Premium
Asymptotic behavior of normalized ground states for the fractional Schrödinger equation with combined L 2 ‐critical and L 2 ‐subcritical nonlinearities
Author(s) -
Chen Ruipeng,
Liu Jiayin
Publication year - 2020
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.6221
Subject(s) - mathematics , complement (music) , schrödinger equation , ground state , mathematical physics , critical mass (sociodynamics) , mathematical analysis , physics , atomic physics , chemistry , social science , biochemistry , complementation , sociology , gene , phenotype
We study the asymptotic behavior of ground states for the fractional Schrödinger equation with combinedL 2 ‐critical andL 2 ‐subcritical nonlinearities FNLS( − Δ ) s u + ω u = a | u | q u + | u | p u i nR N , N ≥ 2 with prescribed mass ‖ u ‖L 22 = c , where a ∈ R , 0 < q < p = 4 s N . We first show that normalized ground states blow up as c ↗ c ∗ : = ‖ R ‖L 22 , where R is the unique positive radial solution to equation (FNLS) with a = 0 . We then give a detailed description for the asymptotic behavior of normalized ground states as c ↗ c ∗ . Moreover, for a < 0 , we prove that all solutions of corresponding evolution equation with initial mass ‖ R ‖L 2exist globally. This result is a complement to the result of a previous study.