# Minimal energy solutions to the fractional Lane–Emden system: Existence and singularity formation

### Woocheol Choi

Incheon National University, Republic of Korea### Seunghyeok Kim

Hanyang University, Seoul, Republic of Korea

## Abstract

In this paper, we study asymptotic behavior of minimal energy solutions to the fractional Lane–Emden system in a smooth bounded domain $\Omega$

for $0 < s < 1$ under the assumption that $(-\Delta)^s$ is the spectral fractional Laplacian and 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

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, generalizing the work of Guerra that studied the corresponding results in the local case $s = 1$. 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.

## Cite this article

Woocheol Choi, Seunghyeok Kim, Minimal energy solutions to the fractional Lane–Emden system: Existence and singularity formation. Rev. Mat. Iberoam. 35 (2019), no. 3, pp. 731–766

DOI 10.4171/RMI/1068