EXISTENCE OF HOMOCLINIC ORBITS OF A CLASS OF SECOND-ORDER QUASILINEAR SCHRÖDINGER EQUATIONS WITH DELAY

Abstract

We study the existence of homoclinic orbits of the second order quasilinear Schrödinger equations u¨(t)−V(t)u(t)+2[u¨(t)u2(t)+u˙2(t)u(t)]+g(t,u(t+τ),u(t),u(t−τ))=h(t). containing both advance and retardation terms. By using critical point theory and variational approaches, we establish two different existence results. The first is based on g which does not satisfy the Ambrosetti–Rabinowitz growth condition. The second is based on g satisfying the Ambrosetti–Rabinowitz growth condition.