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

TL;DR

This paper investigates existence, non-existence, and regularity of solutions for a class of p-Laplacian equations with singular terms in bounded domains, highlighting conditions for solution smoothness and symmetry properties.

Contribution

It provides new existence and regularity results for p-Laplacian equations with singular weights, extending understanding of solution behavior in bounded domains.

Findings

Existence of solutions for t<s.

Non-existence in star-shaped domains when t=s.

Solutions are in C^{1,α} and can be in W^{2,p} under certain conditions.

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.