发表机构
Sun Yat-sen University（中山大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究单位球内临界和超临界薛定谔方程的径向归一化解，证明在径向类中可取得的质量集合为具有有限正上确界的区间，从而否定了Soave提出的所有质量均可取得的问题。
AI 中文摘要
本文研究单位球内非线性薛定谔方程的径向归一化解：\\[-\Delta u+\lambda u=|u|^{p-1}u, \qquad \int_{B_1}|u|^2\\,dx=\rho,\\]其中$\lambda$未知，$p\ge p_c:=\frac{N+2}{N-2}$。Soave \cite{Soave} 证明了对于每个$1<p<p_c$，每个正质量都由无穷多个径向解取得，并明确提出了在Sobolev临界和超临界情形下是否每个质量都能被取得的问题。我们在径向类中否定地回答了这个问题：对于每个$p\ge p_c$，由非平凡径向解取得的质量集合是一个具有有限正上确界的区间。特别地，在$p=p_c$时，可取得的质量集合为$(0,\rho^*_{\rm rad})$或$(0,\rho^*_{\rm rad}]$，其中$0<\rho^*_{\rm rad}<\infty$，并且对于每个$p>p_c$，相同的有限阈值二选一成立。因此，在径向类中，次临界的所有质量现象在整个Sobolev临界和超临界范围内均不成立。
英文摘要
In this paper, we study radial normalized solutions to nonlinear Schrödinger equations in the unit ball \[ -Δu+λu=|u|^{p-1}u, \qquad \int_{B_1}|u|^2\,dx=ρ, \] where $λ$ is unknown and $p\ge p_c:=\frac{N+2}{N-2}$. Soave \cite{Soave} proved that for every $1<p<p_c$ every positive mass is attained by infinitely many radial solutions and explicitly raised the question of whether every mass is attained in the Sobolev critical and supercritical regimes. We answer this question negatively in the radial class: for every $p\ge p_c$, the set of masses attained by nontrivial radial solutions is an interval with finite positive supremum. In particular, at $p=p_c$ the set of attainable masses is $(0,ρ^*_{\rm rad})$ or $(0,ρ^*_{\rm rad}]$, with $0<ρ^*_{\rm rad}<\infty$, and the same finite threshold alternative holds for every $p>p_c$. Thus, in the radial class, the subcritical all masses phenomenon fails throughout the full Sobolev critical and supercritical range.
Comments19 pages
