Singular Structures in some Variational Problem = Structures singulières de quelques problèmes variationnels

This thesis represents the synthesis of the results obtained during these last four years of research and study under the guide and teachings of professor S. Solimini and professor J.-M. Morel. In this thesis we approach some problems of Nonlinear Analysis and of Calculus of Variations which give rise to ``singular'' structures. Our purpose is to show how in some cases these singular structures are an obstacle to overcome to get existence or multiplicity results and in other cases the singularities in the solutions are the object of the study and justify the choice of the functional introduced. The work is divided into two parts: the first one concerns results on a class of problems, which have been faced by researchers in nonlinear analysis in the last twenty years, about the existence and the multiplicity of solutions of elliptic equations, obtained in spite of a lack of compactness due to some concentration phenomena; the second one regards a certain category of irrigation problems in which the trajectories followed by fluid particles give rise to a one-dimensional set and which can be set in the transport theory.