Multiple solutions for a fractional Schrodinger equation with potentials

TL;DR

This paper investigates the existence and multiplicity of solutions for a class of nonlinear fractional Schrödinger equations with sign-changing potentials and sublinear nonlinearities using variational methods.

Contribution

It establishes the existence of at least one solution for general potentials and infinitely many solutions when the potential is nonnegative, introducing new variational techniques.

Findings

Existence of at least one nontrivial solution for sign-changing potentials.

Infinitely many solutions for nonnegative potentials with locally sublinear nonlinearities.

Application of variational and Moser iteration methods to fractional Schrödinger equations.

Abstract

This paper is devoted to study a class of nonlinear fractional Schr\"{o}dinger equations: \begin{equation*} (-\Delta)^{s}u+V(x)u=f(x,u), \quad \text{in}\: \mathbb{R}^{N}, \end{equation*} where $s\in (0,1)$, $\ N>2s$, $(-\Delta)^{s}$ stands for the fractional Laplacian. First, by using a variational approach, we establish the existence of at least one nontrivial solution for the above equation with a general potential $V(x)$ which is allowed to be sign-changing and a sublinear nonlinearity $f(x,u)$. Next, by using variational methods and the Moser iteration technique, we prove the existence of infinitely many solutions with $V(x)$ is a nonnegative potential and the nonlinearity $f(x,u)$ is locally sublinear with respect to $u$.