Positive Blow-up Solutions for a Linearly Perturbed Boundary Yamabe Problem

We consider the problem of prescribing the scalar and boundary mean curvatures via conformal deformation of the metric on a $n-$ dimensional compact Riemannian manifold. We deal with the case of negative scalar curvature $K$ and boundary mean curvature $H$ of arbitrary sign which are non-constant and $\mathfrak D_n=\sqrt{n(n-1)}{|K|}^{-1/2}>1$ at some point of the boundary. It is known that this problem admits a positive mountain pass solution if $n=3$, while no existence results are known for $n\geq 4$. We will consider a perturbation of the geometric problem and show the existence of a positive solution which blows-up at a boundary point which is critical for both prescribed curvatures.