Positive or sign-changing solutions for a critical semilinear nonlocal equation

Abstract

We consider the following critical semilinear nonlocal equation involving the fractional Laplacian $$(-\Delta {)}^{s}u=K(|x|)|u{|}^{2_s^{\ast }-2}u,in{\mathbb{R}}^N,$$ where $K(|x|)$ is a positive radial function, $N>2+2s$, $0<s<1$, and $2_s^{\ast }=\frac{2N}{N-2s}$. Under some asymptotic assumptions on $K(x)$ at an extreme point, we show that this problem has infinitely many non-radial positive or sign-changing solutions.