By the Mountain Pass Theorem,we study existence and multiplicity of positive solutions of p-laplacian equation of the form -△pu =λf(x,u),the nonlinearity f(x,u) grows as uσ at infinity with a singular coefficient,where σ ∈ (p - 1,p* - 1).To manage the asymptotic behavior of its positive solutions with respect to λ,we establish a new Liouville-type theorem for the p-Laplacian operator.