Perturbations of a critical fractional equation

Eduardo Colorado, Arturo de Pablo and Urko Sánchez

Vol. 271 (2014), No. 1, 65–85

DOI: 10.2140/pjm.2014.271.65

Abstract

We deal with the following fractional critical problem:

$$\left\{\begin{array}{c}\hfill \\ \hfill \\ {\left(-\Delta \right)}^{\alpha ∕2}u=|u{|}^{2\alpha ∕\left(N-\alpha \right)}u+f\left(x\right)\hfill & \text{ in }\Omega ,\hfill \\ \hfill \\ \hfill \\ u=0\hfill & \text{ on }\partial \Omega ,\hfill \\ \hfill \\ \hfill \\ \hfill \end{array}\right.$$

where $\Omega \subset {ℝ}^N$ is a regular bounded domain, $0<\alpha <2$ and $N>\alpha$. Under appropriate conditions on the size of $f$, we prove existence and multiplicity of solutions.

Keywords

semilinear elliptic equations, fractional Laplacian, critical problem

Mathematical Subject Classification 2010

Primary: 35A15, 49J35, 35R11

Milestones

Received: 11 April 2013

Revised: 8 July 2013

Accepted: 12 July 2013

Published: 10 September 2014