Synchronized Solutions for Liouville Systems with Sign-Changing Coefficients
Abstract
We consider singular Liouville systems on closed Riemann surfaces with a common sign-changing coefficient. Direct continuation from a positive coefficient is obstructed by the unavoidable appearance of degenerate zeroes along such a homotopy. We instead identify an intrinsic ray in the parameter space on which the system admits synchronized solutions and reduces exactly to a scalar sign-changing mean-field equation. Combining this reduction with the variational existence theory of De Marchis--López-Soriano--Ruiz and the nodal singularity compactness theorem of Wu, we obtain existence for every noncritical parameter on this ray. Positive singular sources are allowed on the nodal curve. In particular, when the surface has nonpositive Euler characteristic and every negative component of the coefficient is a disk, the topological hypothesis required by the scalar min--max construction is automatic.
Related
- Optimization-based numerical methods for the two-and three-dimensional time harmonic Maxwell problem with sign-changing coefficients
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BibTeX
@misc{park2026synchronized,
title = {{Synchronized Solutions for Liouville Systems with Sign-Changing Coefficients}},
author = {Woongbae Park and Lei Zhang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.32812},
eprint = {2609.32812},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.32812},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Synchronized Solutions for Liouville Systems with Sign-Changing Coefficients
AU - Woongbae Park
AU - Lei Zhang
PY - 2026
DA - 2026/09/26
PB - arXiv
DO - 10.48550/arxiv.2609.32812
UR - https://arxiv.org/abs/2609.32812
AB - We consider singular Liouville systems on closed Riemann surfaces with a common sign-changing coefficient. Direct continuation from a positive coefficient is obstructed by the unavoidable appearance of degenerate zeroes along such a homotopy. We instead identify an intrinsic ray in the parameter space on which the system admits synchronized solutions and reduces exactly to a scalar sign-changing mean-field equation. Combining this reduction with the variational existence theory of De Marchis--López-Soriano--Ruiz and the nodal singularity compactness theorem of Wu, we obtain existence for every noncritical parameter on this ray. Positive singular sources are allowed on the nodal curve. In particular, when the surface has nonpositive Euler characteristic and every negative component of the coefficient is a disk, the topological hypothesis required by the scalar min--max construction is automatic.
ER -