Problemas elípticos não lineares envolvendo equações do tipo Kirchhoff

Abstract

In this work we study the existence of weak solutions for four nonlinear elliptic problems of Kirchhoff type. These problems have in common the presence of a function M : R+ [ f0g ! R+; known as Kirchhoff-type function. The first problem deals with a Kirchhoff type equation involving subcritical exponents while the second problem deals with the same equation but with a critical Caffarelli-Kohn-Nirenberg exponent. In the third problem, we study a system of Kirchhoff type equations involving critical Caffarelli-Kohn-Nirenberg exponents. The latter problem involves a Kirchhoff type operator and a nonlinear boundary condition. Due to the presence of the Kirchhoff type operator in the equations, these problems are called nonlocal problems. This phenomenon causes some mathematical difficulties, which makes the study of such a class of problem, particularly interesting. In our studies we used classical variational methods such as The Mountain Pass Theorem and the Krasnoseslkii genus theory.