New type of bubbling solutions to a critical fractional Schr\"{o}dinger equation with double potentials

TL;DR

This paper discovers a new class of infinitely many positive solutions to a critical fractional Schrödinger equation with double potentials, where solutions concentrate at symmetric points on a cylinder, using advanced reduction techniques.

Contribution

It introduces a novel type of bubbling solutions for a fractional Schrödinger equation with double potentials, demonstrating concentration on a cylindrical set with symmetric points.

Findings

Existence of infinitely many solutions concentrating on a cylinder

Solutions can concentrate at symmetric points relative to the origin

Application of a modified finite-dimensional reduction method

Abstract

In this paper, we study the following critical fractional Schr\"odinger equation: \begin{equation} (-\Delta)^s u+V(|y'|,y'')u=K(|y'|,y'')u^{\frac{n+2s}{n-2s}},\quad u>0,\quad y =(y',y'') \in \mathbb{R}^3\times\mathbb{R}^{n-3}, \qquad(0.1)\end{equation} where $n\geq 3$, $s\in(0,1)$, $V(|y'|,y'')$ and $K(|y'|,y'')$ are two bounded nonnegative potential functions. Under the conditions that $K(r,y'')$ has a stable critical point $(r_0,y_0'')$ with $r_0>0$, $K(r_0,y_0'')>0$ and $V(r_0,y_0'')>0$, we prove that equation (0.1) has a new type of infinitely many solutions that concentrate at points lying on the top and the bottom of a cylinder. In particular, the bubble solutions can concentrate at a pair of symmetric points with respect to the origin. Our proofs make use of a modified finite-dimensional reduction method and local Pohozaev identities.