Multiplicity of solutions for a fourth order equation with power-type nonlinearity

Let B be the unit ball in R-N, N >= 3 and n be the exterior unit normal vector on the boundary. We consider radial solutions to Delta(2)u = lambda(1 + sign(p)u)(p) in B, u = 0, partial derivative u/partial derivative n = 0 on partial derivative B where lambda >= 0. For positive p we assume 5 <= N <= 12 and p > N+4/N-4, or N >= 13 and N+4/N-4 < p < p(c), where p(c) is a constant depending on N. For negative p we assume 4 <= N <= 12 and p < p(c), or N = 3 and p(c)(+) < p < p(c), where p(c)(+) is a constant. We show that there is a unique lambda(S) > 0 such that if lambda = lambda(S) there exists a radial weakly singular solution. For lambda = lambda(S) there exist infinitely many regular radial solutions and the number of radial regular solutions goes to infinity as lambda -> lambda(S).