Sobolev versus Holder local minimizers and existence of multiple solutions for a singular quasilinear equation

We investigate the following quasilinear and singular problem, -Delta(p)u=lambda/u delta+u(q) in Omega; (P) i vertical bar partial derivative Omega=0, u>0 in Omega, where Q is an open bounded domain with smooth boundary, 1<p<infinity, p-1<q <= p*-1, lambda>0, and 0<delta<1. As usual, p*=Np/M-p if 1<p<N, p* is an element of (p, infinity) is arbitrarily large if p=N, and p*=infinity if p>N. We employ variational methods in order to show the existence of at least two distinct (positive) Solutions of problem (P) in W-0(1,p) (Omega). While following an approach due to Ambrosetti-Brezis-Cerami, we need to prove two new results of separate interest: a strong comparison principle and a regularity result for solutions to problem (P) in C-1,C-beta((Omega) over bar) with some beta is an element of (0, 1). Furthermore, we show that delta<1 is a reasonable sufficient (and likely optimal) condition to obtain solutions of problem (P) in C-1 (<(Omega)over bar>).