  We consider a kind of Yamabe problem whose scalar curvature vanishes in the
unit ball $\mathbb{B}^n$ and on the boundary $\mathbb{S}^{n-1}$ the mean
curvature is prescribed. By combining critical points at infinity approach with
Morse theory we obtain new existence results in higher dimensional case $n\geq
5$, under suitable pinching conditions on the mean curvature function.
