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Classification of Positive Solutions to a Divergent Equation on the Upper Half Space

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Abstract

In this paper we classify the positive solutions of the divergent equation with Neumann boundary on the upper half space

$$\left\{ {\matrix{{ - {\rm{div}}({t^\alpha }\nabla u) = {t^\beta }f(u),\;\;\;\;(y,t) \in \mathbb{R}_ + ^{n + 1},} \hfill \cr {\mathop {\lim }\limits_{t \to {0^ + }} {t^\alpha }{{\partial u} \over {\partial t}} = 0} \hfill \cr } } \right.$$

by the method of moving spheres and Kelvin transformations, where n ≥ 1, α > 0, β > −1, \({{n - 1} \over {n + 1}}\beta \le \alpha < \beta + 2\), and f : (0, ∞) → (0, ∞) is non-negative continuous function satisfying some conditions. This equation arises from a weighed Sobolev inequality involving divergent operator div(tα∇u) on the upper half space.

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Acknowledgements

We thank the referees for their time and comments.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Shaanxi Normal University, Xi’an, 710119, P. R. China

    Jin Ge Yao & Jing Bo Dou

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  1. Jin Ge Yao
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  2. Jing Bo Dou
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Correspondence to Jing Bo Dou.

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Supported by the National Natural Science Foundation of China (Grant Nos. 12071269, 11971385), Natural Science Basic Research Plan for Distinguished Young Scholars in Shaanxi Province of China (Grant No. 2019JC19), and the Fundamental Research Funds for the Central Universities (Grant No. GK202101008)

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Yao, J.G., Dou, J.B. Classification of Positive Solutions to a Divergent Equation on the Upper Half Space. Acta. Math. Sin.-English Ser. 38, 499–509 (2022). https://doi.org/10.1007/s10114-022-0345-x

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  • DOI: https://doi.org/10.1007/s10114-022-0345-x

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