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Liouville-type theorem for some nonlinear systems in a half-space
© Cao et al.; licensee Springer. 2014
Received: 14 January 2014
Accepted: 25 April 2014
Published: 12 May 2014
In this paper we consider the following Hardy-Littlewood-Sobolev (HLS)-type system of nonlinear equations in the half-space : , , where and is the reflection of x about the boundary . By using the method of moving planes in integral forms, we obtain monotonicity of the positive solution of the integral equations system of the abstract in three cases: the so-called subcritical, critical, and supercritical cases, and we obtain a new Liouville-type theorem of this system under some integrability conditions. In particular, our results unify and generalize many cases of Liouville-type theorems in (Cao and Dai in J. Math. Anal. Appl. 389:1365-1373, 2012; Cao and Dai in J. Inequal. Appl. 2013:37, 2013) and (Li et al. in Complex Var. Elliptic Equ. 2013, doi:10.1080/17476933.2013.854346).
By the method of moving planes in integral forms they derived that the positive solutions of (1.1) are radially symmetric and such solutions are nonexistent under some integrability conditions.
where α is any even number between 0 and n. In fact, their equivalence results are more general than above. Such an equivalence provides a technique for studying the PDEs: one can use the corresponding integral equations to investigate the global properties.
In this paper we want to generalize monotonicity and nonexistence results of positive solutions of an HLS-type system in the whole space to ones in a half-space.
In , the first and the second authors concluded to the nonexistence of (1.4) in the critical case.
Theorem 1.1 ()
Let be a pair of positive solutions of (1.4) in the critical case (1.5). Assume that and , then both u and v are strictly monotonically increasing with the variable .
Theorem 1.2 ()
Let be a pair of positive solutions of (1.4) with the critical case (1.5). Assume that and are nonnegative, then .
In this paper, we further consider the nonnegative solution of the integral equations system (1.4) by using the method of moving planes in integral forms. We prove that the positive solution pair of (1.4) is strictly monotonically increasing with respect to the variable .
Suppose that and is a pair of positive solutions of integral system (1.4), then both u and v are strictly monotonically increasing with respect to the variable .
Theorem 1.3 yields the main result of the paper.
Theorem 1.4 Let be a pair of positive solutions of (1.4) with , and let there exist and such that (1.7), (1.8), and (1.9) hold. Assume that and are nonnegative, then .
To prove Theorem 1.4, we will use the method of moving planes in integral forms to obtain the monotonicity of the positive solutions of system (1.4). Corresponding to the half-space problem (1.4), the Liouville-type Theorem 1.4 for the whole space problem (1.1) was established by Chen and Li .
Remark 1 Theorem 1.4 concerning monotonicity and nonexistence of solutions is true in all three cases: subcritical, critical, and supercritical.
Remark 2 Theorem 1.4 unifies and generalizes some Liouville-type results of positive solutions of other integral systems. In particular, we find some examples to show the existence of such pairs of that satisfy all these conditions (1.7), (1.8), and (1.9) in Theorem 1.4.
In this section, we introduce some lemmas as preliminaries.
where is the reflection of the point x about the .
be the reflection of the point about the plane .
The following lemma states some properties of the function .
Lemma 2.1 (Lemma 2.1 in )
- (i)For any , , we have
- (ii)For any , , we have
Lemma 2.2 (Lemma 3.1 in )
In addition, we need the equivalent form of the Hardy-Littlewood-Sobolev inequality.
Lemma 2.3 (Classical HLS inequality)
3 Proof of main theorems
In this section, by the method of moving planes in integral forms we derive the nonexistence of positive solutions to the integral system (1.4) and obtain a new Liouville-type theorem in a half-space. To prove the theorem, we need some proper match of the exponents in the involving integrals, which will be prepared in Part 1. The moving of planes will be carried out in Part 2.
Part 1. The preparations.
It follows from the intermediate value theorem that there exist and , such that the equality (3.1) holds with and replaced by and while (1.9) remains true. Hence, without loss of generality and for simplicity, in the next step, we may assume (3.1).
Remark The proof of Part 1 is the same as the proof in .
Part 2. The method of moving planes.
To prove Theorem 1.3, we compare and on . The proof consists of two steps.
In the second step, we will move our plane toward the positive direction of the -axis as long as the inequality (3.6) holds.
Now inequality (3.9) implies , and therefore must have measure zero. Similarly, one can show that has measure zero. Therefore (3.6) holds. This completes Step 1.
Step 2. (Move the plane to the limiting position to derive symmetry and monotonicity.)
Now by (3.11), we have , therefore must have measure zero. Similarly, must also have measure zero. This verifies (3.10), therefore both and are symmetric about the plane .
Next, we will show that the plane cannot stop at for some , that is, we will prove that .
Suppose that , Theorem 1.3 shows that the plane entails the symmetric points of the boundary with respect to the plane , and we derive and when x is on the plane . This contradicts the pair of positive solutions of (1.4), thus . Also the monotonicity easily follows from the argument. This completes the proof of Theorem 1.3.
Proof of Theorem 1.4 We know that both and of positive solutions of (1.4) 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 (1.4) does not exist.
This completes the proof of the Theorem 1.4. □
4 Some examples of the pair concerning Liouville-type theorems
One would naturally ask the existence of such pairs of that satisfy all these conditions (1.7), (1.8), and (1.9) in Theorem 1.4, here we present some examples to answer the question.
The first and the second authors in  obtained the following Liouville-type theorem.
Theorem 4.1 ()
Suppose . If the solution u of (4.1) satisfies and is nonnegative, then .
Compared with Theorem 1.4, Theorem 4.1 is the special case where .
Example 2 The first and the second authors in  also considered system (1.4) under the critical case (1.5) and obtained Theorem 1.2. We could find that Theorem 1.2 is coincident with the special case , .
and they showed the Liouville-type theorem as follows.
Theorem 4.2 ()
Suppose that are positive solutions of (4.2) with (4.3). If , and , , then and .
Now consider the special case in (4.2), and system (4.2) reduces to the simple system (1.4). For convenience we rewrite Theorem 4.2 as follows.
Theorem 4.3 Suppose that are positive solutions of (1.4) with p and q are both subcritical, that is, . Assume that and are nonnegative, then .
Theorem 4.3 above is just Theorem 1.4 when , .
Remark 3 Both Theorem 1.2 and Theorem 4.3 are special cases when , in Theorem 1.4, the former concerns the critical case and the latter the subcritical case for system (1.4).
The authors would like to express their gratitude to the referees for valuable comments and suggestions. Besides, this work is partially supported by the National Natural Science Foundation of China (No. U1304101; No. 11171091) and NSF of Henan Provincial Education Committee (No. 132300410141).
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