Existence of homoclinic solutions for a class of second-order Hamiltonian systems with subquadratic growth

Dan Zhang

doi:10.1186/1687-2770-2012-132

Abstract

By properly constructing a functional and by using the critical point theory, we establish the existence of homoclinic solutions for a class of subquadratic second-order Hamiltonian systems. Our result generalizes and improves some existing ones. An example is given to show that our theorem applies, while the existing results are not applicable.

Highlights

Consider the following second-order Hamiltonian system: q(t) – L(t)q(t) + Wq t, q(t) =, t ∈ R, (HS)where q = (q, q, . . . , qn) ∈ Rn, L ∈ C(R, Rn×n) is a symmetric matrix-valued function, and W (t, q) ∈ C (R × Rn, R), Wq(t, q) ∈ C (R × Rn, Rn) is the gradient of W about q
The existence and multiplicity of homoclinic solutions for second-order Hamiltonian systems have been extensively investigated in many papers via variational methods
It is easy to see that (H ) in Theorem . is not satisfied, so we cannot obtain the existence of homoclinic solutions for the Hamiltonian system ( ) by Theorem

Summary

Introduction

Many authors studied the existence and multiplicity of homoclinic solutions for (HS); see [ – ]. ]) Assume that L and W satisfy the following conditions: (H ) L(t) ∈ C(R, Rn×n) is a symmetric matrix for all t ∈ R, and there is a continuous function α : R → R such that α(t) > for all t ∈ R and (L(t)q, q) ≥ α(t)|q| and α(t) → +∞ as |t| → +∞. Assume that the following conditions hold: (H ) W (t, q) ≥ a(t)|q|γ , ∀(t, q) ∈ (R, Rn), where a(t) : R → R+ is a positive continuous function such that a(t) is a constant.

Objectives

Results

Conclusion