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$P_h${ll -div (|Duh|p-2 Duh)=g, & in D Eh uhH1,p0(D Eh). . where $2\le p\le n$ and $E_h$ are random subsets of a bounded open set $D$ of ${\mathbb{R}}^n$ $\left(n\ge 2\right)$. By means of a variational approach, we study the asymptotic behaviour of solutions of $\left(P_h\right)$, characterizing the limit problem for suitable sequences of random sets.