JournalsrlmVol. 18, No. 4pp. 391–412

Some existence results for the Toda system on closed surfaces

  • Andrea Malchiodi

    Scuola Normale Superiore, Pisa, Italy
  • Cheikh Birahim Ndiaye

    Universität Tübingen, Germany
Some existence results for the Toda system on closed surfaces cover
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Abstract

Given a compact closed surface \Sig\Sig, we consider the {\em generalized Toda} system of equations on \Sig\Sig: \Dui=j=12ρjaij(hjeuj\SighjeujdVg1)- \D u_i = \sum_{j=1}^2 \rho_j a_{ij} \left( \frac{h_j e^{u_j}}{\int_\Sig h_j e^{u_j} dV_g} - 1 \right) for i=1,2i = 1, 2, where ρ1,ρ2\rho_1, \rho_2 are real parameters and h1,h2h_1, h_2 are smooth positive functions. Exploiting the variational structure of the problem and using a new minimax scheme we prove existence of solutions for generic values of ρ1\rho_1 and for ρ2<4π\rho_2 < 4 \pi.

Cite this article

Andrea Malchiodi, Cheikh Birahim Ndiaye, Some existence results for the Toda system on closed surfaces. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 18 (2007), no. 4, pp. 391–412

DOI 10.4171/RLM/504