Hardy-Sobolev type equations for $p$-Laplacian, $1<p<2$, in bounded domain

Abstract

We study quasilinear degenerate singular elliptic equations of the type $ -\Delta_p u = \frac{u^{p^*(s)-1}}{|y|^t} $ in a smooth bounded domain $\Omega$ in ${{\mathbb R}^{N}}={{\mathbb R}^{k}}\times{\mathbb R}^{N-k}$, $x=(y,z)\in{{\mathbb R}^{k}}\times{\mathbb R}^{N-k}$, $2 \leq k<N$ and $N\geq 3$, $1<p<2$, $0\leq s\leq p$, $0\leq t\leq s$ and $p^*(s)=\frac{p(n-s)}{n-p}$. We study existence of solutions for $t<s$, non-existence in a star-shaped domain for $t=s$ and $s<k\big(\frac{p-1}{p}\big)$. We also show that the solution is in $C^{1, {\alpha}}(\Omega)$ for some $0< {\alpha}<1$ provided $t<\frac{k}{N}\big(\frac{p-1}{p}\big)$. The regularity of the 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 set of degeneracy in a cylindrically symmetric domain using the method of symmetry.