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

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)