Some nonexistence theorems for semilinear fourth-order equations.

In this paper, we analyse the semilinear fourth-order problem (− Δ)² u = g (u) in exterior domains of ℝ ^N . Assuming the function g is nondecreasing and continuous in [0, + ∞) and positive in (0, + ∞), we show that positive classical supersolutions u of the problem which additionally verify − Δ u > 0 exist if and only if N ≥ 5 and $$\int_0^\delta \displaystyle{{g(s)}\over{s^{(({2(N-2)})/({N-4}))}}} {\rm d}s \lt + \infty$$ for some δ > 0. When only radially symmetric solutions are taken into account, we also show that the monotonicity of g is not needed in this result. Finally, we consider the same problem posed in ℝ ^N and show that if g is additionally convex and lies above a power greater than one at infinity, then all positive supersolutions u automatically verify − Δ u > 0 in ℝ ^N , and they do not exist when the previous condition fails. [ABSTRACT FROM AUTHOR]