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 $\mathring{M}$  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 of  Chang-Qing and Neumann operators. The corresponding BVP has a variational formulation but  the corresponding variational problem, in the case under study, is not compact.To overcome such a difficulty we perform  a systematic study, á  la Bahri, of the so called    "critical points at infinity", compute their Morse indices, determine their contribution to the difference of topology between the sublevel sets of  associated Euler-Lagrange functional and hence extend the  full Morse Theory to this noncompact variational problem. To establish  Morse inequalities we were led to investigate from the topological viewpoint the space of  boundary-weighted barycenters of the underlying manifold, which  arise in the description of the topology of very negative sublevel sets of the related functional. As an application of our approach we derive various existence results and provide a Poincaré-Hopf type criterion for  the prescribed $Q$-curvature problem on compact = four dimensional Riemannian  manifolds with boundary. This is a joint work with Cheik Birahim Ndiaye (Basel/Howard) and Sadok Kallel ( Lille/Sarjah)