Mathematische Annalen

, Volume 356, Issue 4, pp 1599–1612 | Cite as

Liouville theorems for stable solutions of biharmonic problem



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 \(\frac{n+4}{n-4} < q \le \left(\frac{n-8}{n}\right)_+^{-1}\).

Mathematics Subject Classification (1991)

Primary 35B45 Secondary 53C21 35J40 



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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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Department of MathematicsThe Chinese University of Hong KongShatinHong Kong
  2. 2.LMAM, UMR 7122Université de Lorraine-Metz57045 MetzFrance

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