  We consider the minimization problem for an integral functional $J$, possibly
non-convex and non-coercive in $W^{1,1}_0(\Omega)$, where $\Omega\subset\R^n$
is a bounded smooth set. We prove sufficient conditions in order to guarantee
that a suitable Minkowski distance is a minimizer of $J$. The main result is a
necessary and sufficient condition in order to have the uniqueness of the
minimizer. We show some application to the uniqueness of solution of a system
of PDEs of Monge-Kantorovich type arising in problems of mass transfer theory.
