A nonexistence theorem for an equation with critical Sobolev exponent in the half space

In this paper we prove the nonexistence of positive solutions of the equation-Δu=u^2*-1 inR ₊ ^N with certain homogeneous mixed boundary conditions. The proof is based on a monotonicity theorem obtained using the moving plane methods and some recent results of Berestycki and Nirenberg (see BN]). The nonexistence theorem is applied to improve a result of GP] on the characterization of the critical levels of a functional related to some nonlinear elliptic problem with critical Sobolev exponent and mixed boundary conditions.