1. Qualitative properties for elliptic problems with CKN operators
- Author
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Chen, Huyuan and Zheng, Yishan
- Subjects
Mathematics - Analysis of PDEs - 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.
- Published
- 2022