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Given a compact four-dimensional Riemannian manifold $(M, g)$ with boundary, we study the problem of existence of Riemannian metrics on $M$ conformal to $g$ with prescribed $Q$-curvature in the interior $\mathring{M}$Ìý of $M$, and zero $T$-curvature and mean curvature on the boundary $\partial M$ of $M$. This geometric problem is equivalent to solving a fourth-order elliptic boundary value problem (BVP) involving the Paneitz operator with boundary conditions ofÌý Chang-Qing and Neumann operators. The corresponding BVP has a variational formulation butÌý the corresponding variational problem, in the case under study, is not compact.To overcome such a difficulty we performÌý a systematic study, áÌý laÌýBahri, of the so calledÌýÌýÌý "critical points at infinity", computeÌýtheir Morse indices, determine their contribution to the difference ofÌýtopology between the sublevel sets ofÌý associated Euler-LagrangeÌýfunctional and hence extend theÌý full Morse Theory to this noncompact variational problem. To establishÌý Morse inequalities we were led toÌýinvestigate from the topological viewpoint the space ofÌýÌýboundary-weighted barycenters of the underlying manifold, whichÌý arise in the description of the topology of very negative sublevel sets of the related functional.ÌýAs an application of our approach we derive various existence results and provide a Poincaré-Hopf type criterion forÌý the prescribedÌý$Q$-curvature problem on compact = four dimensional RiemannianÌýÌýmanifolds with boundary.ÌýThis is a joint work with Cheik Birahim Ndiaye (Basel/Howard) and Sadok Kallel ( Lille/Sarjah)