In this paper, we investigate the existence of subharmonic weak solutions for the following second-order impulsive Lagrangian system with damped term:
(1.1)
where , , , , , satisfying and , B is a skew-symmetric constant matrix, and are symmetric and continuous matrix-value functions on ℝ satisfying and , and satisfies , where K, W are T-periodic in their first variable, and the following assumption:
-
(A)
is measurable in
t
for every
and continuously differentiable in
x
for a.e.
, and there exist
and
with
such that
for all and a.e. .
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:
(1.2)
where satisfies , 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
-
(F)
and are T-periodic in their first variable with , and that K and W satisfy the following assumptions:
(H1) There exist constants and such that
(H2) for ;
(H3) uniformly for ;
(H4) There exists a function such that
and
(H5) There exist constants and such that
(H6) There exists 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:
(1.3)
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:
(1.4)
where and . 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:
(1.5)
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 , where , 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 . 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:
(1.6)
where is a symmetric and continuous matrix-valued function, B is a skew-symmetric constant matrix and . 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 :
(1.7)
Motivated by [28], in [29], we investigated the following system, more general than system (1.7):
(1.8)
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:
(1.9)
where , , , , , and if , while if .
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 .
Let
and
In this paper, we make the following assumptions:
-
(P)
There exists a constant such that the matrix satisfies
(K1) There exist constants and such that
(K2) for all and a.e. ;
(K3) There exists such that
(W1) uniformly for a.e. ;
(W2) There exist constants and such that
(W3) There exists a function such that
and
where ;
(W4) There exists such that
(W5) There exists a constant such that
(I1) There exist constants () such that
(I2) and for all ;
(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.