Nonexistence results for a class of nonlinear elliptic equations involving critical Sobolev exponents

In this paper we study the problem of bifurcation from the origin of solutions of elliptic Dirichlet problems involving critical Sobolev exponent, defined on a bounded domain $\Omega$ in $\mathbb{R} ^N$: we prove that the first critical case are $N=3, 4$ (not only N=3, as just proved by Brezis and Nirenberg), exhibiting two nonexistence results for a class of elliptic problem in these dimensions.