A topological degree counting for some Liouville systems of mean field type

Abstract

Let A = (aij)n×n be an invertible matrix and A-1 = (aij)n×n be the inverse of A. In this paper, we consider the generalized Liouville system 0.1 Δgui+∑j=1n aijρj(hj eu-j/∫hjeu-j-1) = 0 in M, where 0 < ∑1≤i, j≤n aij ρiρj<8π(N+1) ∑i=1nρi.Equation (0.1) is a natural generalization of the classic Liouville equation and is the Euler-Lagrangian equation of the nonlinear function Φρ: Φρ(u) = 1/2 ∫M∑1≤i,j≤ n aij∇g ui · ∇g uj+∑i=1n∫M ρi ui - ∑i=1n ρi log ∫Mh-i e u-i. The Liouville system (0.1) has arisen in many different research areas in mathematics and physics. Our counting formulas are the first result in degree theory for Liouville systems. © 2010 Wiley Periodicals, Inc.