15 mar 2007

**Abstract.**

We consider an elliptic equation on compact surfaces with motivations from physics (Euler flows, Chern-Simons theory) or differential geometry (Gauss curvature, Q-curvature). Using a Morse-theoretical approach we improve some existence results in the literature which were based on blow-up analysis, and derive a simple and direct proof of a known degree-counting formula for the equation.