| Sumario: | In this work we consider the behaviour for large values of p of the unique positive weak solution u p to Δ p u = u q in Ω, u = +∞ on partial Ω, where q > p - 1. We take q = q(p) and analyze the limit of u p as p → ∞. We find that when q(p)/p → Q the behaviour strongly depends on Q. If 1 < Q < ∞ then solutions converge uniformly in compacts to a viscosity solution of max{-Δ{u}, -|∇ u| +uQ \\} = 0 with u = +∞ on Ω. If Q = 1 then solutions go to ∞ in the whole Ω and when Q = ∞ solutions converge to 1 uniformly in compact subsets of Ω, hence the boundary blow-up is lost in the limit. © 2007 Springer-Verlag.
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