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

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.