O Problema Extremal para Sistemas Elípticos Semilineares

Abstract

In this paper, we discuss some questions related to elliptical systems of the type (...), were (...) is a smooth e bounded domain, (...), the parameters (...) are positives and the functions (...) satisfy growth conditions that will be discussed mainly in Chapters 2 and 3 of this paper. In the first Chapter, we consider the particular case (...), were (...), the number (...),are positives, satisfy (...), and the weight function (...) are positives and belong to (...). In this Chapter we show that the set (...) é um autovalor principal de (...) define a hypersurface in (...) what shall we call principal hypersurface for this system. In the Chapter 2, we use the hypersurface (...) to develop the concept of extremal set for the system (3). Denoting by U the set of (...) such that the system (3) has a positive solution in (...) we show that the extremal set (...) can be written in the form (...) . Still in this Chapter, we have proved some results involving this set and its solutions such as existence, regularity and stability of such solutions. We also prove some qualitative properties of (...) such as continuity, behavior asymptotic and limitation of this set. Finally, in Chapter 3, we investigated the regularity of the extremal solutions for a class of elliptical systems that we will call gradient-type systems. This term is due to the fact that we will be interested in studying the system (3) in the case (...), that is, (...). The functions (...) must satisfy a certain class of conditions which are verified, in particular, to (...), among others. In this part of the work, we extend the regularity result for the extremal solutions due to C. Cowan and M. Fazly in dimensions (...) when (...), more precisely, our regularity result holds for a class of nonlinearities that extends the known result to m = 2.