Calculus of Variations and Geometric Measure Theory

S. Cingolani - M. Gallo - K. Tanaka

Normalized solutions for fractional nonlinear scalar field equations via Lagrangian formulation

created by gallo on 10 May 2022
modified on 13 Nov 2022

[BibTeX]

Published Paper

Inserted: 10 may 2022
Last Updated: 13 nov 2022

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

ArXiv: 2103.10747 PDF
Links: IOP Science

Abstract:

We study existence of solutions for the fractional problem \begin{equation} (Pm) \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 Hsr&(\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