A Variational Analysis of the Toda System on Compact Surfaces

In this paper we consider the following Toda system of equations on a compact surface: {Delta u(1) = 2 rho(1)(h(1)e(u1)/integral(Sigma)h(1)e(u1) dV(g) - 1) - rho(2)(h(2)e(u2)/integral(Sigma)h(2)e(u2) dV(g) - 1), {Delta u(2) = 2 rho(2)(h(2)e(u2)/integral(Sigma)h(2)e(u2) dV(g) - 1) - rho(1)(h(1)e(u1)/integral(Sigma)h(1)e(u1) dV(g) - 1). We will give existence results by using variational methods in a noncoercive case. A key tool in our analysis is a new Moser-Trudinger type inequality under suitable conditions on the center of mass and the scale of concentration of the two components u(1) and u(2). (C) 2012 Wiley Periodicals, Inc.