Two solutions to Kazdan-Warner's problem on surfaces

TL;DR

This paper demonstrates the existence of at least two solutions to the Kazdan-Warner problem on negatively curved surfaces, using variational methods and convex set minimization, when the prescribed function changes sign and has negative average.

Contribution

It introduces two distinct solution methods for the Kazdan-Warner problem on surfaces with negative Euler characteristic, expanding understanding of solution multiplicity.

Findings

Existence of at least two solutions under sign-changing conditions

Application of convex set minimization method

Use of mountain pass variational method

Abstract

In this paper, we study the sign-changing Kazdan-Warner's problem on two dimensional closed Riemannian manifold with negative Euler number $\chi(M)<0$. We show that once, the direct method on convex sets is used to find a minimizer of the corresponding functional, then there is another solution via a use of the variational method of mountain pass. In conclusion, we show that there are at least two solutions to the Kazdan-Warner's problem on two dimensional Kazdan-Warner equation provided the prescribed function changes signs and with this average negative.