Symmetry and Nonexistence of Positive Solutions for Weighted HLS System of Integral Equations on a Half Space

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 about 𝑥 𝑛 -axis, which is much more general than the main result of Zhuo and Li, 2011.

Jin and Li [2] and Chen et al. [3] also discussed the regularity of solutions to (1).
Let   + be the upper half Euclidean space In this paper, we want to consider the similar integral system in the half space   + as (1).More precisely, we discuss the following weighted HLS type system of nonlinear equations in   + : where , V ≥ 0, 0 < ,  < ∞, 0 <  < ,  +  ≥ 0, / < 1/( + 1) < ( −  + )/, and here  * is the reflection point of  about the plane   + .Similar to some integral systems or PDEs systems, the integral system (3) is usually divided into three cases according to the value of exponents (, ).We say that system (3) is in critical case when the pair (, ) satisfies the relation In the special case, where  = 0 and  = 0, system (3) reduces to and system (7) is closely related to the following system of PDEs with Navier boundary conditions: In particular, when  is an even number, the authors ( [4]) proved the equivalence between the two systems (7) and (8) under some mild growth condition.Symmetry of solutions to integral system (8) was established by Zhuo and Li [5].They proved that in critical case 1/( + 1) + 1/( + 1) = ( − )/, any pair of positive solutions of (7) with  ∈  +1 (  + ) and V ∈  +1 (  + ) is rotationally symmetric about some line parallel to   -axis.Under the same integrability conditions, in [6], we obtained the nonexistence of positive solutions of (7).
The general case is that, for  ̸ = 0 and  ̸ = 0 in (3), there are few results concerning symmetry and nonexistence for this doubled weighted system.In this paper, 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.By the method of moving planes, we show that a pair of positive solutions to such system is rotationally symmetric about   -axis.To carry on the moving of planes, we explore global features of the integral equations and estimate certain integral norms.This is the essence of the method of moving planes in integral forms.The readers who are interested in the integral system and the applications of this method may consult [7][8][9][10] and the references therein.
The paper is organized as follows.
(i) If  and  are both supercritical, that is, or (ii) if  and  are both subcritical, that is, then  ≡ 0 and V ≡ 0.
Based on these results and ruling out cases where there are no solutions, we are only interested in critical case (5).In Section 3, by means of method of moving planes in integral form, we establish rotational symmetry of solutions of (3) in critical case (5) as follows.

Proof of Theorem 2
In this section we will prove the nonexistence of positive solutions to the weighted HLS type system (3).These nonexistence results, known as Liouville type theorems, are useful in deriving existence, a priori estimate, regularity, and asymptotic analysis of solutions.
A celebrated result of S. I. Pohozaev is known as the Pohozaev identity.This classical result has many consequences, the most immediate one being the nonexistence of nontrivial bounded solutions to PDE.Here we apply the Pohozaev type identity in integral forms to the integral system (3) (see in [9,11]).
For any  ̸ = 0, there holds By an elementary calculation, Noting  ∈  1 (  ), differentiating both sides of ( 11) with respect to  and letting  = 1, we have Let  +  (0) =   (0) ∩   + be the upper half ball in the half space in   + .Multiplying left side of (13) by   () and integrating on  +  yields Similarly, we also have Since Thus, there exists a sequence {  } such that Let   → ∞; by ( 14), (15), and (17), we have On the other hand, By ( 18) and ( 22), we have hold, it follows that  ≡ 0 and V ≡ 0. This completes the proof of Theorem 2.
Remark 5.In [11], the authors consider another weighted HLS type integral system and showed the Liouville type theorem as follows.
When  =  = 0 in system (26) or  =  = 0 in system (3), the two systems reduce to the simple integral system (7).In this special case, we can find that Theorem 6 is coincident with case (ii) in Theorem 2.

Proof of Theorem 3
In this section, we will consider rotational symmetry of weighted HLS type system (3) in critical case (5).
Firstly, we need the following weighted HLS inequality.
In fact, by Lemma 9 and the mean value theorem, we have, for  ∈ Σ   , where   () is valued between V() and V  (); therefore, on Σ V  , we have The conditions  ∈  +1 (  + ) and V ∈  +1 (  + ) make us able to choose sufficiently negative , so that (44)