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)}H|K|^{-1\over 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.