A critical fractional equation with concave–convex power nonlinearities

Abstract

In this work we study the following fractional critical problem

where $\Omega \subset {\mathbb{R}}^n$ is a regular bounded domain, $\lambda >0$, $0<s<1$ and $n>2s$. Here $(-\mathrm{\Delta }{)}^s$ denotes the fractional Laplace operator defined, up to a normalization factor, by

Our main results show the existence and multiplicity of solutions to problem $(P_{\lambda })$ for different values of $\lambda$. The dependency on this parameter changes according to whether we consider the concave power case ($0<q<1$) or the convex power case ($1<q<2_s^{\ast }-1$). These two cases will be treated separately.