Normalized Solutions to the Fractional Schrödinger Equation with Critical Growth

Abstract

In this paper, we discuss the existence of normalized solutions to the following fractional Schrödinger equation

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

where $N \ge 3$, $s \in (0,1)$, $a>0$, $2_{s}^{*}= 2N/(N-2s)$, $\lambda \in \mathbb {R}$ arises as a Lagrange multiplier, $(-\Delta )^s$ is the fractional Laplace operator and $g: \mathbb {R} \rightarrow \mathbb {R}$ satisfies $L^{2}$-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.