Funções de Green e aplicações a problemas elípticos e poli-harmônicos envolvendo potenciasi de Schrödinger

Abstract

In this work we have studied some problems associated with the laplacian (...), and poly-harmonic elliptic operators (...). In the case of the poly-harmonic operators we are interested in problems of the non-homogeneous type, whose existence of solutions will be proved by the study of their respective Green functions and fixed-point arguments. Some specific characteristics of these problems will also be explored. In the case of the Laplacian operator, we are interested in generalizations of the Faber-Krahn inequality with Neumann-type boundary conditions. This inequality is well known in the case where the boundary conditions are of the Dirichlet type. More precisely, for any bounded domain with fixedvolume, the smallest possible eigenvalue for the Dirichlet problem occurs when the domain is a ball. In the case of the poly-harmonic equation we will find a solution in the anisotropic space (...) and show how the space changes according to the changes of the functions involved in the problem, besides we will speak of some qualitative properties of the solution between them the positivity and symmetry. For the case of system of two (resp. of n) equations we will study the existence of a solution in space (...) and some properties of these solutions as the fact that the components of the solution are even, positive or radial functions depending on the behavior of the functions involved in the problem.