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The Equation Δ<em>u</em>+∇φ•∇<em>u</em>=8<em>πc</em>(1-<em>he</em><sup><em>u</em></sup>) on a Riemann Surface

The Equation Δ<em>u</em>+∇φ•∇<em>u</em>=8<em>πc</em>(1-<em>he</em><sup><em>u</em></sup>) on a Riemann Surface
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Year:    2012

Author:    Meng Wang, Qingyue Liu

Journal of Partial Differential Equations, Vol. 25 (2012), Iss. 4 : pp. 335–355

Abstract

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 $c≤inf_{x∈M^{e^φ(x)}}$.

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Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/jpde.v25.n4.3

Journal of Partial Differential Equations, Vol. 25 (2012), Iss. 4 : pp. 335–355

Published online:    2012-01

AMS Subject Headings:   

Copyright:    COPYRIGHT: © Global Science Press

Pages:    21

Keywords:    Compact Riemann surface

Author Details

Meng Wang

Qingyue Liu

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