In this talk, we will discuss the problem of finding conformal metrics with constant Q-curvature on a given compact four dimensional Riemannian manifold (M,g) with boundary. This will be equivalent to solving a fourth order nonlinear elliptic boundary value problem with boundary condition given by a third-order pseudodifferential operator, and homogeneous Neumann condition which has a variational structure. However when some conformally invariant quantity associated to the problem is large, the Euler-Lagrange functional associated is unbounded from below, implying that we have to find critical points of saddle type. We will show how the search of saddle points leads naturally to consider a new barycentric set of the manifold.