Minimizers and best constants for a weighted critical Sobolev inequality involving the polyharmonic operator

TL;DR

This paper explicitly computes the best constant for a weighted critical Sobolev inequality involving the polyharmonic operator and establishes regularity and classification results for related equations.

Contribution

It provides the explicit value of the best constant and new regularity and classification results for a generalized critical polyharmonic equation.

Findings

Explicit computation of the best constant for the inequality

Regularity results for the generalized critical polyharmonic equation

Classification results in the radial setting

Abstract

Our main goal is to explicitly compute the best constant for the Sobolev-type inequality involving the polyharmonic operator obtained in (Analysis and Applications 22, pp. 1417-1446, 2024). To achieve this goal, we also establish both regularity and classification results for a generalized critical polyharmonic equation in the radial setting.