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It is known that the very weak solution of the distance of u ∈ Ω to the boundary. Here, we show that if f is not in this weighted space L 1 Ω; δ(1+| Ln δ|) , f 0, then its gradient blows up in L(log L) at least. Moreover, we show that there exist a domain Ω of class C ∞ and a function f ∈ L 1 + (Ω, δ) such that the associated very weak solution has its gradient being non integrable on Ω.doi:10.3934/dcds.2013.33.1809 fatcat:x3qstk3zubgnxfxtaj4in6frfu