Classification of positive solutions for an elliptic system with a higher-order fractional Laplacian

Jingbo Dou and Changzheng Qu

Vol. 261 (2013), No. 2, 311–334

DOI: 10.2140/pjm.2013.261.311

Abstract

We discuss properties of solutions to the following elliptic PDE system in  ℝⁿ:

where 0 < α < n, λ_j, μ_j,β_j (j = 1,2) are nonnegative constants and p_i and  q_i (i = 1,2,3,4) satisfy some suitable assumptions. It is shown that this PDE system is equivalent to the integral system

in ℝⁿ. The radial symmetry, monotonicity and regularity of positive solutions are proved via the method of moving plane in integral forms and a regularity lifting lemma. For the special case with

positive solutions of the integral system (or the PDE system) are classified. Furthermore, our symmetry results, together with some known results on nonexistence of positive solutions, imply that, under certain integrability conditions, the PDE system has no positive solution in the subcritical case.

Keywords

system of integral equations, regularity, moving plane method in integral form, classification of solutions

Mathematical Subject Classification 2010

Primary: 35J99, 45K05, 35R09, 35J48, 35B65, 45G15

Milestones

Received: 22 July 2011

Revised: 7 September 2012

Accepted: 26 December 2012

Published: 20 March 2013