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.