On the Leray-Schauder degree of the Toda system on compact surfaces

In this paper we consider the following Toda system of equations on a compact surface: −Δu₁ = 2ρ1 (h₁eᵘ1 ⁻ ¹) − ρ2 (h₂eᵘ² ⁻ ¹) , −Δu₂ = 2ρ2 (h₂eᵘ² ⁻ ¹) − ρ1 (h₁eᵘ1 ⁻ ¹) . Here h₁, h₂ are smooth positive functions and ρ1, ρ2 two positive parameters. In this note we compute the Leray-Schauder degree mod ℤ₂ of the problem for ρᵢ ∈ (4πk, 4π(k + 1)) (k ∈ ℕ). Our main tool is a theorem of Krasnoselskii and Zabreiko on the degree of maps symmetric with respect to a subspace. This result yields new existence results as well as a new proof of previous results in the literature."