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

Malchiodi, Andrea; Ruiz, David

doi:10.1090/S0002-9939-2015-12484-7

On the Leray-Schauder degree of the Toda system on compact surfaces
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Proc. Amer. Math. Soc. 143 (2015), 2985-2990 Request permission

Abstract:

In this paper we consider the following Toda system of equations on a compact surface: \[ \left \{ \begin {array}{ll} - \Delta u_1 = 2 \rho _1 \left ( h_1 e^{u_1}- 1 \right ) - \rho _2 \left (h_2 e^{u_2} - 1 \right ), \\ - \Delta u_2 = 2 \rho _2 \left (h_2 e^{u_2} - 1 \right ) - \rho _1 \left (h_1 e^{u_1} - 1 \right ). & \end {array} \right .\] 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 $\mathbb {Z}_2$ of the problem for $\rho _i \in (4 \pi k, 4 \pi (k+1))$ ($k\in \mathbb {N}$). 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.

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