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

Abstract 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.

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