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Volume 25, Issue 4
The Equation Δu+∇φ•∇u=8πc(1-heu) on a Riemann Surface

Meng Wang & Qingyue Liu

DOI: 10.4208/jpde.v25.n4.3

J. Part. Diff. Eq., 25 (2012), pp. 335-355.

Published online: 2012-12

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  • 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 $c≤inf_{x∈M^{e^φ(x)}}$.

  • Keywords

Compact Riemann surface nonlinear elliptic equation gauss curvature existence of solution

  • AMS Subject Headings

58J05

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

mathdreamcn@zju.edu.cn (Meng Wang)

qyliu@amss.ac.cn (Qingyue Liu)

  • BibTex
  • RIS
  • TXT
@Article{JPDE-25-335, author = {Wang , Meng and Liu , Qingyue}, title = {The Equation Δu+∇φ•∇u=8πc(1-heu) on a Riemann Surface}, journal = {Journal of Partial Differential Equations}, year = {2012}, volume = {25}, number = {4}, pages = {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 $c≤inf_{x∈M^{e^φ(x)}}$.

}, issn = {2079-732X}, doi = {https://doi.org/10.4208/jpde.v25.n4.3}, url = {http://global-sci.org/intro/article_detail/jpde/5190.html} }
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 $c≤inf_{x∈M^{e^φ(x)}}$.

Meng Wang & Qingyue Liu. (2019). The Equation Δu+∇φ•∇u=8πc(1-heu) on a Riemann Surface. Journal of Partial Differential Equations. 25 (4). 335-355. doi:10.4208/jpde.v25.n4.3
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