Multiplicity of Solutions for Quasilinear Differential Models Generated by Instantaneous and Non-Instantaneous Impulses

Abstract

This paper aims to deal with the multiplicity of weak solutions for quasilinear differential models generated by instantaneous and non-instantaneous impulses. By establishing the new variational structure and overcoming the influence of impulsive effects brought by the quasilinear term, some new results are acquired via the gene property, which extends and enriches some previous results. Moreover, an example is given to illustrate the conclusion of the main results.

1. Introduction

In this paper, we are concerned with the following one-dimensional second-order quasilinear differential equation with instantaneous and non-instantaneous impulses as follows.

$$\left\{\begin{array}{c}-u^{″}\left(t\right)+b\left(t\right)u\left(t\right)-{\left(u^2\left(t\right)\right)}^{″}u\left(t\right)=f_i(t,u\left(t\right)),a.e.t\in (s_i,t_{i+1}],i=0,1,2,\dots ,N,\hfill \\ \Delta \left(u^{\prime }\left(t_i\right)\right)=I_i\left(u\left(t_i\right)\right),i=1,2,\dots ,N,\hfill \\ u^{\prime }\left(t\right)={\theta }_i\left(t_i^{+}\right),t\in (t_i,s_i],i=1,2,\dots ,N,\hfill \\ u^{\prime }\left(s_i^{+}\right)=u^{\prime }\left(s_i^{-}\right),i=1,2,\dots ,N,\hfill \\ u\left(0\right)=u\left(T\right)=0,\hfill \end{array}\right.$$

(1)

where $f_i(t,u)=g_i(t,u)+\lambda h_i\left(t\right){\left|u\right|}^{\nu -2}u$, $g_i\in C((s_i,t_{i+1}]\times \mathbb{R},\mathbb{R})$, $b\in L^{\infty }([0,T],\mathbb{R})$, $h\in L^{\infty }((s_i,t_{i+1}],\mathbb{R})$, ${\theta }_i\in L^{\infty }((t_i,s_i],{\mathbb{R}}^{+})$, $\lambda \in {\mathbb{R}}^{+}$, ${\mathbb{R}}^{+}=[0,+\infty )$, $v\in [1,2)$, $s_0=0<t_1<s_1<t_2<\cdots <s_N<t_{N+1}=T$, $I_i\in C(\mathbb{R},\mathbb{R})$, $\Delta \left(u^{\prime }\left(t_i\right)\right)=u^{\prime }\left(t_i^{+}\right)-u^{\prime }\left(t_i^{-}\right)$ and $u^{\prime }\left(t_i^{\pm }\right)={lim}_{t\to t_i^{\pm }}{u}^{\prime }\left(t\right)$, ${\theta }_i\left(t_i^{+}\right)={lim}_{t\to t_i^{+}}{\theta }_i\left(t\right)$.

This problem has a practical background that arises from the standing wave solutions ($\varphi (t,x)=e^{-iwt}u\left(x\right),$$w\in \mathbb{R}$) of a kind of quasilinear Schrödinger equation as follows.

$$i{\partial }_t\varphi =-{\partial }_{xx}\varphi +V\left(x\right)\varphi -{\partial }_{xx}{\left(\right|\varphi |}^2{)\varphi -|\varphi |}^{q-1}\varphi ,x\in \mathbb{R},q>1.$$

(2)

For the theme of existence and multiplicity of standing wave solutions for (2), one can refer to [1,2,3,4] and the references therein. Naturally, an interesting question is whether there is a standing wave solution to (2) with suitable boundary conditions when impulsive effects happen. The multiplicity of solutions of boundary value problems (BVPs for short) to differential equations is an important research topic in the qualitative theory of differential equations. It originated from the practical application in the fields of physics and engineering, etc., and can ensure that appropriate solutions may be found in practical nonlinear problems. Therefore, it has important theoretical significance. By establishing the new variational structure and overcoming the influence of impulsive effects, the multiplicity of weak solutions for Dirichlet BVP (1) is considered via the gene property.

As is known to all that the BVPs of impulsive differential equations are an effective means to describe the discontinuous change of things. It has many practical applications in the scientific and technological fields, such as SIR epidemic models, aerospace technology, controllability, optimization, signal communication, and economic regulation, etc. (see [5,6,7,8] and references therein). Thus, BVPs of differential equations with impulses have attracted the attention of many scholars. For example, Nieto and O’Regan [9] dealt with the instantaneous impulsive Dirichlet BVP

$$\begin{array}{c}\hfill \left\{\begin{array}{c}-u^{″}\left(t\right)+\lambda u\left(t\right)=f(t,u\left(t\right)),a.e.t\in J,\hfill \\ \Delta \left(u^{\prime }\left(t_j\right)\right)=I_j\left(u\left(t_j\right)\right),j=1,2,...,m,\hfill \\ u\left(0\right)=u\left(T\right)=0,\hfill \end{array}\right.\end{array}$$

(3)

and achieved some existing results via employing some critical point theorems. Zhou and Li [10] extended the results of [9] to the case of the variable coefficient. For more articles concerning the second-order Dirichlet BVP with instantaneous impulsive effects, see Zhang and Yuan [11], Sun and Chen [12], etc. It should be mentioned that Shen and Liu [13] investigated the multiplicity of solutions for the Dirichlet BVP (1) by the symmetry mountain pass theorem with instantaneous impulsive effects.

In 2013, Hernández and O’Regan [14] firstly introduced the non-instantaneous impulsive problem, whose impulsive effects keep active on a finite time interval. Since then, more and more scholars have paid attention to this interesting problem (see [15,16] and references therein). Recently, Bai and Nieto [17] made use of the classical Lax–Milgram Theorem to construct the variational structure of the second-order Dirichlet BVP with non-instantaneous impulsive effects and obtained the existence and uniqueness of weak solutions. Khaliq and ur Rehman [18] extended the results of [17] to the case of the fractional Dirichlet BVP with non-instantaneous impulsive effects. Based on Ekeland’s variational principle, Tian and Zhang [19] created a further study on the existence of solutions for the second-order Dirichlet BVP with non-instantaneous and instantaneous impulses as follows.

$$\left\{\begin{array}{c}-u^{″}\left(t\right)=g_i(t,u\left(t\right)),t\in (s_i,t_{i+1}],i=0,1,2,\dots ,N\hfill \\ \Delta \left(u^{\prime }\left(t_i\right)\right)=I_i\left(u\left(t_i\right)\right),i=1,2,\dots N,\hfill \\ u^{\prime }\left(t\right)=u\left(t_i^{+}\right),t\in (t_i,s_i],i=1,2,\dots ,N,\hfill \\ u^{\prime }\left(s_i^{+}\right)=u^{\prime }\left(s_i^{-}\right),i=1,2,\dots ,N,\hfill \\ u\left(0\right)=u\left(T\right)=0.\hfill \end{array}\right.$$

(4)

Zhang and Liu [20] extended the results of [19] to the case of the fractional Dirichlet BVP with non-instantaneous and instantaneous impulsive effects. Moreover, for the topic of the existence of multiple weak solutions for impulsive equations, one can read [21,22] and references therein.

Motivated by the works mentioned above, we are concerned with the multiplicity of weak solutions for the Dirichlet BVP (1). Let us present the characteristics of this paper: First, under the influence of non-instantaneous and instantaneous impulsive effects, a new energy functional is established for the second-order Dirichlet BVP of quasilinear differential equations, which implies that the variational methods can be used to investigate the existence and multiplicity of weak solutions for this problem. Second, the non-instantaneous and instantaneous impulsive effects generated by the quasilinear term ${\left(u^2\right)}^{″}u$ are more complicated than the case of $u^{″}$, which makes this problem more interesting and difficult.

2. Preliminaries

To begin with, we introduce some necessary basic knowledge and signs. Let $C:=C\left(\right[0,T],\mathbb{R})$ with norm ${\parallel u\parallel }_{\infty }={max}_{t\in [0,T]}\left|u\left(t\right)\right|$ and $L^p:=L^p([0,T],\mathbb{R})$ with norm ${\parallel u\parallel }_{L^p}=({\int }_0^T{\left|u\left(t\right)\right|}^p{dt)}^{\frac{1}{p}},1\le p\le \infty$. In the Sobolev space $H_0^1(0,T)$, define the inner product

$$<u,v>={\int }_0^{T}u\left(t\right)v\left(t\right)dt+{\int }_0^T{u}^{\prime }\left(t\right)v^{\prime }\left(t\right)dt,\forall u,v\in H_0^1(0,T),$$

inducing the norm

$$\begin{array}{c}\hfill ∥u∥={\left({\int }_0^T{\left|u\left(t\right)\right|}^2+|u^{\prime }{\left(t\right)|}^{2}dt\right)}^{\frac{1}{2}}.\end{array}$$

(5)

By Poincaré’s inequality ${\parallel u\parallel }_{L^2}\le \frac{1}{\sqrt{\mu }}{\parallel u^{\prime }\parallel }_{L^2}$, where $\mu =\frac{{\pi }^2}{T^2}$ means the first eigenvalue relating to $-u^{″}=\mu u$ with Dirichlet boundary conditions, it follows that $\parallel u^{\prime }{\parallel }_{L^2}^2\le {\parallel u\parallel }^2\le (1+\frac{1}{\mu }){\parallel u^{\prime }\parallel }_{L^2}^2$. Therefore, the norm $\parallel u^{\prime }{\parallel }_{L^2}$ is equivalent to $\parallel u\parallel$. In this paper, assume that ${\mathrm{ess}\inf }_{t\in [0,T]}b\left(t\right)>q$ and $\rho =min\{2{\theta }_{min}^2,q\}>-\mu$, where ${\theta }_{min}={min}_{i=1,2,\dots ,N}{\theta }_i\left(t_i^{+}\right)$, q is a constant. If we consider the following inner product

$$<u,v{>}_{\rho }={\int }_0^T\rho u\left(t\right)v\left(t\right)dt+{\int }_0^T{u}^{\prime }\left(t\right)v^{\prime }\left(t\right)dt,\forall u,v\in H_0^1(0,T),$$

inducing the norm

$$\begin{array}{c}\hfill {∥u∥}_{\rho }={\left({\int }_0^T{\rho \left|u\left(t\right)\right|}^2+|u^{\prime }{\left(t\right)|}^{2}dt\right)}^{\frac{1}{2}},\end{array}$$

(6)

by the Lemma 2.1 in [10] and Poincaré’s inequality, there exists a constant $\vartheta \in (0,1)$ such that

$$\begin{array}{c}\hfill \vartheta \parallel u^{\prime }{\parallel }_{L^2}^2\le {\parallel u\parallel }_{\rho }^2\le (1+\frac{\rho }{\mu }){\parallel u^{\prime }\parallel }_{L^2}^2.\end{array}$$

(7)

Thus, the norms $\parallel u^{\prime }{\parallel }_{L^2}$, ${∥u∥}_{\rho }$ and $\parallel u\parallel$ are equivalent. Moreover, in view of the Sobolev imbedding theorem, we can find a constant $S>0$ such that ${\parallel u\parallel }_{\infty }\le S\parallel u\parallel$. It should be mentioned that for each $u\in H_0^1(0,T)$, u is absolutely continuous and $u^{\prime }\in L^2$. Thus, impulsive effects may occur. Therefore, the following lemma can be established.

Lemma 1.

If a function $u\in H_0^1(0,T)$ is a solution of problem (1), then the following identity

$$\begin{array}{ccc}& & {\int }_0^T{u}^{\prime }\left(t\right)v^{\prime }\left(t\right)dt+{\int }_0^T(2u^{\prime 2}\left(t\right)u\left(t\right)v\left(t\right)+2u^2\left(t\right)u^{\prime }\left(t\right)v^{\prime }\left(t\right))dt+\sum _{i=1}^N{\int }_{t_i}^{s_i}2{\theta }_i^2\left(t_i^{+}\right)u\left(t\right)v\left(t\right)dt\hfill \\ & & +\sum _{i=0}^N{\int }_{s_i}^{t_{i+1}}b\left(t\right)u\left(t\right)v\left(t\right)dt+\sum _{i=1}^N(2u^2\left(t_i\right)+1)I_i\left(u\left(t_i\right)\right)v\left(t_i\right)=\sum _{i=0}^N{\int }_{s_i}^{t_{i+1}}{f}_i(t,u\left(t\right))v\left(t\right)dt\hfill \end{array}$$

(8)

holds for any $v\in H_0^1(0,T)$.

Proof. 

In view of (1), we have

$$\begin{array}{ccc}& & {\int }_0^T(2u^{\prime 2}\left(t\right)u\left(t\right)v\left(t\right)+2u^2\left(t\right)u^{\prime }\left(t\right)v^{\prime }\left(t\right))dt\hfill \\ & & ={\int }_0^T{\left(u\left(t\right)v\left(t\right)\right)}^{\prime }{\left(u^2\left(t\right)\right)}^{\prime }dt\hfill \\ & & =\sum _{i=0}^N{\int }_{s_i}^{t_{i+1}}({\left(u\left(t\right)v\left(t\right)\right)}^{\prime }{\left(u^2\left(t\right)\right)}^{\prime }dt+\sum _{i=1}^N{\int }_{t_i}^{s_i}({\left(u\left(t\right)v\left(t\right)\right)}^{\prime }{\left(u^2\left(t\right)\right)}^{\prime }dt\hfill \\ & & =\sum _{i=0}^N\left(2u^2\left(t\right)u^{\prime }\left(t\right)v\left(t\right)\right){\mid }_{s_i^{+}}^{t_{i+1}^{-}}-\sum _{i=0}^N{\int }_{s_i}^{t_{i+1}}{\left(u^2\left(t\right)\right)}^{″}u\left(t\right)v\left(t\right)dt\hfill \\ & & +\sum _{i=1}^N\left(2u^2\left(t\right)u^{\prime }\left(t\right)v\left(t\right)\right){\mid }_{t_i^{+}}^{s_i^{-}}-\sum _{i=1}^N{\int }_{t_i}^{s_i}{\left(2u\left(t\right)u^{\prime }\left(t\right)\right)}^{\prime }u\left(t\right)v\left(t\right)dt\hfill \\ & & =\sum _{i=1}^N(2u^2\left(t_i\right)u^{\prime }\left(t_i^{-}\right)v\left(t_i\right)-2u^2\left(t_i\right)u^{\prime }\left(t_i^{+}\right)v\left(t_i\right))+2u^2\left(T\right)u^{\prime }\left(T\right)v\left(T\right)-2u^2\left(0\right)u^{\prime }\left(0\right)v\left(0\right)\hfill \\ & & +\sum _{i=1}^N(2u^2\left(s_i\right)u^{\prime }\left(s_i^{-}\right)v\left(s_i\right)-2u^2\left(s_i\right)u^{\prime }\left(s_i^{+}\right)v\left(s_i\right))-\sum _{i=0}^N{\int }_{s_i}^{t_{i+1}}{\left(u^2\left(t\right)\right)}^{″}u\left(t\right)v\left(t\right)dt\hfill \\ & & -\sum _{i=1}^N{\int }_{t_i}^{s_i}2u^{\prime 2}\left(t\right)u\left(t\right)v\left(t\right)dt\hfill \\ & & =-\sum _{i=1}^{N}2I_i\left(u\left(t_i\right)\right)u^2\left(t_i\right)v\left(t_i\right)-\sum _{i=0}^N{\int }_{s_i}^{t_{i+1}}{\left(u^2\left(t\right)\right)}^{″}u\left(t\right)v\left(t\right)dt\hfill \\ & & -\sum _{i=1}^{N}2{\theta }_i^2\left(t_i^{+}\right){\int }_{t_i}^{s_i}u\left(t\right)v\left(t\right)dt.\hfill \end{array}$$

(9)

Similarly, it follows that

$$\begin{array}{c}\hfill {\int }_0^T{u}^{\prime }\left(t\right)v^{\prime }\left(t\right)dt=-\sum _{i=1}^N{I}_i\left(u\left(t_i\right)\right)v\left(t_i\right)-\sum _{i=0}^N{\int }_{s_i}^{t_{i+1}}{u}^{″}\left(t\right)v\left(t\right)dt.\end{array}$$

(10)

Moreover, we can obtain

$$\begin{array}{c}\hfill {\int }_0^{T}b\left(t\right)u\left(t\right)v\left(t\right)dt=\sum _{i=0}^N{\int }_{s_i}^{t_{i+1}}b\left(t\right)u\left(t\right)v\left(t\right)dt+\sum _{i=1}^N{\int }_{t_i}^{s_i}b\left(t\right)u\left(t\right)v\left(t\right)dt,\end{array}$$

(11)

which together with the eqution

$$-u^{″}\left(t\right)+b\left(t\right)u\left(t\right)-{\left(u^2\left(t\right)\right)}^{″}u\left(t\right)=f_i(t,u\left(t\right)),a.e.t\in (s_i,t_{i+1}],$$

(9) and (10) yield (8). □

Definition 1.

A function $u\in H_0^1(0,T)$ is labeled as a weak solution of problem (1), if (8) is satisfied for any $v\in H_0^1(0,T)$.

Define the functional $\mathsf{\Phi }:H_0^1(0,T)\to \mathbb{R}$ by

$$\begin{array}{ccc}\hfill \mathsf{\Phi }\left(u\right)& =& \frac{1}{2}{\int }_0^T{u}^{\prime 2}\left(t\right)dt+\frac{1}{2}\sum _{i=0}^N{\int }_{s_i}^{t_{i+1}}b\left(t\right)u^2\left(t\right)dt+\sum _{i=1}^N{\int }_{t_i}^{s_i}{\theta }_i^2\left(t_i^{+}\right)u^2\left(t\right)dt+{\int }_0^T{u}^{\prime 2}\left(t\right)u^2\left(t\right)dt\hfill \\ & +& \sum _{i=1}^N{\int }_0^{u\left(t_i\right)}(2t^2+1)I_i\left(t\right)dt-\sum _{i=0}^N{\int }_{s_i}^{t_{i+1}}{G}_i(t,u\left(t\right))dt-\frac{\lambda }{\nu }\sum _{i=0}^N{\int }_{s_i}^{t_{i+1}}{h}_i\left(t\right){\left|u\left(t\right)\right|}^{\nu }dt.\hfill \end{array}$$

where $G_i(t,u)={\int }_0^u{g}_i(t,s)ds$. In view of the continuity of $g_i$ and $I_i$, by employing the standard approaches, we can get that $\mathsf{\Phi }\left(u\right)\in C^1(H_0^1(0,T),\mathbb{R})$ and the critical points of $\mathsf{\Phi }\left(u\right)$ are weak solutions of the problem (1).

Next, for obtaining our main results, some knowledge on “genus” will be presented. Let E be a Banach space, $\mathsf{\Phi }\in C^1(E,\mathbb{R})$,

$$\begin{array}{ccc}& & \mathsf{\Sigma }=\{A\subset E\backslash \{0\}:A\mathrm{is} \mathrm{closed} \mathrm{in} E\mathrm{and} \mathrm{symmetric} \mathrm{with} \mathrm{respect} \mathrm{to} 0\},\hfill \\ & & K_{\sigma }=\{u\in E:\mathsf{\Phi }\left(u\right)=\sigma ,{\mathsf{\Phi }}^{\prime }\left(u\right)=0\},{\mathsf{\Phi }}^{\sigma }=\{u\in E:\mathsf{\Phi }\left(u\right)\le \sigma \},\hfill \end{array}$$

where $\sigma \in \mathbb{R}$.

Definition 2

([23]). For $A\in \mathsf{\Sigma }$, if there is an odd map $\varphi \in C(A,{\mathbb{R}}^n\backslash \left\{0\right\})$ such that n is the smallest integer with this property, then the genus of A is n defined by $\gamma \left(A\right)=n$.

Lemma 2

([23]). Assume that $\mathsf{\Phi }\in C^1(E,\mathbb{R})$ meets the (PS)-condition. Moreover, Φ is an even functional. For any $n\in \mathbb{N}$, set

$${\mathsf{\Sigma }}_n=\{A\in \mathsf{\Sigma }:\gamma \left(A\right)\ge n\},{\sigma }_n=\underset{A\in {\mathsf{\Sigma }}_n}{inf}\underset{u\in A}{sup}\mathsf{\Phi }\left(u\right).$$

(i) 

If ${\mathsf{\Sigma }}_n\ne 0$ and ${\sigma }_n\in \mathbb{R}$, then ${\sigma }_n$ is a critical value of Φ;

(ii) 

If there exists $\kappa \in \mathbb{N}$ such that ${\sigma }_n={\sigma }_{n+1}=\cdots ={\sigma }_{n+\kappa }=\sigma \in \mathbb{R},$ and $\sigma \ne \mathsf{\Phi }\left(0\right)$, then $\gamma \left(K_{\sigma }\right)\ge \kappa +1.$

3. Main Results

In order to describe our main results, the following assumptions are given.

(I1)

For any $u\in \mathbb{R}$, $I_i\left(u\right)$ are odd in u and $I_i\left(u\right)u\ge 0$, $i=1,2\dots ,N.$

(I2)

There exist constants ${\alpha }_i>0,$ $d_1>0$ and ${\gamma }_i\in [0,1)$ such that

$$|I_i\left(u\right)|\le {\alpha }_i{\left|u\right|}^{{\gamma }_i}\mathrm{for}\mathrm{any}\left|u\right|\le d_1.$$

(G1)

There exist constants ${\beta }_i>0$, $d_2>0$ and $l\in [0,1)$ such that $g_i(t,u)$ are odd in u, $\forall (t,u)\in (s_i,t_{i+1}]\times [-d_2,d_2]$ and

$$g_i(t,u)\le {\beta }_i{\left|u\right|}^l,\forall (t,u)\in (s_i,t_{i+1}]\times \mathbb{R},i=1,2\dots ,N.$$

(G2)

There exist constants ${\xi }_i>0$, $d_3>0$, $\tau \in [\nu ,{\gamma }_{*}+1)$ and the open sets ${\mathsf{\Omega }}_i\subset (s_i,t_{i+1}]$ such that

$$G_i(t,u)\ge {\xi }_i{\left|u\right|}^{\tau },\forall (t,u)\in {\mathsf{\Omega }}_i\times [-d_3,d_3],i=1,2\dots ,N,$$

where ${\gamma }_{*}={min}_{i=1,2,\dots ,N}{\gamma }_i$. Moreover, ${\int }_{s_i}^{t_{i+1}}{G}_i(t,u)dt>0$.

Let ${\xi }_{min}={min}_{i=1,2,\dots ,N}{\xi }_i$. Now, we state our main results.

Theorem 1.

Assuming that the conditions (I1), (I2), (G1) and (G2) are fulfilled, there exist positive constants ${\xi }_{*},{\lambda }_{*}$ such that if ${\xi }_{min}\in (0,{\xi }_{*})$ and $\lambda \in [0,{\lambda }_{*})$, the Dirichlet BVP (1) has infinitely many nontrivial weak solutions $\left\{u_k\right\}$ satisfying $\mathsf{\Phi }\left(u_k\right)\to 0$ as $k\to +\infty$.

Remark 1.

In (G1), the oddness of $g_i(t,u)$ in u are local.

For obtaining our main results, inspired by [24], by constructing the following truncated functional, the following Lemma 3 can be established.

$$\begin{array}{ccc}\hfill J\left(u\right)& =& \frac{1}{2}{\int }_0^T{u}^{\prime 2}\left(t\right)dt+\frac{1}{2}\sum _{i=0}^N{\int }_{s_i}^{t_{i+1}}b\left(t\right)u^2\left(t\right)dt+\sum _{i=1}^N{\int }_{t_i}^{s_i}{\theta }_i^2\left(t_i^{+}\right)u^2\left(t\right)dt+{\int }_0^T{u}^{\prime 2}\left(t\right)u^2\left(t\right)dt\hfill \\ & -& \frac{\lambda }{\nu }\sum _{i=0}^N{\int }_{s_i}^{t_{i+1}}{h}_i{\left(t\right)\left|u\left(t\right)\right|}^{\nu }dt+\sum _{i=1}^N{\int }_0^{u\left(t_i\right)}(2t^2+1)I_i\left(t\right)dt-\mathrm{Y}(\parallel u\parallel )\sum _{i=0}^N{\int }_{s_i}^{t_{i+1}}{G}_i(t,u\left(t\right))dt,\hfill \end{array}$$

where $\mathrm{Y}\in C^1({\mathbb{R}}^{+},[0,1])$ satisfying

$$\left\{\begin{array}{c}{\mathrm{Y}}^{\prime }\left(t\right)\le 0,\forall t\in [0,T];\hfill \\ \mathrm{Y}\left(t\right)=0,\forall t\ge \frac{\eta }{S};\hfill \\ \mathrm{Y}\left(t\right)=1,\forall t\le \frac{\eta }{2S},\hfill \end{array}\right.$$

(12)

where $\eta =min\{d_1,d_2,d_3\}$. Assume that ${\beta }_{max}<\frac{\vartheta \mu {\eta }^{1-l}}{4T{S}^2(1+\mu )}$, where ${\beta }_{max}={max}_{i=1,2,\dots ,N}{\beta }_i$. Thus, the critical points $\left\{u_n\right\}$ of J satisfying $\parallel u_n\parallel \le \frac{\eta }{2S}$ are the critical points of $\mathsf{\Phi }$. Next, we show that the functional J satisfies the (PS)-condition.

Lemma 3.

Assume that the conditions of Theorem 1 hold, then there exists a positive constant ${\lambda }^{*}$ such that if $\lambda \in [0,{\lambda }^{*})$, $J\left(u\right)$ satisfies the (PS)-condition, i.e., for any $\left\{u_n\right\}\in H_0^1(0,T)$, if

$$\left\{J\left(u_n\right)\right\}isboundedand{J}^{\prime }\left(u_n\right)\to 0asn\to +\infty ,$$

then $\left\{u_n\right\}$ has a convergent subsequence in $H_0^1(0,T)$.

Proof. 

Based on the definition of $J\left(u\right)$, if $\parallel u\parallel \ge \frac{\eta }{S}$, by (I1), we can obtain

$$\begin{array}{ccc}\hfill J\left(u\right)& =& \frac{1}{2}{\int }_0^T{u}^{\prime 2}\left(t\right)dt+\frac{1}{2}\sum _{i=0}^N{\int }_{s_i}^{t_{i+1}}b\left(t\right)u^2\left(t\right)dt+\sum _{i=1}^N{\int }_{t_i}^{s_i}{\theta }_i^2\left(t_i^{+}\right)u^2\left(t\right)dt\hfill \\ & & +{\int }_0^T{u}^{\prime 2}\left(t\right)u^2\left(t\right)dt+\sum _{i=1}^N{\int }_0^{u\left(t_i\right)}(2t^2+1)I_i\left(t\right)dt-\frac{\lambda }{\nu }\sum _{i=0}^N{\int }_{s_i}^{t_{i+1}}{h}_i\left(t\right){\left|u\left(t\right)\right|}^{\nu }dt\hfill \\ & \ge & \frac{1}{2}{\int }_0^T{u}^{\prime 2}\left(t\right)dt+\frac{1}{2}{\int }_0^T\rho u^2\left(t\right)dt-\frac{\lambda h_{max}}{\nu }{\int }_0^T{\left|u\left(t\right)\right|}^{\nu }dt\hfill \\ & \ge & \frac{\vartheta \mu }{2(1+\mu )}{\parallel u\parallel }^2-\frac{\lambda h_{max}T{S}^{\nu }}{\nu }{\parallel u\parallel }^{\nu },\hfill \end{array}$$

which yields that

$$\begin{array}{c}\hfill J\left(u\right)\to +\infty as\parallel u\parallel \to +\infty ,\end{array}$$

(13)

where $h_{max}={max}_{i=1,2,\dots ,N}{\parallel h_i\parallel }_{L^{\infty }}$. Thus, $J\left(u\right)$ is coercive and bounded from below. Moreover, for any $\left\{u_n\right\}\in H_0^1(0,T)$, if $\left\{J\left(u_n\right)\right\}$ is bounded and $J^{\prime }\left(u_n\right)\to 0,$ it follows that $\left\{u_n\right\}$ is bounded in $H_0^1(0,T)$ by (13). Based on the fact that $H_0^1(0,T)$ is a reflexive Banach space, $\left\{u_n\right\}$ has a convergent subsequence (called again $\left\{u_n\right\}$). Since $H_0^1(0,T)$ is compactly embedded into C, so $u_n\rightharpoonup u$ in $H_0^1(0,T)$, $u_n\to u$ uniformly in C. If $\frac{\eta }{2S}<\parallel u_n\parallel \le \frac{\eta }{S}$, by (I1), (G1) and (G2), we have

$$\begin{array}{ccc}\hfill J^{\prime }\left(u_n\right)u_n& =& {\int }_0^T{u^{\prime }}_n^2\left(t\right)dt+\sum _{i=0}^N{\int }_{s_i}^{t_{i+1}}b\left(t\right)u_n^2\left(t\right)dt+2\sum _{i=1}^N{\int }_{t_i}^{s_i}{\theta }_i^2\left(t_i^{+}\right)u_n^2\left(t\right)dt+4{\int }_0^T{u^{\prime }}_n^2\left(t\right)u_n^2\left(t\right)dt\hfill \\ & & +\sum _{i=1}^N(2u^2\left(t_i\right)+1)I_i\left(u\left(t_i\right)\right)u_n\left(t_i\right)-\lambda \sum _{i=0}^N{\int }_{s_i}^{t_{i+1}}{h}_i\left(t\right){\left|u_n\left(t\right)\right|}^{\nu }dt\hfill \\ & & -{\mathrm{Y}}^{\prime }(\parallel u_n\parallel )\parallel u_n\parallel \sum _{i=0}^N{\int }_{s_i}^{t_{i+1}}{G}_i(t,u_n\left(t\right))dt-\mathrm{Y}(\parallel u_n\parallel )\sum _{i=0}^N{\int }_{s_i}^{t_{i+1}}{g}_i(t,u_n\left(t\right))u_n\left(t\right)dt\hfill \\ & \ge & {\int }_0^T{u^{\prime }}_n^2\left(t\right)dt+{\int }_0^T\rho u_n^2\left(t\right)dt-\lambda h_{max}{\int }_0^T|u_n{\left(t\right)|}^{\nu }dt-{\beta }_{max}{\int }_0^T{\left|u_n\left(t\right)\right|}^{l+1}dt\hfill \\ & \ge & \frac{\vartheta \mu }{1+\mu }\parallel u_n{\parallel }^2-\lambda h_{max}T{S}^{\nu }\parallel u_n{\parallel }^{\nu }-{\beta }_{max}T{S}^{l+1}{\parallel u_n\parallel }^{l+1}\hfill \\ & \ge & \frac{\vartheta \mu }{1+\mu }{\left(\frac{\eta }{2S}\right)}^2-\lambda h_{max}T{\eta }^{\nu }-{\beta }_{max}T{\eta }^{l+1},\hfill \end{array}$$

which together with ${\beta }_{max}<\frac{\vartheta \mu {\eta }^{1-l}}{4T{S}^2(1+\mu )}$ yield that there exists a positive constant ${\lambda }^{**}$ such that if $\lambda \in [0,{\lambda }^{**})$, $J^{\prime }\left(u_n\right)u_n>0$. If $\parallel u_n\parallel >\frac{\eta }{S}$, $J^{\prime }\left(u_n\right)u_n\ge {\parallel u_n\parallel }^v(\frac{\vartheta \mu }{1+\mu }{\left(\frac{\eta }{s}\right)}^{2-v}-\lambda h_{max}T{S}^{\nu })$. Hence, there exists a positive constant ${\lambda }^{***}$ such that if $\lambda \in [0,{\lambda }^{***})$, $J^{\prime }\left(u_n\right)u_n>0$. Therefore, $J^{\prime }\left(u_n\right)\nrightarrow 0$ for $\lambda \in [0,{\lambda }^{*})$, where ${\lambda }^{*}=min\{{\lambda }^{**},{\lambda }^{***}\}$. Thus, we just need to deal with the case of $\parallel u_n\parallel \le \frac{\eta }{2S}$. It follows that $|u_n|\le \parallel u_n{\parallel }_{\infty }\le S\parallel u_n\parallel \le \frac{\eta }{2},$ which together with (I2), (G1), $u_n\rightharpoonup u$ in $H_0^1(0,T)$, $u_n\to u$ uniformly in C, $J^{\prime }\left(u_n\right)\to 0asn\to +\infty$ and

$$\begin{array}{ccc}& & {\int }_0^T(u_n^2\left(t\right)u_n^{\prime }\left(t\right)-u^2\left(t\right)u^{\prime }\left(t\right))(u_n^{\prime }\left(t\right)-u^{\prime }\left(t\right))dt\hfill \\ & & ={\int }_0^T{u}_n^{\prime }\left(t\right)(u_n^{\prime }\left(t\right)-u^{\prime }\left(t\right))(u_n^2\left(t\right)-u^2\left(t\right))dt+{\int }_0^T{u}^2\left(t\right){|u_n^{\prime }\left(t\right)-u^{\prime }\left(t\right)|}^{2}dt\hfill \end{array}$$

yields that

$$\begin{array}{ccc}\hfill o\left(1\right)& =& <J^{\prime }\left(u_n\right)-J^{\prime }\left(u\right),u_n-u>={\int }_0^T|u_n^{\prime }\left(t\right)-u^{\prime }{\left(t\right)|}^{2}dt+\sum _{i=0}^N{\int }_{s_i}^{t_{i+1}}b\left(t\right)|u_n{\left(t\right)-u\left(t\right)|}^{2}dt\hfill \\ & & +\sum _{i=1}^N{\int }_{t_i}^{s_i}2{\theta }_i^2\left(t_i^{+}\right){|u_n\left(t\right)-u\left(t\right)|}^{2}dt+{\int }_0^T(u_n^{\prime 2}\left(t\right)u_n\left(t\right)-u^{\prime 2}\left(t\right)u\left(t\right))(u_n\left(t\right)-u\left(t\right))dt\hfill \\ & & +{\int }_0^T(u_n^2\left(t\right)u_n^{\prime }\left(t\right)-u^2\left(t\right)u^{\prime }\left(t\right))(u_n^{\prime }\left(t\right)-u^{\prime }\left(t\right))dt\hfill \\ & & -\lambda \sum _{i=0}^N{\int }_{s_i}^{t_{i+1}}{h}_i\left(t\right)\left(\right|u_n{\left(t\right)|}^{\nu -2}{u}_n\left(t\right)-{\left|u\left(t\right)\right|}^{\nu -2}u\left(t\right))(u_n\left(t\right)-u\left(t\right))dt\hfill \\ & & +\sum _{i=1}^N(2u_n^2\left(t_i\right)+1)I_i\left(u_n\left(t_i\right)\right)(u_n\left(t_i\right)-u\left(t_i\right))-\sum _{i=1}^N(2u^2\left(t_i\right)+1)I_i\left(u\left(t_i\right)\right)(u_n\left(t_i\right)-u\left(t_i\right))\hfill \\ & & -\sum _{i=0}^N{\int }_{s_i}^{t_{i+1}}(g_i(t,u_n\left(t\right))-g_i(t,u\left(t\right)))(u_n\left(t\right)-u\left(t\right))dt\hfill \\ & =& {\int }_0^T|u_n^{\prime }\left(t\right)-u^{\prime }{\left(t\right)|}^{2}dt+{\int }_0^T{u}^2\left(t\right){|u_n^{\prime }\left(t\right)-u^{\prime }\left(t\right)|}^{2}dt+o\left(1\right),\hfill \end{array}$$

which leads to that $u_n\to u$ in $H_0^1(0,T)$, which means that $J\left(u\right)$ satisfies the (PS)-condition. □

Proof. 

Note that $J\in C^1(H_0^1(0,T),\mathbb{R})$ and $J\left(0\right)=0$. If $\parallel u\parallel \ge \frac{\eta }{S}$, based on the definition of $\mathrm{Y}$, we have $J(-u)=J\left(u\right)$. If $\parallel u\parallel \le \frac{\eta }{S}$, we have $\left|u\right|\le {\parallel u\parallel }_{\infty }\le S\parallel u\parallel \le \eta$, which together with (I1) and (G1) yield that $J(-u)=J\left(u\right)$. Thus, $J(-u)=J\left(u\right),\forall u\in H_0^1(0,T).$ Next, in view of Lemma 2, we aim to present that there exists $m>0$ such that

$$\gamma \left(J^{-m}\right)\ge k,k\in {\mathbb{Z}}^{+}.$$

Let k disjoint open sets ${\mathsf{\Lambda }}_i$ satisfy ${\bigcup }_{i=1}^k{\mathsf{\Lambda }}_i\subset {\bigcup }_{i=1}^N{\mathsf{\Omega }}_i$, where $k\ge N.$ Moreover, there has at least one $\iota \in \{1,2,\dots ,k\}$ such that

$${\mathsf{\Lambda }}_{\iota }\subset {\mathsf{\Omega }}_i\subset (s_i,t_{i+1}].$$

Let $u_i\in H_0^1\left({\mathsf{\Lambda }}_i\right)\backslash \left\{0\right\}$, $\parallel u_i\parallel =1$ and

$${\Delta }_k=\mathrm{span}\{u_1,u_2,\dots ,u_k\},{\mathsf{\Pi }}_k=\{u\in {\Delta }_k:\parallel u\parallel =1\}.$$

For any $u\in {\Delta }_k$, in view of the equivalence of the norms on the finite-dimensional space, there exist $c_1>0,c_2>0$ such that $c_1\parallel u\parallel \le {\parallel u\parallel }_{L^{\tau }},$ ${\parallel u\parallel }_{L^{\nu }}\le c_2\parallel u\parallel .$ Moreover, it follows that there exists ${\omega }_i\in \mathbb{R},i=1,2,\dots ,k$ such that $u\left(t\right)={\sum }_{i=1}^k{\omega }_i{u}_i\left(t\right),$

$${\parallel u\parallel }_{L^{\tau }}^{\tau }=\sum _{i=1}^k|{\omega }_i{|}^{\tau }{\int }_{{\mathsf{\Lambda }}_i}|u_i{\left(t\right)|}^{\tau }{dt,\parallel u\parallel }_{L^{\nu }}^{\nu }=\sum _{i=1}^k|{\omega }_i{|}^{\nu }{\int }_{{\mathsf{\Lambda }}_i}{\left|u_i\left(t\right)\right|}^{\nu }dt$$

and

$$\begin{array}{ccc}\hfill {\parallel u\parallel }^2& =& \sum _{i=1}^k{\omega }_i^2{\int }_{{\mathsf{\Lambda }}_i}(|u_i{\left(t\right)|}^2+|u_i^{\prime }{\left(t\right)|}^2)dt=\sum _{i=1}^k{\omega }_i^2.\hfill \end{array}$$

(14)

Let us reorder ${\gamma }_i(i=1,2,\dots ,N)$ as follows.

$$0\le {\tilde{\gamma }}_1\le {\tilde{\gamma }}_2\le \cdots \le {\tilde{\gamma }}_N<1,$$

where ${\tilde{\gamma }}_1={\gamma }_{min}$. Now, we need to consider two cases:

(i) h_i is a negative or sign-changing function; (ii) h_i is a positive function.

For case (i), from (I2), (G2) and (14), for any $u\in {\mathsf{\Pi }}_k$ and $0<\zeta \le r$ that $r=min\{1,\frac{\eta }{2S}\}$, we have

$$\begin{array}{ccc}\hfill J\left(\zeta u\right)& \le & \frac{{(\parallel b\parallel }_{L^{\infty }}+2{\theta }_{max}^2+1){\zeta }^2}{2}{∥u∥}^2+\sum _{i=1}^N{\alpha }_i{S}^{{\gamma }_i+1}{\zeta }^{{\gamma }_i+1}{\parallel u\parallel }^{{\gamma }_i+1}+\sum _{i=1}^{N}2{\alpha }_i{S}^{{\gamma }_i+3}{\zeta }^{{\gamma }_i+3}{\parallel u\parallel }^{{\gamma }_i+3}\hfill \\ & & +S^2{\zeta }^4{\parallel u\parallel }^4-{\zeta }^{\tau }{\xi }_{min}\sum _{i=1}^k|{\omega }_i{|}^{\tau }{\int }_{{\mathsf{\Lambda }}_i}|u_i{\left(t\right)|}^{\tau }dt+\lambda {\zeta }^{\nu }{h}_{max}\sum _{i=1}^k|{\omega }_i{|}^{\nu }{\int }_{{\mathsf{\Lambda }}_i}{\left|u_i\left(t\right)\right|}^{\nu }dt\hfill \\ & \le & \frac{{(\parallel b\parallel }_{L^{\infty }}+2{\theta }_{max}^2+1){\zeta }^2}{2}+\sum _{i=1}^N{\alpha }_i{S}^{{\gamma }_i+1}{\zeta }^{{\gamma }_i+1}+\sum _{i=1}^{N}2{\alpha }_i{S}^{{\gamma }_i+3}{\zeta }^{{\gamma }_i+3}\hfill \\ & & +S^2{\zeta }^4-{\zeta }^{\tau }{\xi }_{min}{c}_1^{\tau }+\lambda {\zeta }^{\nu }{h}_{max}{c}_2^{\nu }\hfill \\ & \le & M{\zeta }^{{\tilde{\gamma }}_1+1}-{\zeta }^{\tau }{\xi }_{min}{c}_1^{\tau }+\lambda {\zeta }^{\nu }{h}_{max}{c}_2^{\nu }\hfill \\ & =& {\zeta }^{\nu }(M{\zeta }^{{\tilde{\gamma }}_1+1-\nu }-{\zeta }^{\tau -\nu }{\xi }_{min}{c}_1^{\tau }+\lambda h_{max}{c}_2^{\nu }),\hfill \end{array}$$

where $M=\frac{{\parallel b\parallel }_{L^{\infty }}+2{\theta }_{max}^2+1}{2}+{\displaystyle \sum _{i=1}^N}{\alpha }_i{S}^{{\gamma }_i+1}+{\displaystyle \sum _{i=1}^N}2{\alpha }_i{S}^{{\gamma }_i+3}+S^2$, ${\theta }_{max}={max}_{i=1,2,\dots ,N}{\theta }_i\left(t_i^{+}\right)$. Define

$$\psi \left(t\right)=M{t}^{{\tilde{\gamma }}_1+1-\nu }-t^{\tau -\nu }{\xi }_{min}{c}_1^{\tau },0<t\le r.$$

Thus, from ${\xi }_{min}\in (0,{\xi }^{*})$ that ${\xi }^{*}=\frac{M{r}^{{\tilde{\gamma }}_1+1-\nu }({\tilde{\gamma }}_1+1-\nu )}{c_1^{\tau }(\tau -\nu )}$, there exists

$$t_0={(\frac{{\xi }_{min}{c}_1^{\tau }(\tau -\nu )}{M({\tilde{\gamma }}_1+1-\nu )})}^{\frac{1}{{\tilde{\gamma }}_1+1-\tau }}\in (0,r],$$

such that for $\tau -{\tilde{\gamma }}_1-1<0$,

$$\psi \left(t_0\right)=\underset{0<t\le r}{min}\psi \left(t\right)=\frac{{\xi }_{min}{c}_1^{\tau }(\tau -{\tilde{\gamma }}_1-1)}{{\tilde{\gamma }}_1+1-\nu }{(\frac{{\xi }_{min}{c}_1^{\tau }(\tau -\nu )}{M({\tilde{\gamma }}_1+1-\nu )})}^{\frac{\tau -\nu }{{\tilde{\gamma }}_1+1-\tau }}<0.$$

Hence, if $\lambda \in [0,{\lambda }^{****})$ and

$${\lambda }^{****}=\frac{{\xi }_{min}\nu c_1^{\tau }({\tilde{\gamma }}_1+1-\tau )}{c_2^{\nu }{h}_{max}({\tilde{\gamma }}_1+1-\nu )}{(\frac{{\xi }_{min}{c}_1^{\tau }(\tau -\nu )}{M({\tilde{\gamma }}_1+1-\nu )})}^{\frac{\tau -\nu }{{\tilde{\gamma }}_1+1-\tau }}>0,$$

we can find constats $m>0,\varpi >0$ such that

$$\begin{array}{c}\hfill J\left(\varpi u\right)<-m\mathrm{for}u\in {\mathsf{\Pi }}_k.\end{array}$$

(15)

Moreover, (15) holds for case (ii), provided that ${\xi }_{min}\in {\mathbb{R}}^{+}\backslash \left\{0\right\},\lambda \in {\mathbb{R}}^{+}.$ Choose $\lambda \in [0,{\lambda }_{*})$, where ${\lambda }_{*}=min\{{\lambda }^{****},{\lambda }^{*}\}$. It implies that J satisfies the (PS)-condition and (15). Let

$${\mathsf{\Pi }}_k^{\varpi }=\{\varpi u:u\in {\mathsf{\Pi }}_k\},\mathsf{\Xi }=\{({\omega }_1,{\omega }_2,\dots ,{\omega }_k)\in {\mathbb{R}}^k:\sum _{i=1}^k{\omega }_i^2<{\varpi }^2\}.$$

From (15), we have

$$J\left(u\right)<-m\mathrm{for}u\in {\mathsf{\Pi }}_k^{\varpi },$$

which implies that ${\mathsf{\Pi }}_k^{\varpi }\subset J^{-m}\in \mathsf{\Sigma }.$ Moreover, based on (15), there exists an odd homeomorphism mapping $\varphi \in C({\mathsf{\Pi }}_k^{\varpi },\partial \mathsf{\Xi })$, which together with the properties of genus (see [23]) yield that

$$\begin{array}{c}\hfill \gamma \left(J^{-m}\right)\ge \gamma \left({\mathsf{\Pi }}_k^{\varpi }\right)=\gamma (\partial \mathsf{\Xi })=k.\end{array}$$

(16)

Let

$${\sigma }_k=\underset{A\in {\mathsf{\Sigma }}_k}{inf}\underset{u\in A}{sup}J\left(u\right).$$

Since $J\left(u\right)$ is coercive and bounded from below, by (16), one has $-\infty <{\sigma }_k\le -m<0$. Based on the above facts, by Lemma 2, the Dirichlet BVP (1) has infinitely many nontrivial weak solutions $\left\{u_k\right\}$ satisfying $\parallel u_k\parallel \le \frac{\eta }{2S}$ and $\mathsf{\Phi }\left(u_k\right)\to 0$ as $k\to +\infty$. □

Next, an example is given to illustrate the conclusion of the main results.

4. Example

Consider the following problem:

$$\left\{\begin{array}{c}-u^{″}\left(t\right)+(t+1)u\left(t\right)-{\left(u^2\left(t\right)\right)}^{″}u\left(t\right)=u^{\frac{1}{9}}\left(t\right)+\lambda {\left|u\left(t\right)\right|}^{-\frac{17}{18}}u\left(t\right),a.e.t\in (s_i,t_{i+1}],i=0,1,2,\dots ,N,\hfill \\ \Delta \left(u^{\prime }\left(t_i\right)\right)=u^{\frac{1}{3}}\left(t_i\right),i=1,2,\dots ,N,\hfill \\ u^{\prime }\left(t\right)={sin}^2\left(t_i^{+}\right),t\in (t_i,s_i],i=1,2,\dots ,N,\hfill \\ u^{\prime }\left(s_i^{+}\right)=u^{\prime }\left(s_i^{-}\right),i=1,2,\dots ,N,\hfill \\ u\left(0\right)=u\left(T\right)=0.\hfill \end{array}\right.$$

(17)

It is not difficult to verify that the conditions (I1), (I2), (G1), (G2) are satisfied.

5. Conclusions

Under the influence of non-instantaneous and instantaneous impulsive effects, by constructing a new energy functional, which makes the variational methods applicable, and overcoming the difficulties brought by the quasilinear term ${\left(u^2\left(t\right)\right)}^{″}u$, the multiplicity of weak solutions for quasilinear differential models generated by instantaneous and non-instantaneous impulses are obtained, which extend and enrich some previous results. In the future, we will develop a further study of the multiplicity of weak solutions to BVPs of differential equations with non-instantaneous and instantaneous impulsive effects when the nonlinear term g satisfies the satisfies superlinear growth.