Concentrating solutions for a class of nonlinear fractional Schr\"odinger equations in $\mathbb{R}^{N}$

TL;DR

This paper proves the existence of positive solutions to a fractional Schr"odinger equation that concentrate at local minima of the potential as a parameter approaches zero, expanding understanding of solutions in fractional quantum models.

Contribution

It establishes the existence of solutions concentrating at potential minima for a class of nonlinear fractional Schr"odinger equations, under asymptotic linearity or superlinearity conditions.

Findings

Positive solutions exist for small epsilon.

Solutions concentrate at local minima of V.

Results apply to both asymptotically linear and superlinear nonlinearities.

Abstract

We deal with the existence of positive solutions for the following fractional Schr\"odinger equation $$ \varepsilon ^{2s} (-\Delta)^{s} u + V(x) u = f(u) \mbox{ in } \mathbb{R}^{N}, $$ where $\varepsilon>0$ is a parameter, $s\in (0, 1)$, $N>2s$, $(-\Delta)^{s}$ is the fractional Laplacian operator, and $V:\mathbb{R}^{N}\rightarrow \mathbb{R}$ is a continuous positive function. Under the assumptions that the nonlinearity $f$ is either asymptotically linear or superlinear at infinity, we prove the existence of a family of positive solutions which concentrates at a local minimum of $V$ as $\varepsilon$ tends to zero.