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

In this paper, by using the mountain pass theorem, we investigate the existence of subharmonic weak solutions for a class of second-order impulsive Lagrangian systems with damped term under asymptotically quadratic conditions. Some new existence criteria are established. Finally, an example is presented to verify our results.

MSC:37J45, 34C25, 70H05.

In this paper, we investigate the existence of subharmonic weak solutions for the following second-order impulsive Lagrangian system with damped term:

where T>0, p\in \mathbb{N}, , u(t)=(u^1(t),\dots ,u^N(t)), I_j:{\mathbb{R}}^N\to \mathbb{R}, q\in C(\mathbb{R},\mathbb{R}) satisfying q(t+T)=q(t) and {\int }_0^{T}q(t)dt=0, B is a skew-symmetric N\times N constant matrix, P(t) and A(t) are symmetric and continuous N\times N matrix-value functions on ℝ satisfying P(t+T)=P(t) and A(t+T)=A(t), and F:\mathbb{R}\times {\mathbb{R}}^N\to \mathbb{R} satisfies F(t,x)=-K(t,x)+W(t,x), where K, W are T-periodic in their first variable, and the following assumption:

1. (A)

   F(t,x) is measurable in t for every x\in {\mathbb{R}}^N and continuously differentiable in x for a.e. t\in [0,T] , and there exist a\in C({\mathbb{R}}^{+},{\mathbb{R}}^{+}) and b\in L^1(0,T;{\mathbb{R}}^{+}) with b(t+T)=b(t) such that

   \begin{array}{c}|F(t,x)|\le a(|x|)b(t),|\mathrm{\nabla }F(t,x)|\le a(|x|)b(t),\hfill \\ |I_j(x)|\le a(|x|),|\mathrm{\nabla }{I}_j(x)|\le a(|x|)\hfill \end{array}

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

Lagrangian systems are applied extensively to study the fluid mechanics, nuclear physics and relativistic mechanics. Especially, as a special case of Lagrangian systems, the following second-order Hamiltonian systems are considered by many authors:

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)

Assume that F satisfies

1. (F)

   F(t,x)=-K(t,x)+W(t,x) and K,W\in C^1(\mathbb{R}\times {\mathbb{R}}^N,\mathbb{R}) are T-periodic in their first variable with T>0, and that K and W satisfy the following assumptions:

(H1) There exist constants b>0 and \gamma \in (1,2] such that

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

(H3) uniformly for t\in [0,T];

(H4) There exists a function g\in L^1([0,T],\mathbb{R}) such that

and

(H5) There exist constants a>0 and d>0 such that

(H6) There exists x_0\in {\mathbb{R}}^N such that

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.

In [16], Nieto introduced a variational formulation for the following one-dimensional damped nonlinear Dirichlet problem with impulses:

and gave the concept of a weak solution for such a problem. They obtained that the weak solutions of problem (1.3) are indeed the critical points of the functional:

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.

For higher dimensional dynamical systems, some interesting results have also been obtained (see [21–23]). In [21], Zhou and Li investigated the second-order Hamiltonian system with impulsive effects:

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.

In recent years, via variational methods, some authors have been interested in studying the existence and multiplicity of periodic solutions and homoclinic solutions for the following Lagrangian systems with damped term:

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].

In 2010, Li et al. [28] investigated the following system, more general than system (1.6), with P(t)\equiv I_N:

Motivated by [28], in [29], we investigated the following system, more general than system (1.7):

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].

In [32], Luo et al. investigated the existence of subharmonic solutions with prescribed minimal period for the following one-dimensional second-order impulsive differential equation:

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\}), , 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}.

Let

and

In this paper, we make the following assumptions:

1. (P)

   There exists a constant m>\frac{1}{2} such that the matrix P(t) satisfies

   P(0)=P(T),(P(t)x,x)>m(x,x)\text{for all }(t,x)\in \mathbb{R}\times \{{\mathbb{R}}^N/\{0\}\};

(K1) There exist constants a>0 and \gamma \in (0,2] such that

(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];

(K3) There exists D_1>0 such that

(W1) uniformly for a.e. t\in [0,T];

(W2) There exist constants b>0 and d>0 such that

(W3) There exists a function h\in L^1(0,T;\mathbb{R}) such that

and

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

(W4) There exists x_0\in {\mathbb{R}}^N such that

(W5) There exists a constant D_2>\frac{\parallel A\parallel }{2}+D_1 such that

(I1) There exist constants l_j>0 (j=1,\dots ,p) such that

(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;

(I3) There exists a constant C such that

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].

Let

Define

and

for each u,v\in H_{kT}^1. Then (H_{kT}^1,〈\cdot ,\cdot 〉) is a Hilbert space. It is well known that

is also a norm on H_{kT}^1. Obviously, if the condition (P) holds, {\parallel u\parallel }_{H_{kT}^1} and \parallel u\parallel are equivalent. Moreover, there exists C_{0k}>0 such that

(see Proposition 1.1 in [1]). Hence, there exist positive constants C_{1k}, C_{2k} such that

For any a,b\in \mathbb{R}, define

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).

Note that Q(t)={\int }_0^{t}q(s)ds. Then, by T-periodicity of q, we have Q(kT)=k{\int }_0^{T}q(t)dt=0. Moreover, obviously, Q(t) is continuous on ℝ. We transform system (1.1) into the following system:

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

By the idea in [15], we take v\in H_{kT}^1 and multiply the two sides of the equality

by v and integrate it from 0 to kT. Then we obtain

Note that P(t+T)=P(t), P(t) is continuous on ℝ and \dot{u}(0)=\dot{u}(kT). By integration by parts and the continuity of v, we obtain

Definition 2.2 u\in H_{kT}^1 is called a subharmonic weak solution of system (1.1) if

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).

Proof Motivated by [15], for j\in \{0,1,2,\dots ,p\}, choose v\in H_{kT}^1 with v(t)=0 for every t\in [0,t_j]\cup [t_{j+1},kT]. Then, by Definition 2.2, we obtain

Choose v\in H_{kT}^1 with v(t)=0 for every t\in [0,T]. Then we obtain

Equations (2.5) and (2.6) imply 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 u satisfies

Multiplying the above equality by v and integrating between 0 and kT, combining the argument of (2.4) and Definition 2.2, we obtain that

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 every k\in \mathbb{N}, define {\phi }_k:H_{kT}^1\to \mathbb{R} by

It follows from assumption (A) and Theorem 1.4 in [1] that the functional {\phi }_k is continuously differentiable and

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])

Let E be a real Banach space, and let \varphi \in C^1(E,\mathbb{R}) satisfy the (PS) condition. If ϕ satisfies the following conditions:

1. (i)

   \varphi (0)=0;

2. (ii)

   There exist constants \rho ,\alpha >0 such that \varphi {|}_{\partial B_{\rho }(0)}\ge \alpha;

3. (iii)

   There exists e\in E/{\overline{B}}_{\rho }(0) such that \varphi (e)\le 0, then ϕ possesses a critical value c\ge \alpha given by

   c=\underset{g\in \mathrm{\Gamma }}{inf}\underset{s\in [0,1]}{max}\varphi (g(s)),

where B_{\rho }(0) is an open ball in E of radius ρ centered at 0, and

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.

Step 1. We prove that {\phi }_k satisfies assumption (ii) of Lemma 2.2. It follows from (W1) and (W2) that there exist , \theta >2 and C_1>0 such that

Choose such that \epsilon {(\frac{\delta }{C_{1k}})}^2-C_1{C}_{1k}^{\theta }{(\frac{\delta }{C_{1k}})}^{\theta }{\int }_0^{kT}{e}^{Q(t)}dt>0. Then it follows from (K1), (I2), (3.1) and (2.1) that for all u\in H_{kT}^1 with \parallel u\parallel =\delta /C_{1k}:={\rho }_k,

Step 2. We prove that {\phi }_k satisfies assumption (iii) of Lemma 2.2. Set \phi (s)=s^{-2}W(t,s{x}_0) for s>0. By the argument in [10], we know that (W3) implies that

and (K2) implies that

It follows from (3.2), (3.3), (W3) and (I1) that for sufficiently large s,