R E S E A R C H
Open Access
A Liouville-type theorem for an integral
system on a half-space
R
n
+
Linfen Cao
1*and Zhaohui Dai
2*Correspondence:
caolf2010@yahoo.com
1College of Mathematics and
Information Science, Henan Normal University, Xinxiang, Henan 453007, P.R. China
Full list of author information is available at the end of the article
Abstract
LetRn
+be ann-dimensional upper half Euclidean space, and let
α
be any real number satisfying 0 <α
<n. In our previous paper (Cao and Dai in J. Math. Anal. Appl. 389:1365-1373, 2012), we considered the single equationu(x) =
Rn
+
1
|x–y|n–α –
1 |x*–y|n–α
ur(y)dy, (.)
wherex*= (x
1,. . .,xn–1, –xn) is the reflection of the pointxabout the
∂
Rn+. We obtained the monotonicity and nonexistence of positive solutions to equation (0.1) under some integrability conditions whenr>nn–α. In (Zhuo and Li in J. Math. Anal. Appl. 381:392-401, 2011), the authors discussed the following system of integral equations inRn+:
u(x) =Rn
+(
1
|x–y|n–α –|x*–y|1n–α)vq(y)dy, v(x) =Rn
+(
1
|x–y|n–α–|x*–y1|n–α)up(y)dy
(.)
withq+11 +p1+1=n–nα. They obtained rotational symmetry of positive solutions of (0.2) about some line parallel toxn-axis under the assumptionu∈Lp+1(R+n) andv∈Lq+1(Rn+). 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
Keywords: Liouville-type theorem; HLS inequality; systems of integral equations; monotonicity; moving planes method
1 Introduction
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 derivea prioriestimates for solutions of boundary value problems.
LetRn
+be then-dimensional upper half Euclidean space
Rn+=x= (x,x, . . . ,xn)∈Rn|xn>
.
In our previous paper [], we studied the integral equation inRn
+:
u(x) =
Rn+
|x–y|n–α –
|x*–y|n–α
ur(y)dy, u(x) > ,x∈Rn
+. (.)
Forr>n–nα ( <α<n), we obtained the following Liouville-type theorem.
Theorem .[] Suppose r>nn–α.If the solution u of(.)satisfies u∈Ln(rα–)(Rn
+)and is nonnegative,then u≡.
The result above motivates us to further study positive solutions of the systems of inte-gral equations inRn
+, ⎧
⎨ ⎩
u(x) =Rn
+(
|x–y|n–α –|x*–y|n–α)vq(y)dy,
v(x) =Rn
+(
|x–y|n–α–|x*–y|n–α)up(y)dy,
(.)
wherepandqsatisfy
q+ +
p+ =
n–α
n .
This is the so-called critical case.
In [] Zhuo and Li discussed regularity and rotational symmetry of solutions for integral system (.).
Theorem . [] Let(u,v)be a pair of positive solutions of(.)with p,q≥.Assume
that u∈Lp+(Rn
+)and v∈Lq+(R+n),then every positive solution(u,v)of(.)is rotationally symmetric about some line parallel to xn-axis.
They also showed close relationships between integral equation (.) and the following PDEs system:
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
(–)αu=vq, u> , inRn
+; (–)αv=up, v> , inRn
+;
u= (–)u=· · ·= (–)α–u= , on∂Rn
+; v= (–)v=· · ·= (–)α–v= , on∂Rn
+,
(.)
whereαis an even number.
Theorem .[] Let(u,v)be a pair of solutions of(.)up to a constant,then(u,v)satisfies (.).
In this paper, we use a simple and more general method to derive that the solution pair (u,v) of (.) is strictly monotonically increasing with respect to the variablexnand further
Theorem . Let(u,v)be a pair of positive solutions of(.)with p,q≥.Assume that u∈Lp+(Rn
+)and v∈Lq+(Rn+),then both u and v are strictly monotonically increasing with respect to the variable xn.
Theorem . Let(u,v)be a pair of positive solutions of(.)with p,q≥.Assume that u∈Lp+(Rn
+)and v∈Lq+(Rn+)are nonnegative,then u=v≡.
2 Properties of the functionG(x,y)
In this section, we introduce some properties of the functionG(x,y) 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, forx,y∈Rn
+, define
G(x,y) =
|x–y|n–α –
|x*–y|n–α,
wherex*= (x
, . . . ,xn–, –xn) is a reflection of the pointxabout the∂Rn+. Letλbe a positive real number. Define
λ=
x= (x,x, . . . ,xn)∈Rn| <xn<λ
,
Tλ=
x∈Rn+|xn=λ
and
λC=Rn+\λ,
the complement ofλinRn+.
Let
xλ= (x,x, . . . ,xn–, λ–xn)
be a reflection of the pointx= (x,x, . . . ,xn) about the planeTλ.
To this end, forx,y∈Rn
+, define
d(x,y) =|x–y|
and
θ(x,y) = ⎧ ⎨ ⎩
xnyn, ifx,y∈Rn+, , x∈/Rn
+ory∈/Rn+.
Then, forx,y∈Rn+,x=y, we have the following expression:
G(x,y) =Hd(x,y),θ(x,y).
HereH: (,∞)×[,∞)→R,
H(s,t) = sγ –
(s+t)γ, γ =
n–α
The following lemma states some properties of the functionG(x,y). Here we present a proof.
Lemma .
(i) For anyx,y∈λ,x=y,we have
Gxλ,yλ>maxGxλ,y,Gx,yλ (.)
and
Gxλ,yλ–G(x,y) >Gxλ,y–Gx,yλ. (.)
(ii) For anyx∈λ,y∈Cλ,it holds
Gxλ,y>G(x,y). (.)
Proof Sincex,y∈λ, it is easy to verify that
dxλ,yλ=d(x,y) <dxλ,y (.)
and
θxλ,yλ>θxλ,y>θ(x,y). (.)
In fact,
θxλ,yλ–θxλ,y= (λ–x
n)(λ–yn) – (λ–xn)yn
= (λ–xn)(λ–yn) > ,
θxλ,y–θ(x,y) = (λ–xn)yn– xnyn
= (λ–xn)yn> ,
i.e.,
θxλ,yλ≥maxθxλ,y,θx,yλ
≥minθxλ,y,θx,yλ≥θ(x,y). (.)
Consider
G(x,y) =H(s,t) = sγ –
(s+t)γ
with
Then, fors,t> , we have
∂H
∂s = (–γ) sγ+ –
(s+t)γ+
< , (.)
∂H
∂t =
γ
(s+t)γ+ > , (.)
∂H
∂t∂s= –
γ(γ+ )
(s+t)γ+ < . (.)
(i) From (.), (.), (.) and (.), we obtain (.). While by (.) and (.), we have
Gxλ,yλ–G(x,y) =
θ(xλ,yλ)
θ(x,y)
∂H(d(x,y),t)
∂t dt
>
θ(xλ,yλ)
θ(x,y)
∂H(d(xλ,y),t)
∂t dt
≥
θ(xλ,y)
θ(x,yλ)
∂H(d(xλ,y),t)
∂t dt
=Hdxλ,y,θxλ,y–Hdx,yλ,θx,yλ
=Gxλ,y–Gx,yλ.
Here we have used the fact thatd(xλ,y) =d(x,yλ).
(ii) Noticing that forx∈λandy∈Cλ, we have
xλ–y<|x–y| and θxλ,y>θ(x,y).
Then (.) follows immediately from (.) and (.).
This completes the proof of Lemma ..
Remark The properties of the functionG(x,y) 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 [, ] 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 (.) and obtain a new Liouville-type theorem on a half-space. To prove the theorems, we need several lemmas.
Letλ> ,
λ=
x∈Rn| <xn<λ
,
˜
λ={xλ|x∈λ}.
Set
uλ(x) =u
xλ and vλ(x) =v
Lemma . Let(u,v)be any pair of positive solutions of(.).For any x∈λ,we have
u(x) –uλ(x)≤
λ
Gxλ,yλ–Gx,yλvq(y) –vq λ(y)
dy, (.)
v(x) –vλ(x)≤
λ
Gxλ,yλ–Gx,yλup(y) –upλ(y)
dy. (.)
Proof Obviously, we have
u(x) =
λ
G(x,y)vq(y)dy+
λ
Gx,yλvqλ(y)dy
+
Cλ\ ˜λ
G(x,y)vq(y)dy,
uλ(x) =
λ
Gxλ,yvq(y)dy+
λ
Gxλ,yλvq λ(y)dy
+
λC\ ˜λ
Gxλ,yvq(y)dy.
Now, by properties (.) and (.) of the functionG(x,y) and the pair of positive solutions of (.), we have
u(x) –uλ(x)≤
λ
Gxλ,yλ–Gx,yλvq(y) –vqλ(y)
dy
+
Cλ\ ˜λ
G(x,y) –Gxλ,yvq(y)dy
≤
λ
Gxλ,yλ–Gx,yλvq(y) –vqλ(y)
dy.
Similarly, we could derive the second inequality in the lemma. This completes the proof
of Lemma ..
Proof of Theorem. To prove Theorem ., we compare (u(x),v(x)) and (uλ(x),vλ(x)) on λ. The proof consists of two steps.
In the first step, we start from the very low end of our regionRn
+,i.e.,xn= . We will
show that forλsufficiently small,
uλ(x)≥u(x) and vλ(x)≥v(x), ∀x∈λ. (.)
In the second step, we will move our planeTλtoward the positive direction ofxn-axis
as long as inequality (.) holds. Step . Define
λu=x|x∈λ,u(x) >uλ(x)
,
and
λv=x|x∈λ,v(x) >vλ(x)
We show that for sufficiently small positiveλ,uλandvλmust both be measure zero. In fact, by Lemma ., it is easy to verify that
u(x) –uλ(x)≤
λ
Gxλ,yλ–Gx,yλvp(y) –vpλ(y)
dy
=
λ\vλ
Gxλ,yλ–Gx,yλvp(y) –vp λ(y)
dy + vλ
Gxλ,yλ–Gx,yλvp(y) –vpλ(y)
dy ≤ vλ
Gxλ,yλ–Gx,yλvp(y) –vp λ(y)
dy
≤
vλ
Gxλ,yλvp(y) –vpλ(y)
dy ≤p λv
|x–y|n–αψ p–
λ (y)
v(y) –vλ(y)
dy ≤p λv
|x–y|n–αv
p–(y)v(y) –v
λ(y)
dy,
whereψλ(y) is valued betweenv(y) andvλ(y). Therefore, onλvwe have
≤vλ(y)≤ψλ(y)≤v(y).
It follows from the Hardy-Littlewood-Sobolev inequality that
uλ–uLp+(u
λ)≤Cv q–(v
λ–v)L(q+)/q(v
λ). (.)
Then by the Hölder inequality,
uλ–uLp+(u
λ)≤Cv q–
Lq+(λv)vλ–vLq+(λv). (.)
Similarly, one can show that
vλ–vLq+(v
λ)≤Cu p–
Lp+(λu)uλ–uLp+(λu). (.)
Combining (.) and (.), we arrive at
uλ–uLp+(u
λ)≤Cv q–
Lq+(λv)u
p–
Lp+(uλ)uλ–uLp+(u
λ). (.)
By the conditions thatu∈Lp+(Rn+) andv∈Lq+(Rn+), we can choose sufficiently small pos-itiveλsuch that
CvqL–q+(v
λ)u p–
Lp+(u
λ)≤
.
Now, inequality (.) impliesuλ–uLp+(u
λ)= , and therefore u
λmust be measure zero.
Step . (Move the plane to the limiting position to derive symmetry and monotonicity.) Inequality (.) provides a starting point to move the planeTλ. Now, we start from the
neighborhood ofxn= and move the plane up as long as (.) holds to the limiting
posi-tion. We will show that the solutionu(x) must be symmetric about the limiting plane and be strictly monotonically increasing with respect to the variablexn. More precisely, define
λ=sup
λ|u(x)≤uμ(x) andv(x)≤vμ(x),∀x∈μ,μ≤λ
.
Suppose that for such aλ, we will show that bothu(x) andv(x) must be symmetric about the planeTλby using a contradiction argument. Assume that onλ, we have
u(x)≤uλ(x) and v(x)≤vλ(x), but u(x)≡uλ(x) or v(x)≡vλ(x).
We show that the plane can be moved further up. More precisely, there exists an > depending onn,α, and the solution (u(x),v(x)) such that
u(x)≤uλ(x) and v(x)≤vλ(x) onλfor allλin [λ,λ+ ). (.)
In the case
v(x)≡vλ(x) onλ,
by Lemma ., we have in factu(x) <uλ(x) in the interior ofλ. Let
λu =x∈λ|u(x)≥uλ(x)
and vλ=x∈λ|v(x)≥vλ(x
.
Then, obviously,λu
has measure zero andlimλ→λ u
λ⊂λu. The same is true for that
ofv. From (.) and (.), we deduce
uλ–uLp+(u
λ)≤Cv q–
Lq+(v
λ)u p–
Lp+(u
λ)uλ–uL
p+(u
λ). (.)
Again, the conditions thatu∈Lp+(Rn
+) andv∈Lq+(Rn+) ensure that one can choose sufficiently small, so that for allλin [λ,λ+ ),
CvqL–q+(v
λ)u p–
Lp+(u
λ)≤
.
Now, by (.), we haveuλ–uLp+(u
λ)= , therefore u
λmust be measure zero. Similarly, vλmust also be measure zero. This verifies (.), therefore bothu(x) andv(x) are sym-metric about the planeTλ. Also, the monotonicity easily follows from the argument. This
completes the proof of Theorem ..
Proof of Theorem. To prove the theorem, firstly we will show that the plane cannot stop atxn=λfor someλ< +∞, that is, we will prove thatλ= +∞.
Suppose thatλ< +∞, the process of Theorem . shows that the planexn= λis the symmetric points of the boundary∂Rn
u(x) = andv(x) = whenxis on the planexn= λ. This contradicts the pair of positive
solutions (u(x),v(x)) of (.), thusλ= +∞.
Besides, we know that bothu(x) andv(x) of positive solutions of (.) are strictly mono-tonically increasing in the positive direction ofxn-axis, butu∈Lp+(R+n) andv∈Lq+(Rn+), so we come to the conclusion that the pair of positive solutions (u(x),v(x)) of (.) does not exist.
This completes the proof of Theorem ..
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
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.
Author details
1College of Mathematics and Information Science, Henan Normal University, Xinxiang, Henan 453007, P.R. China. 2Department of Computer Science, Henan Normal University, Xinxiang, Henan 453007, P.R. China.
Acknowledgements
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.
Received: 13 October 2012 Accepted: 17 January 2013 Published: 4 February 2013 References
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