Existence of Nontrivial Solutions for Periodic Schrödinger Equations with New Nonlinearities

Abstract EN

We study the Schrödinger equation: - Δ u + V x u + f x , u = 0 , u ∈ H 1 ( R N ) , 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.