Proceedings of the Royal Society of Edinburgh: Section A Mathematics

Research Article

Existence of infinitely many homoclinic orbits in Hamiltonian systems

X. H. Tanga1 and Xiaoyan Lina2

a1 School of Mathematical Sciences and Computing Technology, Central South University, Changsha, Hunan 410083, People's Republic of China (tangxh@mail.csu.edu.cn)

a2 School of Mathematical Sciences and Computing Technology, Central South University, Changsha, Hunan 410083, People's Republic of China (tangxh@mail.csu.edu.cn) and Department of Mathematics, Huaihua College, Huaihua, Hunan 418008, People's Republic of China

Abstract

By using the symmetric mountain pass theorem, we establish some new existence criteria to guarantee that the second-order Hamiltonian systems ü(t) − L(t)u(t) + ∇W(t,u(t)) = 0 have infinitely many homoclinic orbits, where t ∈ ℝ, u ∈ ℝN, LC(ℝ, ℝN × N) and WC1(ℝ × ℝN, ℝ) are not periodic in t. Our results generalize and improve some existing results in the literature by relaxing the conditions on the potential function W(t, x).

(Received August 31 2009)

(Accepted January 21 2011)

(Online publication September 26 2011)