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The differential equation $Δu = 8π- 8πh\exp {u}$ on a compact Riemann surface

W. Ding·J. Jost·J. Li·G. Wang
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Abstract

Let $M$ be a compact Riemann surface, $h(x)$ a positive smooth function on $M$. In this paper, we consider the functional $$J(u)={1/2}\int \sb{M}|\bigtriangledown u|\sp 2 + 8π\int\sb{M}u -8π\log\int\sb{M}h\exp {u}$$. We give a sufficient condition under which $J$ achieves its minimum.

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W. Ding, J. Jost, J. Li, G. Wang. 1997-10-07. The differential equation $Δu = 8π- 8πh\exp {u}$ on a compact Riemann surface. https://arxiv.org/abs/dg-ga/9710005

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