Classification of Radial Solutions to Liouville Systems with Singularities

Abstract

Let $A=(a_{ij}{)}_{n\times n}$ be a nonnegative, symmetric, irreducible and invertible matrix. We prove the existence and uniqueness of radial solutions to the following Liouville system with singularity: $$\{arrayll\Delta u_i+\sum _{j=1}^n{a}_{ij}|x{|}^{{\beta }_j}{e}^{u_j(x)}=0,{\mathbb{R}}^2,i=1,...,n{\int }_{{\mathbb{R}}^2}|x{|}^{{\beta }_i}{e}^{u_i(x)}dx<\infty ,i=1,...,narray.$$ where ${\beta }_1,...,{\beta }_n$ are constants greater than -2. If all ${\beta }_i$s are negative we prove that all solutions are radial and the linearized system is non-degenerate.

25 pages