  The objective of this paper is to obtain qualitative characteristics of
multi-bubble solutions to the Lane-Emden-Fowler equations with slightly
subcritical exponents given any dimension $n \ge 3$. By examining the
linearized problem at each $m$-bubble solution, we provide a number of
estimates on the first $(n+2)m$-eigenvalues and their corresponding
eigenfunctions. Specifically, we present a new proof of the classical theorem
due to Bahri-Li-Rey (Calc. Var. Partial Differential Equations 3 (1995) 67-93)
which states that if $n \ge 4$, then the Morse index of a multi-bubble solution
is governed by a certain symmetric matrix whose component consists of a
combination of Green's function, the Robin function, and their first and second
derivatives. Our proof also allows us to handle the intricate case $n = 3$.
