A Priori Estimates and Asymptotic Properties of Solutions for a Class of Fractional Elliptic Systems

WANG Guijun(王贵军)

(School of Mathematics and Statistics,Tianshui Normal University,Tianshui 741001,China)

Abstract: In this paper,first,we make the uniform estimates of solutions for a class of fractional order elliptic systems in bounded and smooth domains.Second,we investigate the asymptotic properties of solutions with prescribed integration in R3.

Key words: Fractional elliptic system;Asymptotic behavior;Uniform priori estimate;Q-curvature

1.Introduction

In this paper,we prove uniform estimates to the following fractional order elliptic systems:

and investigate asymptotic properties of the solutions to the following fractional order elliptic systems:

whereS(R3) is the Schwarz space of rapidly decreasing smooth functions in R3.

In the systems(∗),suppose thatu,v ∈L1(Ω),e3u,e3v ∈Lp′(Ω)(wherep′is the conjugate exponent ofp)so that(∗)has a meaning in the sense of distributions(see Definition 3.1 in[9]).A natural question is whether one can follow that all solutions of (∗) have uniform bounds.As this will be seen in Section 2 (see Theorem 2.4) the answer is positive.It is obvious that we have generalized part works in [10]whenu=vin (∗).

Recently,a lots of works have been devoted to investigate the existence and the asymptotic properties of the solutions of the following equation

whereQis some real function.

WhenQ=6,LIN[11]gave a complete classification ofuin terms of its growth,or of the behavior of ∆uat∞.XU[18]did similar work by using moving spheres methods.In[1]concentration phenomena of solutions of this problem was deeply discussed.Robert and WEI[14]also studied asymptotic behavior of solutions similar this problem fourth order mean field equation with the Dirichlet boundary condition.WEI and XU[16]and Martinazzi[12]also did similar work for higher order conformally invariant equations compared to (∗∗∗).Under the guidance of these works,in Section 3,we consider more general functionsQ1(y),Q2(y)for the problem (∗∗).This generalizes work of [13].First,we obtain the asymptotic behavior of solutions near infinity.Consequently,we prove that all solutions satisfy an identity,which is similar to the well-known Kazdan-Warner condition (see Theorem 3.2).

2.Uniform Estimates for Problem (∗)

SupposeΩ ⊂R3is a bounded domain andhis a solution of

Following the idea of Brezis-Merle[3],Hyder[9]established the following lemma:

where diamΩdenotes the diameter ofΩ.

Using above lemma,we obtain the following result:

Theorem 2.1Supposehis a solution of(2.1)withf ∈L1(Ω).Then for every constantk >0,

ProofLetting,we splitfasf=f1+f2with||f1||1<ϵandf2∈L∞(Ω).Denotehiare the solutions ofk

Next,we prove a result on the regularity of the distribution solutions of (∗).

Corollary 2.1Let (u,v) be a solution of

for allwhereis the closest point toyin∂Ωandν() denotes the unit external normal to∂Ωin the point.

Assumption (H1) yields moving planes methods to get bounds for the functionsuandvnear the boundary.

Theorem 2.3 Suppose that Qi(y),i=1,2,is continuous function with mi ≤Qi ≤Mi for some positive constants mi and Mi,Ω is convex and (H1) holds.Then there exists a positive constant C,depending only on Qi,i=1,2,and Ω,such that

for all (u,v) solutions of (∗).

ProofStep 1 For each (u,v) solution of system (∗) we claim that

where the constantConly depends onQ1,Q2andΩ.

From our basic assumptions forQi,i=1,2,it follows that,there are positive constantsai,i=1,2,witha1×a2andcsuch that

Next,multiplying the equations in (∗) by ϕ1,integrating by parts and using (2.3),we get

The other inequality follows in a similar way.

Step 2 We claim that there existr,δ >0 such that

for each (u,v) solutions of (∗).

Similarly,we can follow that

andξ(y1,y2,y3) is real number betweenu(y1,y2,y3) andu(2λ −y1,y2,y3).

Forλsufficiently small and positive,we know thatΣλhas small measure and so we can use the maximum principle for cooperative elliptic systems in small domains (see [7]) to conclude that

Using similar arguments as in [7]we can also obtain that

Therefore,there existsϵ >0 such thatuandvare increasing inΩϵ.Finally,the conclusion follows in a standard way as in [8].

Step 3 We claim that there existϵ>0 andC >0 which depend only onQ1,Q2andΩsuch that||u||L∞(Ωϵ),||v||L∞(Ωϵ)≤C,for each (u,v) solution of (∗).

The conclusion can be proved by the similar arguments as in [8]and Step 2 above.

Step 4 We claim that our theorem holds.

Denoteα:=inf{ϕ1(y) :y ∈Ωϵ}.Using Step 3,we know thatQ1(y)e3vis bounded inΩϵ.Thus

where we have used Step 1 to estimate the last integral.Using similar argument,we can prove the result forQ2(y)e3u.

Now,we state our main result in this section.

Theorem 2.4Assume thatQi(y),i=1,2,is continuous function withmi ≤Qi ≤Mifor some positive constantsmiandMi,Ωis convex and (H1) holds.Then there exists a constantC >0 such that

for all solutions (u,v) of system (∗).

ProofSince∫Ω Q1(y)e3vdy

We also find that,as a consequence of Theorem 2.3,the solutions (un,vn) of (∗) are bounded inL1(Ω):

A pointy ∈Ωis defined a 2π2regular point with respect toµif there is a functionψ ∈Cc(Ω),0≤ψ ≤1,withψ=1 in a neighborhood ofysuch that

Define

From∫dµ

In fact,for finishing the proof of our theorem,we only need to prove thatSu=Sv=∅.

Next,we prove above conclusion by four steps.

Step 1 We claim that fory0is a regular point for the measureµ(or for the measureν),then there exist constantsρ>0 andC,independent ofn,such that

wherep>1 is a constant depending only onδ.Fromt

On the other hand,from,we follow

Observe that in view of (2.11),by fractional elliptic regularity (see [2]) we get

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From (2.13) and (2.14) we get

Now we go back to the second equation in (∗).By (2.15) and elliptic regularity we get

Using (2.10) and (2.16) we get

which together with (2.15) proves Step 1,taking.

Step 2 We claim thatSµ ⊂ΩµandSν ⊂Ων.

In fact,This follows directly from Step 1 and the definition of the setsΩµ,Sµ,SνandΩν.

Step 3 We claim thatΩµ ⊂SνandΩν ⊂Sµ.

Otherwise,there existsR0>0 and a subsequence,which we denote also by (un),such that

which implies that forR

Thus,there existsR1>0,such that

This means thaty0is a regular point ofµ,which is a contradiction.

Now we observe that there existsR >0 such thaty0is the only non-regular point inBR(y0).

Next,we use (2.17) to prove thaty0∈Sν.In fact,by (2.17) there exists{yn}⊂BR(y0)such thatxn →andv(yn)→+∞.So,one needs to prove=y0.If this was not the case,thenwould be a regular point,which is not possible,sinceunis bounded in a neighborhood of a regular point.

According to similar argument,we can follow thatΩν ⊂Sµ.

As a consequence of Step 2 and Step 3,we conclude that those four sets coincide:

Step 4 We claim thatSµ=∅.

Thus,from the maximum principle we get

Taking the limit,we know thatzn →z,wherezis a solution of the problem

On the other hand the leading term of solution of the problem

Now using the hypothesisQ1(y)e3t ≥Ce3t,we get

which is impossible.

3.Asymptotic Properties of Solutions of Problem (∗∗)

In this section,we consider the asymptotic properties of solutions of (∗∗).

From [3],Brezis-Merle obtain thatuis bounded from above whenusatisfies−∆u=V(y)euand other conditions.This result is used to study the asymptotic properties and classification of solutions for some second order elliptic equation (See [5-6]).Now,one naturally ask: is any solution (u,v) to system (∗∗) with∫R3Q1(y)e3v <+∞and∫R3Q2(y)e3u <+∞bounded from above? We will partially answer this problem and obtain the following result:

Theorem 3.1Suppose thatQi(y),i=1,2 is a positive bounded away from 0 and bounded from above function and (u,v) is aC2solution of (∗∗) with∫R3e3u <+∞,u(y)=◦(|y|2) and∫R3e3v <+∞,v(y)=◦(|y|2).Thenu+∈L∞(R3) andv+∈L∞(R3).

Similar to the proof of Lemma 3.1 and Lemma 3.2 in[19],we obtain the following lemmas:

Lemma 3.1Assume that (u,v) is aC2function on R3such that

(a)Q1e3vandQ2e3uare inL1(R3) with 0

(b) In the sense of weak derivative,u,vrespectively satisfies the following equations:

Then there are two constantsc1,c2>0,respectively depending onv,u,such that|∆u(y)|≤c1on R3and|∆v(y)|≤c2on R3.Whereβ0being given by.In fact,β0=2π2.

Lemma 3.2Assume thatSisC2function on R3such that 0≤(−∆)S(y)≤Aon R3for some constantAand∫R3Q(y)e3S(y)dy=α<∞with 0

Lemma 3.3Assume that (u,v) is a solution of (∗∗).Let

Then there exist two constantsc1,c2such that

Since|y −z| ≤|y|+|z| ≤|y|(|z|+1) for|y|,|z| ≥2 and ln|y −z| ≤ln|y|+cfor|y| ≥4 and|z|≤2,we get

Lemma 3.4Assume that (u,v) is a solution of (∗∗) withu(y)=◦(|y|2) andv(y)=◦(|y|2).Then ∆u(y) and ∆v(y) can be represented by

Sincek=u+w1≤u(y)+βln|y|+cis deduced from Lemma 3.3,we get

Thus ∆k(y0)≤0 for ally0∈R3.By Liouville’s theorem,∆k(y)≡−c1in R3for some constantc1≥0.Hence,we get

Now,we explain thatc1=0.Otherwise,we have ∆u(y)≤−c1<0 for|y| ≥R0whereR0is sufficiently large.Let

whereϵis small such that

for|y| >R0,andAis sufficiently large so thatis achieved by somey0∈R3with|y0|>R0.Applying the maximum principle to (3.5) aty0,we reach a contradiction.Hence,our conclusion holds.

Similarly,we can prove that (3.2) holds.

Proof of Theorem 3.1From Lemmas 3.2,3.4,our conclusion holds.

Now,we consider the asymptotic properties of solutions of equation (∗∗).Following our Theorem 3.1 and the methods of [6],we obtain the following result:

Furthermore,we have the following identity

Lemma 3.5Suppose that (u,v) satisfies the assumptions of Theorem 3.2,then

ProofHere we provew1(y)→β1ln|y|as|y|→∞.We need only to verify that

WriteI=I1+I2+I3,which are the integrals on the regionsD1={z:|y −z| ≤1},D2={z:|y −z| >1 and|z| ≤k}andD3={z:|y −z| >1 and|z| >k}respectively.We may suppose that|y|≥3.

(a) To estimateI1,we simply observe that

Then from the boundedness ofQe3v(see Theorem 3.1) andR3Q1(y)e3v(y)dy <∞,we follow thatI1→0 as|y|→∞.

(b) For each fixedk,in regionD2,we have,as|y|→∞,

HenceI2→0.

(c) To estimateI3→0,we use the fact that for|y −z|>1

Then letk →∞.

Similarly,we getw2(y)→β2ln|y|as|y|→∞.

Lemma 3.6Suppose that (u,v) satisfies the assumptions of Theorem 3.2,then

wherec0andare two constants.

ProofFrom Lemma 3.4,we know ∆(u+w1)=0 in R3.By Theorem 3.1,we getu+∈L∞.So,combing Lemma 3.3,we haveu+w1≤cln|y|+c,sinceu+w1is harmonic function,by the gradient estimates of harmonic functions,we haveu(y)+w1(y)≡c0.Similarly,we knowv(y)+w2(y)≡.

Lemma 3.7Suppose that(u,v)satisfies the assumptions of Theorem 3.2,thenu1(y)≥−β1ln(|y|+1)−c1withβ1>1 andu2(y)≥−β2ln(|y|+1)−c2withβ2>1.

ProofFrom Lemma 3.3 and Lemma 3.6,we get

By above inequality,∫R3e3vdy <+∞and∫R3e3udy <+∞,we getβ1>1,β2>1.

Lemma 3.8Suppose that (u,v) satisfies the assumptions of Theorem 3.2,thenu(y)≤−β1ln(|y|+1)+c1andv(y)≤−β2ln(|y|+1)+c2.

ProofIn fact,for|y −z|≥1,we get

Consequently,

According to the fact (see Lemma 3.5) that

and by the boundedness ofQ1(y),we getI1,I2→0 as|y|→∞andI3is finite.

Thereforew1(y)≥β1ln(|y|+1)−c1.From Lemma 3.6,we haveu(y)≤−β1ln(|y|+1)+c1.Similarly,we getv(y)≤−β2ln(|y|+1)+c2.

Proof of Theorem 3.2By Lemmas 3.7,3.8,then (3.6) and (3.7) hold.From Lemma 3.2 and Theorem 1.1 in [19],we can similarly conclude that (3.8) holds.

4.Conclusions

In this paper,we established a more important uniformL∞norm estimates for all weak solutions of a class of fractional higher order elliptic systems in bounded and smooth domains by using moving planes methods and also obtained the exact asymptotic properties of solutions for fractional higher order elliptic systems in whole space,which is closely connective with differential geometry.