具负Euler特征的方程Δu－K十Ke^（2u）＝0的可解性

Solvability of the Equation ΔU-K+Ke￣(2u)=0with Negative Euler Characteristic

本文证明了下述结果：设Ｍ是紧致２维无边Ｒｉｅｍａｎｎ流形。ｘ（Ｍ）是Ｍ的Ｅｕｌｅｒ示性教．Ｋ为（Ｍ，ｇ）的Ｇａｕｓｓ曲率．则对于给定的Ｋ∈Ｃ∞（Ｍ）具Ｘ（Ｍ）＜０的方程面Δｕ－Ｋ＋Ｋｅ２ｕ＝０有解ｕ∈Ｃ∞（Ｍ），当且仅当ｍｉｎＫ＜０．

In this peper,the following result is proved: Let M be a compact two-dimensional Riemanniat manifold without boundary and let X(M) be the Euler characteristic of M. Let K be the Gaussian curvature of(M, g).Then,for given K ∈C∞(M),the equation ΔU-K+Ke2u=0 with X(M)<0 has a solution U∈C∞(M) and only if minK< 0.