Let and . The well-known classical Hardy–Littlewood–Sobolev inequality  states
for any , , and for .
Hardy and Littlewood also introduced the double weighted inequality, which was later generalized by Stein and Weiss in . It states
where and (see also  for Stein–Weiss inequalities on the Heisenberg group).
One can also write the above Stein–Weiss inequality in another form. Let
where , .
To obtain the best constant in the weighted inequality (1.1), one can maximize the functional
under the constraints . The corresponding Euler–Lagrange equations are the integral system
where and .
Set , then for a proper choice of constants and , system (1.2) becomes
Integral system (1.3) is closely related to the following partial differential equations:
Lei, Li and Ma  established the equivalence between the weighted Hardy–Sobolev type system (1.3) and the corresponding Euler–Lagrange system (1.4). Chen et al.  obtained the symmetry, monotonicity and regularity of solutions of (1.3). In particular, they also obtained the optimal integrability of the solutions for a class of such system. The best constant in a weighted Hardy–Littlewood–Sobolev inequality related to integral system (1.3) was studied in  and solutions of (1.3) for were classified. For more results related to Hardy–Sobolev type equation and system, we refer the reader to [2, 15] and the references therein.
In the special cases when , (1.3) becomes the single equation
The corresponding PDE is the well-known family of semi-linear equations
In particular, when and , equation (1.5) becomes
The classification of the solutions of (1.6) has provided an important ingredient in the study of the well-known Yamable problem and the prescribing scalar curvature. Caffarelli, Gidas and Spruck  classified the solutions of equation (1.6). Wei and Xu  generalized their results to the solutions of the more general equation (1.5) with α being any even number between 0 and n.
Let . We consider the following weighted Hardy–Sobolev type system:
Such and are considerably more complicated when none of , and is zero . Therefore, it is more difficult to deal with these cases.
In this paper, we always assume that () are nonnegative constants and they are not equal to zero simultaneously. We also use the following notations:
where () and .
We first obtain the following regularity of Hardy–Sobolev type system (1.7).
For , , (), if is a pair of solutions of system (1.7), then for any .
We note that and cannot be compared easily, thus one cannot use interpolation to conclude what the spaces or are. Therefore, it is highly nontrivial to derive the integrability of the solutions u and v in Theorem 1.1. One can also see from its proof that the above lower bound for s is the best one can get from the Hardy–Littlewood–Sobolev inequality.
Next, we will establish the of the solutions away from the origin for the solutions u and v. This improves significantly known results in the literature where only the Lipschitz continuity of the solutions of the Hardy–Sobolev system without weight has been established, except in the special case of the single equation and when and the critical exponent (see ).
For , , (), if is a pair of solutions of system (1.7), then and .
It is well known that the moving plane method was invented by Alexandrov in the 1950s. Then the method was further developed by Serrin , Gidas, Ni and Nirenberg , Caffarelli, Gidas and Spruck  and many other authors. In , Chen, Li and Ou developed the method of moving plane in integral forms which is quite different from the traditional moving plane method of differential equations which relies on maximum principles. The moving plane in integral forms can be easily applied to higher order equations without using maximum principles. For more results related to the moving plane in integral forms and related symmetry results and Liouville-type theorems, see, e.g., [4, 7, 12, 13, 14, 16, 17] and the references therein.
To establish the symmetry of the solutions, we use the moving plane in integral forms and the Hardy–Littlewood–Sobolev inequality to prove the following theorem.
For (), assume that is a pair of nonnegative solutions of system (1.7). Then u and v are radially symmetric and strictly decreasing about the origin.
This paper is organized as follows. In Section 2 and Section 3, we prove Theorem 1.1 and Theorem 1.2, respectively. In Section 4, we prove that each pair of solutions of (1.7) is radially symmetric and strictly decreasing about the origin, thus give the proof of Theorem 1.3.
2 Proof of Theorem 1.1
Let V be a topological vector space. Suppose there are two extended norms (i.e., the norm of an element in V might be infinity) defined on V,
Let T be a contraction map from X into itself and from Y into itself. Assume that for any there exists a function such that . Then .
Now, we start our proof of Theorem 1.1. Denote
Assume that for . Define
Denote the norm in the cross product space by
and define the mapping T: by
Consider the equation
It is easy to check that is a pair of solutions of (2.1).
In order to show for all , we carry out the proof by two steps:
T is a contracting map from to itself for sufficiently large a.
F and G belong to .
Then from Lemma 2.1, we conclude the proof.
Step 1. For any with , by the weighted Hardy–Littlewood–Sobolev inequality and the Minkowski inequality, we have
where , and , .
Since , we may choose sufficiently large a such that
By (2.5), we have
which implies that T is a contraction map from to itself.
Step 2. Since the estimates of F and G are similar, we only consider F:
For any , we apply the weighted Hardy–Littlewood–Sobolev inequality and the Minkowski inequality to obtain
Since and for , the constants can be chosen arbitrarily. In view of , we may choose such that
We carry out the process by two cases.
Case 1. If , for any , there exist such that (2.6) is valid. Then we obtain
Similarly, we have
By Lemma 2.1, we conclude that for any .
Case 2. If , for any , we have
By Lemma 2.1 again, we derive that for any . Repeating the above process with replaced by , we can obtain that for any . After a few steps, there exists such that . This returns to Case 1.
Therefore, combining Case 1 and Case 2, we conclude that for any . This completes the proof of Theorem 1.1.
3 Proof of Theorem 1.2
Part I. We show that . Since the estimates of and are similar, we only discuss . The function can be written as
We claim that . Write
We first estimate . If , by the Hölder inequality and for any , we have
If , we also have
Next, it suffices to estimate . For , we have
In virtue of the Hölder inequality, it is easy to see that
We also have
Thus we accomplish the estimate of Part I.
Part II. We show that . One only need to prove that . For any , we choose a ball with radius such that , then
We only show that , . We split into two parts, i.e. can be written as
We show that
For fixed , if h is small enough, we have
where is the i-th unit vector, and ϵ is sufficiently small such that . For such fixed r and ϵ, it is easy to verify
since for any . Consider that
and for any . By the Hölder inequality, we have
Thus, we conclude that
By the standard singular integral estimate (see ), for any , we have
For the estimate of , similarly, we also can split into two parts, i.e. can be written as
We show that
For fixed , if h is small enough, we use the same estimate as (3.10) to obtain
For such fixed r and ϵ, it is easy to verify that
and for any . We have
Similarly, we can also obtain
By Lebesgue’s dominated convergence theorem, we conclude that
Thus we derive that . By the bootstrap technique, we can prove that . Then the proof of Part II is accomplished.
4 Proof of Theorem 1.3
In this section, we will use the method of moving plane in integral forms introduced in  to prove that each pair of solutions of (1.7) is radially symmetric and strictly decreasing about the origin. In order to prove our theorem, we first introduce some notations. For any real number λ, let the moving plane be and denote . Let be the reflection of the point x about the moving plane , and define and .
If is a pair of nonnegative solutions of (1.7), for any and , we have
By (1.7), we have
Since and , we have
Similarly, we can also obtain the second formula. This completes the proof of Lemma 4.1. ∎
Now we start to prove Theorem 1.3, the proof will be carried out by two steps.
Step 1. We compare the values of and , and . For λ sufficiently negative, we are going to show that
We will prove that for sufficiently negative λ, both and must be empty, and hence (4.1) holds.
According to the mean value theorem and Lemma 4.1, for any , we have
For any , we apply the weighted Hardy–Littlewood–Sobolev inequality and the Hölder inequality to the above inequality and obtain
where , and .
Similarly, we can also deduce that
where , and .
In virtue of the condition , we can choose sufficiently negative λ such that
which implies that and must be of measure zero. Therefore, both and must be empty sets.
We will prove
Suppose on the contrary that . We will then show that u and v must be symmetric about the plane , that is,
Otherwise, we may assume on ,
Next, we will show that the plane can be moved further to the right. More precisely, there exists an such that for any ,
Obviously, both and have measure zero, and
This together with the integrability condition ensures that one can choose ε small enough such that, for all ,
thus and must be of measure zero, which verifies (4.9). Finally, we show that the plane cannot stop before hitting the origin. On the contrary, if the plane stops at , then and must be symmetric about the plane , that is
In virtue of , , and , we have
which contradicts (4.10). Since the direction can be chosen arbitrary, we conclude that both and are radially symmetric and strictly decreasing about the origin. This completes the proof of Theorem 1.3.
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About the article
Published Online: 2015-12-08
Published in Print: 2016-02-01
Funding Source: National Natural Science Foundation of China
Award identifier / Grant number: 11371056
Funding Source: National Science Foundation
Award identifier / Grant number: DMS-1301595
Funding Source: Simons Foundation
Award identifier / Grant number: Simons Fellowship
The research of the first and second authors was partly supported by the NNSF of China (grant no. 11371056), the research of the third author was partly supported by US NSF grant DMS-1301595 and a Simons Fellowship from the Simons Foundation.