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Normalized solutions for the fractional Schrödinger equation with a focusing nonlocal perturbation
Author(s) -
Li Gongbao,
Luo Xiao,
Yang Tao
Publication year - 2021
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.7411
Subject(s) - sobolev space , mathematics , lagrange multiplier , perturbation (astronomy) , schrödinger equation , mathematical physics , mathematical analysis , quantum mechanics , physics , mathematical optimization
In this paper, we study the existence and asymptotic properties of solutions to the following fractional Schrödinger equation:( − Δ ) σ u = λ u + | u | q − 2 u + μI α ∗ | u | p| u | p − 2 u inℝ Nunder the normalized constraint∫ℝ Nu 2 = a 2 , where N ≥ 2, σ ∈ (0, 1), α ∈ (0, N ), q ∈ (2 + 4 σ / N , 2 N / N − 2 σ ], p ∈ [2, 1 + 2 σ + α / N ), a > 0, μ > 0,I α ( x ) = | x | α − Nand λ ∈ ℝ appears as a Lagrange multiplier. In the Sobolev subcritical case q ∈ (2 + 4 σ / N , 2 N / N − 2 σ ), we show that the problem admits at least two solutions under suitable assumptions on α , a , and μ . In the Sobolev critical case q = 2 N / N − 2 σ , we prove that the problem possesses at least one ground state solution. Furthermore, we give some stability and asymptotic properties of the solutions. We mainly extend the results of S. Bhattarai published in 2017 on J. Differ. Equ. and B. H. Feng et al published in 2019 on J. Math. Phys. concerning the above problem from L 2 ‐subcritical and L 2 ‐critical setting to L 2 ‐supercritical setting with respect to q , involving Sobolev critical case especially.