*Accepted Paper*

**Inserted:** 7 jan 2002

**Journal:** Boll. Un. Mat. It.

**Year:** 2001

**Abstract:**

We investigate sequences of nonlinear Dirichlet
problems of the form $$ \left \{\eqalign { -& \, div(\vert Du_{h
}
\vert ^{{p}-2} Du_{h)} = g \qquad{\rm in} \,\, D\backslash E_{h} \cr &
u_{h} \in H^{{1,p}}_{0}(D\backslash E_{h).} \cr } \right.$$
where $2\leq{p}\leq{n}$ and $E_h$ are random subsets of a bounded
open set $D$ of $*R*^n$ ($n\geq 2$). By means of a variational
approach, we study the asymptotic behaviour of solutions of the
problems, characterizing the limit problem for suitable sequences
of random sets.

**Keywords:**
p-Laplacian, p-capacity, random set