Multiplicity results for the assigned Gauss curvature problem in R2

TL;DR

This paper investigates the multiplicity of solutions to the Gauss curvature problem with conical singularities on R2, providing new theoretical results and conjectures supported by numerical computations.

Contribution

It introduces new multiplicity results for bounded radial solutions and extends findings to non-radial solutions using symmetry considerations.

Findings

Proved multiplicity results for bounded radial solutions.

Presented conjectures on the number of unbounded solutions.

Extended multiplicity results to non-radial solutions using symmetry.

Abstract

To study the problem of the assigned Gauss curvature with conical singularities on Riemanian manifolds, we consider the Liouville equation with a single Dirac measure on the two-dimensional sphere. By a stereographic projection, we reduce the problem to a Liouville equation on the euclidean plane. We prove new multiplicity results for bounded radial solutions, which improve on earlier results of C.-S. Lin and his collaborators. Based on numerical computations, we also present various conjectures on the number of unbounded solutions. Using symmetries, some multiplicity results for non radial solutions are also stated.