Existence and concentration of solutions for a fractional Schrodinger equations with sublinear nonlinearity

TL;DR

This paper investigates the existence and concentration behavior of solutions to fractional Schrödinger equations with sublinear nonlinearities, demonstrating solutions exist under mild conditions and tend to concentrate on specific sets as a parameter grows large.

Contribution

The study establishes the existence of nontrivial solutions and analyzes their concentration phenomena for fractional Schrödinger equations with sublinear nonlinearities, extending previous results to fractional operators.

Findings

Existence of nontrivial solutions under mild assumptions

Solutions concentrate on the zero set of the potential as parameter increases

Analysis of solution behavior as the parameter tends to infinity

Abstract

This article concerns the fractional elliptic equations \begin{equation*}(-\Delta)^{s}u+\lambda V(x)u=f(u), \quad u\in H^{s}(\mathbb{R}^N), \end{equation*}where $(-\Delta)^{s}$ ($s\in (0\,,\,1)$) denotes the fractional Laplacian, $\lambda >0$ is a parameter, $V\in C(\mathbb{R}^N)$ and $V^{-1}(0)$ has nonempty interior. Under some mild assumptions, we establish the existence of nontrivial solutions. Moreover, the concentration of solutions is also explored on the set $V^{-1}(0)$ as $\lambda\to\infty$.