Annali di Matematica Pura ed Applicata

, Volume 188, Issue 1, pp 171-185

First online:

Supercritical biharmonic equations with power-type nonlinearity

  • Alberto FerreroAffiliated withDipartimento di Matematica, Università di Milano-Bicocca Email author 
  • , Hans-Christoph GrunauAffiliated withFakultät für Mathematik, Otto-von-Guericke-Universität
  • , Paschalis KarageorgisAffiliated withSchool of Mathematics, Trinity College

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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