Infinitely many homoclinic solutions for a second-order Hamiltonian system

In this paper, we study the homoclinic solutions of the following second- order Hamiltonian system mu - L(t)u + Delta W(t, u) = 0 where t is an element of R, u is an element of R-N, L:R -> R(N)x(N) and W : R x R-N -> R. Applying the symmetric Mountain Pass Theorem, we establish a couple of sufficient conditions on the existence of infinitely many homoclinic solutions. Our results significantly generalize and improve related ones in the literature. For example, L( t) is not necessary to be uniformly positive definite or coercive; through W( t, x) is still assumed to be superquadratic near vertical bar x vertical bar = infinity it is not assumed to be superquadratic near x = 0. (C) 2015 WILEY- VCH Verlag GmbH & Co. KGaA, Weinheim