Sobre uma desigualdade do tipo Hardy-Sobolev e aplicações

Abstract

In this work, we present a Hardy-Sobolev type inequality in cylindrical domains and, as a consequence, derive some Sobolev embeddings into weighted Lebesgue spaces. We prove that the attainability of the best constant for these weighted embeddings is equivalent to establishing the existence of ground state solutions for a class of elliptic problems in R. Regularity and behavior of the minimizers are also analyzed. As an application of the obtained embeddings, we prove existence and non-existence results for a class of elliptic problems with critical growth.