  Let $(M,g)$ be a $n-$dimensional compact Riemannian manifold with boundary.
We consider the Yamabe type problem \begin{equation} \left\{ \begin{array}{ll}
-\Delta_{g}u+au=0 & \text{ on }M \\ \partial_\nu u+\frac{n-2}{2}bu= u^{{n\over
n-2}\pm\varepsilon} & \text{ on }\partial M \end{array}\right. \end{equation}
where $a\in C^1(M),$ $b\in C^1(\partial M)$, $\nu$ is the outward pointing unit
normal to $\partial M $ and $\varepsilon$ is a small positive parameter. We
build solutions which blow-up at a point of the boundary as $\varepsilon$ goes
to zero. The blowing-up behavior is ruled by the function $b-H_g ,$ where $H_g$
is the boundary mean curvature.
