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

TL;DR

This paper constructs new non-radial, sign-changing solutions to the fractional critical Schrödinger equation using variational methods, equivariant group actions, and concentration compactness, expanding the known solution space.

Contribution

It introduces a novel approach combining equivariant group actions with concentration compactness to find multiple sign-changing solutions in the energy space.

Findings

Existence of non-radial, sign-changing solutions.

Solutions are obtained via a new variational framework.

Distinct from previous methods by Garrido, Musso, Abreu, Barbosa, and Ramirez.

Abstract

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 $\dot H^s(\R^N)$. The key component is that we use the equivariant group to partion $\dot H^s(\R^N)$ 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 $\dot H^s(\R^N)$. 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}.