Existence of Nontrivial Solutions for Periodic Schrödinger Equations with New Nonlinearities

Shaowei Chen,Dawei Zhang

doi:10.1155/2014/539639

Abstract

We study the Schrödinger equation:-Δu+Vxu+fx,u=0, u∈H1(RN), whereVis1-periodic andfis1-periodic in thex-variables;0is in a gap of the spectrum of the operator-Δ+V. We prove that, under some new assumptions forf, this equation has a nontrivial solution. Our assumptions for the nonlinearityfare very weak and greatly different from the known assumptions in the literature.

Highlights

Introduction and Statement of ResultsIn this paper, we consider the following Schrodinger equation:−Δu + V (x) u + f (x, u) = 0, u ∈ H1 (RN), (1)where N ≥ 1
We study the Schrodinger equation: −Δu + 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
(v) V ∈ C(RN) is 1-periodic in xj for j = 1, . . . , N, 0 is in a spectral gap (−μ−1, μ1) of −Δ + V, and −μ−1 and μ1 lie in the essential spectrum of −Δ + V

Summary

Introduction

In [2], the authors used the dual variational method to obtain a nontrivial solution of (1) with f(x, t) = ±W(x)|t|p−2t, where W is an asymptotically periodic function. In [20], Troestler and Willem firstly obtained nontrivial solutions for (1) with f being a C1 function satisfying the Ambrosetti-Rabinowitz condition: (AR) there exists α > 2 such that, for every u ≠ 0, 0 < αG(x, u) ≤ g(x, u)u, where g(x, u) = −f(x, u), G(x, u) = −F(x, u), and They improved Troestler and Willem’s results and obtained nontrivial solutions for (1) with f only satisfying (f1) and the (AR) condition.

Results

Conclusion