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Liouville theorems for stable solutions of biharmonic problem

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Abstract

We prove some Liouville type results for stable solutions to the biharmonic problem \(\Delta ^2 u= u^q, \,u>0\) in \(\mathbb{R }^n\) where \(1 < q < \infty \). For example, for \(n \ge 5\), we show that there are no stable classical solution in \(\mathbb{R }^n\) when \(\frac{n+4}{n-4} < q \le \left(\frac{n-8}{n}\right)_+^{-1}\).

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Acknowledgments

The research of the first author is supported by General Research Fund from RGC of Hong Kong. The second author is supported by the French ANR project referenced ANR-08-BLAN-0335-01. We both thank the Department of mathematics, East China Normal University for its kind hospitality. J.W. thanks Professor N. Ghoussoub for sharing his idea at a BIRS meeting in March, 2010. D.Y. thanks Professor G. Carron for showing him the reference [12]. Both authors thank the anonymous referee for valuable remarks.

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Authors and Affiliations

  1. Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong

    Juncheng Wei

  2. LMAM, UMR 7122, Université de Lorraine-Metz, 57045 Metz, France

    Dong Ye

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  1. Juncheng Wei
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  2. Dong Ye
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Correspondence to Dong Ye.

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Wei, J., Ye, D. Liouville theorems for stable solutions of biharmonic problem. Math. Ann. 356, 1599–1612 (2013). https://doi.org/10.1007/s00208-012-0894-x

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  • DOI: https://doi.org/10.1007/s00208-012-0894-x

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