Schr\"odinger equations with asymptotically periodic terms

TL;DR

This paper investigates the existence of solutions to Schrödinger equations with asymptotically periodic potentials using variational methods, establishing conditions for nontrivial and ground state solutions.

Contribution

It introduces a novel approach combining variational methods and concentration-compactness to handle asymptotically periodic Schrödinger equations.

Findings

Existence of nontrivial solutions for asymptotically periodic Schrödinger equations.

Existence of ground state solutions for periodic Schrödinger equations.

Application of Mountain Pass Theorem in this context.

Abstract

We study the existence of nontrivial solutions for a class of asymptotically periodic semilinear Schr\"odinger equations in $\mathbb{R}^N$. By combining variational methods and the concentration-compactness principle we obtain a nontrivial solution for asymptotically periodic problem and a ground state solution for the periodic problem. In the proofs we apply the Mountain Pass Theorem and its local version.