  The purpose of this paper is to show that Non-Archimedean Mathematics (NAM),
namely mathematics which uses infinite and infinitesimal numbers, is useful to
model some Physical problems which cannot be described by the usual
mathematics. The problem which we will consider here is the minimization of the
functional E(u,q)=1/2\int_{Omega}|\nabla u(x)|^{2}dx+u(q).
  If Omega is a subset of R^{N} is a bounded open set and u is a C^{2} function
with condition u=0 on the boundary of Omega, this problem has no solution since
inf E(u,q)=-\infty.
  On the contrary, as we will show, this problem is well posed in a suitable
non-Archimedean frame. More precisely, we apply thegeneral ideas of NAM and
some of the techniques of Non Standard Analysis to a new notion of generalized
functions, called ultrafunctions, which are a particular class of functions
based on a Non-Archimedean field. In this class of functions, the above problem
is well posed and it has a solution.
