Compactness results for linearly perturbed Yamabe problem on manifolds with boundary

Let \begin{document}$ (M,g) $\end{document} a compact Riemannian id="M2">\begin{document}$ n $\end{document}-dimensional manifold. It is well know that, under certain hypothesis, in the conformal class of id="M3">\begin{document}$ g there are scalar-flat metrics that have id="M4">\begin{document}$ \partial M as constant mean curvature hypersurface. Also, it known these set. In this paper we prove both case umbilic and non-umbilic boundary, if linearly perturb term id="M5">\begin{document}$ h_{g} with negative smooth function id="M6">\begin{document}$ \alpha, set solutions Yamabe problem still