Sharp estimates, uniqueness and nondegeneracy of positive solutions of the Lane-Emden system in planar domains

TL;DR

This paper derives sharp estimates for positive solutions of the Lane-Emden system in planar domains, establishing uniqueness and nondegeneracy of solutions in convex domains for large exponents.

Contribution

It provides precise asymptotic estimates and convergence rates for multi-bubble solutions, demonstrating uniqueness and nondegeneracy in convex planar domains.

Findings

Sharp estimates of multi-bubble solutions

Convergence rates of local maxima and scaling parameters

Uniqueness and nondegeneracy in convex domains

Abstract

We study the Lane-Emden system $$\begin{cases} -\Delta u=v^p,\quad u>0,\quad\text{in}~\Omega, -\Delta v=u^q,\quad v>0,\quad\text{in}~\Omega, u=v=0,\quad\text{on}~\partial\Omega, \end{cases}$$ where $\Omega\subset\mathbb{R}^2$ is a smooth bounded domain. In a recent work, we studied the concentration phenomena of positive solutions as $p,q\to+\infty$ and $|q-p|\leq \Lambda$. In this paper, we obtain sharp estimates of such multi-bubble solutions, including sharp convergence rates of local maxima and scaling parameters, and accurate approximations of solutions. As an application of these sharp estimates, we show that when $\Omega$ is convex, then the solution of this system is unique and nondegenerate for large $p, q$.