In this paper, our main purpose is to consider the singular (p)-laplacian quasilinear elliptic equation[mbox{div}(|x|^{-ap}|abla u|^{p-2}abla u)=b(x)f(u)hskip 0.4cm mbox{in}hskip 0.2cm Omega,] where (hskip 0.2cm Omegasubseteq{f R}^{mathbf{N}}(Ngeq ap+1)), (p&1), (a&0), (b(x)geq0), (b(x)in? C(overline{Omega})),? and (f) is continuous and non-decreasing on? ([0,+infty)), satisfies (f(0)=0), (f(s)&0) for (s&0). When (Omega=B) (the unit ball in ({f R}^{N}(Ngeq ap+1))), (b(x)in? C(overline{Omega})) is radial, and (f) satisfies the condition [int_{1}^{+infty}frac{1}{f^{frac{1}{p-1}}(t)}dt=+infty,] we give a necessary and sufficient condition for the existence of a positive solution. When (Omega) is an open, connected subset of ({f R}^{N}) with smooth boundary, and (f) satisfies the Keller-Osserman condition [int_{1}^{+infty}[F(s)]^{-frac{1}{p}}ds=+infty,F(s)=int_{0}^{s}f(t)dt.] We establish conditions on the function (b) that are necessary and sufficient for the existence of positive solutions, bounded and unbounded, of the given equation.
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