A new type of bubble solutions for a critical fractional Schr\"odinger equation

TL;DR

This paper constructs a new class of infinitely many solutions for a critical fractional Schrödinger equation, where solutions concentrate on specific points related to the potential's critical points, overcoming challenges from the non-local fractional Laplacian.

Contribution

It introduces a novel type of bubble solutions for the critical fractional Schrödinger equation, utilizing finite-dimensional reduction and Pohozaev identities.

Findings

Existence of infinitely many solutions concentrating on the top and bottom of a cylinder.

Solutions include saddle points of the potential function.

Overcomes difficulties due to the non-local fractional Laplacian.

Abstract

We consider the following critical fractional Schr\"{o}dinger equation \begin{equation*} (-\Delta)^s u+V(|y'|,y'')u = u^{2_s^*-1},\quad u>0,\quad y =(y',y'') \in \mathbb{R}^3\times\mathbb{R}^{N-3}, \end{equation*} where $N\geq 3,s\in(0,1)$, $2_s^*=\frac{2N}{N-2s}$ is the fractional critical Sobolev exponent and $V(|y'|,y'')$ is a bounded non-negative function in $\mathbb{R}^3\times\mathbb{R}^{N-3}$. If $r^{2s}V(r,y'')$ has a stable critical point $(r_0,y_0'')$ with $r_0>0$ and $V(r_0,y_0'')>0$, by using a finite-dimensional reduction method and various local Pohozaev identities, we prove that the problem above has a new type of infinitely many solutions which concentrate at points lying on the top and the bottom of a cylinder. And the concentration points of the bubble solutions include saddle points of the function $r^{2s}V(r,y'')$. We have to overcome some difficulties caused by the non-localness of the fractional Laplacian.