# Maximizers for the variational problems associated with Sobolev type inequalities under constraints

## TL;DR

This paper introduces a novel method to analyze the existence of maximizers for Sobolev inequality-related variational problems, extending results to fractional Laplacian operators and providing new insights in critical and subcritical cases.

## Contribution

It presents a new approach linking supremum attainability in variational problems to special function attainability, applicable to fractional Laplacians and resolving open questions.

## Key findings

- New criteria for existence of maximizers in Sobolev inequalities
- Extension of results to fractional Laplacian operators
- Elementary proof for subcritical case results

## Abstract

We propose a new approach to study the existence and non-existence of maximizers for the variational problems associated with Sobolev type inequalities both in the subcritical case and critical case under the equivalent constraints. The method is based on an useful link between the attainability of the supremum in our variational problems and the attainablity of the supremum of some special functions defined on $(0,\infty)$. Our approach can be applied to the same problems related to the fractional Laplacian operators. Our main results are new in the critical case and in the setting of the fractional Laplacian operator which was left open in the work of Ishiwata and Wadade \cite{Ishiwata,Ishiwata1}. In the subcritical case, our approach provides a new and elementary proof of the results of Ishiwata and Wadade \cite{Ishiwata,Ishiwata1}.