Entire sign-changing solutions to the fractional critical Schr{\"o}dinger equation

We consider the fractional critical Schr{"o}dinger equation (FCSE) \begin{align*} \slaplace{u}-\abs{u}^{2^{\ast}{s}-2}u=0, \end{align*} where $u \in \dot H^s( \R^N)$,$N\geq 2$,$0<s<1$and$2^{\ast}{s}=\frac{2N}{N-2s}$. By virtue of the mini-max theory and the concentration compactness principle with the equivariant group action, we obtain the new type of non-radial, sign-changing solutions of (FCSE) in the energy space ${\dot{H}}^s({\mathbb{R}}^N)$. The key component is that we use the equivariant group to partion ${\dot{H}}^s({\mathbb{R}}^N)$ into several connected components, then combine the concentration compactness argument to show the compactness property of Palais-Smale sequences in each component and obtain many solutions of (FCSE) in ${\dot{H}}^s({\mathbb{R}}^N)$. Both the solutions and the argument here are different from those by Garrido, Musso in \cite{GM2016pjm} and by Abreu, Barbosa and Ramirez in \cite{ABR2019arxiv}.