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 $\Omega$ $$ (-\Delta)^s u = v^p, \ (-\Delta)^s v = u^q, \ u, v > 0 \text{ in } \Omega \quad \text{and} \quad u = v = 0 \text{ on } \partial \Omega $$ 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 $$ (-\Delta)^s u = u^{\frac{n+2s}{n-2s}-\epsilon}, \ u > 0 \text{ in } \Omega \quad \text{and} \quad u = (-\Delta)^{s /2} u = 0\~\text{on}\~\partial \Omega $$ 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.