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

Abstract

Let A = (a ij ) n×n be an invertible matrix and A -1 = (a ij ) n×n be the inverse of A. In this paper, we consider the generalized Liouville system n (0.1) Δ g u i + n/∑a ij ρ j (h j e u j/∫h j e u j-1) = 0 in M, j = 1 where 0 < h j ∈ C 1 (M) and ρ j ∈ ℝ + , and prove that, under the assumptions of (H 1 ) and (H 2 ) (see Introduction), the Leray-Schauder degree of (0.1) is equal to (-χ(M) + 1) ··· (-χ(M) + N)/N! if ρ = (ρ1, ..., ρ n ) satisfies 8πN n/∑ n i=1 ρ i < ∑ 1≤i,j≤n a ij ρ i ρ j < 8π(N + 1) n/∑ n i=1 ρ i . Equation (0.1) is a natural generalization of the classic Liouville equation and is the Euler-Lagrangian equation of the nonlinear function Φ ρ : Φ ρ (u) =1/2 ∫ M ∑ 1≤i,j≤n a ij ∇ g u i · ∇ g u j + n/∑ n i=1 ∫ M ρ i u i - n/∑ n i=1 ρ i log ∫ M h 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.