  We construct the precise boundary trace of positive solutions of $\Delta
u=u^q$ in a smooth bounded domain in $R^N$, for $q$ in the super-critical range
$q\geq (N+1)/(N-1)$. The construction is performed in the framework of the fine
topology associated with the Bessel capacity $C_{2/q,q'}$ on the boundary of
the domain. We prove that the boundary trace is a Borel measure (in general
unbounded), which is outer regular relative to this capacity. We provide a
necessary and sufficient condition for such measures to be the boundary trace
of a positive solution and prove that the corresponding generalized boundary
value problem is well-posed in the class of $\sigma$-moderate solutions.
