Multi-peak positive solutions for a fractional nonlinear elliptic equation

Abstract

In this paper we study the existence of positive multi-peaksolutions to the semilinear equation\begin{eqnarray*} \varepsilon^{2s}(-\Delta)^{s}u + u= Q(x)u^{p-1}, \hskip0.5cm u >0, \hskip 0.2cm u\in H^{s}(\mathbb{R}^{N})\end{eqnarray*}where $(-\Delta)^{s} $ stands for the fractional Laplacian, $s\in(0,1)$, $\varepsilon$ is a positive small parameter, $2 < p <\frac{2N}{N-2s}$, $Q(x)$ is a bounded positive continuous function.For any positive integer $k$, we prove the existence of a positivesolution with $k$-peaks and concentrating near a given local minimumpoint of $Q$. For $s=1$ this corresponds to the result of [22].