Normalized solutions to a kind of fractional Schrödinger equation with a critical nonlinearity

Abstract

In this paper, we study normalized solutions of the fractional Schrödinger equation with a critical nonlinearity

$$\begin{aligned} {\left\{ \begin{array}{ll} (-\Delta )^{s}u=\lambda u+|u|^{p-2}u+|u|^{2_{s}^{*}-2}u\\ \int \limits _{{\mathbb {R}}^{N}}u^{2}=a^{2} \end{array}\right. } \end{aligned}$$

where $N\ge 2$, $s\in (0,1)$, $a>0$, $2<p<2_{s}^{*}=\frac{2N}{2N-2s}$ and $(-\Delta )^{s}$ is the fractional Laplace operator. We prove the existence of the normalized solutions under different conditions on a, p, s and N.