Hardy-Sobolev Type Equations for p-Laplacian, 1 < p < 2, in Bounded Domain

Abstract

We study quasilinear degenerate singular elliptic equation of type -Delta_p u = \frac{u^{p^*(s)-1}}{|y|^t}$ in a smooth bounded domain \Omega in R^n=R^k \times R^{N-k}$, x=(y,z) in R^k \times R^{N-k}, 2 \leq k<N and N \geq 3, 1<p<2, 0\leq s\leq p, 0\leq t\leq s, p^*(s)=\frac{p(n-s)}{n-p}. We study existence of solution for t<s, non-existence in a star-shaped domain for t=s and s<k(\frac{p-1}{p}). We also show that solution is in C^{1,\al}(\Omega) for some 0<\al<1 provided t<\frac{k}{N}(\frac{p-1}{p}). The regularity of solution can be improved to the class W^{2,p}(\Omega) when t<k(\frac{p-1}{p}). We also study some properties of the singular sets in a cylindrically symmetric domain using the method of symmetry.