Existence of nontrivial solutions for periodic schrödinger equations with new nonlinearities

Abstract

We study the Schrödinger equation: - Δ u + V x u + f x, u = 0, u ∈ H1(ℝN), where V is 1 -periodic and f is 1 -periodic in the x -variables; 0 is in a gap of the spectrum of the operator - Δ + V. We prove that, under some new assumptions for f, this equation has a nontrivial solution. Our assumptions for the nonlinearity f are very weak and greatly different from the known assumptions in the literature. © 2014 Shaowei Chen and Dawei Zhang.