Normalized Solutions to the Fractional Schrödinger Equation with Critical Growth

In this paper, we discuss the existence of normalized solutions to the following fractional Schrödinger equation \(\begin{aligned} \left\{ \begin{array}{l} (-\Delta )^s u = \lambda u + g(u) + |u|^{2^{*}_{s}-2}u, \quad x \in \mathbb {R}^{N}, \\ \int _{\mathbb {R}^{N}} u^2 = a^2, \end{array}\right. \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{c}{(-\mathrm{\Delta })}^{s}u=\lambda u+g(u)+{|u|}^{2_s^{\ast }-2}u,x\in {\mathbb{R}}^N,\hfill \\ {\int }_{{\mathbb{R}}^N}{u}^2=a^2,\hfill \end{array}\right)\end{array}$$ where \(N \ge 3\) $N\ge 3$ , \(s \in (0,1)\) $s\in (0,1)$ , \(a>0\) $a>0$ , \(2_{s}^{*}= 2N/(N-2s)\) $2_s^{\ast }=2N/(N-2s)$ , \(\lambda \in \mathbb {R}\) $\lambda \in \mathbb{R}$ arises as a Lagrange multiplier, \((-\Delta )^s\) ${(-\mathrm{\Delta })}^s$ is the fractional Laplace operator and \(g: \mathbb {R} \rightarrow \mathbb {R}\) $g:\mathbb{R}\to \mathbb{R}$ satisfies \(L^{2}\) $L^2$ -supercritical conditions. The proof is based on a constrained minimization method and some characterizations of the mountain pass levels are given in order to prove the existence of ground state normalized solutions.