Annali di Matematica Pura ed Applicata

, Volume 188, Issue 1, pp 171–185 | Cite as

Supercritical biharmonic equations with power-type nonlinearity

  • Alberto Ferrero
  • Hans-Christoph Grunau
  • Paschalis Karageorgis


We study two different versions of a supercritical biharmonic equation with a power-type nonlinearity. First, we focus on the equation Δ2 u = |u| p-1 u over the whole space \({\mathbb{R}^n}\), where n > 4 and p > (n + 4)/(n − 4). Assuming that p < p c, where p c is a further critical exponent, we show that all regular radial solutions oscillate around an explicit singular radial solution. As it was already known, on the other hand, no such oscillations occur in the remaining case pp c. We also study the Dirichlet problem for the equation Δ2 u = λ (1 + u) p over the unit ball in \({\mathbb{R}^n}\), where λ > 0 is an eigenvalue parameter, while n > 4 and p > (n + 4)/(n − 4) as before. When it comes to the extremal solution associated to this eigenvalue problem, we show that it is regular as long as p < p c. Finally, we show that a singular solution exists for some appropriate λ > 0.


Supercritical biharmonic equation Power-type nonlinearity Singular solution Oscillatory behavior Boundedness Extremal solution 

Mathematics Subject Classification (2000)

35J60 35B40 35J30 35J65 


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  1. 1.
    Arioli, G., Gazzola, F., Grunau, H.-Ch., Mitidieri, E.: A semilinear fourth order elliptic problem with exponential nonlinearity. SIAM J. Math. Anal. 36, 1226–1258 (2005). Google Scholar
  2. 2.
    Berchio E. and Gazzola F. (2005). Some remarks on biharmonic elliptic problems with positive, increasing and convex nonlinearities. Electronic J. Differ. Equ. 2005(34): 1–20 MathSciNetGoogle Scholar
  3. 3.
    Brezis H., Cazenave T., Martel Y. and Ramiandrisoa A. (1996). Blow up for u t − Δu = g(u) revisited. Adv. Differ. Equ. 1: 73–90 MATHMathSciNetGoogle Scholar
  4. 4.
    Brezis H. and Vazquez J.L. (1997). Blow up solutions of some nonlinear elliptic problems. Rev. Mat. Univ. Complutense Madrid 10: 443–469 MATHMathSciNetGoogle Scholar
  5. 5.
    Dalmasso R. (1991). Positive entire solutions of superlinear biharmonic equations. Funkcial. Ekvac. 34: 403–422 MATHMathSciNetGoogle Scholar
  6. 6.
    Dávila, J., Dupaigne, L., Guerra, I., Montenegro, M.: Stable solutions for the bilaplacian with exponential nonlinearity. SIAM J. Math. Anal. 39, 565–592 (2007). Google Scholar
  7. 7.
    Ferrero, A., Grunau, H.-Ch.: The Dirichlet problem for supercritical biharmonic equations with power-type nonlinearity. J. Differ. Equ. 234, 582–606 (2007). Google Scholar
  8. 8.
    Fila, M., King, J.R., Winkler, M., Yanagida, E.: Optimal lower bound of the grow-up rate for a supercritical parabolic equation. J. Differ. Equ. 228, 339–356 (2006).
  9. 9.
    Gazzola, F., Grunau, H.-Ch.: Radial entire solutions for supercritical biharmonic equations. Math. Ann. 334, 905–936 (2006). Google Scholar
  10. 10.
    Gidas, B., Spruck, J.: Global and local behavior of positive solutions of nonlinear elliptic equations. Commun. Pure Appl. Math. 34, 525–598 (1981).
  11. 11.
    Joseph, D.D., Lundgren, T.S.: Quasilinear Dirichlet problems driven by positive sources. Arch. Ration. Mech. Anal. 49, 241–269 (1973). Google Scholar
  12. 12.
    Karageorgis, P.: Stability and intersection properties of solutions to the nonlinear biharmonic equation. (submitted)
  13. 13.
    Lions, P.L.: On the existence of positive solutions of semilinear elliptic equations. SIAM Rev. 24, 441–467 (1982). Google Scholar
  14. 14.
    McKenna P.J. and Reichel W. (2003). Radial solutions of singular nonlinear biharmonic equations and applications to conformal geometry. Electronic J. Differ. Equ. 2003(37): 1–13 MathSciNetGoogle Scholar
  15. 15.
    Mignot, F., Puel, J.P.: Sur une classe de problèmes nonlinéaires avec nonlinéarité positive, croissante, convexe. Commun. Partial Differ. Equ. 5, 791–836 (1980). Google Scholar
  16. 16.
    Mitidieri, E., Pohožaev, S.: Apriori estimates and blow-up of solutions to nonlinear partial differential equations and inequalities. Proc. Steklov Inst. Math. 234, 1–362 (2001, translated from Russian)Google Scholar
  17. 17.
    Pohožaev, S.I.: Solvability of an elliptic problem in \({\mathbb{R}^n}\) with a supercritical index of nonlinearity. Dokl. Akad. Nauk SSSR 313, 1356–1360 (1990), English translation in Soviet Math. Dokl. 42, 215–219 (1991)Google Scholar
  18. 18.
    Poláčik, P., Yanagida, E.: On bounded and unbounded global solutions of a supercritical semilinear heat equation. Math. Ann. 327, 745–771 (2003).
  19. 19.
    Serrin J. and Zou H. (1998). Existence of positive solutions of the Lane-Emden system. Atti Semin. Mat. Fis. Univ. Modena 46(suppl): 369–380 MATHMathSciNetGoogle Scholar
  20. 20.
    Swanson, C.A.: The best Sobolev constant. Appl. Anal. 47, 227–239 (1992). Google Scholar
  21. 21.
    Wang, X.: On the Cauchy problem for reaction–diffusion equations. Trans. Am. Math. Soc. 337, 549–590 (1993). Google Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  • Alberto Ferrero
    • 1
  • Hans-Christoph Grunau
    • 2
  • Paschalis Karageorgis
    • 3
  1. 1.Dipartimento di MatematicaUniversità di Milano-BicoccaMilanItaly
  2. 2.Fakultät für MathematikOtto-von-Guericke-UniversitätMagdeburgGermany
  3. 3.School of MathematicsTrinity CollegeDublin 2Ireland

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