Infinitely many homoclinic solutions for a second‐order Hamiltonian system

In this paper, we study the homoclinic solutions of the following second‐order Hamiltonian systemu ̈ − L ( t ) u + ∇ W ( t , u ) = 0 , where t ∈ R , u ∈ R N , L : R → R N × Nand W : R × 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 | x | = ∞ , it is not assumed to be superquadratic near x = 0 .