Normalized solutions for nonlinear Schrödinger equations with $L^2$-critical nonlinearity
Silvia Cingolani, Marco Gallo, Norihisa Ikoma, Kazunaga Tanaka
TL;DR
This work analyzes L^2-normalized solutions to the nonlinear Schrödinger equation -Δu+μu=g(u) in R^N with fixed mass m>0, where g exhibits L^2-critical growth at both 0 and ∞ and the critical exponent is p=1+4/N. The authors develop a Lagrangian variational framework I(λ,u) on R×H^1_r(R^N), establish a mountain-pass structure, and introduce two minimax values underline b and overline b to capture the constrained dynamics; compactness is achieved via a novel PSPC (and PSPC^*) scheme that handles the critical mass case m=m1. Under a perturbation g(s)=|s|^{p-1}s+h(s) with h sublinear at infinity, they prove the existence of a positive radially symmetric solution at m=m1, and they also derive sharp nonexistence results under a Liouville-type condition when the sublinear control is removed. The analysis relies on careful deformation arguments in augmented spaces and uses an odd extension of g to obtain sign information, yielding multiplicity in the balanced (g1) setting and establishing the essential role of the mass constraint in the L^2-critical regime. These results extend the L^2-critical theory to a broader class of nonlinearities with equal asymptotics and provide a robust variational framework for normalized solutions with prescribed mass.
Abstract
We study the following nonlinear Schrödinger equation and we look for normalized solutions $(μ,u)\in {\bf R}\times H^1({\bf R}^N)$ for a given $m>0$ and $N\geq 2$ \[ -Δu + μu = g(u)\quad \text{in}\ {\bf R}^N, \qquad \frac{1}{2}\int_{{\bf R}^N} u^2 dx = m. \] We assume that $g$ has an $L^2$-critical growth, both at the origin and at infinity. That is, for $p=1+\frac{4}{N}$, $g(s)=|s|^{p-1}s +h(s)$, $h(s)=o(|s|^p)$ as $s\sim 0$ and $s\sim\infty$. The $L^2$-critical exponent $p$ is very special for this problem; in the power case $g(s) = |s|^{p-1}s$ a solution exists only for the specific mass $m=m_1$, where $m_1=\frac{1}{2}\int_{{\bf R}^N}ω_1^2\, dx$ is the mass of a least energy solution $ω_1$ of $-Δω+ω=ω^p$ in ${\bf R}^N$. We prove the existence of a positive solution for $m=m_1$ when $h$ has a sublinear growth at infinity, i.e., $h(s)=o(s)$ as $s\sim\infty$. In contrast, we show non-existence results for $h(s)\not=o(s)$ ($s\sim 0$) under a suitable monotonicity condition.