Ultrafunctions and generalized solutions

Abstract

The theory of distributions provides generalized solutions for problems which do not have a classical solution. However, there are problems which do not have solutions, not even in the space of distributions. As model problem you may think of -\Deltau=u^{p-1}, u>0, p\geq(2N)/(N-2) with Dirichlet boundary conditions in a bounded open star-shaped set. Having this problem in mind, we construct a new class of functions called ultrafunctions in which the above problem has a (generalized) solution. In this construction, we apply the general ideas of Non Archimedean Mathematics and some techniques of Non Standard Analysis. Also, some possible applications of ultrafunctions are discussed.