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

2024/08/14 by Li, Ting, Tang, Zhongwei, Wang, Heming +1
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2410.05271

Abstract

In this paper, we study the following critical fractional Schrödinger equation: (-Δ)^s u+V(|y'|,y'')u=K(|y'|,y'')u^{(n+2s)/(n-2s)}, ugt;0, y =(y',y'') ∈ ℝ³×ℝ^n-3, (0.1) where n≥ 3, s∈(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 (r₀,y₀'') with r₀>0, K(r₀,y₀'')>0 and V(r₀,y₀'')>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.

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