Given a non-linear elliptic equation of monotone type in a bounded open set Ω ⊂ Rn, we prove that the asymptotic behaviour, as j → ∞, of the solutions of the Dirichlet problems corresponding to a sequence (Ωj) of open sets contained in Ω is uniquely determined by the asymptotic behaviour, as j → ∞, of suitable non-linear capacities of the sets K \ Ωj, where K runs in the family of all compact subsets of Ω.
