On a singular nonlinear Neumann problem

Abstract

We investigate the solvability of the Neumann problem involving two critical exponents: Sobolev and Hardy-Sobolev. We establish the existence of a solution in three cases: (i) 2 < p + 1 < 2*(s), (ii) p + 1 = 2*(s) and (iii) 2*(s) < p + 1 ≤ 2*, where 2*(s) = 2(N-s)/N-2, 0 < s < 2, and 2* = 2(N-s)/N-2 denote the critical Hardy-Sobolev exponent and the critical Sobolev exponent, respectively.