Published Paper

Inserted: 10 may 2022
Last Updated: 31 may 2023

Journal: Nonlinearity
Volume: 34
Number: 6
Pages: 4017-4056
Year: 2021
Doi: 10.1088/1361-6544/ac0166

Abstract:

We study existence of solutions for the fractional problem \begin{equation} (P_m) \quad \left \{ \begin{aligned} (-\Delta)^{s} u + \mu u &=g(u) & \; \text{in $\mathbb{R}^N$}, \cr \int_{\mathbb{R}^N} u² dx &= m, & \cr u \in H^s_r&(\mathbb{R}^N), & \end{aligned} \right. \label{problemx} \end{equation} where $N\geq 2$, $s\in (0,1)$, $m>0$, $\mu$ is an unknown Lagrange multiplier and $g \in C(\mathbb{R}, \mathbb{R})$ satisfies Berestycki-Lions type conditions. Using a Lagrange formulation of the problem $(P_m)$, we prove the existence of a weak solution with prescribed mass when $g$ has $L^2$ subcritical growth. The approach relies on the construction of a minimax structure, by means of a Pohozaev's mountain in a product space and some deformation arguments under a new version of the Palais-Smale condition introduced in 21,25. A multiplicity result of infinitely many normalized solutions is also obtained if $g$ is odd.

Keywords: fractional Laplacian, Pohozaev identity, nonlinear Schrödinger equation, Radially symmetric solution, Normalized solution, Lagrange multiplier, Prescribed mass