  In this paper, we study a Cherrier-Escobar problem for the extended problem
corresponding to the elliptic Schroedinger-to-Neumann map on a compact
3-dimensional Riemannian manifold with boundary. Using the algebraic
topological argument of Bahri-Coron, we show solvability under the assumption
that the extended problem corresponding to the elliptic Schroedinger-to-Neumann
map has a positive first eigenvalue, a positive Green function, and also
verifies the strong maximum principle.
