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A Liouville-type theorem for an integral system on a half-space
Journal of Inequalities and Applications volume 2013, Article number: 37 (2013)
Let be an n-dimensional upper half Euclidean space, and let α be any real number satisfying . In our previous paper (Cao and Dai in J. Math. Anal. Appl. 389:1365-1373, 2012), we considered the single equation
where is the reflection of the point x about the . We obtained the monotonicity and nonexistence of positive solutions to equation (0.1) under some integrability conditions when . In (Zhuo and Li in J. Math. Anal. Appl. 381:392-401, 2011), the authors discussed the following system of integral equations in :
with . They obtained rotational symmetry of positive solutions of (0.2) about some line parallel to -axis under the assumption and . In this paper, we derive nonexistence results of such positive solutions for (0.2). In particular, we present a simple and more general method for the study of symmetry and monotonicity which has been extensively used in various forms on a half-space.
AMS Subject Classification:35B05, 35B45.
By a Liouville-type theorem, we here mean the statement of nonexistence of nontrivial (bounded or not) solutions on the whole space or on a half-space. In the last two decades, Liouville-type theorems have been widely used, in conjunction with rescaling arguments, to derive a priori estimates for solutions of boundary value problems.
Let be the n-dimensional upper half Euclidean space
In our previous paper , we studied the integral equation in :
For (), we obtained the following Liouville-type theorem.
Theorem 1.1 
Suppose . If the solution u of (1.1) satisfies and is nonnegative, then .
The result above motivates us to further study positive solutions of the systems of integral equations in ,
where p and q satisfy
This is the so-called critical case.
In  Zhuo and Li discussed regularity and rotational symmetry of solutions for integral system (1.2).
Theorem 1.2 
Let be a pair of positive solutions of (1.2) with . Assume that and , then every positive solution of (1.2) is rotationally symmetric about some line parallel to -axis.
They also showed close relationships between integral equation (1.2) and the following PDEs system:
where α is an even number.
Theorem 1.3 
Let be a pair of solutions of (1.2) up to a constant, then satisfies (1.3).
In this paper, we use a simple and more general method to derive that the solution pair of (1.2) is strictly monotonically increasing with respect to the variable and further present the nonexistence of positive solutions of (1.2) under some integrability conditions.
Theorem 1.4 Let be a pair of positive solutions of (1.2) with . Assume that and , then both u and v are strictly monotonically increasing with respect to the variable .
Theorem 1.5 Let be a pair of positive solutions of (1.2) with . Assume that and are nonnegative, then .
2 Properties of the function
In this section, we introduce some properties of the function which is defined on a half-space. By using the properties, one could find a simple and general method for the study of symmetry and monotonicity which has been used in various forms defined in a half-space. More precisely, for , define
where is a reflection of the point x about the .
Let λ be a positive real number. Define
the complement of in .
be a reflection of the point about the plane .
To this end, for , define
Then, for , , we have the following expression:
The following lemma states some properties of the function . Here we present a proof.
For any , , we have(2.1)
For any , , it holds(2.3)
Proof Since , it is easy to verify that
Then, for , we have
From (2.4), (2.5), (2.7) and (2.8), we obtain (2.1).
While by (2.6) and (2.9), we have
Here we have used the fact that .
Noticing that for and , we have
Then (2.3) follows immediately from (2.7) and (2.8).
This completes the proof of Lemma 2.1. □
Remark The properties of the function defined on a half-space are very similar to the properties of Green’s function for a poly-harmonic operator on the ball with Dirichlet boundary conditions. One could find this interesting relation from [3, 4] and .
3 The proof of main theorems
In this section, by using the method of moving planes in integral forms, we derive the nonexistence of positive solutions to integral system (1.2) and obtain a new Liouville-type theorem on a half-space. To prove the theorems, we need several lemmas.
Lemma 3.1 Let be any pair of positive solutions of (1.2). For any , we have
Obviously, we have
Now, by properties (2.2) and (2.3) of the function and the pair of positive solutions of (1.2), we have
Similarly, we could derive the second inequality in the lemma. This completes the proof of Lemma 3.1. □
Proof of Theorem 1.4 To prove Theorem 1.4, we compare and on . The proof consists of two steps.
In the first step, we start from the very low end of our region , i.e., . We will show that for λ sufficiently small,
In the second step, we will move our plane toward the positive direction of -axis as long as inequality (3.3) holds.
Step 1. Define
We show that for sufficiently small positive λ, and must both be measure zero. In fact, by Lemma 3.1, it is easy to verify that
where is valued between and . Therefore, on we have
It follows from the Hardy-Littlewood-Sobolev inequality that
Then by the Hölder inequality,
Similarly, one can show that
Combining (3.5) and (3.6), we arrive at
By the conditions that and , we can choose sufficiently small positive λ such that
Now, inequality (3.7) implies , and therefore must be measure zero. Similarly, one can show that is measure zero. Therefore, (3.3) holds. This completes Step 1.
Step 2. (Move the plane to the limiting position to derive symmetry and monotonicity.)
Inequality (3.3) provides a starting point to move the plane . Now, we start from the neighborhood of and move the plane up as long as (3.3) holds to the limiting position. We will show that the solution must be symmetric about the limiting plane and be strictly monotonically increasing with respect to the variable . More precisely, define
Suppose that for such a , we will show that both and must be symmetric about the plane by using a contradiction argument. Assume that on , we have
We show that the plane can be moved further up. More precisely, there exists an depending on n, α, and the solution such that
In the case
by Lemma 3.1, we have in fact in the interior of . Let
Then, obviously, has measure zero and . The same is true for that of v. From (3.5) and (3.6), we deduce
Again, the conditions that and ensure that one can choose ϵ sufficiently small, so that for all λ in ,
Now, by (3.9), we have , therefore must be measure zero. Similarly, must also be measure zero. This verifies (3.8), therefore both and are symmetric about the plane . Also, the monotonicity easily follows from the argument. This completes the proof of Theorem 1.4. □
Proof of Theorem 1.5 To prove the theorem, firstly we will show that the plane cannot stop at for some , that is, we will prove that .
Suppose that , the process of Theorem 1.4 shows that the plane is the symmetric points of the boundary with respect to the plane , and we derive that and when x is on the plane . This contradicts the pair of positive solutions of (1.2), thus .
Besides, we know that both and of positive solutions of (1.2) are strictly monotonically increasing in the positive direction of -axis, but and , so we come to the conclusion that the pair of positive solutions of (1.2) does not exist.
This completes the proof of Theorem 1.5. □
Cao L, Dai Z:A Liouville-type theorem for an integral equation on a half-space . J. Math. Anal. Appl. 2012, 389: 1365–1373. 10.1016/j.jmaa.2012.01.015
Zhuo R, Li DY: A system of integral equations on half space. J. Math. Anal. Appl. 2011, 381: 392–401. 10.1016/j.jmaa.2011.02.060
Chen W, Zhu J: Radial symmetry and regularity of solutions for poly-harmonic Dirichlet problems. J. Math. Anal. Appl. 2011, 377(2):744–753. 10.1016/j.jmaa.2010.11.035
Boggio T: Sulle Fuzioni di Green d’ordine m . Rend. Circ. Mat. Palermo 1905, 20: 97–135. 10.1007/BF03014033
Cao, L, Chen, W: Liouville type theorems for poly-harmonic Navier problems. Discrete Contin. Dyn. Syst. (2013, to appear)
Most of this work was completed when the first author was visiting Yeshiva University, and she would like to thank the Department of Mathematics for the hospitality. Besides, the authors would like to express their gratitude to Professor Wenxiong Chen for his hospitality and many valuable discussions. This work is partially supported by the National Natural Science Foundation of China (No. 11171091; No. 11001076), NSF of Henan Provincial Education Committee (No. 2011A110008) and Foundation for University Key Teacher of Henan Province.
The authors declare that they have no competing interests.
CL participated in the method of moving planes studies. DZ carried out the applications of inequalities and drafted the manuscript. Both authors read and approved the final manuscript.