Some existence results for the Toda system on closed surfaces (Q931057)

Let \(\Sigma\) be a compact closed surface in the two-dimensional Euclidean space. This paper deals with the generalized Toda system \[ -\Delta u_i=\sum_{j=1}^2\rho_ja_{ij}\left(\frac{h_je^{u_j}}{\int_\Sigma h_je^{u_j}dV_g}-1\right) \] on \(\Sigma\) for \(i=1\), \(2\), where \(\rho_1\), \(\rho_2\) are real parameters and \(h_1\), \(h_2\) are smooth positive functions. The main result of this paper establishes that if \(m\) is an arbitrary positive integer, then the above problem has at least one solution, provided \(\rho_1\in (4\pi m,4\pi (m+1))\) and \(\rho_2<4\pi\). The proof combines the Moser-Trudinger inequality with variational arguments related to the associated energy functional.