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Further study of a weighted elliptic equation

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Abstract

A Liouville type result is established for non-negative entire solutions of a weighted elliptic equation. This provides a positive answer to a problem left open by Du and Guo (2015) and Phan and Souplet (2012) (see (CJ) by Du and Guo (2015) and Conjecture B by Phan and Souplet (2012)). Meanwhile, some regularity results are also obtained. The main results in this paper imply that the number ps is the critical value of the Dirichlet problems of the related equation, even though there are still some open problems left. Our results also apply for the equation with a Hardy potential.

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Acknowledgments

This work was supported by National Natural Science Foundation of China (Grant No. 11571093). The authors thank the referees for valuable comments.

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Authors and Affiliations

  1. Department of Mathematics, Henan Normal University, Xinxiang, 453007, China

    ZongMing Guo & FangShu Wan

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  1. ZongMing Guo
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  2. FangShu Wan
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Correspondence to FangShu Wan.

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Guo, Z., Wan, F. Further study of a weighted elliptic equation. Sci. China Math. 60, 2391–2406 (2017). https://doi.org/10.1007/s11425-017-9134-7

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