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arXiv · 2608.21724

Liouville theorems and symmetry of positive solutions for partially confined nonlinear Schr\"odinger equations

Zhijie Chen·Youjun Wang·Jianjun Zhang·Xuexiu Zhong
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Abstract

We study positive solutions of the partially confined stationary nonlinear Schr\"odinger equation $$-\Delta u+|y|^2u+\lambda u=g(u),\quad (y,z)\in\mathbb{R}^d\times\mathbb{R}^{m},\quad 1\leq d -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.

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Zhijie Chen, Youjun Wang, Jianjun Zhang, Xuexiu Zhong. 2026-08-22. Liouville theorems and symmetry of positive solutions for partially confined nonlinear Schr\"odinger equations. https://arxiv.org/abs/2608.21724

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