On homoclinic orbits for a class of noncoercive superquadratic Hamiltonian systems

Abstract

In this paper, we prove the existence of homoclinic solutions for a class of noncoercive first order Hamiltonian systems J x ̇ − M ( t ) x + u ∗ G ′ ( t , u ( x ) ) = 0 , by the minimax methods in critical point theory, specially, a Generalized Mountain Pass Theorem, when u is a linear operator with adjoint u ∗ and G ( t , y ) satisfies the superquadratic condition G ( t , x ) | y | 2 ⟶ − + ∞ as | y | ⟶ ∞ , uniformly in t , and need not satisfy the global Ambrosetti–Rabinowitz condition.