## Differential and Integral Equations

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

### The mean field equation with critical parameter in a plane domain

#### Abstract

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

**Source**

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

**Dates**

First available in Project Euclid: 21 December 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.die/1356050292

**Mathematical Reviews number (MathSciNet)**

MR2279331

**Zentralblatt MATH identifier**

1212.35128

**Subjects**

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

#### Citation

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