Desigualdade de Sobolev com peso no semi espaço superior e aplicações com condição de Neumann

Abstract

This work is dedicated to the study of a weighted Sobolev inequality and its applica-tions to an elliptic equation, following the work developed by Souza, Felix and Medeiros in [42]. The main objective is to investigate the existence and non-existence of non-trivial solutions to an elliptic semilinear equation with Neumann boundary condition, involving the upper half-plane and weight functions. In this process of studying solutions, weighted Sobolev embedding will be established, so that an adequate variational approach is possible. More specifically, a functional will be introduced in such a way that its critical points are solutions to the problem in question. Finally, it is worth highlighting that the applied methods are based on minimization techniques and the famous Mountain Pass Theorem.