This paper deals with the following prescribed boundary mean curvature problem in
$${\mathbb{B}^N}$$
$$\left\{ {\matrix{{ - \Delta u = 0,\,u > 0,} \hfill & {y \in {\mathbb{B}^N},} \hfill \cr {{{\partial u} \over {\partial \nu }} + {{N - 2} \over 2}u = {{N - 2} \over 2}\tilde K(y){u^{{2^\sharp } - 1}},} \hfill & {y \in {\mathbb{S}^{N - 1}},} \hfill \cr } } \right.$$
where
$$\tilde K(y) = \tilde K(|{y^\prime }|,\tilde y)$$
is a bounded nonnegative function with
$$y = ({y^\prime },\tilde y) \in {\mathbb{R}^2} \ti...
