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Volume 36, Issue 1
Multiple Axially Asymmetric Solutions to a Mean Field Equation on $\mathbb{S}^{2}$

Zhuoran Du, Changfeng Gui, Jiaming Jin & Yuan Li

Anal. Theory Appl., 36 (2020), pp. 19-32.

Published online: 2020-05

[An open-access article; the PDF is free to any online user.]

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  • Abstract

We study the following mean field equation

$$\Delta_{g}u+\rho\left(\frac{e^{u}}{\int_{\mathbb{S}^{2}}e^{u}d\mu}-\frac{1}{4\pi}\right)=0\ \  \mbox{in}\ \ \mathbb{S}^{2},$$

where $\rho$ is a real parameter. We obtain the existence of multiple axially asymmetric solutions bifurcating from $u=0$ at the values $\rho=4n(n+1)\pi$ for any  odd integer $n\geq3$.

  • AMS Subject Headings

35B32, 35J61, 58J55

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

duzr@hnu.edu.cn (Zhuoran Du)

changfeng.gui@utsa.edu (Changfeng Gui)

jiamingjin123@163.com (Jiaming Jin)

liy93@hnu.edu.cn (Yuan Li)

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@Article{ATA-36-19, author = {Du , ZhuoranGui , ChangfengJin , Jiaming and Li , Yuan}, title = {Multiple Axially Asymmetric Solutions to a Mean Field Equation on $\mathbb{S}^{2}$}, journal = {Analysis in Theory and Applications}, year = {2020}, volume = {36}, number = {1}, pages = {19--32}, abstract = {

We study the following mean field equation

$$\Delta_{g}u+\rho\left(\frac{e^{u}}{\int_{\mathbb{S}^{2}}e^{u}d\mu}-\frac{1}{4\pi}\right)=0\ \  \mbox{in}\ \ \mathbb{S}^{2},$$

where $\rho$ is a real parameter. We obtain the existence of multiple axially asymmetric solutions bifurcating from $u=0$ at the values $\rho=4n(n+1)\pi$ for any  odd integer $n\geq3$.

}, issn = {1573-8175}, doi = {https://doi.org/10.4208/ata.OA-0016}, url = {http://global-sci.org/intro/article_detail/ata/16911.html} }
TY - JOUR T1 - Multiple Axially Asymmetric Solutions to a Mean Field Equation on $\mathbb{S}^{2}$ AU - Du , Zhuoran AU - Gui , Changfeng AU - Jin , Jiaming AU - Li , Yuan JO - Analysis in Theory and Applications VL - 1 SP - 19 EP - 32 PY - 2020 DA - 2020/05 SN - 36 DO - http://doi.org/10.4208/ata.OA-0016 UR - https://global-sci.org/intro/article_detail/ata/16911.html KW - Mean field equation, axially asymmetric solutions, bifurcation. AB -

We study the following mean field equation

$$\Delta_{g}u+\rho\left(\frac{e^{u}}{\int_{\mathbb{S}^{2}}e^{u}d\mu}-\frac{1}{4\pi}\right)=0\ \  \mbox{in}\ \ \mathbb{S}^{2},$$

where $\rho$ is a real parameter. We obtain the existence of multiple axially asymmetric solutions bifurcating from $u=0$ at the values $\rho=4n(n+1)\pi$ for any  odd integer $n\geq3$.

Zhuoran Du, Changfeng Gui, Jiaming Jin & Yuan Li. (2020). Multiple Axially Asymmetric Solutions to a Mean Field Equation on $\mathbb{S}^{2}$. Analysis in Theory and Applications. 36 (1). 19-32. doi:10.4208/ata.OA-0016
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