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: $\text{(i)};\ 2\lt p+1\lt 2^(s),$, $\text{(ii)};\ p+1=2^(s)$ and $\text{(iii)};\ 2^(s)\lt p+1 \leq 2^,$ where $2^(s)=\frac{2(N-s)}{N-2},$ $0\lt s\lt 2,$ and $2^=\frac{2N}{N-2}$ denote the critical Hardy-Sobolev exponent and the critical Sobolev exponent, respectively.