Existence of one homoclinic orbit for second order Hamiltonian systems involving certain hypotheses of monotonicity on the nonlinearities

Abstract

Using mainly variational methods and an approximation technique we obtain one homoclinic orbit for a Hamiltonian system type (P)ü+Vu(t,u)=0.Our approach allows us to consider new examples of nonlinearities which do not satisfy the classical Ambrosetti–Rabinowitz condition and neither strict growth hypotheses that allow to use Nehari Manifold. For instance, in our system we can consider a periodic potential V given by V(t,u)=−〈L(t)u,u〉+f(t)|u|2ln(1+|u|),which clearly does not verify those conditions when L(t) is positive definite and symmetric matrix and f is a nontrivial nonnegative continuous function. We notice that f may vanish in some part of its domain.