Local super-quadratic conditions on homoclinic solutions for a second-order Hamiltonian system

Abstract

In this paper, we study the homoclinic solutions of the following second-order Hamiltonian system u ̈ − L ( t ) u + ∇ W ( t , u ) = 0 , where t ∈ R , u ∈ R N , L : R → R N × N and W : R × R N → R . Applying the Mountain Pass Theorem, we prove the existence of nontrivial homoclinic solutions under new weaker conditions. In particular, we use a local super-quadratic condition lim | x | → ∞ W ( t , x ) | x | 2 = ∞ , uniformly in t ∈ ( a , b ) for some − ∞ < a < b < + ∞ , instead of the common one lim | x | → ∞ W ( t , x ) | x | 2 = ∞ , uniformly in t ∈ R , which is essential to show the existence of nontrivial homoclinic solutions for the above system in all existing literature.