Mathematische Annalen

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

Liouville theorems for stable solutions of biharmonic problem

Article

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

Mathematics Subject Classification (1991)

Primary 35B45 Secondary 53C21 35J40 

Notes

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.

References

  1. 1.
    Berchio, E., Gazzola, F.: Some remarks on biharmonic elliptic problems with positive, increasing and convex nonlinearities. Electron J. Diff. Equa. 34, 1–20 (2005)MathSciNetGoogle Scholar
  2. 2.
    Cowan, C., Esposito, P., Ghoussoub, N.: Regularity of extremal solutions in fourth order nonlinear eigenvalue problems on general domains. DCDS-A 28, 1033–1050 (2010)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Dancer, E.N.: Moving plane methods for systems on half spaces. Math. Ann. 342, 245–254 (2008)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Farina, A.: On the classification of solutions of the Lane-Emden equaation on unbouned domains of \(\mathbb{R}^{{N}}\). J. Math. Pures Appl. 87, 537–561 (2007)MathSciNetMATHGoogle Scholar
  5. 5.
    Ferrero, A., Grunau, HCh., Karageorgis, P.: Supercritical biharmonic equations with power-like nonlinearity. Ann. Mat. Pura Appl. 188, 171–185 (2009)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Gazzola, F., Grunau, HCh.: Radial entire solutions for supercritical biharmonic equations. Math. Ann. 334, 905–936 (2006)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Gazzola, F., Grunau, H.Ch., Sweers, G.: Polyharmonic boundary value problems: positivity preserving and nonlinear higher order elliptic equations in bounded domains. In: Lecture Notes in Mathematics, issue no. 1991. Springer, Berlin (2010)Google Scholar
  8. 8.
    Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order, 2nd edn. Springer, Berlin (1983)MATHCrossRefGoogle Scholar
  9. 9.
    Gui, C., Ni, W.M., Wang, X.F.: On the stability and instability of positive steady states of a semilinear heat equation in \({{\bf R}}^{{\bf n}}\). Comm. Pure Appl. Math. XLV, 1153–1181 (1992)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Guo, Z., Wei, J.: Qualitative properties of entire radial solutions for a biharmonic equation with supcritical nonlinearity. Proc. American Math. Soc. 138, 3957–3964 (2010)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Han, Q., Lin, F.H.: Elliptic Partial Differential Equations, Courant Lecture Notes. New York University, New York (1997)Google Scholar
  12. 12.
    Levin, D.: On an analogue of the Rozenblum-Lieb-Cwikel inequality for the biharmonic operator on a Riemannian manifold. Math. Res. Letters 4, 855–869 (1997)MATHGoogle Scholar
  13. 13.
    Lin, C.S.: A classification of solutions to a conformally invariant equation in \(\mathbb{R}^{4}\). Comm. Math. Helv. 73, 206–231 (1998)MATHCrossRefGoogle Scholar
  14. 14.
    Polácik, P., Quittner, P., Souplet, P.: Singularity and decay estimates in superlinear problems via Liouville-type theorems. Part I: Elliptic systems. Duke Math. J. 139, 555–579 (2007)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Rozenblum, G.V.: The distribution of the discrete spectrum for singular differential operators. Dokl. Akad. SSSR 202, 1012–1015 (1972)Google Scholar
  16. 16.
    Serrin, J.: Local behavior of solutions of quasi-linear equations. Acta Math. 111, 247–302 (1964)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Sirakov, B.: Existence results and a priori bounds for higher order elliptic equations and systems. J. Math. Pures Appl. 89, 114–133 (2008)Google Scholar
  18. 18.
    Souplet, P.: The proof of the Lane-Emden conjecture in four space dimensions. Adv. Math. 221, 1409–1427 (2009)MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Wei, J., Xu, X.: Classification of solutions of high order conformally invariant equations. Math. Ann. 313(2), 207–228 (1999)MathSciNetMATHCrossRefGoogle Scholar

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

Personalised recommendations