Liouville theorems and symmetry of positive solutions for partially confined nonlinear Schrödinger equations
Abstract
We study positive solutions of the partially confined stationary nonlinear Schrödinger equation −Δu+|y|2u+λu=g(u), (y,z)∈ℝd×ℝm, 1≤ d<N, m:=N−d. The spectrum of −Δ+ |y|2 is given by [d,∞). We prove that λ≥−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=up, we prove that any nonnegative H1(ℝN) weak solution is trivial whenever 1≤ p≤ max{(N+2)/(N−2), m/(m−2)} for m≥ 3 or p≥ 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.
Related
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BibTeX
@misc{chen2026liouville,
title = {{Liouville theorems and symmetry of positive solutions for partially confined nonlinear Schrödinger equations}},
author = {Zhijie Chen and Youjun Wang and Jianjun Zhang and Xuexiu Zhong},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.21724},
eprint = {2608.21724},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.21724},
}
RIS
TY - UNPB
TI - Liouville theorems and symmetry of positive solutions for partially confined nonlinear Schrödinger equations
AU - Zhijie Chen
AU - Youjun Wang
AU - Jianjun Zhang
AU - Xuexiu Zhong
PY - 2026
DA - 2026/09/05
PB - arXiv
DO - 10.48550/arxiv.2608.21724
UR - https://arxiv.org/abs/2608.21724
AB - We study positive solutions of the partially confined stationary nonlinear Schrödinger equation −Δu+|y|^2u+λu=g(u), (y,z)∈ℝ^d×ℝ^m, 1≤ d<N, m:=N−d. The spectrum of −Δ+ |y|^2 is given by [d,∞). We prove that λ≥−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(ℝ^N) weak solution is trivial whenever 1≤ p≤ max{(N+2)/(N−2), m/(m−2)} for m≥ 3 or p≥ 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.
ER -