  In this paper we compute the Leray Schauder degree for a fourth order
elliptic boundary value problem with exponential nonlinearity and Navier
boundary condition. This will be made by proving a Poincare'-Hopf type theorem.
Moreover by using this result, together with some quantitative results about
the formal set of barycenters, we are able to establish a direct and
geometrically clear degree counting formula for our problem. We remark that
this formula has been proven with complete different methods by Lin Wei and
Wang by using blow-up type estimates.
