In this paper we will be concerned with the equations Δ p ( x ) u = g ( x ) f ( u ), where Ω is a bounded domain, g is a non-negative continuous function on Ω which is allowed to be unbounded on Ω and non-linearity f is a non-negative non-decreasing functions. We show that the equation Δ p ( x ) u = g ( x ) f ( u ) admits a non-negative local weak solution u ∈ W 1 ,p ( x ) loc (Ω) ∩ C (Ω) such that u ( x ) → ∞ ? as x → ? Ω if Δ p ( x ) w = - g ( x ) in the weak sense for some w ∈ W 1 ,p ( x ) 0 (Ω) and f satisfies a generalized Keller-Osserman condition. ? 2000 Mathematics Subject Classification. Primary 35J60; Secondary 58E05. Key words and phrases. elliptic equation, blow-up solutions, p ( x )-Laplacian.
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