Supercritical biharmonic equations with power-type nonlinearity

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 ≥ p_c. 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.