Multiplicity results for fractional Laplace problems with critical growth

Abstract

This paper deals with multiplicity and bifurcation results for nonlinear problems driven by the fractional Laplace operator $(-\Delta {)}^s$ and involving a critical Sobolev term. In particular, we consider $$\begin{cases} (-Δ)^su=γ|u|^{2^*-2}u+f(x,u) & \mbox{in } Ωu=0 & \mbox{in } \mathbb R^n\setminus Ω, \end{cases}$$ where $\Omega \subset {\mathbb{R}}^n$ is an open bounded set with continuous boundary, $n>2s$ with $s\in (0,1)$, $\gamma$ is a positive real parameter, $2^{\ast }=2n/(n-2s)$ is the fractional critical Sobolev exponent and $f$ is a Carathéodory function satisfying different subcritical conditions.