  The Shilkret integral with respect to a completely maxitive capacity is fully
determined by a possibility distribution.
  In this paper, we introduce a weaker topological form of maxitivity and show
that under this assumption the Shilkret integral is still determined by its
possibility distribution for functions that are sufficiently regular. Motivated
by large deviations theory, we provide a Laplace principle for maxitive
integrals and characterize the possibility distribution under certain
separation and convexity assumptions. Moreover, we show a maxitive integral
representation result for weakly maxitive non-linear expectations. The
theoretical results are illustrated by providing large deviations bounds for
sequences of capacities, and by deriving a monotone analogue of Cram\'{e}r's
theorem.
