Subcritical approximation of a Yamabe type non local equation: a Gamma-convergence approach

TL;DR

This paper studies a subcritical approximation of a fractional Sobolev quotient using Gamma-convergence, demonstrating the existence of optimal functions and their energy concentration behavior.

Contribution

It introduces a Gamma-convergence approach to approximate the fractional Sobolev quotient and proves the existence and concentration of optimal functions.

Findings

Optimal functions always exist for the approximation.

Energy concentrates at a single point in the limit.

The approach applies to all fractional orders 0<s<N/2.

Abstract

We investigate a natural approximation by subcritical Sobolev embeddings of the Sobolev quotient for the fractional Sobolev spaces $H^s$ for any $0<s<N/2$, using $\Gamma$-convergence techniques. We show that, for such approximations, optimal functions always exist and exhibit a concentration effect of the $H^s$ energy at one point.