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.