On stable solutions of the biharmonic problem with polynomial growth

We prove the nonexistence of smooth stable solutions to the biharmonic problem δ²u = u^p, u > 0 in ℝ^N for 1 < p <∞and N < 2(1+x₀), where x₀ is the largest root of the equation In particular, as x₀ > 5 when p > 1, we obtain the nonexistence of smooth stable solutions for any N ≤ 12 and p > 1. Moreover, we consider also the corresponding problem in the half-space ℝ₊^N, and the elliptic problem δ²u=λ(u+1)^p on a bounded smooth domain ω with the Navier boundary conditions. We prove the regularity of the extremal solution in lower dimensions.