Supercritical biharmonic equations with power-type nonlinearity

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 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.