Multiplicity of solutions for a fourth order problem with exponential nonlinearity

Let B be the unit ball in R-N, N >= 5 and n be the exterior unit normal vector on the boundary. We consider radial. solutions to Delta(2)u = lambda e(u) in B, u = 0 and partial derivative u/partial derivative n = 0 on partial derivative B, where lambda >= 0. We show that there exists a unique lambda(S) > 0 such that if lambda = lambda(S) there is a radial singular solution. If 5 <= N <= 12 then for lambda = lambda(S) there exist infinitely many regular radial solutions and as lambda -> lambda(S) the number of such solutions goes to infinity. If N >= 13 we prove uniqueness of smooth radial solutions. We derive similar results for the same equation with Navier boundary conditions. (C) 2009 Elsevier Inc. All rights reserved.