Open Access

Existence and multiplicity of weak solutions for a nonlinear impulsive \((q,p)\)-Laplacian dynamical system

Advances in Difference Equations20172017:128

DOI: 10.1186/s13662-017-1145-y

Received: 4 December 2016

Accepted: 17 March 2017

Published: 3 May 2017

Abstract

In this paper, we investigate the existence and multiplicity of nontrivial weak solutions for a class of nonlinear impulsive \((q,p)\)-Laplacian dynamical systems. The key contributions of this paper lie in (i) Exploiting the least action principle, we deduce that the system we are interested in has at least one weak solution if the potential function has sub-\((q,p)\) growth or \((q,p)\) growth; (ii) Employing a critical point theorem due to Ding (Nonlinear Anal. 25(11):1095-1113, 1995), we derive that the system involved has infinitely many weak solutions provided that the potential function is even.

Keywords

\((q,p)\)-Laplacian existence multiplicity nontrivial solution variational methods

MSC

34C25 58E50

1 Introduction and main results

For \(N\in\mathbb{N}\), let \((\mathbb {R}^{N},\langle\cdot,\cdot\rangle,|\cdot|)\) be the N-dimensional Euclidean space. For fixed \(l,k\in\mathbb{N}\), set \(B:=\{1,2,\ldots, l\} \) and \(C:=\{1,2,\ldots, k\}\). If \(f:\mathbb {R}^{n}\rightarrow \mathbb {R}\) is a smooth function, let f stand for the gradient operator. For a smooth function \(f:\mathbb {R}^{n}\times \mathbb {R}^{n}\rightarrow \mathbb {R}\), denote by \(\nabla_{x_{1}}f\) and \(\nabla_{x_{2}}f\) the gradient operator with respect to the first component and the second component, respectively. For a mapping \(f:\mathbb {R}_{+}\rightarrow \mathbb {R}\), \(f(t^{+})\) and \(f(t^{-})\) mean the right-hand side limit and the left-land side limit at t, respectively. For functions \(f:\mathbb {R}^{n}\rightarrow \mathbb {R}^{n}\) and \(g:\mathbb {R}_{+}\rightarrow \mathbb {R}^{n}\), let \(\triangle(f(g(t)))=f(g(t^{+}))-f(g(t^{-}))\).

In this paper, we consider a nonlinear system with impulsive effects on \(\mathbb{H}^{N}:=\mathbb {R}^{N}\times \mathbb {R}^{N}\) for any \(p,q>1\), \(\lambda >0\), \(j\in B\), and \(m\in C\),
$$ \left \{ \textstyle\begin{array}{@{}l} -\frac{d(\Phi_{q}(\dot{u}_{1}(t)))}{dt} +\Phi_{q}({u}_{1}(t))=\lambda\nabla_{u_{1}} F(t,u_{1}(t),u_{2}(t)), \quad \mbox{a.e. } t\in[0,T],\\ -\frac{d(\Phi_{p}(\dot{u}_{2}(t)))}{dt}+\Phi_{p}({u}_{2}(t))=\lambda\nabla_{u_{2}} F(t,u_{1}(t),u_{2}(t)) ,\quad \mbox{a.e. } t\in[0,T],\\ \Delta(\Phi_{q}(\dot{u}_{1}(t_{j})))=\nabla I_{j}(u_{1}(t_{j})), \\ \Delta(\Phi_{p}(\dot{u}_{2}(s_{m})))=\nabla K_{m}(u_{2}(s_{m})), \end{array}\displaystyle \right . $$
(1.1)
with the initial condition \((\dot{u}_{1}(0),\dot{u}_{2}(0))=(u_{1}(0),u_{2}(0))\in\mathbb{H}^{N}\) and the terminal condition \((\dot{u}_{1}(T),\dot{u}_{2}(T))=(0, 0)\in\mathbb{H}^{N}\), where \(\Phi_{\mu}(z):=|z|^{\mu-2}z\) for any \(\mu>1\) and \(z\in \mathbb {R}^{N}\); \(F: \mathbb {R}_{+}\times\mathbb{H}^{N}\to{\mathbb {R}}\); \((t_{j})_{j\in B}\) and \((s_{m})_{m\in C}\) are impulsive times with \(0=t_{0}< t_{1}< t_{2}<\cdots<t_{l}<t_{l+1}=T\), \(0=s_{0}< s_{1}< s_{2}<\cdots<s_{k}<s_{k+1}=T\), and for \(j\in B\) and \(m\in C\), \(I_{j}:{\mathbb {R}}^{N}\to{\mathbb {R}}\) and \(K_{m}:{\mathbb {R}}^{N}\to{\mathbb {R}}\) are continuously differentiable.
For the nonlinear term \(F: [0,T]\times\mathbb{H}^{N}\to{\mathbb {R}}\), we assume that
  1. (A1)

    For fixed \(t\in[0,T]\) and \(x\in\mathbb{H}^{N}\), \(F(\cdot, x) \) is measurable and \(F(t, \cdot) \) is continuously differentiable;

     
  2. (A2)
    There exist \(a_{1},a_{2}\in C({\mathbb {R}}_{+};{\mathbb {R}}_{+})\) and \(b\in L^{1}([0,T];{\mathbb {R}}_{+})\) such that
    $$\begin{aligned}& \bigl\vert F(t, x_{1},x_{2})\bigr\vert \leq\bigl[ a_{1}\bigl(\vert x_{1}\vert \bigr)+a_{2}\bigl( \vert x_{2}\vert \bigr)\bigr]b(t), \\& \bigl\vert \nabla F(t,x_{1},x_{2})\bigr\vert \leq \bigl[a_{1}\bigl(\vert x_{1}\vert \bigr)+a_{2} \bigl(\vert x_{2}\vert \bigr)\bigr]b(t), \\& \bigl\vert I_{j}(x_{1})\bigr\vert \leq a_{1} \bigl(\vert x_{1}\vert \bigr), \qquad \bigl\vert \nabla I_{j}(x_{1})\bigr\vert \leq a_{1}\bigl(\vert x_{1}\vert \bigr), \quad j\in B, \\& \bigl\vert K_{m}(x_{2})\bigr\vert \leq a_{2} \bigl(\vert x_{2}\vert \bigr), \qquad\bigl\vert \nabla K_{m}(x_{2})\bigr\vert \leq a_{2}\bigl(\vert x_{2}\vert \bigr), \quad m\in C \end{aligned}$$
    for all \((x_{1},x_{2})\in{\mathbb{R}}^{N}\times \mathbb {R}^{N}\) and a.e. \(t\in[0, T]\).
     
For \(N=1\), \(p=q=2\), \(F(t,x_{1},x_{2})=F(t,x_{1})\), and \(I_{j}\equiv0\) (\(j\in B\)), system (1.1) reduces to the following second order impulsive differential equation:
$$ \left \{ \textstyle\begin{array}{@{}l} u_{1}''(t) +u_{1}(t)=\lambda_{1}\nabla_{u_{1}} F(t,u_{1}(t)), \quad \mbox{a.e. }t\in[0,+\infty),\\ {u}_{1}'(0)=u_{1}(0),\\ {u}_{1}'(T)=0,\\ \Delta({u}_{1}'(t_{j}))= I_{j}(u_{1}(t_{j})), \quad j\in B. \end{array}\displaystyle \right . $$
(1.2)
Recently, Chen and Sun [2] investigated the following second order impulsive differential equation:
$$ \left \{ \textstyle\begin{array}{@{}l} u_{1}''(t) +u_{1}(t)=\lambda f(t,u_{1}(t)), \quad \mbox{a.e. }t\in[0,+\infty),\\ {u}_{1}'(0^{+})=g(u_{1}(0)),\\ {u}_{1}'(+\infty)=0,\\ \Delta({u}_{1}'(t_{j}))= I_{j}(u_{1}(t_{j})), \quad j\in B, \end{array}\displaystyle \right . $$
(1.3)
where \(f\in C(\mathbb {R}_{+}\times\mathbb {R};\mathbb {R})\), \(g, I_{j}\in C(\mathbb {R};\mathbb {R})\). In [2], the authors not only established the variational structure of equation (1.3) but also obtained that (1.3) enjoys three solutions by using an abstract critical point theorem taken from [3]. More precisely, they obtained the following theorem.

Theoremm A

[2], Theorem 3.1

Suppose that
  1. (H1)

    \(g(u), I_{j}(u)\) are nondecreasing, and \(g(u)u\ge0\), \(I_{j}(u)u\ge0\) for any \(u\in\mathbb {R}\);

     
  2. (H2)
    There exist \(a>0,l\in(0,2)\), \(b\in L^{1}(\mathbb {R}_{+};\mathbb {R}_{+})\), and \(c\in L^{2}(\mathbb {R}_{+};\mathbb {R}_{+})\) such that
    $$F(t,u)\le b(t) \bigl(a+|u|^{l}\bigr), \qquad f(t,u)\le c(t)|u|^{l-1}, $$
    for a.e. \(t\ge0\) and \(u\in\mathbb {R}\), where \(F(t,u):=\int_{0}^{u} f(t,s)\,ds\);
     
  3. (H3)
    There exist \(d,m,M>0\) such that
    $$\frac{d^{2}}{M^{2}}< m^{2}+2\sum_{j=1}^{l} \int_{0}^{me^{-t_{j}}}I_{j}(s)\,ds+2 \int_{0}^{m} g(s)\,ds; $$
     
  4. (H4)
    $$\frac{M^{2}\int_{0}^{+\infty}\max_{|\xi|\le d}F(t,\xi)\,dt}{d^{2}}< \frac{\int _{0}^{+\infty}F(t,me^{-t})\,dt}{m^{2}+2\sum_{j=1}^{l}\int _{0}^{me^{-t_{j}}}I_{j}(s)\,ds+2\int_{0}^{m} g(s)\,ds}. $$
     
Then, for each
$$\lambda\in \biggl[\frac{\frac{m^{2}}{2}+\sum_{j=1}^{l}\int _{0}^{me^{-t_{j}}}I_{j}(s)\,ds+\int_{0}^{m} g(s)\,ds}{\int_{0}^{+\infty }F(t,me^{-t})\,dt}, \frac{d^{2}}{2M^{2}\int_{0}^{+\infty}\max_{|\xi|\le d}F(t,\xi)\,dt} \biggr], $$
(1.3) has at least three classical solutions.

Also, Dai and Zhang [4] showed by using the least action principle that (1.3) has at least one solution if the potential function has subquadratic growth and, by taking advantage of the fountain theorem due to [5], that (1.3) has infinitely many solutions if the potential function is even.

To be precise, they obtained the following theorems.

Theoremm B

[4], Theorem 3.1

Suppose that
  1. (S1)

    \((I_{j})_{j\in B}\) and g satisfy \(\int_{0}^{u}I_{j}(s)\,ds\ge0\) and \(\int_{0}^{u} g(s)\,ds\ge0\), \(u\in\mathbb {R}\), respectively;

     
  2. (S2)
    There exist \(a>0\), \(\alpha\in(1,2)\), and \(b\in L^{1}(\mathbb {R}_{+};\mathbb {R}_{+})\) such that
    $$F(t,u)\le b(t) \bigl(a+\vert u\vert ^{\alpha}\bigr) $$
    for a.e. \(t\ge0\) and all \(u\in\mathbb {R}\).
     
Then, for \(\lambda>0\), (1.3) has at least one classical solution.

Theoremm C

[4], Theorem 3.2

Besides (S1) above, for a.e. \(t\ge0\) and all \(u\in\mathbb {R}\), assume that
  1. (S3)
    There exist \(\alpha\in(1,2)\) and \(d\in L^{\frac{2}{2-\alpha }}(\mathbb {R}_{+};\mathbb {R}_{+})\) such that
    $$F(t,u)\ge d(t)|u|^{\alpha}; $$
     
  2. (S4)
    There exist \(\gamma\in(0,1)\) and \(h_{1},h_{2}\in L^{1}(\mathbb {R}_{+};\mathbb {R}_{+})\) such that
    $$f(t,u)\le h_{1}(t)|u|^{\gamma}+h_{2}(t); $$
     
  3. (S5)
    There exist \(\gamma_{j}>\alpha-1\), \(\theta>\alpha-1\), and \(q_{j} ,q>0, j\in B\), such that
    $$I_{j}(u)\le q_{j}|u|^{\gamma_{j}},\qquad g(u)\le q|u|^{\theta}; $$
     
  4. (S6)

    \(f(t, u)\), \(I_{j}(u)\), and \(g(u)\) are odd about u.

     
Then, for any \(\lambda>0\), (1.3) has infinitely many solutions.
Recently, by applying the least action principle and saddle point theorem, [68] investigated the existence of periodic solutions for the following dynamical systems:
$$ \left \{ \textstyle\begin{array}{@{}l} \frac{d}{dt}\Phi_{q}(\dot{u}_{1}(t))=\nabla_{u_{1}} F(t,u_{1}(t),u_{2}(t)),\quad \mbox{a.e. }t\in[0,T],\\ \frac{d}{dt}\Phi_{p}(\dot{u}_{2}(t))=\nabla_{u_{2}} F(t,u_{1}(t),u_{2}(t)),\quad \mbox{a.e. }t\in[0,T],\\ u_{1}(0)-u_{1}(T)=\dot{u}_{1}(0)-\dot{u}_{1}(T)=0,\\ u_{2}(0)-u_{2}(T)=\dot{u}_{2}(0)-\dot{u}_{2}(T)=0 \end{array}\displaystyle \right . $$
(1.4)
and
$$ \left \{ \textstyle\begin{array}{@{}l} \frac{d}{dt}\Phi_{q}(\dot{u}_{1}(t))+\nabla_{u_{1}} F(t,u_{1}(t),u_{2}(t))=0,\quad \mbox{a.e. }t\in[0,T],\\ \frac{d}{dt}\Phi_{p}(\dot{u}_{2}(t))+\nabla_{u_{2}} F(t,u_{1}(t),u_{2}(t))=0,\quad \mbox{a.e. }t\in[0,T],\\ u_{1}(0)-u_{1}(T)=\dot{u}_{1}(0)-\dot{u}_{1}(T)=0,\\ u_{2}(0)-u_{2}(T)=\dot{u}_{2}(0)-\dot{u}_{2}(T)=0, \end{array}\displaystyle \right . $$
(1.5)
respectively. Subsequently, by variational approach, Yang and Chen [9, 10] discussed the existence and multiplicity of periodic solutions for the following two classes of nonlinear \((q,p)\)-Laplacian dynamical systems with impulsive effects:
$$ \left \{ \textstyle\begin{array}{@{}l} \frac{d(\rho_{1}(t)\Phi_{q}(\dot{u}_{1}(t)))}{dt}-\rho_{2}(t)\Phi_{\lambda}(u_{1}(t))+\nabla_{u_{1}} F(t,u_{1}(t),u_{2}(t))=0,\quad \mbox{a.e. }t\in[0,T],\\ \frac{d(\gamma_{1}(t)\Phi_{p}(\dot{u}_{2}(t)))}{dt}-\gamma_{2}(t)\Phi_{\eta}(u_{2}(t))+\nabla_{u_{2}} F(t,u_{1}(t),u_{2}(t))=0, \quad \mbox{a.e. }t\in[0,T],\\ u_{1}(0)-u_{1}(T)=\dot{u}_{1}(0)-\dot{u}_{1}(T)=0,\\ u_{2}(0)-u_{2}(T)=\dot{u}_{2}(0)-\dot{u}_{2}(T)=0,\\ \Delta(\rho_{1}(t_{j})\Phi_{q}(\dot{u}_{1}(t_{j})))=\nabla I_{j}(u_{1}(t_{j})), \quad j\in B,\\ \Delta(\gamma_{1}(s_{m})\Phi_{p}(\dot{u}_{2}(s_{m})))=\nabla K_{m}(u_{2}(s_{m})),\quad m\in C, \end{array}\displaystyle \right . $$
(1.6)
and
$$ \left \{ \textstyle\begin{array}{@{}l} \frac{d(\Phi_{q}(\dot{u}_{1}(t)))}{dt}=\nabla_{u_{1}} F(t,u_{1}(t),u_{2}(t)),\quad \mbox{a.e. }t\in[0,T],\\ \frac{d(\Phi_{p}(\dot{u}_{2}(t)))}{dt}=\nabla_{u_{2}} F(t,u_{1}(t),u_{2}(t)),\quad \mbox{a.e. }t\in[0,T],\\ u_{1}(0)-u_{1}(T)=\dot{u}_{1}(0)-\dot{u}_{1}(T)=0,\\ u_{2}(0)-u_{2}(T)=\dot{u}_{2}(0)-\dot{u}_{2}(T)=0,\\ \Delta(\Phi_{q}(\dot{u}_{1}(t_{j})))=\nabla I_{j}(u_{1}(t_{j})), \quad j\in B,\\ \Delta(\Phi_{p}(\dot{u}_{2}(s_{m})))=\nabla K_{m}(u_{2}(s_{m})), \quad m\in C, \end{array}\displaystyle \right . $$
(1.7)
respectively, where \(p,q,\lambda,\eta>1\) and \(\rho_{1},\rho_{2},\gamma_{1},\gamma_{2}\in C([0,T];\mathbb {R}_{+})\).

Motivated by [2, 4, 610], in this paper, we are interested in the existence and multiplicity of a nontrivial weak solution for system (1.1) by using the least action principle and a critical point theorem due to Ding [1]. To be precise, we obtain the following results.

Theorem 1.1

Suppose that
  1. (HIK1)
    For \(x_{1},x_{2}\in\mathbb {R}^{N}\),
    $$\begin{aligned} \sum_{j=1}^{l} I_{j}(x_{1}) \ge0, \qquad \sum_{m=1}^{k} K_{m}(x_{2})\ge0; \end{aligned}$$
     
  2. (HF1)
    There exist \(\alpha_{1}\in[0,q)\), \(\alpha_{2}\in[0,p)\), \(a_{1}>0\), and \(d_{1}\in L^{1}([0,T];\mathbb {R}_{+})\) such that
    $$\begin{aligned} F(t,x_{1},x_{2})\le d_{1}(t) \bigl(a_{1}+|x_{1}|^{\alpha_{1}}+|x_{2}|^{\alpha _{2}} \bigr), \quad\forall(x_{1},x_{2})\in\mathbb {R}^{N} \times\mathbb {R}^{N}. \end{aligned}$$
     
Then, for each \(\lambda>0\), system (1.1) has at least one weak solution in \(X_{q}\times X_{p}\), where, for \(s>1\),
$$\begin{aligned} X_{s}=\bigl\{ u:[0,T]\to{\mathbb {R}}^{N}| u\textit{ is absolutely continuous and }\dot{u}\in L^{s}[0,T]\bigr\} . \end{aligned}$$

Remark 1.1

There exist examples satisfying Theorem 1.1. For example, let \(q=4\), \(p=3\),
$$ I_{j}(x_{1})=\bigl(\vert x_{1} \vert +c_{1}\bigr)^{\xi_{1}}\quad (j\in B), \qquad K_{m}(x_{2})=\ln \bigl(|x_{2}|+c_{2}\bigr)^{\xi_{2}} \quad (m\in C), $$
(1.8)
where \(c_{1},c_{2},\xi_{1},\xi_{2}>0\), and for all \(t\in[0,T]\),
$$F(t,x_{1},x_{2})=\sin t |x_{1}|^{3}+ \cos t |x_{2}|^{2}, $$
or
$$F(t,x_{1},x_{2})=t^{2}\ln\bigl(1+|x_{1}|^{2} \bigr)+\frac{e^{t} |x_{2}|^{4}}{1+|x_{2}|^{2}}. $$

Theorem 1.2

In addition to (HIK1), we assume that
  1. (HF2)
    There exist \(a_{2}>0\) and \(d_{2}\in L^{1}([0,T];\mathbb {R}_{+})\) such that
    $$\begin{aligned} F(t,x_{1},x_{2})\le d_{2}(t) \bigl(a_{2}+|x_{1}|^{q}+|x_{2}|^{p} \bigr),\quad \forall (x_{1},x_{2})\in\mathbb {R}^{N} \times\mathbb {R}^{N}. \end{aligned}$$
     
Then, for each \(0<\lambda<\min\{\frac{1}{q(D_{0}(q))^{q}},\frac {1}{p(D_{0}(p))^{p}}\}\), (1.1) has at least one weak solution in \(X_{q}\times X_{p}\), where
$$D_{0}(s):= \biggl(T^{-\frac{1}{s-1}}+\frac{(s-1)T}{2^{\frac{s}{s-1}}\cdot (2s-1)} \biggr)^{\frac{s-1}{s}}, \quad s=p,q. $$

Remark 1.2

There exist examples satisfying Theorem 1.2. For example, let \(q=4\), \(p=3\), and \(I_{j}\), \(K_{m}\) defined by (1.8). For all \(t\in[0,T]\), let
$$F(t,x_{1},x_{2})=e^{t} |x_{1}|^{4}+ \sin t |x_{2}|^{3}, $$
or
$$F(t,x_{1},x_{2})=t^{2}|x_{1}|^{3} \ln\bigl(1+|x_{1}|^{2}\bigr)+\bigl(t^{3}+1 \bigr)|x_{2}|^{3}. $$

Theorem 1.3

Along with (HIK1) and (HF2), for \(x_{1},x_{2}\in\mathbb {R}^{N}\), \(j\in B, m\in C\), and \(t\in[0,T]\), we suppose that
  1. (HIK2)
    There exist \(\nu_{1}\ge q\), \(\nu_{2}\ge q\), and \(\delta_{0}>0\) such that
    $$I_{j}(x_{1})\le d_{3}|x_{1}|^{\nu_{1}}, \qquad K_{m}(x_{2})\le d_{4}|x_{2}|^{\nu_{2}}, \quad |x|\le\delta_{0}; $$
     
  2. (HIK3)

    \(I_{j}(x_{1})=I_{j}(-x_{1})\), \(K_{m}(x_{2})=K_{m}(-x_{2})\), \(I_{j}(0)=0\), \(K_{m}(0)=0\);

     
  3. (HF3)
    There exist \(\mu_{1}\in(1,q)\), \(\mu_{2}\in(1,p)\), \(d_{5}>0\), and \(\delta_{1}>0\) such that
    $$\begin{aligned} F(t,x_{1},x_{2})\ge d_{5}\bigl(|x_{1}|^{\mu_{1}}+|x_{2}|^{\mu_{2}} \bigr), \quad |x_{1}|\le \delta_{1},|x_{2}|\le \delta_{1} ; \end{aligned}$$
     
  4. (HF5)

    \(F(t,x_{1},x_{2})=F(t,-x_{1},-x_{2}), F(t,0,0)\equiv0 \).

     
Then, for each \(0<\lambda<\min\{\frac{1}{q(D_{0}(q))^{q}}, \frac {1}{p(D_{0}(p))^{p}}\}\), (1.1) has infinitely many weak solutions in \(X_{q}\times X_{p}\).

Remark 1.3

There exist examples satisfying Theorem 1.3. For example, let \(q=4\), \(p=3\), and
$$I_{j}(x_{1})=c_{3}|x_{1}|^{5} \quad(j\in B),\qquad K_{m}(x_{2})=c_{4} |x_{2}|^{4}\quad (m\in C), $$
where \(c_{3},c_{4}>0\). For all \(t\in[0,T]\), let
$$F(t,x_{1},x_{2})=\bigl(e^{t}+1\bigr) |x_{1}|^{3}+\bigl(t^{2}+1\bigr) |x_{2}|^{2}. $$
If we take \(\nu_{1}=4.5\), \(\nu_{2}=3.5\), \(\mu_{1}=3.5\), and \(\mu_{2}=2.5\), it is easy to see that the example satisfies Theorem 1.3.

2 Variational structure and some preliminaries

For \(u\in X_{s}\) with \(s=q,p\), define
$$\begin{aligned} \|u\|_{X_{s}}= \biggl( \int_{0}^{T}\bigl|{u}(t)\bigr|^{s}\,dt+ \int_{0}^{T}\bigl|\dot {u}(t)\bigr|^{s}\,dt \biggr)^{1/s}. \end{aligned}$$
Set
$$\begin{aligned} \|u\|_{s}:= \biggl( \int_{0}^{T}\bigl|u(t)\bigr|^{s}\,dt \biggr)^{1/s} \quad\mbox{and}\quad \|u\|_{\infty}:=\max _{t\in[0,T]}{\bigl|u(t)\bigr|}. \end{aligned}$$
Set \(X:=X_{q}\times X_{p}\) and define the norm \(\|(u_{1},u_{2})\|_{X}=\|u_{1}\| _{X_{q}}+\|u_{2}\|_{X_{p}}\). Obviously, X is a reflexive Banach space. Let
$$\begin{aligned} \mathcal{C}=\bigl\{ u:[0,T]\to{\mathbb {R}}^{N}| u \mbox{ is continuous} \bigr\} . \end{aligned}$$
\(X_{s}\) embeds into \(\mathcal{C}\) continuously and, according to [11], Lemma 2.4,
$$\begin{aligned} \|u\|_{\infty}\le D_{0}(s) \|u\|_{X_{s}} \quad\mbox{for any } u\in X_{s}. \end{aligned}$$
(2.1)

Lemma 2.1

[12], Proposition 1.2

If \(u_{k}\) converges to u weakly, then \(u_{k}\) uniformly converges to u on \([0,T]\).

If \(u\in X_{s}\), then u is absolutely continuous, whereas need not be continuous. Hence, it is possible that \(\Delta\Phi_{s}(\dot{u}(t))=\Phi_{s}(\dot{u}(t^{+}))-\Phi_{s}(\dot{u}(t^{-}))\neq0\), which leads to impulse effects.

Following the idea [13], multiplying by \(v_{1}\in X_{q}\) on both sides of the first equation in (1.1) and integrating from 0 to T yields that
$$ \int_{0}^{T} \biggl[-\frac{d}{dt}\bigl(\bigl\vert \dot{u}_{1}(t)\bigr\vert ^{q-2}\dot {u}_{1}(t)\bigr)+\bigl\vert u_{1}(t)\bigr\vert ^{q-2}u_{1}(t)-\lambda\nabla_{x_{1}} F \bigl(t,u_{1}(t),u_{2}(t)\bigr) \biggr]v_{1}(t)\,dt=0. $$
(2.2)
Since \(v_{1}\) is continuous, \(v_{1}(t_{j}^{-})=v_{1}(t_{j}^{+})=v_{1}(t_{j})\). Combining \(\dot{u}_{1}(T)=0\) with \(\dot{u}_{1}(0)=u_{1}(0)\) implies that
$$\begin{aligned} & \int_{0}^{T} \biggl(\frac{d(\Phi_{q}(\dot{u}_{1}(t))}{dt}, v_{1}(t) \biggr)\,dt \\ &\quad= \sum_{j=0}^{l} \int_{t_{j}}^{t_{j+1}} \biggl(\frac{d(\Phi_{q}(\dot {u}_{1}(t)))}{dt},v_{1}(t) \biggr)\,dt \\ &\quad= \sum_{j=0}^{l} \bigl[ \bigl( \Phi_{q}\bigl(\dot{u}_{1}\bigl(t_{j+1}^{-}\bigr) \bigr), v_{1}\bigl(t_{j+1}^{-}\bigr) \bigr)- \bigl( \Phi_{q}\bigl(\dot{u}_{1}\bigl(t_{j}^{+}\bigr) \bigr), v_{1}\bigl(t_{j}^{+}\bigr) \bigr) \bigr]\,dt \\ &\qquad{} -\sum_{j=0}^{l} \int_{t_{j}}^{t_{j+1}} \bigl(\Phi_{q}\bigl( \dot{u}_{1}(t)\bigr), \dot {v}_{1}(t) \bigr)\,dt \\ &\quad= \bigl(\Phi_{q}\bigl(\dot{u}_{1}(T) \bigr),v_{1}(T)\bigr)-\bigl(\Phi_{q}\bigl( \dot{u}_{1}(0)\bigr),v_{1}(0)\bigr) \\ &\qquad{} -\sum_{j=1}^{l}\bigl(\Delta \Phi_{q}\bigl(\dot{u}_{1}(t_{j})\bigr), v_{1}(t_{j})\bigr)- \int _{0}^{T} \bigl(\Phi_{q}\bigl( \dot{u}_{1}(t)\bigr), \dot{v}_{1}(t) \bigr)\,dt \\ &\quad= -\bigl(\Phi_{q}\bigl(u_{1}(0)\bigr),v_{1}(0) \bigr) -\sum_{j=1}^{l}\bigl(\Delta \Phi_{q}\bigl(\dot {u}_{1}(t_{j})\bigr), v_{1}(t_{j})\bigr)- \int_{0}^{T} \bigl(\Phi_{q}\bigl( \dot{u}_{1}(t)\bigr), \dot {v}_{1}(t) \bigr)\,dt \\ &\quad= -\bigl(\Phi_{q}\bigl(u_{1}(0)\bigr),v_{1}(0) \bigr) -\sum_{j=1}^{l}\bigl(\nabla I_{j}\bigl(u_{1}(t_{j})\bigr), v_{1}(t_{j})\bigr)- \int_{0}^{T} \bigl(\Phi_{q}\bigl( \dot{u}_{1}(t)\bigr), \dot{v}_{1}(t) \bigr)\,dt, \end{aligned}$$
which, together with (2.2), further leads to
$$\begin{aligned} & \bigl(\Phi_{q}\bigl(u_{1}(0)\bigr),v_{1}(0) \bigr) +\sum_{j=1}^{l}\bigl(\nabla I_{j}\bigl(u_{1}(t_{j})\bigr), v_{1}(t_{j})\bigr)+ \int_{0}^{T} \bigl(\Phi_{q}\bigl( \dot{u}_{1}(t)\bigr), \dot{v}_{1}(t) \bigr)\,dt \\ &\quad{} + \int_{0}^{T}\bigl|u_{1}(t)\bigr|^{q-2} \bigl(u_{1}(t),v_{1}(t)\bigr)\,dt-\lambda \int_{0}^{T}\bigl(\nabla _{x_{1}} F \bigl(t,u_{1}(t),u_{2}(t)\bigr), v_{1}(t)\bigr)\,dt=0. \end{aligned}$$
Analogously, for any \(v_{2}\in X_{p}\),
$$\begin{aligned} & \bigl(\Phi_{p}\bigl(u_{2}(0)\bigr),v_{2}(0) \bigr) +\sum_{m=1}^{k}\bigl(\nabla K_{m}\bigl(u_{2}(t_{m})\bigr), v_{2}(t_{m})\bigr)+ \int_{0}^{T} \bigl(\Phi_{p}\bigl( \dot{u}_{2}(t)\bigr), \dot{v}_{2}(t) \bigr)\,dt \\ &\quad{} + \int_{0}^{T}\bigl|u_{2}(t)\bigr|^{p-2} \bigl(u_{2}(t),v_{2}(t)\bigr)\,dt-\lambda \int_{0}^{T}\bigl(\nabla _{x_{2}} F \bigl(t,u_{1}(t),u_{2}(t)\bigr), v_{2}(t)\bigr)\,dt=0. \end{aligned}$$

With the two equalities above in hand, we present the notion of weak solutions for (1.1).

Definition 2.1

For any \(v=(v_{1},v_{2})\in X_{q}\times X_{p}\), if
$$\begin{aligned} & \bigl(\Phi_{q}\bigl(u_{1}(0)\bigr),v_{1}(0) \bigr) +\sum_{j=1}^{l}\bigl(\nabla I_{j}\bigl(u_{1}(t_{j})\bigr), v_{1}(t_{j})\bigr)+ \int_{0}^{T} \bigl(\Phi_{q}\bigl( \dot{u}_{1}(t)\bigr), \dot{v}_{1}(t) \bigr)\,dt \\ &\quad{} + \int_{0}^{T}\bigl|u_{1}(t)\bigr|^{q-2} \bigl(u_{1}(t),v_{1}(t)\bigr)\,dt-\lambda \int_{0}^{T}\bigl(\nabla _{x_{1}} F \bigl(t,u_{1}(t),u_{2}(t)\bigr), v_{1}(t)\bigr)\,dt=0 \end{aligned}$$
and
$$\begin{aligned} &\bigl(\Phi_{p}\bigl(u_{2}(0)\bigr),v_{2}(0) \bigr) +\sum_{m=1}^{k}\bigl(\nabla K_{m}\bigl(u_{2}(t_{m})\bigr), v_{2}(t_{m})\bigr)+ \int_{0}^{T} \bigl(\Phi_{p}\bigl( \dot{u}_{2}(t)\bigr), \dot{v}_{2}(t) \bigr)\,dt \\ &\quad{} + \int_{0}^{T}\bigl|u_{2}(t)\bigr|^{p-2} \bigl(u_{2}(t),v_{2}(t)\bigr)\,dt-\lambda \int_{0}^{T}\bigl(\nabla _{x_{2}} F \bigl(t,u_{1}(t),u_{2}(t)\bigr), v_{2}(t)\bigr)\,dt=0, \end{aligned}$$
then \(u=(u_{1},u_{2})\in X_{q}\times X_{p}\) is called a weak solution of (1.1).
For \(u=(u_{1},u_{2})\in X_{q}\times X_{p}\), define the functional \(\varphi: X\to\mathbb {R}\) by
$$\begin{aligned} \varphi(u) ={}& \varphi(u_{1},u_{2}) \\ ={}& \frac{1}{q} \int_{0}^{T} \bigl|\dot{u}_{1}(t)\bigr|^{q}\,dt+ \frac{1}{p} \int_{0}^{T} \bigl|\dot {u}_{2}(t)\bigr|^{p}\,dt- \lambda \int_{0}^{T} F\bigl(t,u_{1}(t),u_{2}(t) \bigr)\,dt \\ &{} +\frac{1}{q} \int_{0}^{T} \bigl|u_{1}(t)\bigr|^{q}\,dt+ \frac{1}{p} \int_{0}^{T} \bigl|u_{2}(t)\bigr|^{p}\,dt \\ &{} +\sum_{j=1}^{l} I_{j} \bigl(u_{1}(t_{j})\bigr)+\sum_{m=1}^{k} K_{m}\bigl(u_{2}(s_{m})\bigr) \\ &{} +\frac{1}{q} \bigl|u_{1}(0)\bigr|^{q}+\frac{1}{p} \bigl|u_{2}(0)\bigr|^{p} \\ = {}& \phi(u_{1},u_{2})+\psi(u_{1},u_{2}), \end{aligned}$$
where
$$\begin{aligned} \phi(u_{1},u_{2}): = {}& \frac{1}{q} \int_{0}^{T} \bigl|\dot{u}_{1}(t)\bigr|^{q}\,dt+ \frac {1}{p} \int_{0}^{T} \bigl|\dot{u}_{2}(t)\bigr|^{p}\,dt- \lambda \int_{0}^{T} F\bigl(t,u_{1}(t),u_{2}(t) \bigr)\,dt \\ &{} +\frac{1}{q} \int_{0}^{T} \bigl|u_{1}(t)\bigr|^{q}\,dt+ \frac{1}{p} \int_{0}^{T} \bigl|u_{2}(t)\bigr|^{p}\,dt \\ &{} +\frac{1}{q} \bigl|u_{1}(0)\bigr|^{q}+\frac{1}{p} \bigl|u_{2}(0)\bigr|^{p}, \\ \psi(u_{1},u_{2}) : = {}& \sum_{j=1}^{l} I_{j}\bigl(u_{1}(t_{j})\bigr)+\sum _{m=1}^{k} K_{m}\bigl(u_{2}(s_{m}) \bigr). \end{aligned}$$
By virtue of (A1) and (A2), by following the argument of [12], Theorem 1.4, one has \(\phi\in C^{1}(X_{q}\times X_{p},{\mathbb{R}})\). Thanks to continuous differentiability of \((I_{j})_{j\in B}\) and \((K_{m})_{m\in C}\), we have \(\psi\in C^{1}(X_{q}\times X_{p},{\mathbb{R}})\). As a consequence, \(\varphi\in C^{1}(X,{\mathbb{R}})\) and, for all \((v_{1},v_{2})\in X_{q}\times X_{p}\),
$$\begin{aligned} & \bigl\langle \varphi'(u_{1},u_{2}), (v_{1},v_{2}) \bigr\rangle \\ &\quad= \int_{0}^{T}\bigl(\Phi_{q}\bigl( \dot{u}_{1}(t)\bigr),\dot{v}_{1}(t)\bigr)\,dt+ \int_{0}^{T}\bigl(\Phi_{p}\bigl(\dot {u}_{2}(t)\bigr),\dot{v}_{2}(t)\bigr)\,dt \\ &\qquad{} + \int_{0}^{T} \bigl(\Phi_{q} \bigl(u_{1}(t)\bigr),v_{1}(t)\bigr)\,dt+ \int_{0}^{T}\bigl(\Phi _{p} \bigl(u_{2}(t)\bigr),v_{2}(t)\bigr)\,dt \\ &\qquad{} +\bigl(\Phi_{q}\bigl(u_{1}(0) \bigr),v_{1}(0)\bigr) +\bigl(\Phi_{p}\bigl(u_{2}(0) \bigr),v_{2}(0)\bigr) \\ &\qquad{} -\lambda \int_{0}^{T}\bigl(\nabla_{x_{1}} F \bigl(t,u_{1}(t),u_{2}(t)\bigr),v_{1}(t)\bigr)\,dt- \lambda \int_{0}^{T}\bigl(\nabla_{x_{2}} F \bigl(t,u_{1}(t),u_{2}(t)\bigr),v_{2}(t)\bigr)\,dt \\ &\qquad{} +\sum_{j=1}^{l} \bigl(\nabla I_{j}\bigl(u_{1}(t_{j})\bigr), v_{1}(t_{j})\bigr)+\sum_{m=1}^{k} \bigl(\nabla K_{m}\bigl(u_{2}(s_{m})\bigr), v_{2}(s_{m})\bigr). \end{aligned}$$
Definition 2.1 shows that the critical point of φ is the weak solution of system (1.1).

The following lemma plays a crucial role in achieving the critical point of φ.

Lemma 2.2

[14]

Assume that \(\varphi\in C^{1}(E,\mathbb {R})\) is bounded from below (above) and satisfies the (PS) condition. Then
$$c=\inf_{u\in E}\varphi(u)\quad \Bigl(c=\sup_{u\in E} \varphi(u)\Bigr) $$
is a critical value of φ.

Lemma 2.3

[1]

Let E be an infinite dimensional Banach space, and let \(\varphi\in C^{1}(E,{\mathbb {R}})\) with \(\varphi(0)=0\) be even and satisfy (PS). If \(E=E_{1}\oplus E_{2}\), where \(E_{1}\) is finite dimensional, and φ satisfies that
(\(\varphi_{1}\)): 

φ is bounded from above on \(E_{2}\);

(\(\varphi_{2}\)): 

for each finite dimensional subspace \(\tilde{E}\subset E\), there are positive constants \(\rho=\rho(\tilde{E})\) and \(\sigma=\sigma(\tilde{E})\) such that \(\varphi\ge0\) on \(B_{\rho}\cap \tilde{E}\) and \(\varphi|_{\partial B_{\rho}\cap\tilde{E}}\ge\sigma\), where \(B_{\rho}=\{x\in E;\|x\|\le\rho\}\),

then φ possesses infinitely many nontrivial critical points.

3 Proofs of theorems

Proof of Theorem 1.1

It follows from (HIK1), (HF1), and (2.1) that
$$\begin{aligned} \varphi(u) = & \varphi(u_{1},u_{2}) \\ = & \frac{1}{q} \int_{0}^{T} \bigl|\dot{u}_{1}(t)\bigr|^{q}\,dt+ \frac{1}{p} \int_{0}^{T} \bigl|\dot {u}_{2}(t)\bigr|^{p}\,dt- \lambda \int_{0}^{T} F\bigl(t,u_{1}(t),u_{2}(t) \bigr)\,dt \\ &{} +\frac{1}{q} \int_{0}^{T} \bigl|u_{1}(t)\bigr|^{q}\,dt+ \frac{1}{p} \int_{0}^{T} \bigl|u_{2}(t)\bigr|^{p}\,dt \\ &{} +\sum_{j=1}^{l} I_{j} \bigl(u_{1}(t_{j})\bigr)+\sum_{m=1}^{k} K_{m}\bigl(u_{2}(s_{m})\bigr) +\frac{1}{q} \bigl|u_{1}(0)\bigr|^{q}+\frac{1}{p} \bigl|u_{2}(0)\bigr|^{p} \\ \ge& \frac{1}{q}\|u_{1}\|_{X_{q}}^{q}+ \frac{1}{p}\|u_{2}\|_{X_{p}}^{p}-\lambda \int_{0}^{T}d_{1}(t) \bigl(a_{1}+\bigl|u_{1}(t)\bigr|^{\alpha_{1}}+\bigl|u_{2}(t)\bigr|^{\alpha_{2}} \bigr)\,dt \\ \ge& \frac{1}{q}\|u_{1}\|_{X_{q}}^{q}+ \frac{1}{p}\|u_{2}\|_{X_{p}}^{p}-\lambda\| u_{1}\|_{\infty}^{\alpha_{1}} \int_{0}^{T} d_{1}(t)\,dt \\ & -\lambda\|u_{2}\|_{\infty}^{\alpha_{2}} \int_{0}^{T} d_{1}(t)\,dt-\lambda a_{1} \int _{0}^{T} d_{1}(t)\,dt \\ \ge& \frac{1}{q}\|u_{1}\|_{X_{q}}^{q}+ \frac{1}{p}\|u_{2}\|_{X_{p}}^{p}-\lambda \bigl(D_{0}(q)\bigr)^{\alpha_{1}}\|u_{1} \|_{X_{q}}^{\alpha_{1}} \int_{0}^{T} d_{1}(t)\,dt \\ &{} -\lambda\bigl(D_{0}(p)\bigr)^{\alpha_{2}}\|u_{2} \|_{X_{p}}^{\alpha_{2}} \int_{0}^{T} d_{1}(t)\,dt-\lambda a_{1} \int_{0}^{T} d_{1}(t)\,dt. \end{aligned}$$
Owing to \(\alpha_{1}\in[0,q)\) and \(\alpha_{2}\in[0,p)\), we readily obtain that \(\varphi(u)\to+\infty\) as \(\|u\|_{X}\to \infty\), i.e., φ satisfies the coercive condition on X. So φ is bounded below on X.
Hereinafter, we claim that φ satisfies the (PS) condition. If \(\{\varphi(u_{1n},u_{2n})\}\) is bounded and \(\|\varphi'(u_{1n},u_{2n})\|\to 0\) as \(n\to\infty\), then there exists a positive constant \(D_{1}\) such that
$$\begin{aligned} \bigl|\varphi(u_{1n},u_{2n})\bigr|\le D_{1}, \qquad \bigl\| \varphi'(u_{1n},u_{2n})\bigr\| \le D_{1}, \quad\forall n\in\mathbb {N}. \end{aligned}$$
Since φ satisfies a coercive condition on X, we infer that \(\|u_{1n}\|_{X_{q}}\) and \(\|u_{2n}\|_{X_{p}}\) is bounded. Next, in light of the reflexive property of \(X_{s}\), there exists a subsequence, still denoted by \(\{u_{n}=(u_{1n},u_{2n})\}\), such that
$$\begin{aligned} u_{1n}\rightharpoonup u_{1} \quad\mbox{on } X_{q},\qquad u_{2n}\rightharpoonup u_{2} \quad \mbox{on } X_{p}. \end{aligned}$$
Thus, Lemma 2.1 gives that
$$\begin{aligned} u_{1n}\to u_{1} \quad \mbox{in } C\bigl(0,T;{ \mathbb {R}}^{N}\bigr) \quad\mbox{and}\quad u_{2n}\to u_{2} \quad\mbox{in } C\bigl(0,T;{\mathbb {R}}^{N}\bigr). \end{aligned}$$
Following the argument in [1517], we can derive that \(\|u_{n}-u\|_{X}\to0\), where \(u=(u_{1},u_{2})\). Consequently, φ satisfies the (PS) condition. Thus, with the help of Lemma 2.2, we deduce that φ has at least one critical point on X. Hence system (1.1) has at least one solution on X. □

Proof of Theorem 1.2

By (HIK1), (HF2), and (2.1), it follows that
$$\begin{aligned} \varphi(u) = & \varphi(u_{1},u_{2}) \\ = & \frac{1}{q} \int_{0}^{T} \bigl|\dot{u}_{1}(t)\bigr|^{q}\,dt+ \frac{1}{p} \int_{0}^{T} \bigl|\dot {u}_{2}(t)\bigr|^{p}\,dt- \lambda \int_{0}^{T} F\bigl(t,u_{1}(t),u_{2}(t) \bigr)\,dt \\ &{} +\frac{1}{q} \int_{0}^{T} \bigl|u_{1}(t)\bigr|^{q}\,dt+ \frac{1}{p} \int_{0}^{T} \bigl|u_{2}(t)\bigr|^{p}\,dt \\ &{} +\sum_{j=1}^{l} I_{j} \bigl(u_{1}(t_{j})\bigr)+\sum_{m=1}^{k} K_{m}\bigl(u_{2}(s_{m})\bigr) \\ &{} +\frac{1}{q} \bigl|u_{1}(0)\bigr|^{q}+\frac{1}{p} \bigl|u_{2}(0)\bigr|^{p} \\ \ge& \frac{1}{q}\|u_{1}\|_{X_{q}}^{q}+ \frac{1}{p}\|u_{2}\|_{X_{p}}^{p}-\lambda \int_{0}^{T}\bigl[d_{2}(t) \bigl(a_{2}+\bigl|u_{1}(t)\bigr|^{q}+\bigl|u_{2}(t)\bigr|^{p} \bigr)\bigr]\,dt \\ \ge& \frac{1}{q}\|u_{1}\|_{X_{q}}^{q}+ \frac{1}{p}\|u_{2}\|_{X_{p}}^{p}-\lambda\| u_{1}\|_{\infty}^{q} \int_{0}^{T} d_{2}(t)\,dt \\ &{} -\lambda\|u_{2}\|_{\infty}^{p} \int_{0}^{T} d_{2}(t)\,dt-\lambda a_{2} \int_{0}^{T} d_{2}(t)\,dt \\ \ge& \frac{1}{q}\|u_{1}\|_{X_{q}}^{q}+ \frac{1}{p}\|u_{2}\|_{X_{p}}^{p}-\lambda \bigl(D_{0}(q)\bigr)^{q}\|u_{1} \|_{X_{q}}^{q} \int_{0}^{T} d_{2}(t)\,dt \\ &{} -\lambda\bigl(D_{0}(p)\bigr)^{p}\|u_{2} \|_{X_{p}}^{p} \int_{0}^{T} d_{2}(t)\,dt-\lambda a_{2} \int _{0}^{T} d_{2}(t)\,dt. \end{aligned}$$
(3.1)
In view of \(\lambda<\min\{\frac{1}{q(D_{0}(q))^{q}},\frac{1}{p(D_{0}(p))^{p}}\} \), one has \(\varphi(u)\to+\infty\) as \(\|u\|_{X}\to\infty\), that is, φ satisfies the coercive condition on X. Hence φ is bounded below on X. By carrying out a similar argument to derive Theorem 1.1, we get that system (1.1) has at least one solution in X. □

Proof of Theorem 1.3

Keep in mind that φ and −φ have the same critical points. Let \(\Theta=-\varphi\). In the sequel, we aim at verifying that all conditions in Lemma 2.3 are fulfilled by Θ. In fact, from (HIK3) and (HF5), we find that Θ is even and \(\Theta(0)=0\). Taking (HIK1), (HF2), and (3.1) into account, we obtain that \(\Theta(u)\to-\infty\) as \(\|u\|_{X}\to\infty\). Hence Θ is bounded above on X so that Θ satisfies (\(\varphi _{1}\)) in Lemma 2.3.

Assume that \(\tilde{X}\subset X\) is finite-dimensional. For any \(u=(u_{1}, u_{2})\in\tilde{X}=\tilde{X}_{q}\times\tilde{X}_{p}\), where \(\tilde{X}_{q}\subset X_{q}\) and \(\tilde{X}_{p}\subset X_{p}\), we deduce that \(\|u_{1}\|_{\mu_{1}}\) is equivalent to \(\|u_{1}\|_{X_{q}}\), and \(\| u_{2}\|_{\mu_{2}}\) is equivalent to \(\|u_{2}\|_{X_{p}}\). Hence there exist constants \(d_{6},d_{7}>0\) such that
$$\begin{aligned} \|u_{1}\|_{\mu_{1}}\ge d_{6} \|u_{1}\|_{X_{q}},\qquad \|u_{2}\|_{\mu_{2}}\ge d_{7}\|u_{2}\|_{X_{p}}. \end{aligned}$$
(3.2)
Let \(\rho_{0}=\min\{\frac{\min\{\delta_{0},\delta_{1}\}}{D_{0}(q)},\frac{\min\{ \delta_{0},\delta_{1}\}}{D_{0}(p)}\}\). For any \(\rho\in(0,\rho_{0})\), if \(\|u\| _{X}=\rho\), then \(\|u_{1}\|_{\infty}\le D_{0}(q)\|u_{1}\|_{X_{q}}\le D_{0}(q)\rho\le \min\{\delta_{0},\delta_{1}\}\) and \(\|u_{2}\|_{\infty}\le D_{0}(p)\|u_{2}\| _{X_{q}}\le D_{0}(p)\rho\le\min\{\delta_{0},\delta_{1}\}\). Thus it follows from (HIK2), (HF3), (3.2), and Hölder’s inequality that
$$\begin{aligned} \Theta(u) = {}& {-}\varphi(u_{1},u_{2}) \\ = {}& {-}\frac{1}{q} \int_{0}^{T} \bigl|\dot{u}_{1}(t)\bigr|^{q}\,dt- \frac{1}{p} \int_{0}^{T} \bigl|\dot {u}_{2}(t)\bigr|^{p}\,dt+ \lambda \int_{0}^{T} F\bigl(t,u_{1}(t),u_{2}(t) \bigr)\,dt \\ &{} -\frac{1}{q} \int_{0}^{T} \bigl|u_{1}(t)\bigr|^{q}\,dt- \frac{1}{p} \int_{0}^{T} \bigl|u_{2}(t)\bigr|^{p}\,dt \\ &{} -\sum_{j=1}^{l} I_{j} \bigl(u_{1}(t_{j})\bigr)-\sum_{m=1}^{k} K_{m}\bigl(u_{2}(s_{m})\bigr) \\ &{} -\frac{1}{q} \bigl|u_{1}(0)\bigr|^{q}-\frac{1}{p} \bigl|u_{2}(0)\bigr|^{p} \\ \ge{}& -\frac{1}{q}\|u_{1}\|_{X_{q}}^{q}- \frac{1}{p}\|u_{2}\|_{X_{p}}^{p} +\lambda d_{5} \int_{0}^{T}\bigl(\bigl|u_{1}(t)\bigr|^{\mu_{1}}+\bigl|u_{2}(t)\bigr|^{\mu_{2}} \bigr)\,dt \\ &{} -\sum_{j=1}^{l} d_{3}\bigl|u_{1}(t_{j})\bigr|^{\nu_{1}}- \sum_{m=1}^{k} d_{4}\bigl|u_{2}(s_{m})\bigr|^{\nu _{2}} -\frac{1}{q} \|u_{1}\|_{\infty}^{q}- \frac{1}{p} \|u_{2}\|_{\infty}^{p} \\ \ge{}& -\frac{1}{q}\|u_{1}\|_{X_{q}}^{q}- \frac{1}{p}\|u_{2}\|_{X_{p}}^{p} +\lambda d_{5} d_{6}^{\mu_{1}}\|u_{1} \|_{X_{q}}^{\mu_{1}} +\lambda d_{5} d_{7}^{\mu_{2}} \|u_{2}\|_{X_{p}}^{\mu_{2}} \\ &{} -ld_{3}\|u_{1}\|_{\infty}^{\nu_{1}}-k d_{4}\|u_{2}\|_{\infty}^{\nu_{2}} \\ &{} -\frac{1}{q} \bigl(D_{0}(q)\bigr)^{q} \|u_{1}\|_{X_{q}}^{q}-\frac{1}{p} \bigl(D_{0}(p)\bigr)^{p}\|u_{2}\| _{X_{p}}^{p} \\ \ge{}& -\frac{1}{q}\|u_{1}\|_{X_{q}}^{q}- \frac{1}{p}\|u_{2}\|_{X_{p}}^{p} +\lambda d_{5} d_{6}^{\mu_{1}}\|u_{1} \|_{X_{q}}^{\mu_{1}} +\lambda d_{5} d_{7}^{\mu_{2}} \|u_{2}\|_{X_{p}}^{\mu_{2}} \\ &{} -ld_{3}\bigl(D_{0}(q)\bigr)^{\nu_{1}} \|u_{1}\|_{X_{q}}^{\nu_{1}}-k d_{4} \bigl(D_{0}(p)\bigr)^{\nu_{2}}\| u_{2} \|_{X_{p}}^{\nu_{2}} \\ &{} -\frac{1}{q} \bigl(D_{0}(q)\bigr)^{q} \|u_{1}\|_{X_{q}}^{q}-\frac{1}{p} \bigl(D_{0}(p)\bigr)^{p}\|u_{2} \|_{X_{p}}^{p}. \end{aligned}$$
Observing that \(\mu_{1}\in(1,q)\) and \(\mu_{2}\in(1,p)\), we take sufficiently small \(\rho\in(0,\rho_{0})\) such that \(\Theta(u)\ge0\) on \(B_{\rho}\cap \tilde{X}\) and \(\Theta(u)> 0\) on \(\partial B_{\rho}\cap \tilde{X}\). Therefore, Θ satisfies (\(\varphi_{2}\)) in Lemma 2.3. Then, according to Lemma 2.3, Θ has infinitely many critical points in X so that (1.1) has infinitely many solutions in X. □

Declarations

Acknowledgements

This work is supported by the National Natural Science Foundation of China (NO: 61304011) and by the Hunan Provincial Natural Science Foundation of China (NO: 2016JJ3139).

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
School of Mathematics and Statistics, Central South University

References

  1. Ding, YH: Existence and multiplicity results for homoclinic solutions to a class of Hamiltonian systems. Nonlinear Anal. 25(11), 1095-1113 (1995) MathSciNetView ArticleMATHGoogle Scholar
  2. Chen, H, Sun, J: An application of variational method to second-order impulsive functional differential equation on the half-line. Appl. Math. Comput. 217, 1863-1869 (2010) MathSciNetMATHGoogle Scholar
  3. Bonanno, G, Marano, SA: On the structure of the critical set of non-differentiable functions with a weak compactness condition. Appl. Anal. 89, 1-10 (2010) MathSciNetView ArticleMATHGoogle Scholar
  4. Dai, B, Zhang, D: The existence and multiplicity of solutions for second-order impulsive differential equations on the half-line. Results Math. 63, 135-149 (2013) MathSciNetView ArticleMATHGoogle Scholar
  5. Zou, W: Variant fountain theorems and their applications. Manuscr. Math. 104, 343-358 (2001) MathSciNetView ArticleMATHGoogle Scholar
  6. Pasca, D: Periodic solutions of a class of nonautonomous second-order differential systems with \((q,p)\)-Laplacian. Bull. Belg. Math. Soc. Simon Stevin 17, 841-850 (2010) MathSciNetMATHGoogle Scholar
  7. Pasca, D, Tang, CL: Some existence results on periodic solutions of nonautonomous second-order differential systems with \((q,p)\)-Laplacian. Appl. Math. Lett. 23, 246-251 (2010) MathSciNetView ArticleMATHGoogle Scholar
  8. Pasca, D, Tang, CL: Some existence results on periodic solutions of ordinary \((q,p)\)-Laplacian systems. J. Appl. Math. Inform. 29(1-2), 39-48 (2011) MathSciNetMATHGoogle Scholar
  9. Yang, X, Chen, H: Periodic solutions for a nonlinear \((q,p)\)-Laplacian dynamical system with impulsive effects. J. Appl. Math. Comput. 40, 607-625 (2012) MathSciNetView ArticleMATHGoogle Scholar
  10. Yang, X, Chen, H: Periodic solutions for autonomous \((q,p)\)-Laplacian system with impulsive effects. J. Appl. Math. 2011, Article ID 378389 (2011) MathSciNetMATHGoogle Scholar
  11. Zhang, X, Tang, X: Non-constant periodic solutions for second order Hamiltonian system involving the p-Laplacian. Adv. Nonlinear Stud. 13, 945-964 (2013) MathSciNetView ArticleMATHGoogle Scholar
  12. Mawhin, J, Willem, M: Critical Point Theory and Hamiltonian Systems. Springer, New York (1989) View ArticleMATHGoogle Scholar
  13. Nieto, JJ, O’Regan, D: Variational approach to impulsive differential equations. Nonlinear Anal., Real World Appl. 10, 680-690 (2009) MathSciNetView ArticleMATHGoogle Scholar
  14. Lu, WD: Variational Methods in Differential Equations. Scientific Publishing House in China (2002) (in Chinese)
  15. Xu, B, Tang, CL: Some existence results on periodic solutions of ordinary p-Laplacian systems. J. Math. Anal. Appl. 333, 1228-1236 (2007) MathSciNetView ArticleMATHGoogle Scholar
  16. Zhang, X, Tang, X: Periodic solutions for an ordinary p-Laplacian system. Taiwan. J. Math. 15, 1369-1396 (2011) MathSciNetMATHGoogle Scholar
  17. Yang, X, Chen, H: Existence of periodic solutions for a damped vibration problem with \((q, p)\)-Laplacian. Bull. Belg. Math. Soc. Simon Stevin 21, 51-66 (2014) MathSciNetMATHGoogle Scholar
  18. Luo, Z, Xiao, J, Xu, Y: Subharmonic solutions with prescribed minimal period for some second-order impulsive differential equations. Nonlinear Anal. 75, 2249-2255 (2012) MathSciNetView ArticleMATHGoogle Scholar
  19. Zeidler, E: Nonlinear Functional Analysis and Its Applications, Vol. III. Springer, Berlin (1985) View ArticleMATHGoogle Scholar
  20. Zhang, X: Subharmonic solutions for a class of second-order impulsive Lagrangian systems with damped term. Bound. Value Probl. 2013, Article ID 218 (2013) MathSciNetView ArticleMATHGoogle Scholar

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