## Advances in Differential Equations

- Adv. Differential Equations
- Volume 13, Number 7-8 (2008), 601-640.

### Symmetry and asymmetry: the method of moving spheres

Qinian Jin, YanYan Li, and Haoyuan Xu

#### Abstract

This paper consists of two parts. The first part concerns a question raised by Véron on the symmetry property of positive solutions of the semilinear elliptic equation $$ \Delta u+\frac{c}{|x|^2} u +u^{(n+2)/(n-2)}=0 \quad \mbox{in } \mathbb {R}^n\setminus \{0\}. $$ The second part concerns some nonlinear elliptic equations on the unit sphere ${\mathbb S}^n$. By the method of moving spheres and the global bifurcation theory, we obtain various symmetry, asymmetry, and non-existence results.

#### Article information

**Source**

Adv. Differential Equations, Volume 13, Number 7-8 (2008), 601-640.

**Dates**

First available in Project Euclid: 18 December 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.ade/1355867331

**Mathematical Reviews number (MathSciNet)**

MR2479025

**Zentralblatt MATH identifier**

1201.35099

**Subjects**

Primary: 35J60: Nonlinear elliptic equations

Secondary: 35A25: Other special methods 35B05: Oscillation, zeros of solutions, mean value theorems, etc.

#### Citation

Jin, Qinian; Li, YanYan; Xu, Haoyuan. Symmetry and asymmetry: the method of moving spheres. Adv. Differential Equations 13 (2008), no. 7-8, 601--640. https://projecteuclid.org/euclid.ade/1355867331