Qualitative properties for elliptic problems with CKN operators

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 operato...

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Bibliographic Details
Published in:arXiv.org 2022-06
Main Authors: Chen, Huyuan, Zheng, Yishan
Format: Article
Language:English
Subjects:
Liouville theorem
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Summary: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.
ISSN:2331-8422