GROUND STATE SOLUTIONS FOR SCHRÖDINGER EQUATIONS WITH PERIODIC POTENTIALS

Abstract

In this paper, we study a class of Schrödinger equations including multiple different periodic potentials, this type of equation has a strong physical background and has become a hot topic in current research, especially its widespread application in the theory of Bose-Einstein condensates. Under some appropriate assumptions, we prove the existence of ground state solutions using the variational methods and the concentration compactness principle. Additionally, defining the equation on an unbounded domain and excluding semi-trivial solutions are relatively difficult parts. In the proofs we apply the variant of the Mountain Pass Theorem where it is considered the Gerami condition instead of the PalaisSmale condition