Qualitative properties for elliptic problems with CKN operators

TL;DR

This paper investigates the fundamental properties of a Hardy-type elliptic operator with critical gradient terms, deriving solutions and Liouville theorems relevant to the Caffarelli-Kohn-Nirenberg inequality.

Contribution

It provides a detailed analysis of the operator's fundamental solutions and establishes Liouville theorems for related Lane-Emden equations, advancing understanding of critical elliptic operators.

Findings

Derived fundamental solutions in weighted distributional form

Established Liouville theorems for Lane-Emden equations with the operator

Classified isolated singular solutions in bounded domains

Abstract

The purpose of this paper is to study basic property of the operator $$\mathcal{L}_{\mu_1,\mu_2} u=-\Delta +\frac{\mu_1 }{|x|^2}x\cdot\nabla +\frac{\mu_2 }{|x|^2},$$ which generates at the origin due to the critical gradient and the Hardy term, where $\mu_1,\mu_2$ 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 $\Omega \subset \mathbb{R}^N$ ($N \geq 2$) containing the origin.