TY - JOUR
T1 - The Equation Δu+∇φ•∇u=8πc(1-heu) on a Riemann Surface
AU - Wang , Meng
AU - Liu , Qingyue
JO - Journal of Partial Differential Equations
VL - 4
SP - 335
EP - 355
PY - 2012
DA - 2012/12
SN - 25
DO - http://doi.org/10.4208/jpde.v25.n4.3
UR - https://global-sci.org/intro/article_detail/jpde/5190.html
KW - Compact Riemann surface
KW - nonlinear elliptic equation
KW - gauss curvature
KW - existence of solution
AB - Let M be a compact Riemann surface, h(x) a positive smooth function on M, and f(x) a smooth function on M which satisfies that $∫_Me^φdV_g=1$. In this paper, we consider the functional $J(u)=½∫_M|∇u|^2e^φdV_g+8πc∫_Mue^φdV_g-8πclog∫_Mhe^{u+φ}dV_g$. We give a sufficient condition under which J achieves its minimum for $c≤inf_{x∈M^{e^φ(x)}}$.