A NECESSARY CONDITION FOR CERTAIN INTEGRAL EQUATIONS WITH NEGATIVE EXPONENTS?

Jiankai XU

College of Sciences;College of Computer Science,Hunan Agriculture University,Changsha 410128,China

E-mail:jiankaixu@126.com

Zhong TAN

School of Mathematical Sciences,Xiamen University,Xiamen 361005,China

E-mail:ztan85@163.com

Weiwei WANG

College of Mathematics and Computer Science,Fuzhou University,Fuzhou 361000,China

E-mail:wei.wei.84@163.com

Zepeng XIONG

The First Middle School of Longhui,Longhui 422200,China

E-mail:770685723@qq.com

Abstract This paper is devoted to studying the existence of positive solutions for the following integral system

Key words integral equations;Lane-Emden system;conformal invariance;positive solu

1 Introduction

In this article,we investigate the existence of positive solutions for a non-linear integral equations of the form:

where λ >0,0

The well known Lane-Emden system,which arises from the chemical,biological and physical studies and has attracted several researchers’attention,can be written as follows

here u(x),v(x)≥0,0

(1.2)is called critical.We also say that system(1.2)is supercritical,or subcritical if(p,q)satis fi es that

The famous Lane-Emden conjecture states that system(1.3)does not admit a positive solution under the subcritical condition.That is to say that(1.3)is a corresponding dividing curve with the property that(1.2)admits positive solutions if and only if(p,q)satis fi es the critical condition or supercritical condition.

Also,system(1.2)has the natural extension as follows:

Under certain regularity conditions,system(1.5)is equivalent to the following integral system,which is closely related to the problem of fi nding the sharp constant in the Hardy-Littlewood-Sobolev inequality

here λ ∈ (0,n),n ≥ 3 and p,q>1.System(1.5)or(1.6)is not only a natural extension of(1.2),but also has own interest which provides an important way to study the Lane-Emden conjecture.Similarly,the value of exponents(p,q)in(1.6)is also divided into three cases and the corresponding Lane-Emden conjecture becomes the Hardy-Littlewood-Sobolev type integral equations conjecture,namely,system(1.6)does not admit a positive solution if and only if these parameters(p,q)satisfy the following inequality

Now,we recall some results which are closely related to our topic.In 1998,by the shooting method and the Pohozaev identity,Serrin and Zou[21]showed the existence of a positive solution of(1.2),when(p,q)satis fi es that

Later on,Mitidieri[17]proved that the Lane-Emden conjecture holds with additional assumption that(u,v)is a pair of radial solution of(1.2).Therefore,for radial case of system(1.2),Sobolev hyperbola(1.3)is the dividing curve for the existence and nonexistence of positive solutions.As for the non-radial solutions of(1.2),the Lane-Emden conjecture is still open except for n≤4.We refer the readers to[11,17,19,20,23,24,27],among numerous references,for more information.When α (α

then system(1.5)has no radial non-negative solutions.With the same assumption to the parameter α,Lei and Li[9]proved that system(1.5)admits a pair of positive radial solutions(u,v),provided

Moreover,with the help of degree theory,Li and Villavert in[12,13]considered the more general abstract model and established the existence result under suitable conditions.For higher order system(1.5)or(1.6)with the general parameter λ=n?α∈(0,n),Lieb[15]proved that the existence of positive solution for(1.6)in the critical case.Subsequently,Caristi,Dambrosio and Mitidieri[1],under certain smooth condition assumptions,proved the conjecture for the Hardy-Littlewood-Sobolev type integral equations.That is,if(u,v)∈C2(Rn)×C2(Rn)is a pair of nonnegative radial solutions of(1.6)and λ

is the dividing curve in the(p,q)-plane for the existence and non-existence of Hardy-Littlewood-Sobolev type integral system.

Based on the above,it is natural and interesting to ask whether there exists a corresponding dividing curve in the(p,q)-plane such that(1.1)for λ ∈ (0,∞)and p,q>0,admits positive solutions if and only if(p,q)is on or above the curve?The main purpose of this paper is to address this question.Our main result can be formulated as follows.

Theorem 1.1Suppose that(u,v)is a pair of positive solutions of system(1.1)with λ ∈ (0,∞)and 0

Remark 1.2Comparing Theorem 1.1 with the results of the well-known Lane-Emden system(1.2)and its natural extension to Hardy-Littlewood-Sobolev type integral equations(1.6),system(1.1)has the same radial symmetry solution,provided the exponents(p,q)lies on the hyperbola curve.However,as the pair of parameters(p,q)is not on the hyperbola curve,there is obvious di ff erence between(1.1)and(1.6).Precisely,by Theorem 1.1,we can see that system(1.1)has no positive solution,if(p,q)is not on the hyperbola.But as for Lane-Emden system(1.2)and(1.6),the system has positive solutions when(p,q)is under the supercritical conditions.The essential reasons for the di ff erence between(1.1)and(1.6),comes from the integrability and asymptotic behavior of each system.

Remark 1.3From an analytical point of view,a natural and interesting question that raised from the above result is whether there exist positive solutions on hyperbola(1.8)for system(1.1)?The conjecture is not true.An example is that it is easy to see that for p,q near 1,from the asymptotic estimates(2.1)and(2.2),there is no such solution on hyperbola(1.8).On the other hand,as p=q>1 and max{p,q}>(n+ λ)/λ,by[7]and[10],system(1.1)admits a pair of radial positive solution.

The rest of this paper is organized as follows.After recalling and establishing some technical lemmas in Section 2,we will prove Theorem 1.1 in Section 3.Throughout this paper,we always use the letter C to denote positive constants that may vary at each occurrence but are independent of the essential variables.

2 Preliminary

In this section,we will recall and establish some standard ingredients needed in the proof of our theorem.These results essentially follow from[7,10,14,25].Here,for completeness,we will present the corresponding proofs.

Lemma 2.1For n≥1,let u(x),v(x)be a pair of positive Lebesgue measurable solutions of(1.1)with λ>0,q>0 and p>0.Then the following properties hold

?equivalents of pointwise

?integrability of u(x)and v(x)

Furthermore,

Hence,the fi rst inequality in(2.1)follows from(2.11)and(2.12).Similarly,we can get the fi rst inequality in(2.2).

Next,we will prove the second inequality in(2.1)and(2.2).By(2.10),there existˉx such that|x|∈[1,2]and v(ˉx)<∞,u(ˉx)<∞.Thus when|y|≥4,we conclude that

and

which,together with the fi rst inequality in(2.1)and p>0,q>0,yields that

Similarly,

This is(2.5)and completes the proof of Lemma 2.1. ?

By(2.1)and(2.2)with p,q>0,we have

Thus when 0<λ<1,by Hardy-Littlewood-Sobolev inequality,(2.1),(2.2)and(2.13),we have

where 1/r+1/s+(1?λ)/n=2,r>1,s>1.

On the other hand,for λ ∈ [1,∞),note that

It follows from(2.8)and(2.9)that

which combining with(2.19)implies that

Therefore,we can di ff erentiate IIu(x)and IIv(x)under the integral for|x|

Next,we verify the smooth property of Iu(x)and Iv(x).Note that

We conclude that Iu(x),Iv(x)∈ C1(BR).This,together with IIu(x),IIv(x)∈ C∞(BR),(2.20)and(2.21),implies that(u(x),v(x))∈C1(BR)×C1(BR).Meanwhile,by chain rule of derivatives and the arbitray of R,it is easy to check that v?q(x)and u?p(x)are derivative on Rn.Therefore,we can improve the regularity of Iu(x)and Iv(x)to C2in x∈BR(0)which with(2.20)and(2.21),implies that(u(x),v(x))∈ C2(Rn)×C2(Rn).Similarly,by the bootstrapping arguments,we eventually get that u(x),v(x)∈C∞(Rn).The proof of Lemma 2.3 is completed.?

3 Proof of Theorem 1.1

Proof of Theorem 1.1By Lemma 2.2,f(t)=t1?p(p 6=1)being a C1function and the chain rule of weak derivatives,we conclude,in the sense of distribution,that

To pass to the limit on the above,we need to build up a prior estimate of E1+E2.Letbe a vector-value function from Rnto Rn,given by

It is easy to check that

This together with(2.5),(3.2)and(3.5)–(3.7)yields that

Theorem 1.1 is proved.?