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Symmetry and regularity of positive solutions to integral systems with Bessel potential
Journal of Inequalities and Applications volume 2014, Article number: 222 (2014)
In this paper, we are concerned with the symmetry and regularity of positive solutions of the following integral system: , , , where is the α th-order Bessel kernel, , , and . We show that every positive solution triple of the system is radially symmetric and monotonic decreasing about some point by the moving planes method in integral forms. Moreover, by the regularity lifting method, we prove that belongs to and which is then locally Hölder continuous.
In this paper, we are concerned with the symmetry and regularity of positive solutions of the following integral system:
where is the α th-order Bessel kernel, , , , and
Problem (1.1) is related to the α th-order Bessel potentials . The α th-order Bessel kernel is given by
Suppose with , the α th-order Bessel potential of f is defined by
where ∗ denotes the convolution of functions.
The symmetry of the solutions to nonlinear elliptic problems are in general investigated by the moving planes method, which was first used by Alexanderoff for differential equations, and developed by Serrin , Gidas et al. , Caffarelli et al.  etc. In particular, it was considered in [2, 4] the radial symmetry and monotonicity of nonnegative solutions of nonlinear elliptic equations by the moving planes method. Such a method is based on the maximum principle, and hence it cannot be directly applied to problems in the absence of the maximum principle. Especially for nonlinear integral equations or systems, one needs a replacement of the maximum principle. It was observed by Chen et al. in [5, 6] that when studying the radial symmetry and monotonicity of nonnegative solutions of integral equations, one can use a Hardy-Littlewood-Sobolev (HLS) type inequality, instead of the maximum principle, in the moving planes method. They show in [5, 6] that nonnegative solutions of integral equations
are radially symmetric. Since then, radial symmetry and monotonicity of positive solutions for integral equations and integral systems with the Riesz potential have been extensively studied; see [5–7] and  etc.
To study the properties of positive solutions for integral equations with Bessel kernel, Ma and Chen  establish a HLS type inequality for the Bessel potentials and obtain the radial symmetry and monotonicity of positive solutions of integral equations by using the moving planes method in integral forms. Later, they  studied the following integral system with Bessel kernel:
If , this system is the stationary Dirac-Schrödinger system, and if , system (1.4) becomes the stationary Schrödinger system. More equations associated with Bessel potential can be found in [11–14] etc.
Recently, Chen and Yang established in  the HLS type inequality for the Bessel potentials with double weights, and they showed that positive solutions pairs of the integral system
are Hölder continuous, radially symmetric, and strictly decreasing about the origin. Moreover, in , the authors consider the radial symmetry and uniqueness of positive solution pairs of integral system
In this paper, we will investigate the regularity and symmetry as well as Hölder continuity of solutions to problem (1.1), which can be deduced to neither problem (1.5) nor (1.6). Firstly, using the moving planes method in integral forms, we have the following symmetric result.
Theorem 1.1 Under condition (1.2), any positive solution triple to system (1.1) is radially symmetric and monotone decreasing about some point in .
Secondly, by the regularity lifting method, we show the boundedness of each component of the solution triple. To state it precisely, we have the following.
Theorem 1.2 Let triple be a positive solution to integral system (1.1), and p, q, r satisfies (1.2). Then .
In the proof of Theorem 1.2, we first lift the integrability of a suitable cut-off function of the solution by the regularity lifting method to some , and then we show that they are actually in . Finally, we assert that the solution triple is locally Hölder continuous.
Theorem 1.3 u, v, and w are locally Hölder continuous.
In Section 2, we show that is radially symmetric by the moving planes method. Then, using the regularity lifting method, we prove in Section 3. In the last section, we prove Theorem 1.3.
2 Proof of Theorem 1.1
This section is devoted to proving the symmetry and monotonicity of positive solutions to system (1.1). Assume that is a given real number, we define , . For , let , and define
Lemma 2.1 For any positive solution of (1.1), we have
Proof Let . Since , , and is radially symmetric in , it follows from (1.1) that
Substituting x by , we obtain
Similarly, we have (2.2) and (2.3). The proof is complete. □
To prove Theorem 1.1, we need Hardy-Littlewood-Sobolev’s inequality for the Bessel potential, which can be found in .
Lemma 2.2 Let , , . In addition . Then there exists a positive constant C independent of and such that the following inequality holds:
where , , , which means .
Proof of Theorem 1.1 First, we show that there exists a negative number λ, such that
We derive from Lemma 2.1 that
Observing that for , , , and that is decreasing, if , we obtain
Since condition (1.2) holds, we can choose . By Hardy-Littlewood-Sobolev’s inequality and Hölder’s inequality, we have
Similarly, choosing , we get
On the other hand, using the fact that , we can choose λ small enough, such that
That is, , , and are zero-measure sets. Define
Obviously , by the same method in , we can get and the proof of Theorem 1.1 is complete. □
3 Proof of Theorem 1.2
In this section, we show that any solution triple of (1.1) in belongs to . To this purpose, we will use the regularity method developed in , which we will state as follows. Let Z be a given vector space, and be two norms on Z. Define a new norm by
Suppose that Z is complete with respect to the norm . Let X and Y be the completion under and , respectively. Here one can choose s such that . According to what one needs, it is easy to see that . The following regularity lifting theorem was obtained in .
Lemma 3.1 (Regularity lifting)
Let T be a contracting map from X into itself and from Y into itself. Assume that and that there exists a function such that , then f also belongs to Z.
Proof of Theorem 1.2 Let be a triple of solution to integral systems (1.1). We first show by Lemma 3.1 that , a cut-off function of defined below, belongs to for satisfying
then we prove that . For any sufficient large positive real number ξ, define
Similarly, we may define and . Let
Denote , . , , and , can be defined in the same way.
By (1.1), we have
Now, we show that is a contraction map from into for , , satisfying
We may verify that , , according to (1.4) and (3.6). Choosing such that
we may infer by (1.2) that
By Hardy-Littlewood-Sobolev’s inequality and Hölder’s inequality, we find
In the same way, choosing and , we obtain
Since , one can choose ξ sufficiently large so that
In other words, is a contraction map from into itself for
In particular, for , , , we see that is also a contraction map from into itself.
We may assume, without loss of generality, that . Then , and . Similarly, . Choosing large such that , we also have . This is possible because we may require . By Lemma 3.1, we conclude that
Next, we claim that . We estimate only, the estimations for and can be done in the same way. From (3.7), we need to show to get for any .
Condition (1.2) implies . Thus, for any , we have , which means there exists a t such that . By using Hardy-Littlewood-Sobolev’s inequality and Hölder’s inequality, we get
Hence, for all . The fact can be done in the same way. As for , it is trivial.
By the fact that , and , , belong to , which is obvious by their definitions, we obtain .
Now, we claim that for any .
It is easy to see from the definition that . So we need only to show . Since
if , , then and
If , then
Now we estimate and , respectively,
where we use the fact that . Therefore, . Similarly, we have .
As for , we have
we infer .
Thus, we have shown that for any . Similarly, we can deduce that for any . Consequently, for any .
Finally, we show that . Since , , , and , we only need to verify . By (3.2)-(3.5) and , it is sufficient to verify that , where
Choosing large enough such that , that is,
If , , then . Thus,
where we use the fact that .
If , then
Now we estimate , , respectively. By Hölder’s inequality, we have
since . On the other hand, we deduce
by the fact that .
Inequalities (3.13)-(3.16) imply that . Now we estimate . For any ,
By Hölder’s inequality,
Now we estimate . Using the fact that , we can choose a such that . Hence, Hölder’s inequality implies that
Consequently, both N and K belong to , so we have .
By (1.2), (3.6), (3.12), and the assumption that , we see that , satisfies
Inequalities (3.19) and (3.20) allow us to show in the same way that . Consequently, . Theorem 1.2 is proved. □
4 Hölder continuity
We will show in this section that solution triples of (1.1) are Hölder continuous by regularity lifting Theorem II in . We first recall the theorem.
Let V be a Hausdorff topological vector space. Suppose there are two extended norms defined on V, . Let
We also assume that X is complete and that the topology in V is weaker than the topology of X and the weak topology of Y, which means that the convergence in X or weak convergence in Y will imply convergence in V. The pair of spaces described as above is called an XY-pair, if whenever the sequence with in X and will imply .
It is known from Remark 3.3.5 in  that if for , for , and V is the space of distributions, where U can be any subset of or itself, then is an XY-pair.
Lemma 4.1 (Regularity lifting II )
Let X, Y be Banach spaces contained in some larger topological space V satisfying properties described above, and that and be closed subsets of X and Y, respectively. Suppose that is an XY-pair, is a contraction:
and is shrinking:
Moreover, assume that
Then there exists a unique solution u of equation
and, more importantly,
Proof of Theorem 1.3 Since
the solution triples of (1.1) are solutions of the following system:
Hence, it is sufficient to prove that the solutions of (4.1) are Hölder continuous.
For any , denote . Let and , where . By Theorem 1.2, , so we may define
For every , we define
Furthermore, we define
Obviously, a solution of (4.1) is a solution of the equation
where . Write
We will show for small that
is a contracting operator and
is a shrinking operator. Furthermore,
This then will yield by Lemma 4.1.
We first show that is a contracting operator from to . We have . By the mean value theorem, we have
where we use the fact that
Choosing small so that , we see that is a contracting operator from to . Similarly, and are also contracting operators from to . Therefore, is a contracting operator from to .
Next, we verify that is a shrinking operator from to . We only show it for ; it can be done in the same way for and . For any and we have
For , , we have and . So both and are regular in for . In particular, there exists such that . Hence,
If , ; if , . Therefore,
On the other hand, by the mean value theorem,
Choosing ε sufficiently small, we obtain
Similarly, we have
for , that is, T is a shrinking operator from to .
Now, we show that is Hölder continuous. We will show that is finite. Indeed,
For any , we have