Non-degeneracy and local uniqueness of positive solutions to the Lane-Emden problem in dimension two

TL;DR

This paper proves the non-degeneracy and local uniqueness of multi-spike positive solutions to the Lane-Emden problem in two dimensions for large p, using asymptotic analysis, ODE theory, and Pohozaev identities.

Contribution

It establishes the non-degeneracy and local uniqueness of solutions for general domains, improving previous asymptotic estimates in the Lane-Emden problem.

Findings

Proved non-degeneracy of solutions.

Established local uniqueness of multi-spike solutions.

Enhanced asymptotic estimates for large p solutions.

Abstract

We are concerned with the Lane-Emden problem \begin{equation*} \begin{cases} -\Delta u=u^{p} &{\text{in}~\Omega},\\[0.5mm] u>0 &{\text{in}~\Omega},\\[0.5mm] u=0 &{\text{on}~\partial \Omega}, \end{cases} \end{equation*} where $\Omega\subset \mathbb R^2$ is a smooth bounded domain and $p>1$ is sufficiently large. Improving some known asymptotic estimates on the solutions, we prove the non-degeneracy and local uniqueness of the multi-spikes positive solutions for general domains. Our methods mainly use ODE's theory, various local Pohozaev identities, blow-up analysis and the properties of Green's function.