Existence of sign-changing solution for a problem involving the fractional Laplacian with critical growth nonlinearities

Abstract

We study the existence of least energy sign-changing solution for the fractional equation $(-\Delta {)}^{s}u=|u{|}^{2_s^{\ast }-2}u+\lambda f(x,u)$ in a smooth bounded domain $\Omega$ of ${\mathbb{R}}^N,$ $u=0$ in ${\mathbb{R}}^N\setminus \Omega$, where $s\in (0,1)$ and $2_s^{\ast }$ is the fractional critical Sobolev exponent. The proof is based on constrained minimization in a subset of Nehari manifold, containing all the possible sign-changing solutions of the equation.