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

2022/05/30 by Chen, Zhijie, Li, Houwang, Zou, Wenming
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2205.15055

Abstract

We study the Lane-Emden system \begincases -Δu=v^p, ugt;0,\quadin~Ω, -Δv=u^q, vgt;0,\quadin~Ω, u=v=0,\quadon~∂Ω, \endcases where Ω⊂ℝ² is a smooth bounded domain. In a recent work, we studied the concentration phenomena of positive solutions as p,q→+∞ and |q-p|≤ Λ. 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 Ω is convex, then the solution of this system is unique and nondegenerate for large p, q.

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