Existência e multiplicidade de soluções para uma classe de problemas envolvendo operadores fracionários

Abstract

In this work, we study existence and multiplicity of weak solutions for three problems involving fractional operators, with emphasis on critical growth nonlinearities. The first problem deals with the existence of sign-changing solution for an equation involving the fractional Laplacian and fractional critical Sobolev exponents. In the second problem, we study the existence of signed and sign-changing solutions for an equation involving the fractional p-Laplacian with a Kirchhoff term and fractional subcritical and critical Hardy-Sobolev exponent. The last problem approaches existence and multiplicity of positive solutions for an equation involving the fractional p-Laplacian with a Kirchhoff term, fractional subcritical and critical Hardy-Sobolev exponent and weight with indefinite signal. The presence of critical exponents generates additional mathematical dificulties in obtaining solutions due to lack of compactness of the Sobolev embedding. In our studies, we used variational methods such as the Mountain Pass Theorem, constraint minimization on Nehari sets and the fibering method.