Two-dimensional solutions of a mean field equation on flat tori

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:269

Article

Du, Zhuoran¹  Gui, Changfeng² 

[1] Hunan Univ, Sch Math, Changsha 410082, Hunan, Peoples R China

[2] Univ Texas San Antonio, Dept Math, San Antonio, TX 78249 USA

关键词: Mean field equation;    Torus;    Bifurcation;    Symmetry;   

DOI  :  10.1016/j.jde.2020.07.012

来源: Elsevier

【 摘 要 】

We study the mean field equation on the flat torus T-sigma := C/(Z + Z sigma) Delta u + rho(e(u)/integral(T sigma)e(u) - 1/vertical bar T-sigma vertical bar) = 0, where rho is a real parameter. For a general flat torus, we obtain the existence of two-dimensional solutions bifurcating from the trivial solution at each eigenvalue (up to a multiplicative constant vertical bar T-sigma vertical bar) of Laplace operator on the torus in the space of even symmetric functions. We further characterize the subset of all eigenvalues through which only one bifurcating curve passes. Finally local convexity near bifurcating points of the solution curves are obtained. (C) 2020 Elsevier Inc. All rights reserved.

【 授权许可】

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