Multi-peak positive solutions for a fractional nonlinear elliptic equation

Xudong Shang; Jihui Zhang

doi:10.3934/dcds.2015.35.3183

Article Contents

2015, Volume 35, Issue 7: 3183-3201. Doi: 10.3934/dcds.2015.35.3183

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Multi-peak positive solutions for a fractional nonlinear elliptic equation

Xudong Shang^1, and
Jihui Zhang^1,

1.
Jiangsu Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023

Received: August 31, 2014

Revised: October 31, 2014

Published: May 2017

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Abstract

In this paper we study the existence of positive multi-peak solutions 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 positive solution with $k$-peaks and concentrating near a given local minimum point of $Q$. For $s=1$ this corresponds to the result of [22].

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Mathematics Subject Classification: Primary: 35A15, 35J60; Secondary: 35B38.

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