On homoclinic orbits for a class of noncoercive superquadratic Hamiltonian systems

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