Existence of Complete conformal metrics of negative Ricci curvature on manifolds with boundary

We show that on a compact Riemannian manifold with boundary there exists $u \in C^{\infty}(M)$ such that, $u_{|\partial M} \equiv 0$ and $u$ solves the $\sigma_k$-Ricci problem. In the case $k = n$ the metric has negative Ricci curvature. Furthermore, we show the existence of a complete conformally related metric on the interior solving the $\sigma_k$-Ricci problem. By adopting results of Mazzeo-Pacard, we show an interesting relationship between the complete metrics we construct and the existence of Poincar\'e-Einstein metrics. Finally we give a brief discussion of the corresponding questions in the case of positive curvature.