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

TL;DR

This paper investigates the qualitative properties of multi-bubble solutions to nonlinear elliptic equations with critical exponents, providing new insights into eigenvalues, eigenfunctions, and Morse index calculations across different dimensions.

Contribution

It offers a novel proof of the Morse index characterization for multi-bubble solutions, extending results to the case when the dimension is three.

Findings

Eigenvalue estimates for linearized problems at multi-bubble solutions

A new proof of the Morse index theorem for n ≥ 4

Extension of Morse index analysis to the case n = 3

Abstract

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 $n = 3$.