Existence of explosive positive solutions of quasilinear elliptic equations

In this paper, our main purpose is to consider the quasilinear equationdiv(|∇u|p-2∇u)=m(x)f(u)on a domain Ω ⊆ RN, N ⩾ 3, where f is a nonnegative, nondecreasing continuous function which vanishes at the origin, and m is a nonnegative continuous function with the property that any zero of m is contained in a bounded domain in Ω such that m is positive on its boundary. For Ω bounded, we show that a nonnegative solution u satisfying u(x) → ∞ as x → ∂Ω exists. For Ω un-boundary (including Ω = RN), we show that a similar result holds where u(x) → ∞ as ∣x∣ → ∞ within Ω and u(x) → ∞ as x → ∂Ω.