2020/12/24 by Ma, Li · 1 citation
#35J60 #53C20 #58G99 #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics
paper · doi:10.48550/arxiv.2012.13071
In this paper, we study the sign-changing Kazdan-Warner's problem on two dimensional closed Riemannian manifold with negative Euler number χ(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.