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# Liouville-type theorem for some nonlinear systems in a half-space

- Linfen Cao
^{1}Email author, - Zhaohui Dai
^{2}and - Wenyan Li
^{3}

**2014**:173

https://doi.org/10.1186/1029-242X-2014-173

© Cao et al.; licensee Springer. 2014

**Received: **14 January 2014

**Accepted: **25 April 2014

**Published: **12 May 2014

## Abstract

In this paper we consider the following Hardy-Littlewood-Sobolev (HLS)-type system of nonlinear equations in the half-space ${R}_{+}^{n}$: $u(x)={\int}_{{R}_{+}^{n}}(\frac{1}{{|x-y|}^{n-\alpha}}-\frac{1}{{|{x}^{\ast}-y|}^{n-\alpha}}){v}^{q}(y)\phantom{\rule{0.2em}{0ex}}dy$, $v(x)={\int}_{{R}_{+}^{n}}(\frac{1}{{|x-y|}^{n-\alpha}}-\frac{1}{{|{x}^{\ast}-y|}^{n-\alpha}}){u}^{p}(y)\phantom{\rule{0.2em}{0ex}}dy$, where $p,q>1$ and ${x}^{\ast}$ is the reflection of *x* about the boundary $\{{x}_{n}=0\}$. 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).

**MSC:**35B05, 35B45.

## Keywords

## 1 Introduction

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 ${R}^{n}$ to ones in a half-space.

*i.e.*

In [3], the first and the second authors concluded to the nonexistence of (1.4) in the critical case.

**Theorem 1.1** ([3])

*Let* $(u,v)$ *be a pair of positive solutions of* (1.4) *in the critical case* (1.5). *Assume that* $u\in {L}^{p+1}({R}_{+}^{n})$ *and* $v\in {L}^{q+1}({R}_{+}^{n})$, *then both* *u* *and* *v* *are strictly monotonically increasing with the variable* ${x}_{n}$.

**Theorem 1.2** ([3])

*Let* $(u,v)$ *be a pair of positive solutions of* (1.4) *with the critical case* (1.5). *Assume that* $u\in {L}^{p+1}({R}_{+}^{n})$ *and* $v\in {L}^{q+1}({R}_{+}^{n})$ *are nonnegative*, *then* $u=v\equiv 0$.

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 $(u,v)$ of (1.4) is strictly monotonically increasing with respect to the variable ${x}_{n}$.

**Theorem 1.3**

*Assume that*$1<p,q<\mathrm{\infty}$,

*and that there exist*${p}_{1}\ge 1$

*and*${q}_{1}\ge 1$

*such that*

*Suppose that* $u\in {L}^{{p}_{1}}({R}_{+}^{n})$ *and* $v\in {L}^{{q}_{1}}({R}_{+}^{n})$ *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 *${x}_{n}$.

Theorem 1.3 yields the main result of the paper.

**Theorem 1.4** *Let* $(u,v)$ *be a pair of positive solutions of* (1.4) *with* $1<p,q<\mathrm{\infty}$, *and let there exist* ${p}_{1}\ge 1$ *and* ${q}_{1}\ge 1$ *such that* (1.7), (1.8), *and* (1.9) *hold*. *Assume that* $u\in {L}^{{p}_{1}}({R}_{+}^{n})$ *and* $v\in {L}^{{q}_{1}}({R}_{+}^{n})$ *are nonnegative*, *then* $u=v\equiv 0$.

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 [1].

**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 $({p}_{1},{q}_{1})$ that satisfy all these conditions (1.7), (1.8), and (1.9) in Theorem 1.4.

## 2 Preliminaries

In this section, we introduce some lemmas as preliminaries.

where ${x}^{\ast}=({x}_{1},\dots ,{x}_{n-1},-{x}_{n})$ is the reflection of the point *x* about the $\partial {R}_{+}^{n}$.

*λ*be a positive real number. Define

be the reflection of the point $x=({x}_{1},{x}_{2},\dots ,{x}_{n})$ about the plane ${T}_{\lambda}$.

The following lemma states some properties of the function $G(x,y,\alpha )$.

**Lemma 2.1** (Lemma 2.1 in [4])

- (i)
*For any*$x,y\in {\mathrm{\Sigma}}_{\lambda}$, $x\ne y$,*we have*$G({x}^{\lambda},{y}^{\lambda},\alpha )>max\{G({x}^{\lambda},y,\alpha ),G(x,{y}^{\lambda},\alpha )\}$

*and*

- (ii)
*For any*$x\in {\mathrm{\Sigma}}_{\lambda}$, $y\in {\mathrm{\Sigma}}_{\lambda}^{C}$,*we have*$G({x}^{\lambda},y,\alpha )>G(x,y,\alpha ).$

**Lemma 2.2** (Lemma 3.1 in [3])

*Let*$(u,v)$

*be any pair of positive solutions of*(1.4),

*for any*$x\in {\mathrm{\Sigma}}_{\lambda}$,

*we have*

In addition, we need the equivalent form of the Hardy-Littlewood-Sobolev inequality.

**Lemma 2.3** (Classical HLS inequality)

*Let*$g\in {L}^{\frac{nr}{n+\alpha p}}({R}^{n})$

*for*$\frac{n}{n-\alpha}<r<\mathrm{\infty}$.

*Define*

*Then*

## 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*.

*Step*1. For convenience, we may assume equality in (1.7):

It follows from the intermediate value theorem that there exist ${\overline{p}}_{1}\in ({p}_{1},{p}_{2})$ and ${\overline{q}}_{1}\in ({q}_{1},{q}_{2})$, such that the equality (3.1) holds with ${p}_{1}$ and ${q}_{1}$ replaced by ${\overline{p}}_{1}$ and ${\overline{q}}_{1}$ while (1.9) remains true. Hence, without loss of generality and for simplicity, in the next step, we may assume (3.1).

*Step*2. Under the conditions of the theorem, there exist two non-empty open intervals ${I}_{s}$ and ${I}_{r}$ such that for any $r\in {I}_{r}$ and a corresponding $s\in {I}_{s}$, we have

**Remark** The proof of Part 1 is the same as the proof in [1].

Part 2. *The method of moving planes*.

To prove Theorem 1.3, we compare $(u(x),v(x))$ and $({u}_{\lambda}(x),{v}_{\lambda}(x))$ on ${\mathrm{\Sigma}}_{\lambda}$. The proof consists of two steps.

*i.e.*near ${x}_{n}=0$. We will show that for

*λ*sufficiently small,

In the second step, we will move our plane ${T}_{\lambda}$ toward the positive direction of the ${x}_{n}$-axis as long as the inequality (3.6) holds.

*Step*1. Define

*λ*, ${\mathrm{\Sigma}}_{\lambda}^{u}$, and ${\mathrm{\Sigma}}_{\lambda}^{v}$ must have measure zero. In fact, by Lemma 2.2, it is easy to verify that

*λ*, such that

Now inequality (3.9) implies ${\parallel {u}_{\lambda}-u\parallel}_{{L}^{p+1}({\mathrm{\Sigma}}_{\lambda}^{u})}=0$, and therefore ${\mathrm{\Sigma}}_{\lambda}^{u}$ must have measure zero. Similarly, one can show that ${\mathrm{\Sigma}}_{\lambda}^{v}$ 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.)

*n*,

*α*, and the solution $(u(x),v(x))$ such that

*v*. From (3.7) and (3.8), we deduce

*ϵ*sufficiently small, so that for all

*λ*in $[{\lambda}_{0},{\lambda}_{0}+\u03f5)$,

Now by (3.11), we have ${\parallel {u}_{\lambda}-u\parallel}_{{L}^{{p}_{1}}({\mathrm{\Sigma}}_{\lambda}^{u})}=0$, therefore ${\mathrm{\Sigma}}_{\lambda}^{u}$ must have measure zero. Similarly, ${\mathrm{\Sigma}}_{\lambda}^{v}$ must also have measure zero. This verifies (3.10), therefore both $u(x)$ and $v(x)$ are symmetric about the plane ${T}_{{\lambda}_{0}}$.

Next, we will show that the plane cannot stop at ${x}_{n}={\lambda}_{0}$ for some ${\lambda}_{0}<+\mathrm{\infty}$, that is, we will prove that ${\lambda}_{0}=+\mathrm{\infty}$.

Suppose that ${\lambda}_{0}<+\mathrm{\infty}$, Theorem 1.3 shows that the plane ${x}_{n}=2{\lambda}_{0}$ entails the symmetric points of the boundary $\partial {R}_{+}^{n}$ with respect to the plane ${T}_{{\lambda}_{0}}$, and we derive $u(x)=0$ and $v(x)=0$ when *x* is on the plane ${x}_{n}=2{\lambda}_{0}$. This contradicts the pair of positive solutions $(u(x),v(x))$ of (1.4), thus ${\lambda}_{0}=+\mathrm{\infty}$. 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 $u(x)$ and $v(x)$ of positive solutions of (1.4) are strictly monotonically increasing in the positive direction of ${x}_{n}$-axis, but $u\in {L}^{{p}_{1}}({R}_{+}^{n})$ and $v\in {L}^{{q}_{1}}({R}_{+}^{n})$, so we come to the conclusion that the pair of positive solutions $(u(x),v(x))$ of (1.4) does not exist.

This completes the proof of the Theorem 1.4. □

## 4 Some examples of the pair $({p}_{1},{q}_{1})$ concerning Liouville-type theorems

One would naturally ask the existence of such pairs of $({p}_{1},{q}_{1})$ 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.

**Example 1**In the special case where $p=q$ and $u(x)=v(x)$, system (1.4) becomes the following single integral equation:

The first and the second authors in [4] obtained the following Liouville-type theorem.

**Theorem 4.1** ([4])

*Suppose* $p>\frac{n}{n-\alpha}$. *If the solution* *u* *of* (4.1) *satisfies* $u\in {L}^{\frac{n(p-1)}{\alpha}}({R}_{+}^{n})$ *and is nonnegative*, *then* $u\equiv 0$.

Compared with Theorem 1.4, Theorem 4.1 is the special case where ${p}_{1}={q}_{1}=\frac{n(p-1)}{\alpha}$.

**Example 2** The first and the second authors in [3] 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 ${p}_{1}=p+1$, ${q}_{1}=q+1$.

**Example 3**In [5], the authors discussed the more general integral system

*p*and

*q*are both subcritical, that is,

and they showed the Liouville-type theorem as follows.

**Theorem 4.2** ([5])

*Suppose that* $u(x),v(x)\in {C}^{1}({R}^{n})$ *are positive solutions of* (4.2) *with* (4.3). *If* ${\int}_{{R}_{+}^{n}}\frac{{u}^{p+1}}{{|x|}^{t}}\phantom{\rule{0.2em}{0ex}}dx<\mathrm{\infty}$, ${\int}_{{R}_{+}^{n}}\frac{{v}^{q+1}}{{|x|}^{s}}\phantom{\rule{0.2em}{0ex}}dx<\mathrm{\infty}$ *and* $\alpha -s>1$, $\alpha -t>1$, *then* $u\equiv 0$ *and* $v\equiv 0$.

Now consider the special case $s=t=0$ 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* $u(x),v(x)\in {C}^{1}({R}^{n})$ *are positive solutions of* (1.4) *with* *p* *and* *q* *are both subcritical*, *that is*, $p,q<\frac{n+\alpha}{n-\alpha}$. *Assume that* $u\in {L}^{p+1}({R}_{+}^{n})$ *and* $v\in {L}^{q+1}({R}_{+}^{n})$ *are nonnegative*, *then* $u=v\equiv 0$.

Theorem 4.3 above is just Theorem 1.4 when ${p}_{1}=p+1$, ${q}_{1}=q+1$.

**Remark 3** Both Theorem 1.2 and Theorem 4.3 are special cases when ${p}_{1}=p+1$, ${q}_{1}=q+1$ in Theorem 1.4, the former concerns the critical case and the latter the subcritical case for system (1.4).

## Declarations

### Acknowledgements

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).

## Authors’ Affiliations

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## Copyright

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