Asymptotic behavior of normalized ground states for the fractional Schrödinger equation with combined L 2 ‐critical and L 2 ‐subcritical nonlinearities

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.