Minimal energy solutions to the fractional Lane-Emden system, I: Existence and singularity formation

TL;DR

This paper investigates minimal energy solutions to the fractional Lane-Emden system in bounded domains, focusing on existence, singularity formation, and limits as parameters approach critical Sobolev hyperbola, with implications for related inequalities.

Contribution

It establishes the existence of minimal energy solutions for the fractional Lane-Emden system and introduces a new approach for extremal functions of the Hardy-Littlewood-Sobolev inequality.

Findings

Existence of minimal energy solutions in convex domains.

Analysis of pointwise limits as parameters approach criticality.

A novel method for extremal function existence in Sobolev inequalities.

Abstract

This is the first of two papers which study asymptotic behavior of minimal energy solutions to the fractional Lane-Emden system in a smooth bounded domain $\Omega$ \[(-\Delta)^s u = v^p, \quad (-\Delta)^s v = u^q \text{ in } \Omega \quad \text{and} \quad u = v = 0 \text{ on } \pa \Omega \quad \text{for } 0 < s < 1\] under the assumption that the subcritical pair $(p,q)$ approaches to the critical Sobolev hyperbola. If $p = 1$, the above problem is reduced to the subcritical higher-order fractional Lane-Emden equation with the Navier boundary condition \[(-\Delta)^s u = u^{\frac{n+2s}{n-2s}-\ep} \text{ in } \Omega \quad \text{and} \quad u = (-\Delta)^{s \over 2} u = 0 \quad \text{for } 1 < s < 2.\] The main objective of this paper is to deduce the existence of minimal energy solutions, and to examine their (normalized) pointwise limits provided that $\Omega$ is convex. As a by-product of our study, a new approach for the existence of an extremal function for the Hardy-Littlewood-Sobolev inequality is provided.