Multiplicity of solutions for a fourth order problem with exponential nonlinearity

Let B   be the unit ball in RNRN, N⩾5N⩾5 and n be the exterior unit normal vector on the boundary. We consider radial solutions toΔ2u=λeuin B,u=0and∂u∂n=0on∂B, where λ⩾0λ⩾0. We show that there exists a unique λS>0λS>0 such that if λ=λSλ=λS there is a radial singular solution. If 5⩽N⩽125⩽N⩽12 then for λ=λSλ=λS there exist infinitely many regular radial solutions and as λ→λSλ→λS the number of such solutions goes to infinity. If N⩾13N⩾13 we prove uniqueness of smooth radial solutions. We derive similar results for the same equation with Navier boundary conditions.