Normalized ground states for the critical fractional NLS equation with a perturbation

In this paper, we study normalized ground states for the following critical fractional NLS equation with prescribed mass: $$\begin{aligned} {\left\{ \begin{array}{ll} (-\Delta )^{s}u=\lambda u +\mu |u|^{q-2}u+|u|^{2_{s}^{*}-2}u,&{}x\in \mathbb {R}^{N}, \\ \int _{\mathbb {R}^{N}}u^{2}dx=a^{2},\\ \end{array}\right. } \end{aligned}$$ ( - Δ ) s u = λ u + μ | u | q - 2 u + | u | 2 s ∗ - 2 u , x ∈ R N , ∫ R N u 2 d x = a 2 , where $$(-\Delta )^{s}$$ ( - Δ ) s is the fractional Laplacian, $$02s$$ N > 2 s , $$20$$ a > 0 , $$\mu \in \mathbb {R}$$ μ ∈ R . By using Jeanjean’s trick in Jeanjean (Nonlinear Anal 28:1633–1659, 1997), and the standard method which can be found in Brézis and Nirenberg (Commun Pure Appl Math 36:437–477, 1983) to overcome the lack of compactness, we first prove several existence and nonexistence results for a $$L^{2}$$ L 2 -subcritical (or $$L^{2}$$ L 2 -critical or $$L^{2}$$ L 2 -supercritical) perturbation $$\mu |u|^{q-2}u$$ μ | u | q - 2 u , then we give some results about the behavior of the ground state obtained above as $$\mu \rightarrow 0^{+}$$ μ → 0 + . Our results extend and improve the existing ones in several directions.