Radially Symmetric Solutions to a Dirichlet Problem Involving Critical Exponents

In this paper we answer, for N = 3,4, the question raised in [1] on the number of radially symmetric solutions to the boundary value problem -Δu(x) = λu(x) + u(x)|u(x)|^{4/(N-2)}, x ε B: = x ε RN:{|x| \u3c 1}, u(x)=0, x ε ∂B, where Δ is the Laplacean operator and λ\u3e0. Indeed, we prove that if N = 3,4, then for any λ\u3e0 this problem has only finitely many radial solutions. For N = 3,4,5 we show that, for each λ\u3e0, the set of radially symmetric solutions is bounded. Moreover, we establish geometric properties of the branches of solutions bifurcating from zero and from infinity