Annali di Matematica Pura ed Applicata

, Volume 188, Issue 1, pp 171–185

Supercritical biharmonic equations with power-type nonlinearity

Authors

    • Dipartimento di MatematicaUniversità di Milano-Bicocca
  • Hans-Christoph Grunau
    • Fakultät für MathematikOtto-von-Guericke-Universität
  • Paschalis Karageorgis
    • School of MathematicsTrinity College
Article

DOI: 10.1007/s10231-008-0070-9

Cite this article as:
Ferrero, A., Grunau, H. & Karageorgis, P. Annali di Matematica (2009) 188: 171. doi:10.1007/s10231-008-0070-9

Abstract

We study two different versions of a supercritical biharmonic equation with a power-type nonlinearity. First, we focus on the equation Δ2u = |u|p-1u over the whole space \({\mathbb{R}^n}\), where n > 4 and p > (n + 4)/(n − 4). Assuming that p < pc, where pc 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 ppc. We also study the Dirichlet problem for the equation Δ2u = 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 < pc. Finally, we show that a singular solution exists for some appropriate λ > 0.

Keywords

Supercritical biharmonic equationPower-type nonlinearitySingular solutionOscillatory behaviorBoundednessExtremal solution

Mathematics Subject Classification (2000)

35J6035B4035J3035J65

Copyright information

© Springer-Verlag 2008