Radial Symmetry and Nondegeneracy for Singular Liouville Systems with Small Singular Exponents
Abstract
We extend the classification (Lin and Zhang, Ann. Inst. H. Poincaré Anal. Non Linéaire \textbf{27}, 2010) to finite-mass singular Liouville systems with sufficiently small positive singular exponents. For symmetric, entrywise nonnegative, irreducible and invertible coupling matrices, every solution whose mass vector lies near a fixed compact subset of the regular mass surface is radial about the singular point and is uniquely determined by its component masses up to dilation. We also establish nondegeneracy modulo dilation. To overcome the failure of the moving-plane argument for positive exponents, we use compactness and a local reduction based on integral identities. The results also hold for nonzero nonnegative exponent vectors and for small mixed-sign vectors satisfying an explicit integral inequality.
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BibTeX
@misc{cheng2026radial,
title = {{Radial Symmetry and Nondegeneracy for Singular Liouville Systems with Small Singular Exponents}},
author = {Zetao Cheng and Haoyu Li and Lei Zhang},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.03456},
eprint = {2611.03456},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.03456},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Radial Symmetry and Nondegeneracy for Singular Liouville Systems with Small Singular Exponents
AU - Zetao Cheng
AU - Haoyu Li
AU - Lei Zhang
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.03456
UR - https://arxiv.org/abs/2611.03456
AB - We extend the classification (Lin and Zhang, Ann. Inst. H. Poincaré Anal. Non Linéaire \textbf{27}, 2010) to finite-mass singular Liouville systems with sufficiently small positive singular exponents. For symmetric, entrywise nonnegative, irreducible and invertible coupling matrices, every solution whose mass vector lies near a fixed compact subset of the regular mass surface is radial about the singular point and is uniquely determined by its component masses up to dilation. We also establish nondegeneracy modulo dilation. To overcome the failure of the moving-plane argument for positive exponents, we use compactness and a local reduction based on integral identities. The results also hold for nonzero nonnegative exponent vectors and for small mixed-sign vectors satisfying an explicit integral inequality.
ER -