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arXiv · 2102.09523

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

Massimo Grossi·Isabella Ianni·Peng Luo·Shusen Yan
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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.

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Massimo Grossi, Isabella Ianni, Peng Luo, Shusen Yan. 2021-02-18. Non-degeneracy and local uniqueness of positive solutions to the Lane-Emden problem in dimension two. https://arxiv.org/abs/2102.09523

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