The existence and multiplicity of the normalized solutions for fractional Schrödinger equations involving Sobolev critical exponent in the L 2 -subcritical and L 2 -supercritical cases

Abstract This paper is devoted to investigate the existence and multiplicity of the normalized solutions for the following fractional Schrödinger equation: (P) (− Δ) s u + λ u = μ ∣ u ∣ p − 2 u + ∣ u ∣ 2 s ∗ − 2 u, x ∈ R N, u > 0, ∫ R N ∣ u ∣ 2 d x = a 2, \array{l (-) ^su+ u= | u| ^p-2u+| u| ^{2ₒ^ -2}u, 1emx {R}^N, 1. 0em\\ u 0, 1em {ₑ^N}| u| ^2dx=a^2, 1. 0emarray. where 0 s 1 0 s 1, a a, μ > 0 0, <jats: inline-graphic xmlns: xlink="http: //ww