Subharmonic solutions for a class of second-order impulsive Lagrangian systems with damped term

for all $x\in {\mathbb{R}}^N$ and a.e. $t\in [0,T]$.

where $F:\mathbb{R}\times {\mathbb{R}}^N\to \mathbb{R}$ satisfies $F(t+T,x)=F(t,x)$, and the existence and multiplicity of periodic solutions, subharmonic solutions and homoclinic solutions are obtained via variational methods. We refer readers to [1–14]. Especially, in 2010, under the asymptotically quadratic conditions, Tang and Jiang [10] obtained the following interesting result.

Theorem A (see [10], Theorem 1.1)

(H2) $(\mathrm{\nabla }K(t,x),x)\le 2K(t,x)$ for $(t,x)\in [0,T]\times {\mathbb{R}}^N$;

(H3) ${lim\hspace{0.17em}sup}_{|x|\to 0}\frac{W(t,x)}{{|x|}^2}<b$ uniformly for $t\in [0,T]$;

Then system (1.2) has a nontrivial T-periodic solution.

In recent years, variational methods have been applied to study the existence and multiplicity of solutions for impulsive differential equations and lots of interesting results have been obtained, see [15–20].

In [15], Nieto and O’Regan considered a one-dimensional Dirichlet boundary value problem with impulses. They obtained that the solutions of the impulsive problem minimize some (energy) functional and the critical points of the functional are indeed solutions of the impulsive problem.

where $G(t)={\int }_0^{t}g(t)dt$ and $F(t,v)={\int }_0^{v}f(t,s)ds$. In [17] and [18], the authors also dealt with some one-dimensional impulsive problems with damped term by variational methods.

By using the least action principle and the saddle point theorem, they obtained some existence results of solutions under sublinear condition and some reasonable conditions. In [22], system (1.5) with $F(t,u)=\frac{1}{2}A(t)u\cdot u-\lambda W(t,u)-\mu G(t,u)$, where $\lambda ,\mu \in \mathbb{R}$, was also investigated. By using variational methods, the authors obtained that system (1.5) has at least three weak solutions. In [23], the authors investigated system (1.5) with $F(t,u)=\frac{1}{2}A(t)u\cdot u-W(t,u)$. They obtained that system (1.5) has infinitely many solutions under the assumptions that nonlinear term is superquadratic, asymptotically quadratic and subquadratic, respectively.

where $P(t)$ is a symmetric and continuous $N\times N$ matrix-valued function, B is a skew-symmetric $N\times N$ constant matrix and $F:\mathbb{R}\times {\mathbb{R}}^N\to \mathbb{R}$. They obtained some interesting results. We refer readers to [24–27].

By variational methods, under superquadratic or subquadratic conditions, we obtained that system (1.8) has infinitely many solutions. One can see more details of our results and more research background of system (1.8) in [29].

where $f\in C(\mathbb{R}\times \mathbb{R},\mathbb{R})$, $Z_0=Z^{+}\cup Z^{-}$, $J^{\prime }=\mathbb{R}\mathrm{\setminus }\{t_m\mid m\in Z_0\}$, $I_m\in C(\mathbb{R},{\mathbb{R}}^{+}\cup \{0\})$, $0<t_1<t_2<\cdots <t_p<T$, $I_{m+p}=I_m$ and $t_m=t_{m+p}-T$ if $m\in Z^{+}$, while $t_m=t_{m+p+1}-T$ if $m\in Z^{-}$.

In this paper, motivated by [10, 15, 16, 21, 28, 29] and [32], we focus on the existence of subharmonic weak solutions for system (1.1), which is of impulsive conditions, and we study the problem under asymptotically quadratic conditions. To the best of our knowledge, there are few papers that consider such a problem for system (1.1). We call a solution u subharmonic if u is kT-periodic for some $k\in \mathbb{N}$.

(K2) $(\mathrm{\nabla }K(t,x),x)\le 2K(t,x)$ for all $x\in {\mathbb{R}}^N$ and a.e. $t\in [0,T]$;

(W1) ${lim}_{|x|\to 0}\frac{W(t,x)}{{|x|}^2}<a$ uniformly for a.e. $t\in [0,T]$;

where $Q(t)={\int }_0^{t}q(s)ds$;

(I2) ${\sum }_{j=1}^p{e}^{Q(t_j)}{I}_j(0)=0$ and ${\sum }_{j=1}^p{e}^{Q(t_j)}{I}_j(x)\ge 0$ for all $x\in {\mathbb{R}}^N$;

This paper is organized as follows. In Section 2, we present the definition of a subharmonic classical solution, a subharmonic weak solution and the variational structure for system (1.1) and make some preliminaries. In Section 3, we present our main theorems and their proofs. In Section 4, an example is given to verify our main theorems.

In this section, we present the variational structure of system (1.1), which is motivated by [15–17, 28] and [29].

If $u\in H_{kT}^1$, then $\dot{u}(t^{+})-\dot{u}(t^{-})=0$ may not hold, which leads to impulsive effects.

Definition 2.1 Assume that $u\in H_{kT}^1\cap H^2[0,t_1)\cap ({\bigcap }_{j=1}^{p-1}{H}^2(t_j,t_{j+1}))\cap H^2(t_p,T]\cap H^2[T,kT]$ and the limits $\dot{u}(t_j^{+})$ and $\dot{u}(t_j^{-})$ ($j=1,2,\dots ,p$) exist. If u satisfies system (1.1), then we say that u is a subharmonic classical solution of system (1.1).

Remark 2.1 In [32], impulsive effects may occur periodically in $t_j$, $j\in \{1,\dots ,p\}$. In order to obtain a sequence of distinct subharmonic weak solutions (see Theorem 3.2 below), different from [32], in Definition 2.1, we assume that the impulsive effects only occur in $t_j$, $j=1,\dots ,p$, which belong to $(0,T)$. In other words, u is absolutely continuous on ℝ and $\dot{u}$ is absolutely continuous on $[0,t_1)\cup ({\bigcup }_{j=1}^{p-1}(t_j,t_{j+1}))\cup (t_p,T]\cup [T,kT]$. Moreover, note that $u(t)=u(t+kT)$. Then it is easy to see that $\dot{u}(0)=\dot{u}(kT)$.

Then system (2.2) is equivalent to system (1.1) and its solutions are the solutions of system (1.1).

holds for any $v\in H_{kT}^1$.

Lemma 2.1 If $u\in H_{kT}^1$ is a subharmonic weak solution of system (1.1), then u is a subharmonic classical solution of system (1.1).

Hence, $\mathrm{\Delta }(P(t_j)\dot{u}(t_j))=\mathrm{\nabla }{I}_j(u(t_j))$ for every $j=1,2,\dots ,p$. This completes the proof. □

for $u,v\in H_{kT}^1$. Obviously, if $u_0\in H_{kT}^1$ is a critical point of ${\phi }_k$, i.e., ${\phi }_k^{\prime }(u_0)=0$, then $u_0$ is a subharmonic weak solution of system (1.1).

We will use the following mountain pass theorem to prove our results.

Lemma 2.2 (see [30])

Remark 2.2 As shown in [31], a deformation lemma can be proved by replacing the usual (PS)-condition with the condition (C), and it turns out that Lemma 2.2 holds true under the condition (C). We say that ϕ satisfies the condition (C), i.e., for every sequence $\{u_n\}\subset E$, $\{u_n\}$ has a convergent subsequence if $\varphi (u_n)$ is bounded and $(1+\parallel u_n\parallel )\parallel {\varphi }^{\prime }(u_n)\parallel \to 0$ as $n\to \mathrm{\infty }$.

Theorem 3.1 Assume that (P), (K1), (K2), (W1)-(W4) and (I1)-(I3) hold. Then, for every $k\in \mathbb{N}$, system (1.1) has at least one kT-periodic weak solution in $H_{kT}^1$.

Proof We use Lemma 2.2 to prove the theorem. Let $E=H_{kT}^1$.