Normalized ground states for NLS equations with mass critical nonlinearities
Silvia Cingolani, Marco Gallo, Norihisa Ikoma, Kazunaga Tanaka
TL;DR
This work advances the theory of normalized solutions to nonlinear Schrödinger equations with mass-critical nonlinearities by analyzing the Lagrangian minimax structure and its relation to $L^2$-prescribed mass. It provides explicit constructions of nonlinearities that realize different signs of the minimax levels $\underline b$ and $\overline b$, proving existence, nonexistence, or multiplicity of minimal-energy normalized states under precise variational conditions. A key contribution is the $L^2$-minimization framework, including a Legendre-type relation $d(m_1)=\inf_{\mu>0}(a(\mu)-\mu m_1)$ and results on existence and compactness of minimizers, together with a stability theory showing robustness under small perturbations of the nonlinearity. The stability results rely on zero-mass problem continuity and a uniform 2D-path class, yielding persistence of positive normalized solutions under perturbations that keep $\overline b>0$, with implications for constructing robust nonlinearities in critical regimes.
Abstract
We study normalized solutions $(μ,u)\in \mathbb{R} \times H^1(\mathbb{R}^N)$ to nonlinear Schrödinger equations $$ -Δu + μu = g(u)\quad \hbox{in}\ \mathbb{R}^N, \qquad \frac{1}{2}\int_{\mathbb{R}^N} u^2 dx = m, $$ where $N\geq 2$ and the mass $m>0$ is given. Here $g$ has an $L^2$-critical growth, both at the origin and at infinity, that is $g(s)\sim |s|^{p-1}s$ as $s\sim 0$ and $s\sim\infty$, where $p=1+\frac{4}{N}$. We continue the analysis started in [Cingolani-Gallo-Ikoma-Tanaka, 2024], where we found two (possibly distinct) minimax values $\underline{b} \leq 0 \leq \overline{b}$ of the Lagrangian functional. In this paper we furnish explicit examples of $g$ satisfying $\underline{b}<0<\overline{b}$, $\underline{b}=0<\overline{b}$ and $\underline{b}<0=\overline{b}$; notice that $\underline{b}=0=\overline{b}$ in the power case $g(t)=|t|^{p-1}t$. Moreover, we deal with the existence and non-existence of a solution with minimal energy. Finally, we discuss the assumptions required on $g$ to obtain the existence of a positive solution for perturbations of $g$.