Year: 2026
Type: preprint
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.
Source: arXiv (Cornell University)
Authors: Park, Woongbae, Zhang, Lei
Institutions: University of Florida, University of Rochester
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Funder: Science Foundation Ireland
OpenAlex ID: W7214777479