New type of bubbling solutions to a critical fractional Schrödinger equation with double potentials

Abstract

In this paper, we study the following critical fractional Schrödinger equation: \begin{equation} (-Δ)^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 $n\ge 3$, $s\in (0,1)$, $V(|y^{\prime }|,y^{\prime \prime })$ and $K(|y^{\prime }|,y^{\prime \prime })$ are two bounded nonnegative potential functions. Under the conditions that $K(r,y^{\prime \prime })$ has a stable critical point $(r_0,y_0^{\prime \prime })$ with $r_0>0$, $K(r_0,y_0^{\prime \prime })>0$ and $V(r_0,y_0^{\prime \prime })>0$, 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.

arXiv admin note: substantial text overlap with arXiv:2307.02272 by other authors