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

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 Ωunder 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 (−∆) s u = u n+2s n−2s −ǫ in Ω and u = (−∆) s 2 u = 0 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 Ω 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.

Asymptotic behavior of least energy solutions to the Lane-Emden system near the critical hyperbola

“…Proof. Consult the proof of Lemma 4.3 of [6]. It works in our case as well, once the order s of the fractional Laplacian (−∆) s is taken to be 1.…”