Small order asymptotics for nonlinear fractional problems

TL;DR

This paper investigates the behavior of solutions to nonlinear boundary value problems involving the fractional Laplacian as the order parameter approaches zero, revealing convergence to a problem involving the logarithmic Laplacian.

Contribution

It introduces a novel analysis of the limiting behavior of fractional Laplacian solutions as the order tends to zero, utilizing variational methods and a new logarithmic Sobolev inequality.

Findings

Solutions converge to a nontrivial solution of the logarithmic Laplacian problem.

Established uniform energy estimates for solutions as s approaches zero.

Developed a new logarithmic-type Sobolev inequality applicable to this context.

Abstract

We study the limiting behavior of solutions to boundary value nonlinear problems involving the fractional Laplacian of order $2s$ when the parameter $s$ tends to zero. In particular, we show that least-energy solutions converge (up to a subsequence) to a nontrivial nonnegative least-energy solution of a limiting problem in terms of the logarithmic Laplacian, i.e., the pseudodifferential operator with Fourier symbol $\ln(|\xi|^2)$. These results are motivated by some applications of nonlocal models where a small value for the parameter $s$ yields the optimal choice. Our approach is based on variational methods, uniform energy-derived estimates, and the use of a new logarithmic-type Sobolev inequality.