Normalized solutions for the fractional Schrödinger equation with a focusing nonlocal perturbation.

In this paper, we study the existence and asymptotic properties of solutions to the following fractional Schrödinger equation: (−Δ)σu=λu+|u|q−2u+μIα∗|u|p|u|p−2uinℝNunder the normalized constraint ∫ℝNu2=a2,where N ≥ 2, σ ∈ (0, 1), α ∈ (0, N), q ∈ (2 + 4σ/N, 2N/N − 2σ], p ∈ [2, 1 + 2σ + α/N), a > 0, μ > 0, Iα(x)=|x|α−N and λ∈ℝ appears as a Lagrange multiplier. In the Sobolev subcritical case q ∈ (2 + 4σ/N, 2N/N − 2σ), we show that the problem admits at least two solutions under suitable assumptions on α, a, and μ. In the Sobolev critical case q=2N/N−2σ, 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 L2‐subcritical and L2‐critical setting to L2‐supercritical setting with respect to q, involving Sobolev critical case especially. [ABSTRACT FROM AUTHOR]