Morse index, topological degree and local uniqueness of multi-spikes solutions to the Lane-Emden problem in dimension two

TL;DR

This paper analyzes multi-spike solutions to the Lane-Emden problem in two dimensions, computing their Morse index and topological degree, which leads to new local uniqueness results for these solutions.

Contribution

It extends classical Morse index and topological degree results to the 2D case and investigates the concentration behavior of solutions.

Findings

Computed Morse index for multi-spike solutions in 2D

Derived the total topological degree of solutions

Established a new local uniqueness result

Abstract

We consider multi-spike positive solutions to the Lane-Emden problem in any bounded smooth planar domain and compute their Morse index, extending to the dimension $N=2$ classical theorems due to Bahri-Li-Rey (1995) and Rey (1999) when $N\geq 4$ and $N=3$, respectively. Furthermore, by deeply investigating their concentration behavior, we also derive the total topological degree. The Morse index and the degree counting formula yield a new local uniqueness result.