Existence of solutions to asymptotically periodic Schrödinger equations

Abstract

We show the existence of a nonzero solution for the semilinear Schrödinger equation -∆u + V(x)u = ƒ(x, u). The potential V is periodic and 0 belongs to a gap of σ(-∆ + V). The function ƒ is superlinear and asymptotically periodic with respect to x variable. In the proof we apply a new critical point theorem for strongly indefinite functionals proved in [3].