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

Authors: Andrea Malchiodi and David Ruiz
Journal: Proc. Amer. Math. Soc. 143 (2015), 2985-2990
MSC (2010): Primary 35J47, 35J61, 58J20
Published electronically: February 16, 2015
MathSciNet review: 3336622
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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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Additional Information

Andrea Malchiodi
Affiliation: Mathematics Institute, University of Warwick, Zeeman Building, Coventry CV4 7AL, United Kingdom — and — Scuola Internazionale Superiore Di Studi Avanzati (SISSA), via Bonomea 265, 34136 Trieste, Italy
MR Author ID: 655662

David Ruiz
Affiliation: Departamento de Análisis Matemático, University of Granada, 18071 Granada, Spain

Keywords: Geometric PDEs, Leray-Schauder degree
Received by editor(s): November 29, 2013
Received by editor(s) in revised form: February 14, 2014
Published electronically: February 16, 2015
Additional Notes: The first author was supported by the FIRB project Analysis and Beyond, the PRIN Variational Methods and Nonlinear PDE’s and by the University of Warwick. Both authors were supported by the Spanish Ministry of Science and Innovation under Grant MTM2011-26717. The second author was also supported by J. Andalucia (FQM 116).
Communicated by: Lei Ni
Article copyright: © Copyright 2015 American Mathematical Society