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 language | English |
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Pages (from-to) | 171-185 |
Number of pages | 15 |
Journal | Annali di Matematica Pura ed Applicata |
Volume | 188 |
Issue number | 1 |
DOIs | |
Publication status | Published - Jan 2009 |
Externally published | Yes |
Keywords
- Boundedness
- Extremal solution
- Oscillatory behavior
- Power-type nonlinearity
- Singular solution
- Supercritical biharmonic equation