Supercritical biharmonic equations with power-type nonlinearity

Alberto Ferrero, Hans Christoph Grunau, Paschalis Karageorgis

Research output: Contribution to journalArticlepeer-review

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 ℝn, 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 p ≥ pc. We also study the Dirichlet problem for the equation Δ2 u = λ (1 + u)p over the unit ball in ℝn, 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.

Original languageEnglish
Pages (from-to)171-185
Number of pages15
JournalAnnali di Matematica Pura ed Applicata
Volume188
Issue number1
DOIs
Publication statusPublished - Jan 2009
Externally publishedYes

Keywords

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

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