Existence and asymptotic behavior of blow-up solutions to weighted quasilinear equations

Given a bounded domain Ω we consider local weak blow-up solutions to the equation Δpu = g(x)f (u) on Ω. The non-linearity f is a non-negative non-decreasing function and the weight g is a non-negative continuous function on Ω which is allowed to be unbounded on Ω. We show that if Δpw =−g(x) in the weak sense for some w ∈ W 1,p 0 (Ω) and f satisfies a generalized Keller– Osserman condition, then the equation Δpu = g(x)f (u) admits a non-negative local weak solution u ∈ W 1,p loc (Ω) ∩ C(Ω) such that u(x)→∞ as x →∂Ω. Asymptotic boundary estimates of such blow-up solutions will also be investigated.  2004 Elsevier Inc. All rights reserved.