Noname manuscript No. (will be inserted by the editor) Ground states and dynamics of Bose-Einstein condensation with higher order interactions Weizhu Bao · Yongyong Cai · Xinran Ruan 7 Received: date/Accepted: date 1 0 2 Abstract We analyze the ground states and dynamics of a Bose-Einstein n condensate in the presence of a higher order interaction (HOI), modeled by a a modified Gross-Pitaevskii equation (MGPE). In fact, due to the appearance J oftheHOI,thegroundstatestructuresbecomeveryrichandcomplicated.We 5 establish the existence and uniqueness as well as non-existence results under different parameter regimes and obtain their limiting behaviors and/or struc- ] h tures under different combinations of HOI and contact interaction strengths. p Different structures of ground states are identified for the MGPE with either - h a harmonic potential or a box potential. Finally, the dynamics of the center- t of-massisinvestigatedandananalyticalsolutionoftheMGPEisconstructed. a m [ Keywords Bose-Einstein condensation higher order interaction Gross- · · Pitaevskii equation ground state dynamics 1 · · v 5 Mathematics Subject Classification (2000) 35Q55 35A01 81Q99 4 · · 2 1 0 This workwas supported by the Ministryof Education of Singapore grant R-146-000-223- . 112 (W. Bao and X. Ruan) and the Natural Science Foundation of Chinagrant U1530401 1 (Y.Cai). 0 7 W.Bao 1 DepartmentofMathematics,NationalUniversityofSingapore,Singapore119076,Singapore : E-mail:[email protected] v Url: http://www.math.nus.edu.sg/˜bao/ i X Y.Cai r BeijingComputational ScienceResearchCenter,No.10WestDongbeiwangRoad,Haidian a District,Beijing100193, P.R.China E-mail:[email protected] X.Ruan DepartmentofMathematics,NationalUniversityofSingapore,Singapore119076,Singapore E-mail:[email protected] 2 WeizhuBaoetal. 1 Introduction Bose-Einstein condensation (BEC) has been thoroughly studied since its first experimentalrealizationin1995[1,14]basedonthemean-fieldGross-Pitaevskii equation(GPE)[31,6,2]. Inthe derivationofthe GPE,one key assumptionis that the binary interactionbetweenthe particles canbe welldescribedby the shape-independent approximation (or pseudopotential approximation), i.e. a Diracfunctionastheinteractionkernel,wheretheinteractionstrengthischar- acterized by the s-wave scattering length [16]. It is well-known that such an approximationisvalidinlowenergies(orlowdensities)andbecomeslessvalid inhighenergies(orhighdensities)[31,38,33].Therefore,numerouseffortshave beendevotedtotheimprovementofthepseudopotentialapproximationforthe two-body interaction, which would lead better mean field theory towards the understanding of recent BEC experiments [38]. Recently,a higher order interaction(HOI) correctionto the pseudopential approximationhas been proposedand analyzed[12,16,35]. As a consequence, at temperature T much smaller than the criticaltemperature T , a BEC with c an HOI can be described by the wave function ψ := ψ(x,t) whose evolu- tion is governed by the modified Gross-Pitaevskii equation (MGPE) in three dimensions (3D) [12,16,35] ~2 i~∂ ψ = 2+V˜(x)+g˜ (ψ 2+g˜ 2 ψ 2) ψ. (1.1) t 0 1 −2m∇ | | ∇ | | (cid:20) (cid:21) Here, t is time, x = (x,y,z)T R3 is the Cartesian coordinate vector, ~ is the Planck constant, m is the m∈ass of the particle, g˜ = 4π~2as is the contact 0 m interaction strength with a being the s-wave scattering length (positive for s repulsiveinteractionandnegativeforattractiveinteraction),g˜ = a2s asre is 1 3 − 2 theHOIstrengthwithr beingtheeffectiverangeofthetwo-bodyinteraction e and r = 2a for hard sphere potential, V˜(x) is a given real-valued exter- e 3 s naltrappingpotential.Intypicalcurrentexperiments,the followingharmonic potential is commonly used m V˜(x)= ω2x2+ω2y2+ω2z2 , x=(x,y,z)T R3, (1.2) 2 x y z ∈ (cid:2) (cid:3) where ω > 0, ω > 0 and ω > 0 are trapping frequencies in x-, y- and x y z z-directions, respectively. The wave function ψ is normalized as ψ 2 := ψ(x,t)2dx=N, (1.3) k k R3| | Z where N is the total number of particles in the BEC. In order to nondimensionalize the MGPE (1.1) with (1.2) and (1.3), we introduce [31,6,2] t x x3/2 t˜= , x˜ = , ψ˜(x˜,t˜)= s ψ(x,t), (1.4) t x N1/2 s s TitleSuppressedDuetoExcessiveLength 3 where t = 1 and x = ~ with ω = min ω ,ω ,ω are the scaling s ω0 s mω0 0 { x y z} parametersofdimensionlesqstimeandlengthunits,respectively.Plugging(1.4) into (1.1), multiplying by t2s , and then removing all˜, we obtain the m(xsN)1/2 following dimensionless MGPE in 3D for a BEC 1 i∂ ψ = 2+V(x)+g ψ 2 g 2 ψ 2 ψ, (1.5) t 0 1 −2∇ | | − ∇ | | (cid:20) (cid:21) where g = 4πNas, g = 4πNa2s(3re−2as), γ = ωx, γ = ωy and γ = ωz, and 0 xs 1 6x3s x ω0 y ω0 z ω0 the dimensionless trapping potential is given by 1 V(x)= γ2x2+γ2y2+γ2z2 , x R3. (1.6) 2 x y z ∈ (cid:0) (cid:1) Whenthetrappingpotentialin(1.6)isstronglyanisotropic,e.g.max γ ,γ x y { } γ for a quasi-2D BEC or γ min γ ,γ for a quasi-1D BEC, similar z x y z ≪ ≪ { } to the dimension reduction of the conventionalGPE for BEC [6,2,10,31], the MGPE (1.5) in 3D can be formally reduced to two dimensions (2D) or one dimension (1D) for the disk-shapedor cigar-shapedBEC [38], respectively.In fact, the resulting MGPE can be written in a unified form in d-dimensions (d = 1,2,3) with x Rd (denoted as x = x R for d = 1, x = (x,y)T R2 for d=2 and x=(x∈,y,z)T R3 for d=3) a∈s ∈ ∈ 1 i∂ ψ = 2+V(x)+β ψ 2 δ 2 ψ 2 ψ, (1.7) t −2∇ | | − ∇ | | (cid:20) (cid:21) where 1(γ2x2+γ2y2+γ2z2), d=3, 2 x y z V(x)= 1(γ2x2+γ2y2), d=2, (1.8) 2 x y 1γ2x2, d=1; 2 x and β and δ are two dimensionless real constants for describing the contact interaction and HOI strengths, respectively. For other potentials such as box potential, optical lattice potential and double-well potential, we refer to [6,31,4] and references therein. Thus, in the subsequent discussion, we will treat the external potential V(x) in (1.7) as a general real-valued function and the parameters β and δ as arbitrary real constants. In addition, without loss of generality, we assume V(x) 0 for x Rd inthe restofthispaper.ThedimensionlessMGPE(1.7)conser≥vesthe ∈ total mass, i.e. N(t):= ψ(,t) 2 = ψ(x,t)2dx ψ(,0) 2 =1, t 0, (1.9) k · k ZRd| | ≡k · k ≥ and the energy per particle E(ψ)= 1 ψ 2+V(x)ψ 2+ β ψ 4+ δ ψ 2 2 dx. (1.10) ZRd(cid:20)2|∇ | | | 2| | 2 ∇| | (cid:21) (cid:12) (cid:12) (cid:12) (cid:12) 4 WeizhuBaoetal. Theoretically,otherhigherordertermscanbeincludedintheMGPE(1.7) as the higher order corrections of the two-body interaction [12]. Here, we fo- cus on the current MGPE (1.7) to understand the effect of the HOI to the conventionalGPE for BEC. In fact, the MGPE (1.7) has been found in other applications (in a generalized form), such as the modelling of ultrashort laser pulses in plasmas [11,15], the description of the thin-film superfluid conden- sates [21] and the study of the Heisenberg ferromagnets [36]. The MGPE (1.7) with δ =0 has been thoroughly studied in the literature and we refer the readers to [6,2,31] and references therein. However, there have been only a few mathematical results for the MGPE (1.7), including the local well-posedness of the Cauchy problem [32,29], existence of solutions to the time independent version of (1.7) [24,25], the stability of standing waves [13], a spectral method for (1.7) [27], etc. To the best of our knowledge, all the known mathematical results for the MGPE (1.7) are not based on the application in BEC and thus have different setups in the trapping potentials and/or parameter regimes. On the contrary, some physical studies for the MGPE (1.7) have been carried out with the application in BEC, such as the ground state properties [18,37], the dynamical instabilities [33,34], etc. Very recently, we have studied the dimension reduction of the MGPE from 3D to lowerdimensions[35]. Here,wewill presentsome mathematicalresults onthe ground states and dynamics of the MGPE (1.7) for BEC with HOI. The paper is organized as follows. In section 2, we establish existence and uniqueness as well as non-existence results of ground states under different parameter regimes. In section 3, we study the asymptotic profiles of ground states in different parameter regimes. In section 4, we derive some dynamical properties of the MGPE. Some conclusions are drawn in section 5. 2 Existence and uniqueness for ground states Introduce the function space X = φ H1(Rd) φ 2 = φ 2+ φ 2+ V(x)φ(x)2dx< . (cid:26) ∈ (cid:12)k kX k k k∇ k ZRd | | ∞(cid:27) (cid:12) The groundstateφ :=(cid:12) φ (x) ofBECmodelledby the MGPE(1.7)is defined g (cid:12) g astheminimizeroftheenergyfunctional(1.10)undertheconstraint(1.9),i.e. φ :=argmin E(φ), (2.1) g φ∈S where S is defined as S := φ X φ =1, E(φ)< . (2.2) { ∈ |k k ∞} Since S is a nonconvex set, the problem (2.1) is a nonconvex minimization problem. In addition, the ground state φ is a solution of the following non- g linear eigenvalue problem, i.e. Euler-Lagrangeequation of the problem (2.1) 1 µφ= 2+V(x)+β φ2 δ 2 φ2 φ, (2.3) −2∇ | | − ∇ | | (cid:20) (cid:21) TitleSuppressedDuetoExcessiveLength 5 underthenormalizationconstraintφ S,wherethecorrespondingeigenvalue ∈ (or chemical potential) µ:=µ(φ) can be computed as µ=E(φ)+ β φ4+ δ φ2 2 dx. (2.4) ZRd(cid:18)2| | 2 ∇| | (cid:19) (cid:12) (cid:12) (cid:12) (cid:12) The following embedding results hold [6]. Lemma 2.1 Under the assumption that V(x) 0 for x Rd is a confining ≥ ∈ potential, i.e. lim essinf|x|≥RV(x)=+ , we have that the embedding X ֒ R→∞ ∞ → Lp(Rd) is compact provided that the exponent p satisfies p [2,6), d=3, ∈ p [2, ), d=2, (2.5)  ∈ ∞ p [2, ], d=1.  ∈ ∞ In 2D, i.e. d=2, let C be the best constant in the following inequality [39] b f 2 f 2 C := inf k∇ kL2(R2)k kL2(R2) =π (1.86225...). (2.6) b 06=f∈H1(R2) kfk4L4(R2) · For the existence and uniqueness of the ground states in (2.1), we have Theorem 2.1 (Existence and uniqueness) Suppose V(x) 0 satisfying the ≥ confining condition, i.e. lim V(x) = + , then there exists a minimizer |x|→∞ ∞ φ S of (2.1) if one of the following conditions holds g ∈(i) δ >0 when d=1,2,3 for all β R; (ii) δ =0 when d=1 for all β R,∈when d=3 for β 0, and when d=2 ∈ ≥ for β > C . b − Furthermore, eiθφ is also a ground state of (2.1) for any θ [0,2π). In g ∈ particular, the ground state can be chosen as positive and the positive ground state is unique if δ 0 and β 0. In contrast, there exists no ground state of ≥ ≥ (2.1) if one of the following holds (i’) δ <0; (ii’) δ =0 and β <0 when d=3; and δ =0 and β < C when d=2. b The results also apply to the bounded connected open do−main Ω Rd case, ⊂ i.e. V(x)=+ when x / Ω. In such case, for any δ >0, there exists C >0 Ω ∞ ∈ (depending on Ω) such that when β δ/C , the positive ground state φ of Ω g ≥− (2.1) is unique. Proof The case with δ =0 is well-known [23,6] and thus is omitted here. (i)Inordertoprovethe existence,weassumeδ >0.Bythe inequality[22] φ(x) φ(x), a.e. x Rd, (2.7) |∇| ||≤|∇ | ∈ we deduce E(φ) E(φ), (2.8) ≥ | | 6 WeizhuBaoetal. where equality holds iff φ=eiθ φ for some constant θ [0,2π). It suffices to | | ∈ considerthe realnon-negativeminimizersof (2.1).Onthe otherhand,forany φ S, denote ρ= φ2, Nash inequality and Young inequality imply that ∈ | | 4 φ4dx C ρ(x)dx d+2 φ2 d2+d2 C +ε ρ 2, ε>0. ZRd| | ≤(cid:20) ZRd (cid:21) k∇| | k ≤ ε k∇ k ∀ Thus we can conclude that E(φ) (φ S) is bounded from below ∈ E(φ) 1 φ2+V(x)φ2+ δ φ2 2 dx C. ≥ZRd(cid:18)2|∇ | | | 4 ∇| | (cid:19) − (cid:12) (cid:12) Taking a nonnegative minimizing sequence φ (cid:12)∞ (cid:12) S, we find the φ is { n}n=1 ⊂ n uniformly bounded in X and there exists φ X and a subsequence (denote ∞ ∈ as the original sequence for simplicity) such that φ ֒ φ in X. (2.9) n ∞ → Lemma 2.1 ensures that φ φ in Lp with p given in the lemma. We also n ∞ → have φ 2 ֒ φ 2 in L2. Hence we know φ S with φ being n ∞ ∞ ∞ ∇| | → ∇| | ∈ nonnegative. Under the condition δ >0, we get E(φ ) liminfE(φ )=minE(φ), (2.10) ∞ n ≤ n→∞ φ∈S which shows that φ is a ground state. ∞ For the case β 0 and δ 0, we can prove the uniqueness of the nonneg- ≥ ≥ ative ground state. In order to do so, denote ρ = φ2, then for φ = √ρ S, | | ∈ the energy is 1 β δ E(√ρ)= √ρ2+V(x)ρ+ ρ2+ ρ2 dx. (2.11) ZRd(cid:20)2|∇ | 2 2|∇ | (cid:21) ThesumoffirstthreetermsintheenergyE(√ρ)isstrictlyconvexinρ[23,6], andthe lasttermis alsoconvexbecause itis quadratic inρ andδ 0.Hence, ≥ we know E(√ρ) is strictly convex in ρ and the uniqueness of the nonnegative ground state follows [23,6]. In addition, from regularity results (see details in Theorem 2.2 below) and maximal principle [23,22], we can deduce that the nonnegative ground state is strictly positive. (ii) We prove the nonexistence when δ < 0. Choosing a non-negative smooth function ϕ(x) S with compact support and denoting ϕ (x) = ε ∈ ε−d/2ϕ(x/ε) S, we have ∈ E(ϕ )= 1 ϕ2+V(εx)ϕ2+ β ϕ4+ δ ϕ2 2 dx. ε ZRd(cid:20)2ε2|∇ | | | 2εd| | 2ε2+d ∇| | (cid:21) (cid:12) (cid:12) (2.12) (cid:12) (cid:12) From the above equation, we see that lim E(ϕ ) if δ < 0 and thus ε ε→0+ → −∞ there exists no ground state. TitleSuppressedDuetoExcessiveLength 7 (iii) In the case with V(x)=+ for x / Ω, we know φ H1(Ω). Using ∞ ∈ g ∈ 0 Sobolev inequality, there exists C >0 such that Ω f C f , f H1(Ω). (2.13) k kL2(Ω) ≤ Ωk∇ kL2(Ω) ∀ ∈ 0 Denoteρ= φ2,thenforφ=√ρ S,weclaimtheenergyE(√ρ)isconvexin | | ∈ ρ for β δ/C . To see this, we only need examine the case β [ δ/C ,0). ≥− Ω ∈ − Ω For any √ρj ∈S (j =1,2) with ρj ∈H01(Ω) and θ ∈[0,1], we have θE(√ρ )+(1 θ)E(√ρ ) E( θρ +(1 θ)ρ ) 1 2 1 2 − − − 1 θ(1 θ) β ρ ρ 2+δ p(ρ ρ ) 2 1 2 1 2 ≥ 2 − k − k k∇ − k 1 (cid:0) (cid:1) θ(1 θ) δ (ρ ρ ) 2+δ (ρ ρ ) 2 =0, 1 2 1 2 ≥ 2 − − k∇ − k k∇ − k (cid:0) (cid:1) where we used the fact √ρ 2 is convex in ρ. This shows E(√ρ) is convex k∇ k when β δ . The uniqueness follows. In the general whole space case, the ≥−CΩ energy functional E(√ρ) is no longer convex and the uniqueness when β < 0 isingeneralnotclear.We remarkherethatsomerecentresultswereobtained by Guo et al. in [19] about the uniqueness when δ = 0 with β < 0 and β is small. | |(cid:3) Concerning the ground state of (2.1), we have the following properties. Theorem 2.2 Let δ >0 and φ S bethe nonnegativeground stateof (2.1), g ∈ we have the following properties: (i) There exists α>0 and C >0 such that φ (x) Ce−α|x| for x Rd. g (ii) If V(x) L∞(Rd), we have φ is once c|ontinu|o≤usly differentiabl∈e and ∈ loc g φ is Ho¨lder continuous with order 1. In particular, if V(x) C∞, φ is g g ∇ ∈ smooth. Proof (i)WeshowtheL∞ boundofφ byaMoser’siterationandDe Giorgi’s g iterationfollowing[25].Fromthefactthatφ S minimizestheenergy(1.10), g ∈ it is easy to check that φ satisfies the Euler-Lagrange equation (2.3), which g shows that for any test function ϕ C∞(Rd), the following holds for φ = φ ∈ 0 g and µ=µ(φ ) g 1 φ ϕ+V(x)φϕ+2δφ φ (φϕ) dx= β φ2φϕ+µφϕ dx. ZRd(cid:20)2∇ ·∇ ∇ ·∇ (cid:21) ZRd − | | (cid:2) (2(cid:3).14) Using the Moser and De Giorgi iterations, we will prove that any weak solution φ X E(φ) < of (2.14) is bounded and decays exponentially ∈ ∩{ ∞} as x . In detail, we first observe that by an approximation argument, | | → ∞ the testfunction ϕ canbe any functions inX suchthat ϕ2 φ2dx< Rd| | |∇ | ∞ and φ2 ϕ2dx< . Rd| | |∇ | ∞ R Firstly, we show that for all q 1, (1+φ2q) φ2dx < . Choosing R ≥ Rd |∇ | ∞ q =12,since φ2 L2andφ H1,wecangetthatφ Lp(Rd)forp [2,q ] 0 ∇ ∈ ∈ R ∈ ∈ 0 8 WeizhuBaoetal. and d=1,2,3. Let M >0 and M, φ(x)>M, φ (x)= φ(x), φ(x) M, x Rd, M  | |≤ ∈ M, φ(x)< M,  − − and take ϕ = φM q0−4φM as the test function. Plugging ϕ = φM q0−4φM | | | | into (2.14), we obtain 1 (q 3) +2δφ2 φ q0−4 φ φ dx+2δ φφ φ q0−4 φ2dx 0 M M M M − ZRd(cid:18)2 (cid:19)| | ∇ ·∇ ZRd | | |∇ | + V(x)φφ φ q0−4dx= β φ2φ+µφ φ q0−4φ dx. M M M M ZRd | | ZRd − | | | | (cid:0) (cid:1) Letting M , we get →∞ 2(q 2)δ φ2q˜ φ2dx+ V(x)φ2q˜dx β φq0 + µ φq0−2 dx, 0 − ZRd| | |∇ | ZRd | | ≤ZRd | || | | || | (cid:0) ((cid:1)2.15) which shows φq˜ φ2dx < with q˜= q0 1. So φq˜+1 L2 and for Rd| | |∇ | ∞ 2 − ∇ ∈ q =6q˜=3q =36,φ Lp(Rd)forp [2,q ]andd=1,2,3.Then,theMoser it1eration can0Rcontinue∈with qj = 3jq0∈, and1φ Lqj(Rd) (it is obvious when ∈ d=1,2) which verifies our claim. In particular φ Lp for any p [2, ). Secondly,weshowthatφ L∞(Rd)andlim|x|→∈∞φ(x)=0by∈De G∞iorgi’s ∈ iteration. Denoting f = β φ2φ+µφ and choosing the test function ϕ(x) = − | | (ξ(x))2(φ(x) k) with k 0 in (2.14), where (g(x)) = max g(x),0 and + + − ≥ { } ξ(x) is a smooth cutoff function, we have 1 +2δφ2+2δφ(φ k) ξ 2 (φ k) 2+V(x)ξ 2φ(φ k) dx + + + ZRd(cid:20)(cid:18)2 − (cid:19)| | |∇ − | | | − (cid:21) = (1+4δφ2)(φ k) ξ (φ k) ξ+fξ2(φ k) dx. + + + ZRd − − ∇ − ·∇ − (cid:2) (cid:3) Cauchy inequality gives that (1+4δφ2)(φ k) ξ (φ k) ξdx + + ZRd− − ∇ − ·∇ ε (1+φ2) (φ k) 2dx+C (1+φ2) ξ 2(φ k)2 dx. ≤ ZRd |∇ − +| εZRd |∇ | − + Now choosing sufficiently small ε > 0 and defining the function Φ (x) = k (1+φ)(φ k) , we can get + − Φ 2 ξ 2dx C ξ 2Φ2dx+C f (φ k) ξ2dx, (2.16) ZRd|∇ k| | | ≤ ZRd|∇ | k ZRd| | − + and (ξΦ )2dx C ξ 2Φ2dx+C f (φ k) ξ2dx. (2.17) ZRd|∇ k | ≤ ZRd|∇ | k ZRd| | − + TitleSuppressedDuetoExcessiveLength 9 Sincef = β φ2φ+µφ Lq(Rd)forany2 q < ,wecanproceedtoobtain − | | ∈ ≤ ∞ L∞ bound of φ by De Giorgi’s iteration. Let B (x) be the ball centered at x r with radius r >0, and we use B for short to denote the ball centered at the r origin.For0<r <R 1,wechooseC∞ nonnegativecutofffunctionξ(x)=1 ≤ 0 for x B (x ) and ξ(x) = 0 for x / B (x ) such that ξ(x) 2 . Since ∈ r 0 ∈ R 0 |∇ | ≤ R−r for large q, f (φ k)+ξ2dx ξf Lq (φ k)+ξ L6 Φkξ >0 65−1q, (2.18) ZRd| | − ≤k k k − k |{ }| where A denotes the Lebesgue measure of the set A, for any ε>0, we have | | by Ho¨lder inequality and Sobolev inequality in 2D and 3D f (φ k) ξ2dx + ZRd| | − C ξf Lq ((φ k)+ξ) Φkξ >0 65−1q ≤ k k k∇ − k|{ }| ≤εk∇((φ−k)+ξ)k2+Cεkξfk2Lq|{Φkξ >0}|35−q2 ≤4ε(k∇(Φkξ)k2+2kΦk∇ξk2)+Cεkξfk2Lq|{Φkξ >0}|35−2q. Thus, from the above inequality and (2.17), we arrive at k∇(ξΦk)k2 ≤C kΦk∇ξk2+kξfk2Lq|{Φkξ >0}|35−q2 . (2.19) (cid:16) (cid:17) Since ξΦ H1(B (x )), we conclude by Sobolev inequality that, k ∈ 0 1 0 kξΦkk2 ≤kξΦkk2L6|{Φkξ >0}|1−62 ≤C(d)k∇(ξΦk)k2|{Φkξ >0}|32. (2.20) By choosing q =3, (2.19) and (2.20) imply that kξΦkk2 ≤C kΦk∇ξk2|{Φkξ >0}|32 +kfk2L3(B1(x0))|{Φkξ >0}|35 . (2.21) (cid:16) (cid:17) Denote A(k,r)= xx B (x ), φ(x)>k . (2.22) r 0 { | ∈ } For k >0 and 0<r <R 1, we have ≤ 2 ZA(k,r)Φ2kdx≤C"|A(R(k−,Rr))|23 ZA(k,R)Φ2kdx+|A(k,R)|53 kfk2L3(B1(x0))#. (2.23) We claim that there exists C˜ > 0, such that for k = C˜ kfkL3(B1(x0))+ k(1+ h φ)φkL2(B1(x0)) , i Φ2dx=0. (2.24) k ZA(k,12) Taking h>k >k and 0<r<1, we find A(h,r) A(k,r) with 0 ⊂ Φ2dx Φ2dx. (2.25) h ≤ k ZA(h,r) ZA(k,r) 10 WeizhuBaoetal. In addition, since Φ =(1+φ)(φ k) , we have k + − 1 A(h,r) = B (x ) φ k h k Φ2dx. (2.26) | | | r 0 ∩{ − ≥ − }|≤ (h k)2 k − ZA(k,r) Choosing 1 r <R 1, from (2.23), we get 2 ≤ ≤ Φ2dx h ZA(h,r) 1 ≤C (R−r)2 ZA(h,R)Φ2hdx+kfk2L3(B1(x0))|A(h,R)|!|A(h,R)|23 5 C 1 + kfk2L3(B1(x0)) Φ2dx 3 , ≤ (h−k)34 (R−r)2 (h−k)2 ! ZA(k,R) k ! and kΦhkL2(Br(x0)) ≤ (h−Ck)23 (cid:18)R1−r + kfkLh3−(B1k(x0))(cid:19)kΦkkL352(BR(x0)). (2.27) Denote the function χ(k,r)=kΦkkL2(Br(x0)). (2.28) For some value of k >0 to be determined later, we define 1 1 1 k = 1 k, r = + , l=0,1,2,..., (2.29) l − 2l l 2 2l+1 (cid:18) (cid:19) then k k = k and r r = 1 . From (2.27), we find l− l−1 2l l−1− l 2l+1 χ(kl,rl)≤C 2l+1+ 2lkfkL3k(B1(x0))!2k2323l(χ(kl−1,rl−1))35 ≤2CkfkL3(Bk153(x0))+k253l(χ(kl−1,rl−1))53. Then, we prove that there exists γ >1 such that χ(k ,r ) 0 0 χ(k ,r ) , l=0,1,2,... (2.30) l l ≤ γl We will argue by induction. When l =0, it is obvious true. Suppose (2.30) is true for l 1 with l 1, i.e. − ≥ χ(k0,r0) 5 γ35(χ(k0,r0))23 χ(k0,r0) χ(kl−1,rl−1)≤ γl−1 ⇒(χ(kl−1,rl−1))3 ≤ γ23l · γl .