On a Sobolev-type inequality and its minimizers

TL;DR

This paper establishes a critical Sobolev-type inequality for weighted Sobolev spaces, investigates extremal functions, and applies these results to prove the existence of solutions for certain elliptic equations involving polyharmonic operators.

Contribution

It introduces a new Sobolev-type inequality for weighted spaces and demonstrates the existence of extremal functions and solutions for related elliptic problems.

Findings

Proved a new critical Sobolev-type inequality for weighted spaces.

Established the existence of extremal functions for the inequality.

Proved the existence of weak solutions for critical semilinear elliptic equations.

Abstract

Critical Sobolev-type inequality for a class of weighted Sobolev spaces on the entire space is established. We also investigate the existence of extremal function for the associated variational problem. As an application, we prove the existence of a weak solution for a general class of critical semilinear elliptic equations related to the polyharmonic operator.