Differential and Integral Equations

The mean field equation with critical parameter in a plane domain

Yilong Ni

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Consider the mean field equation with parameter $\lambda=8\pi$ in a bounded smooth domain $\Omega$. Denote by $E_{8\pi}(\Omega)$ the infimum of the associated functional $I_{8\pi}(\Omega)$. We prove that if $|\Omega|=\pi$, then $E_{8\pi}(\Omega)\ge E_{8\pi}(B_1)$ and equality holds if and only if $\Omega$ is a ball. We also give a sufficient condition for the existence of a minimizer for $I_{8\pi}(\Omega)$.

Article information

Differential Integral Equations, Volume 19, Number 12 (2006), 1333-1348.

First available in Project Euclid: 21 December 2012

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35J60: Nonlinear elliptic equations
Secondary: 35J20: Variational methods for second-order elliptic equations 35J25: Boundary value problems for second-order elliptic equations 47J30: Variational methods [See also 58Exx] 49J10: Free problems in two or more independent variables


Ni, Yilong. The mean field equation with critical parameter in a plane domain. Differential Integral Equations 19 (2006), no. 12, 1333--1348. https://projecteuclid.org/euclid.die/1356050292

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