Year: 2024
Type: preprint
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.
Source: arXiv (Cornell University)
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Awards: G124829060, 非局所非線形楕円型方程式に対する特異摂動解析, Study of solution structures of elliptic equations: supercritical, critical and subcritocal cases, Mathematical Analysis on Interactions of Derivative Type in the Classical Field Theory, G465505792, G5488169599, 特異性や制約条件を持つ非線形楕円型方程式の解構造の研究, Study of the strucure of solutions to nonlinear elliptic equations with various effects
OpenAlex ID: W4404347103