_chb@csu.edu.cn}

\thanks{Submitted  March 15, 2015. Published August 10, 2015.}
\subjclass[2010]{34A37, 34B37}
\keywords{Boundary value problems; impulsive effects; nontrivial solutions; 
\hfill\break\indent Morse theory; local linking}

\begin{abstract}
 In this article we study a class of boundary value problems with
 impulsive effects. First by using Morse theory in combination with
 local linking arguments, the existence result of at least two
 nontrivial solutions are obtained. Next we prove that the problems
 have $k$ distinct pairs of solutions by using the Clark theorem.
 Recent results from the literature are improved and extended.
\end{abstract}

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


\section{Introduction and statement of main results}

In this article, we consider the  impulsive boundary value problem
\begin{equation}
\begin{gathered}
-(\rho(x)\Phi_{p}(u'(x)))'+s(t)\Phi_{p}(u(x))=f(x,u(x)),\quad \text{a.e. }  x\in(a,b),\\
\Delta(\rho(x_j)\Phi_{p}(u'(x_j)))=\iota_j(u(x_j)),\quad  j=1,2,\ldots,m,\\
\alpha_1u'(a^{+})-\alpha_2u(a)=0,\quad \beta_1u'(b^{-})+\beta_2u(b)=0,
\end{gathered} \label{e1.1}
\end{equation}
 where $\Phi_{p}(u)=|u|^{p-2}u$,  $p>1$,  $\rho, s\in L^{\infty}[a,b]$  with
$\operatorname{ess\,inf}_{[a,b]}\rho>0$,  $\operatorname{ess\,inf}_{[a,b]}s>0$,
 $0<\rho(a), \rho(b)<\infty$, $\alpha_1, \alpha_2, \beta_1, \beta_2>0$,
$a=x_{0}<x_1<x_2<\dots<x_{m}<x_{m+1}=b$, $u'(x_j^{+}))$ and
$u'(x_j^{-}))$ denotes the right and left limit of $u'(x_j)$ at $x=x_j$,
respectively, $\iota_j\in C(\mathbb{R},\mathbb{R})$, $j=1,2,\ldots,m$,
$f\in C([a,b]\times \mathbb{R},\mathbb{R})$.

Since many evolution processes exhibit impulsive effects in the real world, 
the theory of impulsive differential equations has developed rapidly 
in recent years. For the significance, it is important to study the 
solvability of impulsive differential equations. We refer some recent 
works on the theory of impulsive differential equations that developed 
by a large number of mathematicians \cite{a2,c3,l2,n1,n3,n4,z1,z2}.
 Classical approaches to such problems include fixed point theory, 
topological degree theory and comparison method and so on. 
More recently, variational method is one of the most promising techniques 
for differential equations, especially for the boundary value problems of 
impulsive differential equations, and the literature on this technique 
has grown extensively, see \cite{b1,b2,c1,l5,n2,s3,s5,t2,t3,z3,z4}
and the references therein.

Morse theory and local linking arguments are powerful tools in modern 
nonlinear analysis \cite{c2,f1,j1,s1,s6},
especially for the problems
with resonance \cite{l1,s2}. However, to the best of our knowledge, 
there are few papers dealing with the existence of solutions for impulsive
 boundary value problems by using Morse theory. Recently, in \cite{a1}, 
the authors considered the following problem
\begin{equation}
\begin{gathered}
-u''=f(x,u),\quad x\in (0,1)\setminus \{x_1,x_2,\dots x_{m}\},\\
\Delta u'(x_j)= \iota_j(u(x_j)), \quad j=1, 2,\ldots,m,\\
u(0)=u(1)=0. 
\end{gathered} \label{e1.2}
\end{equation}
They obtained the existence of one nontrivial solution for  \eqref{e1.2}
when the impulses are asymptotically linear near zero via computing
the critical groups at zero.

Inspired by the above facts, the goal of this paper is to consider the 
multiplicity of nontrivial solutions for \eqref{e1.1}.
Under some suitable assumptions, by using Morse theory in combination 
with local linking arguments, the existence result of at least 
two nontrivial solutions are obtained. Next we prove that the problems 
have $k$ distinct pairs of solutions by using the Clark theorem.

Before stating our main results, we present the following assumptions 
on $\iota_j$ $(j=1,2,\dots,m)$:
\begin{itemize}
\item[(I1)] $\iota_j(t)t\geq 0$ and there exist $a_j>0$ and
 $0\leq \gamma_j<p-1$ such that
$$
|\iota_j(t)|\leq a_j|t|^{\gamma_j},\quad j=1,2,\dots,m;
$$

\item[(I2)]  $\iota_j(-t)=-\iota_j(t)$, $j=1,2,\dots,m.$
\end{itemize}

\begin{remark}\label{re1.1} \rm
From  condition (I1), we can see that
 $$
|I_j(t)|\leq a_j|t|^{\gamma_j+1} \quad\text{and}\quad
 I_j(t)\geq0 \quad (j=1,2,\dots,m),
$$
here and in the sequel $I_j(t)=\int_{0}^{t}\iota_j(s)ds$.
\end {remark}

Furthermore, we assume that the nonlinearity $f(x,u)$ satisfies the
 conditions:
\begin{itemize}
\item[(F1)] there exist $c_1>0$ and $0\leq\alpha<p-1$ such that 
$$
|f(x,u)|\leq c_1|u|^{\alpha}, \quad \forall(x,u)\in [a,b]\times \mathbb{R};
$$

\item[(F2)] there exist small constants $0<r<r_{0}$, $c_2>0$, 
$0<c_{3}<\frac{1}{pS_{p}^{p}}$, $1<\gamma<\max\{\gamma_j+1\}$ such that
$$
c_{3}|u|^{p}>F(x,u)\geq c_2 |u|^{\gamma},\quad  r\leq|u|\leq r_{0} \quad 
\text{a.e. } x\in[a,b],
$$
here and in the sequel $F(x,u)=\int_{0}^{u}f(x,s)ds$,
furthermore, $S_{p}$ is the Sobolev constant from 
$W^{1,p}([a,b])$ to $L^{p}([a,b])$;

\item[(F3)] $f(x,-u)=-f(x,u)$.
\end{itemize}
Now, we are ready to state the main results of this article.

\begin{theorem}\label{the1.1}
Assume that {\rm (I1), (F1), (F2)} hold. Then \eqref{e1.1} has at least
two nontrivial solutions.
\end {theorem}

\begin{theorem}\label{the1.2}
Assume that {\rm (I1), (I2), (F1)--(F3)} hold. Then  \eqref{e1.1} has at 
least $k$ distinct pairs of solutions.
\end{theorem}

The remainder of this article is organized as follows. 
In Section \ref{sec2}, some preliminary results are presented. 
In Section \ref{sec3}, we give the proof of our main result. 
Finally, an example is given to demonstrate the applicability of our 
main results in Section \ref{sec4}. Furthermore,  we want to point 
out that a similar approach can be used to study different elliptic problems, 
such as in the paper \cite{d1}.

\section{Preliminaries and variational setting} \label{sec2}

Throughout this article, $C$, $C_{i}$ denotes positive constants which may vary; 
$\to$ denotes the strong and $\rightharpoonup$ the weak convergence;
$B_{r}$ denotes the ball of radius $r$ and $E^{*}$ denotes the dual space of $E$.

The Sobolev space $E=W^{1,p}([a,b])$ is equipped with the norm
$$
\|u\|=\Big(\int_{a}^{b}\rho(x)|u'(x)|^{p}+s(x)|u(x)|^{p}\Big)^{1/p},
$$
which is equivalent to the usual one.

 As usual, for $1\leq p< +\infty$, we let
\begin{gather*}
\|u\|_{p}=\Big(\int_{a}^{b}|u(x)|^{p}dx\Big)^{1/p},\quad
 u\in L^{p}([a,b]),
\\
\|u\|_{\infty}=\text{max}_{x\in [a,b]} |u(x)|, \quad u\in C([a,b]).
\end{gather*}

\begin{lemma}[{\cite[Lemma 2.6]{t2}}]  \label{lem2.1}
For $u\in E$, then we have $\|u\|_{\infty}\leq C_1\|u\|$,
where
\begin{equation*}
C_1=2^{1/q}\max\Big\{\frac{1}{(b-a)^{1/p}
(\operatorname{ess\,inf}_{[a,b]}s)^{1/p}},
\frac{(b-a)^{1/q}}{(\operatorname{ess\,inf}_{[a,b]}\rho)^{1/p}}\Big\} ,\quad
 \frac{1}{p}+\frac{1}{q}=1.
\end{equation*}
\end{lemma}

Now we begin describing the variational formulation of problem \eqref{e1.1}.
Consider the functional $\varphi:E\to \mathbb{R}$ defined by
\begin{equation}
\begin{aligned}
 \varphi(u)&=\frac{\|u\|^{p}}{p}
+\sum_{j=1}^{m}I_j(u(x_j))+\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 -\int_{a}^{b}F(x,u)dx.
\end{aligned}\label{e2.1}
\end{equation}
Since $f$ and $\iota_j(j=1,2,\dots,m)$ are continuous, we deduce that $\varphi$
is of class $C^{1}$ and its derivative is given by
\begin{equation}
\begin{aligned}
\varphi'(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}(\frac{\alpha_2u(a)}{\alpha_1})v(a)
  +  \rho(b)\Phi_{p}(\frac{\beta_2u(b)}{\beta_1})v(b)
  +  \sum_{j=1}^{m}\iota_j(u(x_j))v(x_j)  \\
&\quad  -  \int_{a}^{b}f(x,u(x))v(x)dx,
\end{aligned}\label{e2.2}
\end{equation}
for all $u,v\in E$. Then we can infer that $u\in E$ is a critical point of
$\varphi$ if and only if it is a solution of \eqref{e1.1}.

We will use Morse theory in combination with local linking arguments to 
obtain the critical points of $\varphi$. Now, it is necessary to recall 
the following definitions and results.

\begin{definition}\label{def2.1} \rm
Let $E$ be a real reflexive Banach space. We say that $\varphi$ satisfies 
the (PS)-condition, i.e. every sequence $\{u_{n}\}\subset E$ satisfying 
$\varphi(u_{n})$ bounded and $\lim_{n\to\infty}\varphi'(u_{n})=0$ contains 
a convergent subsequence.
\end{definition}

Let $E$ be a real Banach space and $\varphi\in C^{1}(E,\mathbb{R})$. 
$K=\{u\in E:\varphi'(u)=0\}$, then the q-th critical group of $\varphi$ 
at an isolated critical point $u\in K$ with $\varphi(u)=c$ is defined by
$$
C_{q}(\varphi,u):=H_{q}(\varphi^{c}\cap U,\varphi^{c}\cap U\setminus \{u\}), \quad 
q\in \mathbb{N}:=\{0,1,2,\dots\},
$$
where $\varphi^{c}=\{u\in E: \varphi (u)\leq c\}$, $U$ is a neighborhood 
of $u$, containing the unique critical point, $H_{*}$ is the singular 
relative homology with coefficient in an Abelian group $G$.

We say that $u\in E$ is a homological nontrivial critical point of 
$\varphi$ if at least one of its critical groups is nontrivial.
Now, we present the following propositions which will be used later.

\begin{proposition}[{\cite[Proposition 2.1]{l3}}] \label{pro2.1}
Assume that $\varphi$ has a critical point $u=0$ with $\varphi(0)=0$. 
Suppose that $\varphi$ has a local linking at 0 with respect to $E=V\oplus W$, 
$k=\dim V<\infty$; that is, there exists $\rho >0$ small such that
\begin{equation*} %\label{1.1}
\begin{gathered}
\varphi(u)\leq 0,\quad u\in V,\quad \|u\|\leq\rho;\\
\varphi(u)> 0,\quad u\in W, \quad  0<\|u\|\leq\rho.\\
\end{gathered}
\end{equation*}
Then $C_{k}(\varphi,0)\ncong0$, hence 0 is a homological nontrivial 
critical point of $\varphi$.
\end{proposition}

\begin{proposition}[{\cite[Theorem 2.1]{l3}}] \label{pro2.2}
Let $E$ be a real Banach space and let $\varphi\in C^{1}(E,\mathbb{R})$ 
satisfy the (PS)-condition and is bounded from below. If  $\varphi$ 
has a critical point that is homological nontrivial and is not a
 minimizer of  $\varphi$, then $\varphi$ has at least three critical points.
\end{proposition}


\begin{proposition}[{\cite[Theorem 9.1]{r1}}] \label{pro2.3}
 Let $E$ be a real Banach space, $\varphi\in C^{1}(E,\mathbb{R})$ with 
$\varphi$ even, bounded from below, and satisfying (PS)-condition. 
Suppose $\varphi(0)=0$, there is a set $K\subset E$ such that $K$ 
is homeomorphic to $S^{j-1}$ by an odd map, and $\sup_{K}\varphi<0$. 
Then $\varphi$ possesses at least $j$ distinct pairs of critical points.
\end{proposition}

\section{Proof of main results}\label{sec3}

In this section, we  prove Theorems \ref{the1.1} and \ref{the1.2}. 
To complete the proof, we need the following lemmas.

\begin{lemma} \label{lem3.1}
Suppose that $\varphi$ satisfies {\rm (I1), (F1)}, then $\varphi$ satisfies 
the (PS)-condi\-tion.
\end{lemma}

\begin{proof}
 We first prove that $\varphi$ is coercive. It follows from $(I1)$ and (F1) that
\begin{align*}
\varphi(u)  
&=  \frac{\|u\|^{p}}{p}  +  \sum_{j=1}^{m}I_j(u(x_j))  
 +  \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}  
 -   \int_{a}^{b}F(x,u)dx\\
&\geq \frac{\|u\|^{p}}{p}+\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}
 -\int_{a}^{b}c_1|u|^{\alpha+1}dx\\
&\geq \frac{\|u\|^{p}}{p}+\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}-C_2\|u\|^{\alpha+1}
\end{align*}
Since $\alpha+1<p$, it follows that $\varphi(u)\to+\infty$ as $\|u\|\to\infty$.

Suppose that $\{u_{n}\}$ is a (PS) sequence, then $\{u_{n}\}$ is bounded, there
exists a constant $M>0$ such that
\begin{equation}
\|u_n\|\leq M,\quad\forall n\in \mathbb{N}. \label{e3.1}
\end{equation}
Going to a subsequence, if necessary, we can assume that
$u_n\rightharpoonup u_0$ in $E.$ Hence, by compact embedding theorem of
Sobolev space, we have
$$
u_n\to u_0 \text{ in } L^{p}([a,b]),\quad
u_n\to u_0  \text{ a.e. }  x\in[a,b].
$$
By \eqref{e2.2}, we have
\begin{equation}
\begin{aligned}
&   (\varphi'(u_{n})-\varphi'(u_{0}),u_{n}-u_{0})\\
&=  \int_{a}^{b}\rho(x)(\Phi_{p}(u'_{n}(x))
 -  \Phi_{p}(u_{0}'(x)))(u'_{n}(x)-u_{0}'(x))dx\\
&\quad +  \int_{a}^{b}s(x)(\Phi_{p}(u_{n}(x))
 -  \Phi_{p}(u_{0}(x)))(u_{n}(x)  -  u_{0}(x))dx\\
&\quad +\rho(a)(\Phi_{p}(\frac{\alpha_2u_{n}(a)}{\alpha_1})
 -  \Phi_{p}(\frac{\alpha_2u_{0}(a)}{\alpha_1}))(u_{n}(a)
 -  u_{0}(a))\\
& \quad +\rho(b)(\Phi_{p}(\frac{\beta_2u_{n}(b)}{\beta_1})
  -  \Phi_{p}(\frac{\beta_2u_{0}(b)}{\beta_1}))(u_{n}(b)  -  u_{0}(b))\\
&\quad +\sum_{j=1}^{m}(\iota_j(u_{n}(x_j))-\iota_j(u_{0}(x_j)))(u_{n}(x_j)-u_{0}(x_j))\\
&\quad -\int_{a}^{b}(f(t,u_{n}(x))  -  f(t,u_{0}(x)))(u_{n}(x)
 -  u_{0}(x))dx.
\end{aligned}\label{e3.2}
\end{equation}
If $p\geq2$, it is easy to show that for any $x,y\in\mathbb{R}$,
there exists $c_{p}>0$  such that
\[
      (|x|^{p-2}x-|y|^{p-2}y)(x-y)\geq c_{p}|x-y|^{p},\quad p\geq 2.
\]
Combining this inequality with \eqref{e3.2}, we have
\begin{align*}
 c_{p}\|u_{n}-u_{0}\|^{p}
&\leq \|\varphi'(u_{n})-\varphi'(u_{0})\|\|u_{n}-u_{0}\|\\
&\quad -\rho(a)(\Phi_{p}(\frac{\alpha_2u_{n}(a)}{\alpha_1})
 -\Phi_{p}(\frac{\alpha_2u_{0}(a)}{\alpha_1}))(u_{n}(a)-u_{0}(a)) \\
&\quad -\rho(b)(\Phi_{p}(\frac{\beta_2u_{n}(b)}{\beta_1})
 -\Phi_{p}(\frac{\beta_2u_{0}(b)}{\beta_1}))(u_{n}(b)-u_{0}(b))\\
&\quad-\sum_{j=1}^{l}(\iota_j(u_{n}(x_j))-\iota_j(u_{0}(x_j)))(u_{n}(x_j)
 -u_{0}(x_j))\\
&\quad+\int_{a}^{b}(f(x,u_{n}(x))-f(x,u_{0}(x)))(u_{n}(x)-u_{0}(x))dx.
\end{align*}
It follows directly that $u_n\to u_0$ in $E$.

If $1<p<2$, by the results of \cite{b1}, there exists $d_{p}>0$ such that
\begin{align*}
&\int_{a}^{b}\rho(x)(\Phi_{p}(u'_{n}(x)) 
  -  \Phi_{p}(u_{0}'(x)))(u'_{n}(x)  -  u_{0}'(x))dx \\
& +  \int_{a}^{b}s(x)(\Phi_{p}(u_{n}(x))  -  \Phi_{p}(u_{0}(x)))\\
&\geq\frac{d_{p}2^{p-2}\|u_{n}-u_{0}\|^{2}}{(\|u_{n}\|+\|u_{0}\|)^{2-p}}
\end{align*}
Similarly, we can obtain that $u_{n}\to u_{0}$ in $E$, i.e. 
$\varphi$ satisfies the (PS)-condition.
\end{proof}

We choose an orthogonal basis $\{e_j\}$ of $E$ and define 
$X_j:=\operatorname{span}\{e_j\}$, $j=1,2,\dots$, 
$Y_{k}:=\oplus_{j=1}^{k} X_j$, $Z_{k}=\overline{\oplus_{j=k+1}^{\infty} X_j}$, 
then $E=Y_{k}\oplus Z_{k}$.

\begin{lemma} \label{lem3.2}
Suppose that $\Phi$ satisfies {\rm (I1), (F2)}, then there exists 
$k_{0}\in \mathbb{N}$ such that $C_{k_{0}}(\varphi, 0)\ncong0$.
\end{lemma}

\begin{proof}  
Since $F(x,0)=0$ and $I_j(0)=0 (j=1,2,\dots,m)$, then the zero function 
is a critical point of $\varphi$. So we only need to prove that $\varphi$ 
has a local linking at 0 with respect to $E=Y_{k}\oplus Z_{k}$.
\smallskip

\noindent\textbf{Step 1.}
 Take $u\in Y_{k}$, since $Y_{k}$ is finite dimensional, we have that for given
 $r_{0}$, there exists $0<\rho<1$ small such that
$$
u\in Y_{k}, \quad \|u\|\leq\rho\Rightarrow |u|<r_{0}, \quad x\in [a,b]
$$
For $0<r<r_{0}$, let $\Omega_1=\{x\in[a,b]:|u(x)|<r\}$, 
$\Omega_2=\{x\in[a,b]:r\leq|u(x)|\leq r_{0}\}$, 
$\Omega_{3}=\{x\in[a,b]:|u(x)|>r_{0}\}$, then 
$[a,b]=\bigcup_{i=1}^{3}\Omega_{i}$. For the sake of simplicity, 
let $G(x,u)=F(x,u)-c_2|u|^{\gamma}$. 
Therefore, it follows form (I1) and (F2) that
\begin{align*}
\varphi(u)&\leq\frac{1}{p}\|u\|^{p}+\sum_{j=1}^{m}a_j|u|^{\gamma_j+1}
 +\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-\int_{a}^{b}c_2|u|^{\gamma}dx
-\Big(\int_{\Omega_1}+\int_{\Omega_2}+\int_{\Omega_{3}}\Big)G(x,u)dx\\
&\leq\frac{1}{p}\|u\|^{p}+\sum_{j=1}^{m}a_j\|u\|^{\gamma_j+1}
 +\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-\int_{a}^{b}c_2|u|^{\gamma}dx-\int_{\Omega_1}G(x,u)dx.
\end{align*}

Note that the norms on $Y_{k}$ are equivalent to each other, 
$\|u\|_{p}$ is equivalent to $\|u\|$ and $\int_{\Omega_1}G(x,u)dx\to0$ as $r\to0$.
Since $0<\gamma<max\{\gamma_j+1\}<p$, then $\Phi(u)\leq0$, for all
 $u\in Y_{k}$ with $\|u\|\leq\rho$.
\smallskip

\noindent\textbf{Step 2.}
Take $u\in Z_{k}$, Since the embedding $E\hookrightarrow L^{p}([a,b])$ is compact.
  We have that for given $\varepsilon>0$, there exists $0<\rho<1$ small such that
$$
u\in Z_{k}, \; \|u\|\leq\rho\Rightarrow |u|<\varepsilon, \quad x\in [a,b].
$$
Therefore, it follows from (I1) and (F2) that
$$ 
\varphi(u)\geq\frac{1}{p}\|u\|^{p}-\int_{a}^{b}c_{3}|u|^{p}dx
\geq\frac{1}{p}\|u\|^{p}-\frac{1}{p}\|u\|^{p}>0.
$$
The proof is complete.
\end{proof}


\begin{proof}[Proof of Theorem \ref{the1.1}]
  By Lemma \ref{lem3.1}, $\varphi$ satisfies the (PS)-condition and is bounded 
from below. By Lemma \ref{lem3.2} and Proposition \ref{pro2.1}, 
the trivial solution $u=0$ is homological nontrivial and is not a minimizer. 
Then it follows immediately from Proposition \ref{pro2.2} that  \eqref{e1.1}
has at least two nontrivial solutions. 
\end{proof}

\begin{proof}[Proof of Theorem \ref{the1.2}]
  By (I2) and (F3), we can easily check the functional $\varphi$ is even. 
Lemma \ref{lem3.1} shows that $\varphi$ satisfies the (PS)-condition and is
 bounded from below. For $\rho>0$, let $K=S_{\rho}=\{u\in Y_{k}:\|u\|=\rho\}$.
 Thus, just as shown in the proof of Lemma \ref{lem3.2}, if $\rho>0$ 
is small enough, we have that
$$
\sup_{K}\varphi(u)\leq 0.
$$
By the definition of $Y_{k}$, we have $\dim Y_{k}=k$, then by 
Proposition \ref{pro2.3}, we have that $\varphi$ has at least $k$ 
distinct pairs of critical points. Therefore, \eqref{e1.1} has at least
$k$ distinct pairs of solutions. 
\end{proof}

\section{An example}\label{sec4}

In this section, we  illustrate our main results with an example.
In problem \eqref{e1.1}, let $p=2$, $\rho(x)=s(x)=1$,
\begin{gather*}
f(x,u)=\frac{1+\sin^{2}x}{1+e^{|x|}}\cdot\frac{2n-2}{n}|u|^{-\frac{2}{n}}u, \\
\iota_j(u)=\frac{2n-1}{n}|u|^{-\frac{1}{n}}u(j=1,2,\dots,m),
\end{gather*}
then 
$$
F(x,u)=\frac{1+\sin^{2}x}{1+e^{|x|}}|u|^{\frac{2n-2}{n}}, \quad 
I_j(u)=|u|^{\frac{2n-1}{n}}.
$$
When $n$ is an integer (large enough), we know that $f$  satisfies the conditions 
(F1) and (F2) and impulses $\iota_j$ $(j=1,2,\dots,m)$ fulfill (I1). 
By Theorem \ref{the1.1},  the problem has at least two nontrivial solutions.
 Furthermore, we can show that the nonlinearity $f$ and the impulses 
$\iota_j$ $(j=1,2,\dots,m)$ are all even. Thus by Theorem \ref{the1.2}, 
the problem has $k$ distinct pairs of solutions.

\subsection*{Acknowledgments}
This research was supported by Natural Science Foundation of China 11271372,
by the Fundamental Research Funds for the Central Universities of
Central South University 2015zzts010 and by the Mathematics and
Interdisciplinary Sciences project of CSU.
The authors are grateful to the anonymous referees  for their valuable comments 
and suggestions.


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