Revista Matemática Iberoamericana


Full-Text PDF (552 KB) | Metadata | Table of Contents | RMI summary
Volume 35, Issue 5, 2019, pp. 1367–1414
DOI: 10.4171/rmi/1086

Published online: 2019-06-04

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

Vincenzo Ambrosio[1]

(1) Università Politecnica delle Marche, Ancona, Italy

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.

Keywords: Fractional Laplacian, concentrating solutions, penalization technique, variational methods

Ambrosio Vincenzo: Concentrating solutions for a class of nonlinear fractional Schrödinger equations in $\mathbb R^N$. Rev. Mat. Iberoam. 35 (2019), 1367-1414. doi: 10.4171/rmi/1086