  Let M,g a compact Riemannian n-dimensional manifold. It is well know that,
under certain hypothesis, in the conformal class of g there are scalar-flat
metrics that have the boundary of M as a constant mean curvature hypersurface.
Also, under certain hypothesis, it is known that these metrics are a compact
set. In this paper we prove that, both in the case of umbilic and non-umbilic
boundary, if we linearly perturb the mean curvature term with a negative smooth
function, the set of solutions of Yamabe problem is still a compact set.
