For a nonlinear two point boundary value problem

The present paper, using a branch point approach, offers an explanation of a different behavior with respect to the existence of positive solution of the nonlinear elliptic problem investigated by H. Brezis e L. Nirenberg. ∆u + u^p + λu = 0 in Ω \subset R^N u = 0 on \partial Ω where p = N+2/N-2 is the critical Sobolev exponent, in the cases N = 3 and N = 4. Precisely when Ω is a ball a connection is showed between such phenomenon and the interval of existence of a solution of two point boundary value problem of radially symmetric solutions.