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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 $\mathring{M}$ of $M$, and zero$T$-curvature and mean curvature on the boundary $\partial M$ of $M$. Thisgeometric problem is equivalent to solving a fourth-order elliptic boundaryvalue problem (BVP) involving the Paneitz operator with boundary conditions ofChang-Qing and Neumann operators. The corresponding BVP has a variationalformulation but the corresponding variational problem, in the case under study,is not compact. To overcome such a difficulty we perform a systematic study,\`a la Bahri, of the so called {\it critical points at infinity}, compute theirMorse indices, determine their contribution to the difference of topologybetween the sublevel sets of associated Euler-Lagrange functional and henceextend the full Morse Theory to this noncompact variational problem. Toestablish Morse inequalities we were led to investigate from the topologicalviewpoint the space of boundary-weighted barycenters of the underlyingmanifold, which arise in the description of the topology of very negativesublevel sets of the related functional. As an application of our approach wederive various existence results and provide a Poincar\'e-Hopf type criteriumfor the prescribed $Q$-curvature problem on compact four dimensional Riemannianmanifolds with boundary.

The resonant boundary $Q$-curvature problem and boundary-weighted barycenters