Existence of solutions to asymptotically periodic Schrodinger equations

Abstract

We show the existence of a nonzero solution for the semilinear Schrodinger equation $-\Delta u+V(x)u=f(x,u)$. The potential V is periodic and 0 belongs to a gap of $\sigma(-\Delta +V)$. The function f 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].

Keywords

• Strongly indefinite functionals
• Schrodinger equation
• asymptotically periodic problem