On a supercritical Hardy-Sobolev type inequality with logarithmic term and related extremal problem

Our main goal is to investigate supercritical Hardy-Sobolev type inequalities with a logarithmic term and their corresponding variational problem. We prove the existence of extremal functions for the associated variational problem, despite the loss of compactness. As an application, we show the existence of weak solution to a general class of related elliptic partial differential equations with a logarithmic term.