  For the prescribed scalar curvature equation on $S^n$ ($n \ge 6$), we
consider the situation where the number of bubbles tends to infinity in the
Lyapunov-Schmidt (finite dimension) reduction method. In an outstanding paper
by Wei and Yan, the special case where the bubbles are arranged "evenly" (close
to a great circle) is considered. Here we are concerned with the generic
scenario, where the bubbles are "planted" (arranged) in general position, in
adjacent to the critical points of the prescribed function. The main interest
of this article is to extract key information in the finite dimensional reduced
functional, as well as in its first partial derivatives. Overall, there are two
main contributions in the "expansions", namely, interaction between bubbles and
curvature involvement. As each bubble becomes infinitesimally close to other
bubbles in the neighborhood, the errors in the expansions are carefully
estimated in terms of the geometric parameters. This lays the foundation for
our next step on constructing blow-up sequences of solutions with dense blow-up
points (clustered blow-up), where the (non-radial symmetric) prescribed scalar
curvature function has non-isolated critical points.
