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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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On the Leray-Schauder degree of the Toda system on compact surfaces
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by Andrea Malchiodi and David Ruiz PDF
Proc. Amer. Math. Soc. 143 (2015), 2985-2990 Request permission


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
  • Email:,
  • David Ruiz
  • Affiliation: Departamento de Análisis Matemático, University of Granada, 18071 Granada, Spain
  • Email:
  • 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
  • © Copyright 2015 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 143 (2015), 2985-2990
  • MSC (2010): Primary 35J47, 35J61, 58J20
  • DOI:
  • MathSciNet review: 3336622