Vol. 261, No. 2, 2013

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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
Abstract

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

{
(− Δ )α∕2u = λ1up1 + μ1vp2 + β1up3vp4,
(− Δ )α∕2v = λ2uq1 + μ2vq2 +β2uq3vq4,

where 0 < α < n, λj, μjj (j = 1,2) are nonnegative constants and pi and  qi (i = 1,2,3,4) satisfy some suitable assumptions. It is shown that this PDE system is equivalent to the integral system

(       ∫     p1        p2        p3   p4
||{u(x) =    λ1u--(y)+-μ1v-(y)+-β1u--(y)v--(y)dy,
∫ℝn           |x− y|n−α
||(v(x) =    λ2uq1(y)-+-μ2vq2(y)+-β2uq3(y)vq4(y)dy
ℝn           |x− y|n−α

in n. 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

                                  n +α
p1 = p2 = q1 = q2 = p3 + p4 = q3 + q4 =--,
n − α

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
Authors
Jingbo Dou
School of Statistics
Xi’an University of Finance and Economics
710100 Xi’an
China
Changzheng Qu
Center for Nonlinear Studies
Ningbo University
315211 Ningbo
China