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

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- by Andrea Malchiodi and David Ruiz PDF
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## 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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## 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: A.Malchiodi@warwick.ac.uk, malchiod@sissa.it
**David Ruiz**- Affiliation: Departamento de Análisis Matemático, University of Granada, 18071 Granada, Spain
- Email: daruiz@ugr.es
- 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: https://doi.org/10.1090/S0002-9939-2015-12484-7
- MathSciNet review: 3336622