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
Article

Abstract

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.

Keywords

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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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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