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:(Formula presented). 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.

In this paper we consider the following Toda system of equations on a compact surface: { -Delta u(1) = 2 rho(1) (h(1)e(u1) - 1) - rho(2) (h(2)e(u2) - 1), -Delta u(2) = 2 rho(2) (h(2)e(u2) - 1) - rho(1) (h(1)e(u1) - 1). Here h(1), h(2) are smooth positive functions and rho(1), rho(2) two positive parameters. In this note we compute the Leray-Schauder degree mod Z(2) of the problem for rho(i) is an element of (4 pi k, 4 pi(k + 1)) (k is an element of 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.