Multiple semiclassical states for fractional Schrodinger equations with asymptotically linear nonlinearities

TL;DR

This paper studies multiple solutions to a fractional Schrödinger equation with asymptotically linear nonlinearities, showing how the number of solutions relates to the potential's minimum points and introducing new methods to handle the nonlinearities.

Contribution

It introduces novel techniques involving the Nehari manifold to establish multiple solutions for fractional Schrödinger equations with asymptotically linear nonlinearities.

Findings

Multiple positive solutions are linked to the topology of the potential's minimum set.

Construction of multiple concentrating solutions at global minimum points.

Development of new methods to handle asymptotically linear nonlinearities.

Abstract

In this paper, we consider the singularly perturbed fractional Schr\"{o}dinger equation \begin{equation*} \epsilon^{2\alpha}(-\Delta)^\alpha u+V(x)u=f(u),\quad x\in \mathbb{R}^N, \end{equation*} where $\epsilon>0$ is a small parameter, $\alpha\in(0,1)$, $(-\Delta)^\alpha$ is the fractional Laplacian operator of order $\alpha$, $V$ possesses global minimum points, and $f$ is asymptotically linear at infinity. We investigate the relationship between the number of positive solutions and the topology of the set where the potential $V$ attains its global minimum. We also construct multiple concentrating solutions if $V$ has several strict global minimum points. In particular, some new tricks and the method of Nehari manifold dependent on a suitable restricted set are introduced to overcome the difficulty resulting from the appearance of asymptotically linear nonlinearity.