Blow-up rate and local uniqueness for fractional Schrödinger equations with nearly critical growth

2021/04/11 by Daniele Cassani, Cassani, Daniele, Youjun Wang +1
Mathematics · #35A15 #35B40 #35J60 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Functional Analysis (math.FA) #Nonlinear Partial Differential Equations #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2104.04931

openalex publication_date 2021/04/11 · openalex created_date 2021/04/26 · openalex updated_date 2026/07/28

Abstract

We study quantitative aspects and concentration phenomena for ground states of the following nonlocal Schrödinger equation (-Δ)^s u+V(x)u= u^{2_s^*-1-ε} in ℝ^N, where ε>0, s∈ (0,1), 2^*_s:=(2N)/(N-2s), N>4s. We show that the ground state u_ε blows up and precisely with the following rate ‖u_ε‖_L^∞(ℝ^N)∼ ε^-(N-2s)/(4s), as ε→ 0⁺. We also localize the concentration points and, in the case of radial potentials V, we prove local uniqueness of sequences of ground states which exhibit a concentrating behavior.

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