Symmetry and Asymmetry: The Method of Moving Spheres

TL;DR

This paper explores symmetry properties and nonexistence of solutions for nonlinear elliptic equations on Euclidean space and spheres, using the method of moving spheres, and also establishes multiplicity results via bifurcation theory.

Contribution

It introduces the application of the method of moving spheres to analyze symmetry and nonexistence, and employs global bifurcation theory to prove solution multiplicity.

Findings

Solutions exhibit symmetry properties under certain conditions.

Nonexistence results are established for specific equations.

Multiple solutions are found using bifurcation analysis.

Abstract

We consider some nonlinear elliptic equations on ${\mathbb R}^n$ and ${\mathbb S}^n$. By the method of moving spheres, we obtain the symmetry properties of solutions and some nonexistence results. Moreover, by the global bifurcation theory, we obtain a multiplicity result for a class of semilinear elliptic equations.