- Open Access
A Liouville-type theorem for an integral system on a half-space
© Cao and Dai; licensee Springer 2013
- Received: 13 October 2012
- Accepted: 17 January 2013
- Published: 4 February 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.
- Liouville-type theorem
- HLS inequality
- systems of integral equations
- moving planes method
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.
For (), we obtained the following Liouville-type theorem.
Theorem 1.1 
Suppose . If the solution u of (1.1) satisfies and is nonnegative, then .
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.
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 .
where is a reflection of the point x about the .
the complement of in .
be a reflection of the point about the plane .
The following lemma states some properties of the function . Here we present a proof.
- (i)For any , , we have(2.1)
- (ii)For any , , it holds(2.3)
- (ii)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 .
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.
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 second step, we will move our plane toward the positive direction of -axis as long as inequality (3.3) holds.
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.)
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. □
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.
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