Non-local equations and optimal Sobolev inequalities on compact manifolds

TL;DR

This paper establishes an optimal Sobolev inequality for fractional Sobolev spaces on compact manifolds and applies it to prove the existence of solutions for non-local equations with critical non-linearity.

Contribution

It introduces an optimal Sobolev inequality for fractional Sobolev spaces on compact manifolds and uses it to analyze non-local equations with critical non-linearity.

Findings

Proved a Sobolev inequality with an optimal constant for fractional Sobolev spaces.

Demonstrated existence of solutions for non-local equations with critical non-linearity.

Extended classical Sobolev inequalities to fractional settings on manifolds.

Abstract

This paper deals with fractional Sobolev spaces on a compact Riemannian manifold. We prove a Sobolev inequality in the critical range with an optimal constant for these fractional Sobolev spaces. We use this result to study the existence of a non-trivial solution for equations driven by a non-local integro-differential operator $\mathcal{L}_{\mathcal{K}}$ with critical non-linearity.