Normalized solutions for fractional nonlinear scalar field equations via Lagrangian formulation

TL;DR

This paper establishes the existence of normalized solutions for fractional nonlinear scalar field equations using a Lagrangian approach, introducing a new variational framework and proving multiple solutions under certain symmetry conditions.

Contribution

It develops a novel Lagrangian formulation and minimax variational method to find normalized solutions for fractional equations with prescribed mass.

Findings

Proved existence of solutions with prescribed mass for fractional nonlinear equations.

Established a minimax variational structure using Pohozaev's mountain.

Obtained infinitely many solutions when the nonlinearity is odd.

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} u^2 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.