Some properties of solutions for a nonlinear integral system

R E S E A R C H

Open Access

Some properties of solutions for

a nonlinear integral system

Xiaoying Wang

_{, Junjie Li}

_{and Jiankai Xu}

*_{Correspondence:}

[email protected]

3_{College of Sciences, Hunan}

Agriculture University, Hunan, 410128, China

Full list of author information is available at the end of the article

Abstract

In this paper, a nonlinear integral system is considered in critical space. Some important properties of positive solutions such as symmetry, monotonicity, integrability, and asymptotic behaviors, are obtained. Moreover, by comparison and analysis, we discover that those properties are an important tool to characterize the tightness of the system.

Keywords: integral system; regularity lifting lemma; weighted Hardy-Littlewood-Sobolev inequality

1 Introduction

In this paper, we consider the following nonlinear integral system involving the weighted Riesz potentials:

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

u(x) =_Rnv

p_(y)uq_(y)wr_(y)

|x–y|λ _|y_|βdy,

v(x) =_Rn

vr_(y)up_(y)wq_(y)

|x–y|λ _|y_|βdy,

w(x) =_Rn v

q_(y)ur_(y)wp_(y)

|x–y|λ _|y_|βdy,

(.)

where  <λ<n,  <β+λ<n, andp,q,r≥ satisfyingp+q+r= (n–λ– β)/λ. Recently, there has been tremendous interest in studying integral systems. Because the integral equation(s) not only formulate abstractly many laws and relations in science, en-gineering, economics, and other fields of applied science, but also they provide a special technique to investigate the global properties of the corresponding differential equation(s) due to the fact that the integral equation(s), under certain integrability conditions, is (are) equivalent to differential equation(s). We recall some related background and investiga-tions as follows.

Asu(x) =v(x) =w(x) and  <β<n–λ,p+q+r= (n–β)/λ– , the system (.) can be reduced to the following single equation:

u(x) =

Rn

up+q+r_(y)

|x–y|λ|_y|βdy. (.)

Lu and Zhu in [] obtained some regularity results and showed that every positive solu-tionu(x) is radially symmetric about the origin and strictly decreasing inLn/λ_(Rn_).

Sub-sequently, under the same conditions, Lei in [] showed that for  <p+q+r≤(n–β)/λ,

equation (.) has no positive solution and that forp+q+r> (n–β)/λ, all of positive solution of (.),u∈Ln/λ_(Rn_{) is bounded and decays fast with rate|x|}–λ_.

Whenβ= ,λ=n–  andw(x) =v(x), the system (.) can be rewritten as

⎧ ⎨ ⎩

u(x) =_Rnv q(y)up(y)

|x–y|λ dy,

v(x) =_Rnu p_(y)vp_(y)

|x–y|λ dy.

(.)

The equations (.), under some integrability condition, is equivalent to the following dif-ferential system:

⎧ ⎨ ⎩

–u(x) =vq_(x)up_{(x) in}Rn_,

–v(x) =up_(x)vq_{(x) in}Rn_,

(.)

which are closely related to stationary Schrödinger system with critical exponents for the Bose-Einstein condensate. Li and Ma in [] showed that for n≥, ≤p, q ≤

(n+ )/(n– ), and p +q= (n+ )/(n– ), any positive solution pair (u,v) of system

(.) inLn/(n–)_(Rn_)×Ln/(n–)_(Rn_{) is radially symmetric and unique.}

Later on, Zhao and Lei in [] considered the following weighted nonlinear system:

⎧ ⎨ ⎩

u(x) =_Rn

vq(y)up(y)

|x–y|λ_|y|β dy,

v(x) =_Rnu p_(y)vp_(y)

|x–y|λ_|y|β dy.

(.)

The authors in [] showed that asλ+β<n, ≤β<λandp+q= (n–λ–β)/(λ+β), any

positive solution pair (u,v) of equation (.) is inLn/(λ+β)_(Rn_)×Ln/(λ+β)_(Rn_{). Additionally,}

if (n–λ)(λ+β)≥nβ, then the positive solution pair (u,v) of system (.) is bounded and satisfies

lim

|x|→∞|x| λ

u(x) =

Rn

uq_(y)vp_(y)

|y|β dy, _|xlim_|→∞|x| λ

v(x) =

Rn

up_(y)vq_(y)

|y|β dy. (.)

An analogous integral system of (.) is

⎧ ⎨ ⎩

u(x) =_|x_|α

Rn v q_(y)

|x–y|λ_|y|βdy,

v(x) = 

|x|β

Rn

up_(y)

|x–y|λ_|y|αdy.

(.)

It is closely related to the best constant in the weighted Hardy-Littlewood-Sobolev in-equality.

Assume that  <r,s<∞,  <λ<n,α+β≥, /r+ /s+ (λ+α+β)/n= , and  – /r–

λ/n≤α/n<  – /r, the well-known weighted Hardy-Littlewood-Sobolev inequality states that

_Rn

Rn|x|

–α

f(x)g(y)|x–y|–λ|y|–βdx dy≤C(λ,α,β,n)fLr_(Rn₎g_Ls_(Rn₎

To find the best constantC(λ,α,β,n) in the above inequality, one can maximize the functional

J(f,g) =

Rn

Rn|x|

–α

f(x)g(y)|x–y|–λ|y|–βdx dy,

under the constraints

fLr_(Rn₎=g_Ls_(Rn₎= .

The corresponding Euler-Lagrange equations are the following integral system:

⎧ ⎨ ⎩

λrfr–(x) =_|x_|α

Rn_|x–yg(y)_|λ_|y|βdy, λsgs–(x) =_|x_|β

Rn_|x–yf(y)_|λ_|y|αdy.

(.)

Here f,g ≥, and λr=λs=J(f,g). After a nonlinear transformation,i.e. u=cfr–,

v=cgs–, the integral system (.) becomes (.). By the method of moving plane in

inte-gral form, Jin and Li in [] showed that all of the positive solutions of (.) are radially sym-metric. Subsequently, they used the regularity lifting lemma and showed, whenα,β> , that  <λ<nandp,q>  satisfy

 p+ +

 q+ =

¯

λ

n,

 p+ –

λ

n<

α

n <  p+ .

Then

u∈Lr Rn, v∈Ls Rn,

provided that, for (q+ )(λ+ β)≥n,

 r∈

max{α,βq+λ¯–n}

n ,

λ+α

n

, 

s ∈

β

n,

min{λ+β,p(λ+α) +λ¯–n} n

, (.)

and for (q+ )(λ+ β) < n

 r∈

α

n,

min{λ+α,q(λ+β) +λ¯–n} n

, 

s ∈

max{β,pλ+λ¯–n}

n ,

β+λ

n

. (.)

Furthermore, with the help of the integrability and symmetry of positive solution for (.), Li, Lim, Lei and Ma in [–] obtained the sharp asymptotic estimates as follows.

Whenα+β≥, the pair of solutions (u,v) of (.) have the following asymptotic be-haviors at the origin:

u(x) A

and

v(x)

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

A

|x|β, ifλ+α(p+ ) <n, A|ln|x||

|x|β , ifλ+α(p+ ) =n, A

|x|α(p+)+β+λ–n, ifλ+α(p+ ) >n.

At the same time, (u,v) at infinity admits the following asymptotic behaviors:

u(x) B

|x|λ+α, ifλq+β(q+ ) >n, (.)

and

v(x)

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

B

|x|β+λ, ifλp+α(p+ ) >n, B|ln|x||

|x|β+λ , ifp+α(p+ ) =n, B

|x|(α+λ)(p+)+β–n, ifp+α(p+ ) <n.

(.)

Here, we use the notationw(x)C/|x|tto denote thatlimx→ or∞|x|tw(x) =Cfor a

func-tionw(x), a real numbert, and a non-zero real numberC.

In this paper, we will study some important properties of (.), such as symmetry, mono-tonicity, integrability, and asymptotic behaviors. At the same time, by comparison and analysis, we observe that the related properties are important tools to characterize the tightness of the system. Specifically, our main results can be formulated as follows.

Theorem . Let(u,v,w)be a pair of positive solutions of(.)and p+q+r= (n–λ– β)/λ.Assume that(u(x),v(x),w(x))∈Ls_(Rn_)×Ls_(Rn_)×Ls_(Rn_{)for s=n(p+q+r– )/(n–} β–λ).Then:

R. (u,v,w)is radially symmetric and monotone decreasing about the origin.

R. (u,v,w)admits the same integral interval:

u(x),v(x),w(x)∈Lτ Rn, ∀τ∈

n

λ,∞

. (.)

R. We have the optimal decay estimates

lim

|x|→∞

|x|λu(x)=

Rn

vp_(y)uq_(y)wr_(y)

|y|β dy; (.)

lim

|x|→∞

|x|λv(x)=

Rn

vr_(y)up_(y)wq_(y)

|y|β dy; (.)

lim

|x|→∞

|x|λ_w(x)=

Rn

vq_(y)ur_(y)wp_(y)

|y|β dy. (.)

a new way to obtain the asymptotic result. Precisely, the difference between systems (.) and (.) leads to the thoroughly different asymptotic behavior at infinity. Indeed, by (.) and (.), we have learned that the asymptotic behavior ofuandvin (.) is completely different, however, for the system (.), we know from (.), (.), and (.) that the so-lution (u,v,w) has the same decay estimate at infinity. This implies that the triplet (u,v,w) in (.) is tighter than those of system (.).

The other significant difference between (.) and (.) is the corresponding optimal in-tegrable intervals of solutions. From (.), we know that the triplet (u,v,w) in (.) admits the same optimal integrable intervals. However, for the pair of solutions (u,v) in system (.), the optimal integrable interval ofuis different from the one ofv.

Remark . We remark that the integral system (.) considered in [] can be regarded as the special cases of (.) withv=wandr=p. At the same time, the system (.) including its reduced model (.) studied in [], have similar radial symmetry, monotonicity, and asymptotic behavior to our system (.). However, the solution spaceLs(Rn)×Ls(Rn)× Ls_(Rn_{) withs=n(p+q+r– )/(n–λ) = n/λin our theorems is different fromL}s_(Rn_)×

Ls_(Rn_{) withs= n/(λ+β) in [] since the indexp+q+r= (n–λ– β)/λin our paper is}

different from onep+q= (n–λ–β)/(λ+β) in []. Moreover, the method for the upper bound estimate of integrable interval to system (.) in our paper is completely different from the one in []. By contrast, our technique is more simple than the one used in [].

The rest of this paper is organized as follows. In Section , we will consider Rand the

proofs of Rwill be given in Section . Finally, we will build up the sharp asymptotic

esti-mates of (.) in Section .

Throughout this paper, we always use the letterCto denote positive constants that may vary at each occurrence but are independent of the essential variables.

2 Radial symmetry

To obtain our results, in this section, we will use the method of moving plane in the inte-gral forms recently introduced by Chenet al.in [] and prove the radial symmetry and monotonicity of positive solutions of system (.). First of all, we introduce some necessary lemma.

For a given real numberμ∈R, define

μ

x= (x, . . . ,xn)∈Rn:x≥μ

,

and letxμ_(μ–x

,x, . . . ,xn),uμ(x)u(xμ),vμ(x)v(xμ), andwμ(x)w(xμ).

Lemma . Let(u(x),v(x),w(x))be a solution of system(.),then

uμ(x) –u(x)

=

μ



|x–y|λ– 

|xμ_–y|λ



|yμ|β v

p

μuqμwrμ(y) –vpuqwr

+



|xμ_–y|λ– 

|x–y|λ



|y|β– 

|yμ|β

vpuqwr

dy

vμ(x) –v(x) = μ 

|x–y|λ– 

|xμ_–y|λ



|yμ|β v

r

μu

p

μw

q

μ(y) –v

r_up_wq

+



|xμ_–y|λ– 

|x–y|λ



|y|β– 

|yμ_|β

vrupwq

dy

B(x) +B(x); (.)

and

wμ(x) –w(x)

=

μ



|x–y|λ– 

|xμ_–y|λ



|yμ|β v

q

μu

r

μw

p

μ(y) –v

q_ur_wp

+



|xμ_–y|λ– 

|x–y|λ



|y|β– 

|yμ_|β

vqurwp(y)

dy

C(x) +C(x). (.)

Proof By a direct calculation, it is easy to check that

u(x) =

μ

vp_uq_wr

|x–y|λ|_y|βdy+

μ



|xμ_–y|λ vpμuqμwr

μ(y)

|yμ|β dy and

uμ(x) =

μ

vp_uq_wr

|xμ_–y|λ|_y|βdy+

μ



|x–y|λ vpμuqμwr

μ(y)

|yμ|β dy,

(.) is a direct result of the above equations. Similarly, we get (.) and (.). This

com-pletes the proof of Lemma ..

Proof of R Now, we turn to the first part of Theorem .. The proof is made up of two

steps. In Step , we compare the values ofu(x) withuμ(x),v(x) withvμ(x) andw(x) with wμ(x) onμ, respectively, and show that for sufficiently negativeμ< , we have

u(x)≥uμ(x), v(x)≥vμ(x) and w(x)≥wμ(x) ∀x∈μ–{}. (.) In Step , we continuously move the planex=μalong thexdirection from near negative

infinity to the right as long as (.) holds. By moving this plane in this way, we finally show that the plane will stop at the origin. Next we turn our attention to Step .

Step: SinceA(x) <  and|yμ|>|y|for anyy∈μ, we have uμ(x) –u(x)

≤

μ



|x–y|λ– 

|xμ_–y|λ



|yμ|β v

p

μuqμwrμ(y) –vpuqwr

dy ≤ vμ 

|x–y|λ– 

|xμ_–y|λ



|yμ|β v

p

μ–vp

uqμwrμ(y)

dy + μu 

|x–y|λ– 

|xμ_–y|λ



|yμ|β v

p_uq

μ–uq

wrμ(y)

+

μw



|x–y|λ– 

|xμ_–y|λ



|yμ|β v

p_uq_wr

μ(y) –wr

dy

A,(x) +A,(x) +A,(x). (.)

Similarly, we conclude that

vμ(x) –v(x)

≤

μ



|x–y|λ– 

|xμ_–y|λ



|yμ_|β v

r

μupμwqμ(y) –vrupwq

dy ≤ vμ 

|x–y|λ– 

|xμ_–y|λ



|yμ|β v

r

μ–v

r_up

μw

q

μ(y)

dy + μu 

|x–y|λ– 

|xμ_–y|λ



|yμ_|β v

r_up

μ–up

wqμ(y)

dy + μw 

|x–y|λ– 

|xμ_–y|λ



|yμ|β v

r_up_wq

μ(y) –w

q_dy

B,(x) +B,(x) +B,(x) (.)

and

wμ(x) –w(x)

≤

μ



|x–y|λ– 

|xμ_–y|λ



|yμ|β v

q

μu

r

μw

p

μ(y) –v

q_ur_wp_dy

≤

vμ



|x–y|λ– 

|xμ_–y|λ



|yμ|β v

q

μ–v

q_ur

μw

p

μ(y)

dy + μu 

|x–y|λ– 

|xμ_–y|λ



|yμ_|β v

q_ur

μ–ur

wpμ(y)

dy + μw 

|x–y|λ– 

|xμ_–y|λ



|yμ|β v

q_ur_wp

μ(y) –w

p_dy

C,(x) +C,(x) +C,(x), (.)

whereμu={x∈μ|u(x) <uμ(x)},μv={x∈μ|v(x) <vμ(x)}andμw={x∈μ|w(x) < wμ(x)}.

Sincep+q+r= (n–λ– β)/λand  <β+λ<n,

sn(p+q+r– )

n–λ–β =

n

λ >

n

λ.

By the weighted Hardy-Littlewood-Sobolev inequality and the Hölder inequality, we con-clude that

A,(x)_Ls₍u_μ)≤C(n,β,λ)vpμ–vp

uqμwrμ

L

ns n+(n–λ–β)s_(μ)v

≤C(n,β,λ,p,q)uqμvp–μ wrμ

L

n

n–λ–β_(μ)vμ–vLs(μ)v

≤C(n,β,λ,p,q)vμp–_Ls₍v_μ)wμr_Ls_(μ)uμq_Ls_(μ)vμ–vLs₍v

Similarly, we have

A,(x)_Ls₍v

μ)≤C(n,p,q,β,λ)v p

Ls_(μ)uq–_Ls_(μ)uwμr_Ls_(μ)uμ–uLs₍u

μ) (.)

and

A,(x)_Ls₍v_μ)≤C(n,p,q,β,λ)vp_Ls_(μ)uq_Ls_(μ)wμr–_Ls_(μ)wμ–wLs_(μ)w. (.)

This together with (.), implies that

uμ–uLs_(μ)u ≤A_,(x)

Ls_(μ)u +A,(x)_Ls₍u_μ)+A,(x)_Ls_(μ)u

≤Cvμp–_Ls₍v

μ)wμ r

Ls_(μ)uμq_Ls_(μ)vμ–vLs_(μ)v +vp_Ls_(μ)uq–_Ls_(μ)u wμr_Ls_(μ)uμ–uLs₍u

μ)

+vp_Ls_(μ)uq_Ls_(μ)wμr–_Ls_(μ)wμ–wLs₍w

μ)

. (.)

Sinceu,v,w∈Ls(Rn), we can chooseN>  large enough, such that for anyμ≤–N< , C(n,β,λ,p,q)vμ_Lp–s₍v_μ)wμr_Ls_(μ)uμq_Ls_(μ)+vp_Ls_(μ)uq–_Ls_(μ)uwμr_Ls_(μ)

+vp_Ls_(μ)u_Lqs_(μ)wμr–_Ls_(μ)

≤ 

,

which, combined with (.), implies that

uμ–uLs_(μ)u

≤ 

uμ–uLs(μu)+



vμ–vLs(vμ)+



wμ–wLs(wμ). (.)

With the same method, we have

vμ–vLs_(μ)v

≤ 

uμ–uLs(μu)+



vμ–vLs(vμ)+



wμ–wLs(wμ) (.)

and

wμ–wLs₍w_μ)

≤ 

uμ–uLs(μ)u + 

vμ–vLs(vμ)+ 

wμ–wLs(wμ). (.)

This together with (.), (.), and (.) leads to

uμ–uLs₍u

μ)+vμ–vLs(μv)+wμ–wLs(wμ)= .

Step: We will continuously move the planex=μto the right as long as (.) holds.

Indeed, suppose that atx=μ< , we have, for anyx∈μ u(x)≥uμ(x), v(x)≥vμ(x), and w(x)≥wμ(x), but

u(x) ≡uμ(x) or v(x) ≡vμ(x) or w(x) ≡wμ(x).

Next, we will show that the plane can be moved further to the right. Precisely, there exists andepending onn,α,λ,βand the solution (u(x),v(x),w(x)) itself such that for∀x∈μ,

μ∈[μ,μ+),

u(x)≥uμ(x), v(x)≥vμ(x), and w(x)≥wμ(x). (.)

Under the assumption thatv(x) ≡vμ(x) orw(x) ≡w_μ(x) on_μ, by (.), (.), (.), and

the non-negativity ofu,v,w, we haveu(x) >uμ(x) in the interior of_μ.

Let

_μu=

x∈μ|u(x)≤uμ(x)

,_μv=

x∈μ|v(x)≤vμ(x)

and_μw=

x∈μ|w(x)≤w_μ(x)

.

From the analysis mentioned above, it is easy to check that the_μu is a zero measure

set inRn. Similarly, we also havem{v_μ}=  andm{_μw}= . This together with the

integrability conditions (u,v,w)∈Ls_(Rn_{) ensures that one can choosesmall enough such}

that for allμ∈[μ_,μ₊₎

C(n,β,λ,p,q)vμp–_Ls₍v

μ)wμ r

Ls_(μ)uμ_Lqs_(μ)+v

p Ls_(μ)u

q– Ls₍u

μ)wμ r Ls_(μ) +vp_Ls_(μ)u_Lqs_(μ)wμr–_Ls₍w

μ)

+vμr–_Ls_(μ)v uμp_Ls_(μ)wμp_Ls_(μ) +vr_Ls_(μ)uμp–_Ls_(μ)uwμq_Ls_(μ)+vr_Ls_(μ)up_Ls_(μ)wμ

q– Ls_(μ)

+vμq–_Ls₍v

μ)uμ r

Ls_(μ)wμ_Lps_(μ)+v

q

Ls_(μ)uμr–_Ls₍u_μ)wμp_Ls_(μ) +vq_Ls_(μ)u_Lrs_(μ)wμp–_Ls₍w

μ)

≤ 

.

So (.) and (.) hold. Thus we also have

uμ(x) –u(x)_Ls₍u_μ)=vμ(x) –v(x)_Ls_(μ)v =wμ(x) –w(x)_Ls₍v_μ)= , (.)

which implies that the measures of u_μ,_μv, and_μw must be zero. This verifies (.). Finally, we show that the plane cannot stop before hitting the origin. On the contrary, if the plane stops atx=μ< , thenu(x),v(x), andw(x) must be symmetric about the plane

x=μ,i.e.,

On the other hand, noting that|xμ_{|>|x|for anyx∈}

μ, we have

u(x) –uμ(x)

=

Rn

vp(y)uq(y)wr(y)

|x–y|λ 

|y|βdy–

Rn

vp(y)uq(y)wr(y)

|xμ_–y|λ 

|y|βdy

≥

Rn



|x–y|λ– 

|xμ

–y|λ

vp_(y)uq_(y)wr_(y)

|y|β dy

=

_μ



|x–y|λ– 

|xμ

–y|λ

vp_uq_wr_(y)

|y|β – vp_μu

q

μw

r

μ(y)

|yμ_|β

dy

>

_μ



|x–y|λ– 

|xμ

–y|λ

_vp_uq_wr_{(y) –v}p

μ(y)u

q

μw

r

μ(y)

|y|β

dy

= , (.)

which obviously contradicts with (.). Since the direction is arbitrary, we derive thatu andvare radially symmetric about the origin and decreasing. This completes the proof

of R.

3 Integrability

In this section, we will apply the regularity lifting lemma to obtain the integrable inter-vals of the solutions of system (.). On the other hand, a new skill is adapted to get the uniformly bound of positive solutions. Here, for completeness, we first of all give the reg-ularity lifting lemma as follows.

Lemma .(cf.[]) LetX andY be both Banach spaces with norm · _X and · _Y, respectively.The subspace ofX andY,Z=X∩Yis endowed with a new norm by

· Z= p

· p

X+ · pY, p∈[,∞].

Suppose thatT is a contraction map from a Banach spaceXinto itself and from a Banach spaceYinto itself.If f ∈Xand there exists a function g∈Z=X∩Ysuch that f=Tf+g, then f also belongs toZ.

Theorem . For s=n(p+q+r– )/(n–λ),let(u(x),v(x),w(x))∈Ls_(Rn_)×Ls_(Rn_)×Ls_(Rn₎

be a positive solutions of(.).Then

u(x),v(x),w(x)∈Lτ Rn, ∀τ∈

n

λ,∞

. (.)

Proof For every fixed real numberA> , set

fA(x) ⎧ ⎨ ⎩

f(x), iff(x)≥A, or|x| ≥A;

, otherwise,

and

By (.), we have

f(x) =u(x) +v(x) +w(x)

=

Rn

up(y)vq(y)wr(y) +uq(y)vr(y)wp(y) +ur(y)vp(y)wq(y) dy

|y|β|_x–y|λ

≤C(p,q,r)

Rn

fp+q+r_(y)

|x–y|λ 

|y|βdy. (.)

On the other hand, denoteB(x) as follows:

B(x) f(x) Rn

fp+q+r_(y)

|x–y|λ _|y_|βdy

, ∀x∈Rn.

Therefore, this, together with (.), implies thatB(x) is a bounded positive function inRn

and

f(x) =B(x)

Rn

fp+q+r_(y)

|x–y|λ 

|y|βdy, ∀x∈R

n_.

To obtain the integrability off(x), define the following functional:

MA(g)(x)B(x)

Rn

f_Ap+q+r–(y)g(y)

|x–y|λ 

|y|β dy, ∀x∈R

n_,

and

R(x)B(x)

Rn

(f–fA)p+q+r(y)

|x–y|λ 

|y|βdy, ∀x∈R

n_.

Obviously, by (.),f(x) is a positive solution of the following equation:

f(x) =MA(f)(x) +R(x). (.)

To obtain the integrability off(x), we first of all build up a prior estimate ofMA(g)(x)

andR(x). Observe thatf(x)∈L(p+nq–+λr––)βn₍Rn_{), by the weighted Hardy-Littlewood-Sobolev}

inequality, we conclude that for∀r∈(n/λ, +∞)

MA(g)(x)_r≤C(β,λ,n)fAp+q+r–g nr n+(n–λ–β)r

≤C(β,λ,n)fAp+q+r–(p+q+r–)n n–λ–β

gr. (.)

Hence, according to the definition offA, we can choose a real numberAlarge enough such

that

fAp+q+r–(p+q+r–)n n–λ–β

≤

,

which together with (.) implies thatMA(g) is a contraction mapping fromLr(Rn) to itself.

∀r∈(n/λ,∞), we have

_R(x)_r≤C B(x)_∞(f–fA)p+q+r nr n+(n–λ–β)r

.

Now, we are in a position to apply Lemma . to obtain the sharp integrability off. Taking X=L(p+nq–+λr––)βn₍Rn_{) and}Y_=Lr₍Rn_),r∈_(n/λ,∞_{), in Lemma ., this together with [(p+q+}

r– )n]/(n–λ–β) = n/λ>n/λimplies that

f ∈Lτ Rn, τ ∈

n

λ, +∞

. (.)

Next, we consider the end-point case. By (.), it suffices to obtain the boundedness of M(f)(x) andR(x). Indeed, as|x|> Aand|y|<A, we have|x–y|>Aand

R(x) =B(x)

Rn

(f –fA)p+q+r(y)

|x–y|λ 

|y|βdy

≤B(x)Ap+q+r–λ

|y|<A



|y|βdy≤C n,B(x)∞

Ap+q+r+n–λ–β. (.)

At the same time, by Hölder’s inequality, we derive that for|x| ≤A

R(x)≤B(x)Ap+q+r–λ

|y|<A



|x–y|λ 

|y|βdy

≤B(x)_∞Ap+q+r–λ

|y|<A



|y|β

λ+β β

dy

β β+λ

|y–x|<A



|y–x|λ

λ+β λ

dy

λ λ+β

=C B∞Ap+q+r–β–λ+n. (.)

Next we discuss the boundedness ofMA(f)(x). At first, we decomposeMA(f)(x) as follows:

MA(f)(x) =B(x)

Rn

f_Ap+q+r(y)

|x–y|λ |y|

–β dy

=B(x)

|y|<A

+

|y|≥A Rn

f_Ap+q+r(y)

|x–y|λ 

|y|βdy

MA,(f)(x) +MA,(f)(x).

Note that

n(p+q+r– ) (n–β–λ)(p+q+r)=

n

n–λ– β and n–

nβ λ+ β > .

Therefore when|x|> Aand|y|<A, it is easy to verify that|x–y|>Aand

MA,(f)(x)≤

|y|<A

f_Ap+q+r(y)

|x–y|λ 

|y|βdy

≤ 

Aλ

|y|<A

f n(p+q+r–)

n–λ–β A (y)dy

(p+q+r)(n–λ–β)

n(p+q+r–)

|y|<A

|y|–β×λ+nβ_dy λ+β

n

Similarly, for|x|< A, sinceβ+λ<n, there exists a positive real numberεsuch that

 <ε<n–β–λ

n .

Let /s=β/n+ε, /s=λ/n+εand /s= (n–λ–β)/n, then, together with Hölder’s

inequality, we have

MA,(f)(x)≤C(A,n)

|y|<A

|y|–βs_dy 

s_

×

|x–y|<A

|x–y|–λs_dy 

s_

·

|y|<A

f(p+q+r)s A (y)dy



s_

. (.)

Next, we turn our attention toMA,(f)(x). After a basic calculation, we derive

MA,(f)(x)≤C

|y|≥A

f_Ap+q+r(y)

|x–y|λ 

|y|βdy

≤

(Rn_\B

A())∩(Rn\BA(x))

f_Ap+q+r(y)

|x–y|λ 

|y|βdy+  Aβ

|x–y|≤A

f_Ap+q+r(y)

|x–y|λ dy

MA,,(f)(x) +MA,,(f)(x). (.)

Take the parametersr,s,τas follows:

 r

=λ+β n ,

 s

= β

n, 

τ

=n– λ– β

n .

By Hölder’s inequality, we conclude that

MA,,(f)(x) =

 Aβ

|x–y|≤A

f_Ap+q+r(y)

|x–y|λ dy

≤C(n,A)

|x–y|<A

f(p+q+r)s A (y)dy



s

×

|x–y|<A

|x–y|–λr_dy 

r_

Aτn_{. (.)}

Noting thatβ<n, then there exists a positive numberεsmall enough satisfying

n–β–λ

n +ε<

n– β–λ

n and ε<

β+λ

n .

Set

 s =

β+λ

n –ε and

 s =

n– (β+λ)

n +ε.

Then

 s(p+q+r)=

n– (β+λ)

n +ε

× λ

n– β–λ< λ

and with Hölder’s inequality

MA,,(f)(x) =

(Rn_\B

A())∩(Rn\BA(x))

f_Ap+q+r(y)

|x–y|λ 

|y|βdy

≤

Rn_\B A(x)

f_Ap+q+r(y)

|x–y|λ+βdy+

Rn_\B A()

f_Ap+q+r(y)

|y|β+λ dy

≤

Rn_\B A()

f_A(p+q+r)s(y)



s

Rn_\B A()



|y|(β+λ)s dy 

s

+

Rn_\B A(x)

f_A(p+q+r)s(y)



s

Rn_\B A(x)



|x–y|(β+λ)s dy 

s

. (.)

Obviously, (.) directly follows from (.) and (.)-(.). This completes the proof of

Theorem ..

Remark . The integrability of (u(x),v(x),w(x)) in (.) is optimal. Indeed, as|x| ≥ and

|y| ≤, we have

u(x)≥

|y|≤

vp_(y)uq_(y)wr_(y)

(|x|+|y|)λ 

|y|βdy

≥C(λ,p,q,r)|x|–λ

|y|<|

y|–βdy (.)

and

Rn

u(x)_τ ≥C(λ,β,p,q,r)

|y|≥

|y|–λτdy

/τ

=∞, ∀τ≤n/λ.

Remark . Note thatMA(g)(x) is not a contraction mapping fromL∞(Rn) toL∞(Rn),

therefore the regularity lifting lemma is unavailable and we have to look for a new way to obtain the bound estimate.

4 Asymptotic behavior

In this section, we show R, which implies the sharp decay rates ofuandvat infinity. The

proof is made up of two propositions.

Proposition . The following improper integrals are convergent:

u∞

Rn

vp(y)uq(y)wr(y)|y|–β_{dy, (.)}

v∞

Rnv

r_(y)up_(y)wq_(y)|y|–β

dy, (.)

w∞

Rnv

q_(y)ur_(y)wp_(y)|y|–β

dy. (.)

Proof Note that

n(p+q+r– ) (n–β–λ)(p+q+r)=

n

n–λ– β and n–

This, together with the integrability off(x) and Hölder’s equality, implies that

S∞

Rn

vp(y)uq(y)wr(y) +vr(y)up(y)wq(y) +vq(y)ur(y)wp(y)|y|–βdy

≤C(p,q,r)

|y|<A

+

|y|≥A

fp+r+q(y)|y|–βdy

≤C(p,q,r,β,λ)An–βfp+q+r_∞ +fp+q+rn(p+q+r–)

n–λ–β

Aλ_,

whereA>  is a real number large enough. This completes the proof of Proposition ..

Proposition .

lim

|x|→∞

|x|λu(x)=

Rnv

p_(y)uq_(y)wr_(y)|y|–β

dy, (.)

lim

|x|→∞

|x|λv(x)=

Rnv

r_(y)up_(y)wq_(y)|y|–β

dy, (.)

lim

|x|→∞

|x|λw(x)=

Rnv

q_(y)ur_(y)wp_(y)|y|–β

dy. (.)

Proof Define first of allA(x),A(x),A(x), respectively, as follows:

A(x)

BR()

vp_(y)uq_(y)wr_(y)

|y|β

|x|λ

|x–y|λdy, (.)

A(x)

(Rn_\B R())\B|x|

 (x)

vp_(y)uq_(y)wr_(y)

|y|β

|x|λ

|x–y|λdy, (.)

A(x)

B|x|

 (x)

vp_(y)uq_(y)wr_(y)

|y|β

|x|λ

|x–y|λdy. (.)

Observe that wheny∈BR() and|x| ≥R, by (.), we conclude that

vp_(y)uq_(y)wr_(y)

|y|β

_|x|x_–|λ_y|λ– 

≤C(λ)v

p_(y)uq_(y)wr_(y)

|y|β ∈L

 _Rn_,

and with Lebesgue’s dominated convergence theorem,

lim

|x|→∞

BR()

vp(y)uq(y)wr(y)

|y|β

_|

x|λ

|x–y|λ– 

dy

= ,

which means that

lim

R→∞|xlim|→∞A(x) =

Rn

vp_(y)uq_(y)wr_(y)

|y|β =u∞. (.)

To obtain (.), it suffices to provelim_|x|→∞A(x) =  andlim|x|→∞A(x) = . Noting that

y∈(Rn_\B

R())\B|x|

 (x) and|x|> R, by (.), we have A(x)≤C(λ)

(Rn_\B R())

vp(y)uq(y)wr(y)

Now, we turn toA(x). By Theorem . and the result of Rin Theorem ., we conclude

that, for anyτ∈(n/λ,∞),

fτ |x|

|x|/<y<|x|

dy≤

|x|/<y<|x|

fτ(y)dy=fττ,

which implies thatf(|x|)|x|nτ ≤Cfor|x| ≥Rand A(x)≤

B|x|

 (x)

fp+q+r(y)

|y|β

|x|λ

|x–y|λdy

≤C(λ)|x|λ–β_fp+q+r

|x|  B|x|

 (x)

|x–y|–λ_dy

=C(λ,n,β)|x|n–β–(p+qτ+r)n_.

On the other hand, noting that

n–β–(p+q+r)nλ

n=λ+β–n< ,

and taking /τ=λ/n–εwithεbeing a small positive number, we derive that

n–β–(p+q+r)n

τ < ,

and

lim

|x|→∞

A(x)= .

Then, combining with (.) and (.), we obtain (.). Similarly, we can get (.) and

(.). Therefore, this completes the proof of Proposition ..

Competing interests

The authors declare they have no competing interests.

Authors’ contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Author details

1_{School of Mathematics and Physics, North China Electric Power University, Beijing, 102206, China.}2_{School of Applied}

Mathematics, Xiamen University of Technology, Xiamen, 361024, China.3_{College of Sciences, Hunan Agriculture}

University, Hunan, 410128, China.

Acknowledgements

The first author was partially supported by the National Natural Science Foundation of China (Grant: U1430103), by the fund for Beijing Higher Education Young Elite Teacher Project (Grant: YETP0724) and by the Fundamental Research Funds for the Central Universities (Grant: 13MS35). The third author was partially supported by the National Natural Science Foundation of China (Grant: 11126148).

Received: 18 August 2015 Accepted: 1 December 2015 References

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