发表机构
Tsinghua University; South China University of Technology; Chongqing Jiaotong University; South China Normal University（清华大学; 华南理工大学; 重庆交通大学; 华南师范大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对带有部分简谐受限的非线性薛定谔方程，结合谱分析、变分法与移动平面法，证明了正解存在的必要条件、零质量情形下的无解性、正解的存在性及对称性。
AI 中文摘要
本文研究带有部分简谐受限的定态非线性薛定谔方程的正解：$$-\nabla u + \big( x_1^2 + \boldsymbol{\times} + x_d^2 \big) u + \boldsymbol{\times} u = g(u), \boldsymbol{\times} u \boldsymbol{\times} H^1(\boldsymbol{R}^N),$$其中 $1 \boldsymbol{d} \boldsymbol{N-1}$，非线性项 $g(u)\boldsymbol{0}$ 满足标准正则性与次临界增长条件。此前研究仅关注 $\boldsymbol{\times} \boldsymbol{0}$ 的情形，本文通过对薛定谔算子 $H=-\nabla+|y|^2, y=(x_1,\boldsymbol{\times},x_d)$ 进行谱分析，结合变分法与移动平面法，证明以下结论：1. 正解存在的必要条件为 $\boldsymbol{\times} \boldsymbol{-d}$；当 $\boldsymbol{\times} \boldsymbol{-d}$ 且零质量情形下，只要 $N-d \boldsymbol{4}$，则不存在正解。2. 当 $\boldsymbol{\times} \boldsymbol{-d}$ 时，该方程至少存在一个正解；此外，所有正解关于前 $d$ 个受限变量 $y=(x_1,\boldsymbol{\times},x_d)$ 必为径向对称且递减，关于剩余 $N-d$ 个不受限变量，经平移后也为径向对称且递减。本文谨以此文纪念邹文明教授六十华诞。
英文摘要
We study positive solutions of the partially confined stationary nonlinear Schrödinger equation $$-Δu+|y|^2u+λu=g(u),\quad (y,z)\in\mathbb{R}^d\times\mathbb{R}^{m},\quad 1\leq d<N,\quad m:=N-d.$$ The spectrum of $-Δ+ |y|^2$ is given by $[d,\infty)$. We prove that $λ\geq-d$ is necessary for positive solutions in the classes considered here, and the threshold $λ=-d$ yields several Liouville type results. In particular, for the pure-power equation $$(-Δ+ |y|^2-d)u=u^p,$$ we prove that any nonnegative $H^1(\mathbb{R}^N)$ weak solution is trivial whenever $1\leq p\leq \max\{(N+2)/(N-2), m/(m-2)\}$ for $m\geq 3$ or $p\geq 1$ for $m=1,2$. The proof combines a half-space oscillator gap, moving planes at the spectral threshold, a Picone inequality, a strict Gaussian second-moment inequality, and anisotropic Pohozaev identities. For $λ>-d$, we establish the existence of positive solutions under some standard assumptions. Furthermore, every positive solution decaying at infinity is radially symmetric and strictly decreasing in the confined variables and, up to one common translation, radially symmetric and strictly decreasing in the free variables. \vskip 0.2in Dedicated to our supervisor Prof. Wenming Zou on the occasion of his 60th birthday.
Comments35 pages
