Existence of explosive positive solutions of quasilinear elliptic equations

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In this paper, our main purpose is to consider the quasilinear equation div ( | ∇ u | p - 2 ∇ u ) = m ( x ) f ( u ) on a domain Ω ⊆ R N , 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 = R N ), we show that a similar result holds where u( x) → ∞ as ∣ x∣ → ∞ within Ω and u( x) → ∞ as x → ∂ Ω.