The existence and multiplicity of homoclinic orbits are considered for a class of subquadratic second order Hamiltonian systems

In this paper, we will study the existence and multiplicity of homoclinic orbits for the second order Hamiltonian systems of the type:

Hamiltonian system theory, a classical as well as a modern study area, widely consists in mathematical sciences, life sciences, and various aspects of social science. Lots of mechanical and field theory models even exist in the form of Hamiltonian system. Solutions of Hamiltonian system can be divided into periodic solution, subharmonic solution, homoclinic orbit, heteroclinic orbit, and so on. Homoclinic orbit was discovered among the models of nonlinear dynamical system and affects the nature of the whole system significantly. Starting from the Poincaré era, the study of homoclinic orbits in the nonlinear dynamical system exploits Perturbation method mainly. It is not until the recent decade that variational principle has been widely used. The existence and multiplicity of homoclinic orbits for the second order Hamiltonian systems have been extensively investigated in many recent papers (see [

Assume that

Then, (

Subsequently, the condition (

The authors in [

Motivated by these papers, we also consider the subquadratic case. Main results are the following theorems.

Assume that (

there exists a constant

there exists a constant

Then, (

Theorem

Assume that (

Then, (

Theorem

Assume that (

there exist constants

Then, (

Theorem

We should point out that there exist errors in the proofs of [

Let

Then,

Obviously,

In the following, we always denote by

Assume that

Under the conditions of (

Let

Then, by the Hölder inequality and (

In fact, for any given

It follows from (

Therefore, for any

On the other hand, one has

Hence, there exists

proving the weak continuity of

Under the conditions of (

Since the proof is exactly similar to the proof of Lemma

By Lemmas

Let

We divide our proof into three steps.

By (

In order to prove Theorems

Let

Assume that

the compact group

Besides, let

Then,

Let

In the following, we will check that all conditions in Lemma

We claim that

Since

Similar to the third step in the proof of Theorem

If not, there exists a sequence

To be specific, by negation, we have

for all

Then, one gets

It follows from (

At last, it is standard to verify that condition (

Since the proof of Theorem

The author declares that there is no conflict of interests regarding this work.

The paper is supported by General Project of Educational Department in Sichuan (no. 13ZB0182) and Doctor Research Foundation of Southwest University of Science and Technology (no. 11zx7130).