Qualitative properties of multi-bubble solutions for nonlinear elliptic equations involving critical exponents

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≥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 and unified proof of the classical theorems due to Bahri-Li-Rey (1995) [2] and Rey (1999) [24] which state that if n≥4 or n=3, respectively, 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.