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 in Omega and u = v = 0 on partial derivative 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 n+2s/n-2s-epsilon, u > 0 in Omega and u - (-Delta)(s/2) u = 0 on partial derivative 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.