Qualitative properties for elliptic problems with CKN operators

2022/06/09 by Chen, Huyuan, Zheng, Yishan
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2206.04247

Abstract

The purpose of this paper is to study basic property of the operator L_μ1,μ2 u=-Δ+(μ₁ )/(|x|²)x⋅∇ +(μ₂ )/(|x|²), which generates at the origin due to the critical gradient and the Hardy term, where μ₁,μ₂ are free parameters. This operator arises from the critical Caffarelli-Kohn-Nirenberg inequality. We analyze the fundamental solutions in a weighted distributional identity and obtain the Liouville theorem for the Lane-Emden equation with that operator, by using the classification of isolated singular solutions of the related Poisson problem in a bounded domain Ω⊂ ℝ^N (N ≥ 2) containing the origin.

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