RADIALLY SYMMETRICAL 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 -DELTAu(x) = lambdau(x) + u(x)\u(x)\4/(N-2), x is-an-element-of B : = {x is-an-element-of R(N): \\x\\ < 1}, u(x) = 0, x is-an-element-of partial derivative B, where DELTA is the Laplacean operator and lambda > 0. Indeed, we prove that if N = 3, 4, then for any lambda > 0 this problem has only finitely many radial solutions. For N = 3, 4, 5 we show that, for each lambda > 0, the set of radially symmetric solutions is bounded. Moreover, we establish geometric properties of the branches of solutions bifurrating from zero and from infinity.