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

Abstract

In this paper, we study normalized ground states for the following critical fractional NLS equation with prescribed mass:

$$\begin{aligned} {\left\{ \begin{array}{ll} (-\Delta )^{s}u=\lambda u +\mu |u|^{q-2}u+|u|^{2_{s}^{*}-2}u,&{}x\in \mathbb {R}^{N}, \\ \int _{\mathbb {R}^{N}}u^{2}dx=a^{2},\\ \end{array}\right. } \end{aligned}$$

where $(-\Delta )^{s}$ is the fractional Laplacian, $0<s<1$, $N>2s$, $2<q<2_{s}^{*}=2N/(N-2s)$ is a fractional critical Sobolev exponent, $a>0$, $\mu \in \mathbb {R}$. 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 $L^{2}$-subcritical (or $L^{2}$-critical or $L^{2}$-supercritical) perturbation $\mu |u|^{q-2}u$, then we give some results about the behavior of the ground state obtained above as $\mu \rightarrow 0^{+}$. Our results extend and improve the existing ones in several directions.