Multiplicity of solutions for a fourth order problem with exponential nonlinearity

Let B be the unit ball in , N 5 and n be the exterior unit normal vector on the boundary. We consider radial solutions to where λ 0. We show that there exists a unique λS>0 such that if λ=λS there is a radial singular solution. If 5 N 12 then for λ=λS there exist infinitely many regular radial solutions and as λ→λS 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