Normalized solutions for fractional nonlinear scalar field equations via Lagrangian formulation

Abstract

We study existence of solutions for the fractional problem \begin{equation*} (P_m) \quad \left \{ \begin{aligned} (-Δ)^{s} u + μu &=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 $N\ge 2$, $s\in (0,1)$, $m>0$, $\mu$ 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.

To be published in Nonlinearity (accepted)

References in corpus (1)

• Numerical study of fractional Nonlinear Schrödinger equations

Cited by in corpus (4)

• On global minimizers for a mass constrained problem
• On fractional Schrödinger equations with Hartree type nonlinearities
• Infinitely many free or prescribed mass solutions for fractional Hartree equations and Pohozaev identities
• Radial and non-radial multiple solutions to a general mixed dispersion NLS equation