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

Abstract

We study the Lane-Emden system $$\{\begin{array}{l}-\Delta u=v^p,u>0,\text{in}\Omega ,-\Delta v=u^q,v>0,\text{in}\Omega ,u=v=0,\text{on}\partial \Omega ,\end{array}$$ 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 $|q-p|\le \Lambda$. 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 $p,q$.

arXiv comment63 pages. This is a revised version of arXiv:2205.15055v1. We fix a gap in the previous version and add some details of the proof