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

$$\ddot{u}(t)-V(t)u(t)+2[\ddot{u}(t)u^2(t)+{\dot{u}}^2(t)u(t)]+g(t,u(t+\tau ),u(t),u(t-\tau ))=h(t).$$

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.