# Concentrating solutions for a class of nonlinear fractional Schrödinger equations in $\mathbb R^N$

### Vincenzo Ambrosio

Università Politecnica delle Marche, Ancona, Italy

## Abstract

We deal with the existence of positive solutions for the following fractional Schrödinger equation:

$\varepsilon ^{2s} (-\Delta)^{s} u + V(x) u = f(u) \quad \mbox{in } \mathbb{R}^{N},$

where $\varepsilon>0$ is a parameter, $s\in (0, 1)$, $N\geq 2$, $(-\Delta)^{s}$ is the fractional Laplacian operator, and $V\colon \mathbb{R}^{N}\rightarrow \mathbb{R}$ is a positive continuous 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.

## Cite this article

Vincenzo Ambrosio, Concentrating solutions for a class of nonlinear fractional Schrödinger equations in $\mathbb R^N$. Rev. Mat. Iberoam. 35 (2019), no. 5, pp. 1367–1414

DOI 10.4171/RMI/1086