Abstract
We consider system of integral equations related to the weighted Hardy-Littlewood-Sobolev (HLS) inequality in a half space. By the Pohozaev type identity in integral form, we present a Liouville type theorem when the system is in both supercritical and subcritical cases under some integrability conditions. Ruling out these nonexistence results, we also discuss the positive solutions of the integral system in critical case. By the method of moving planes, we show that a pair of positive solutions to such system is rotationally symmetric aboutxn-axis, which is much more general than the main result of Zhuo and Li, 2011.
Highlights
In [1], Jin and Li studied the weighted HLS system of nonlinear equations in Rn: u (x) = 1 |x|α ∫ Rn x 1 − yλVq (y) yβ dy, (1) V (x)
By the method of moving planes, we show that a pair of positive solutions to such system is rotationally symmetric about xn-axis, which is much more general than the main result of Zhuo and Li, 2011
Symmetry of solutions to integral system (8) was established by Zhuo and Li [5]
Summary
In [1], Jin and Li studied the weighted HLS system of nonlinear equations in Rn:. 1 − yλ up (y) yα dy, where 0 < λ < n and 1/(p + 1) + 1/(q + 1) = (λ + α + β)/n. Symmetry of solutions to integral system (8) was established by Zhuo and Li [5] They proved that in critical case 1/(p + 1) + 1/(q + 1) = (n − α)/n, any pair of positive solutions of (7) with u ∈ Lp+1(R+n ) and V ∈ Lq+1(R+n ) is rotationally symmetric about some line parallel to xn-axis. By the Pohozaev type identity in integral form, we present a Liouville type theorem when the system (3) is in both supercritical and subcritical cases under some integrability conditions Based on these nonexistence results, we discuss the positive solutions of (3) in critical case.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
