JournalsrlmVol. 18 , No. 4DOI 10.4171/rlm/504

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

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.