Existence and symmetry of solutions for critical fractional Schr¨odinger equations with bounded potentials Xia Zhanga, Binlin Zhangb,∗ and Duˇsan Repovˇsc 7 1 a DepartmentofMathematics, HarbinInstituteofTechnology, Harbin150001, P.R.China 0 2 b DepartmentofMathematics,HeilongjiangInstituteofTechnology, Harbin150050, P.R.China n c FacultyofEducationandFacultyofMathematics andPhysics,UniversityofLjubljana, a J Kardeljevaploˇsˇcad 16,SI-1000Ljubljana,Slovenia 9 ] P A Abstract . h This paper is concerned with the following fractional Schro¨dinger equations involvingcritical exponents: at (−∆)αu+V(x)u=k(x)f(u)+λ|u|2∗α−2u in RN, m where (−∆)α is the fractional Laplacian operator with α ∈ (0,1), N ≥ 2, λ is a positive real parameter [ and 2∗α =2N/(N −2α) is the critical Sobolev exponent, V(x) and k(x) are positive and bounded functions 1 satisfying some extra hypotheses. Based on the principle of concentration compactness in the fractional v Sobolev space and the minimax arguments, we obtain the existence of a nontrivial radially symmetric weak 1 solution for the above-mentioned equations without assuming the Ambrosetti-Rabinowitz condition on the 0 subcritical nonlinearity. 1 2 Keywords: fractional Schr¨odinger equations; critical Sobolev exponent; Ambrosetti-Rabinowitz condi- 0 tion; concentration compactness principle . 1 2010 MSC: 35A15, 35J60, 46E35. 0 7 1 1 Introduction and main result : v Xi In this paper, we study the solutions of the following Schr¨odinger equations involving a critical nonlinearity: ar (−∆)αu+V(x)u=k(x)f(u)+λ|u|2∗α−2u in RN, (1.1) drivenbythefractionalLaplacianoperator(−∆)α oforderα∈(0,1),whereN ≥2,λisapositiverealparameter and 2∗ =2N/(N −2α) is the critical Sobolev exponent. α The fractional Laplacian operator (−∆)α, which (up to normalization constants), may be defined as u(x)−u(y) (−∆)αu(x):=P.V. dy, x∈RN, ZRN |x−y|N+2α where P.V. stands for the principal value. It may be viewed as the infinitesimal generators of a L´evy stable diffusionprocesses(see[1]). This operatorarisesinthe descriptionofvariousphenomenainthe appliedsciences, such as phase transitions, materials science, conservation laws, minimal surfaces, water waves, optimization, ∗Corresponding author. E-mail address:[email protected] (X. Zhang), [email protected](B. Zhang), du- [email protected](D.Repovˇs) 1 plasma physics and so on, see [13] and references therein for more detailed introduction. Here we would like to pointoutsomeinterestingmodelsinvolvingthefractionalLaplacian,suchas,thefractionalSchr¨odingerequation (see[14,15,22,23,24]),the fractionalKirchhoffequation(see[16,32,33,46,47]), thefractionalporousmedium equation (see [9, 45]), the fractional Yamabe problem (see [34]) and so on, have attracted recently considerable attention. Asamatteroffact,theliteratureonfractionaloperatorsandtheirapplicationstopartiallydifferential equations is quite large, here we would like to mention a few, see for instance [2, 11, 12, 26, 27, 35]. In what follows, let us sketch the related advance involving the fractional Schr¨odinger equations with critical growth in resent years. In [37], Shang and Zhang studied the existence and multiplicity of solutions for the critical fractional Schr¨odinger equation: ε2α(−∆)αu+V(x)u=λf(u)+|u|2∗α−2u in RN. (1.2) Based on variational methods, they showed that problem (1.2) has a nonnegative ground state solution for all sufficiently large λ and small ε. In this paper, the following monotone condition was imposed on the continuous subcritical nonlinearity f: f(t)/t is strictly increasing in (0,+∞). (1.3) t Observe that (1.3) implies 2F(t) < f(t)t, where F(t) := f(ξ)dξ. Moreover, Shen and Gao in [36] obtained 0 the existence of nontrivial solutions for problem (1.2) under various assumptions on f(t) and potential function R V(x), inwhichthe authorsassumedthe well-knownAmbrosetti-Rabinowitzcondition((AR) conditionforshort) on f: there exists µ>2 such that 0<µF(t)≤f(t)t for any t>0. (1.4) Seealsorecentpapers[38,42]onthefractionalSchr¨odingerequations(1.2). In[44],TengandHewereconcerned with the following fractional Schr¨odinger equations involving a critical nonlinearity: (−∆)αu+u=P(x)|u|p−2u+Q(x)|u|2∗α−2u in RN. (1.5) where2<p<2∗,potentialfunctionsP(x)andQ(x)satisfycertainhypotheses. Usingthes-harmonicextension α technique of Caffarelli and Silvestre [10], the concentration-compactness principle of Lions [29] and methods of Br´ezis and Nirenberg [4], the author obtained the existence of ground state solutions. On fractional Kirchhoff problems involving critical nonlinearity, see for example [3, 31] for some recent results. Last but not least, fractionalelliptic problems withcriticalgrowth,ina boundeddomain, havebeen studiedby someauthorsinthe last years, see [6, 7, 18, 28, 39, 41] and references therein. On the other hand, Feng in [17] investigated the following fractional Schr¨odinger equations: (−∆)αu+V(x)u=λ|u|p−2u in RN, (1.6) where 2 < p < 2∗, V(x) is a positive continuous function. By using the fractional version of concentration α compactness principle ofLions [29], the author obtainedthe existence of groundstate solutions to problem(1.6) for some λ > 0. Zhang et al. in [48] considered the following fractional Schr¨odinger equations with a critical nonlinearity: (−∆)αu+u=λf(u)+|u|2∗α−2u in RN. (1.7) Based on another fractional version of concentration compactness principle (see [30, Theorem 1.5]) and radially decreasing rearrangements,they obtained the existence of a ground state solution for (1.7) which is nonnegative and radially symmetric for any λ∈[λ ,∞), where λ >0. ∗ ∗ Inspired by the above works, we are interested in non autonomous cases (1.1), that is, V(x) is not only a constant. To this end, we assume the following conditions on the potential V: (V1) V ∈C1(RN,R) and ∇V(x)·x≤0 for any x∈RN; 2 (V2) V is radially symmetric, i.e. V(x) = V(|x|) for any x ∈ RN and there exist positive constants V and V 1 2 such that V ≤V(x)≤V for any x∈RN. 1 2 Moreover,the following assumptions are imposed on the coefficient k: (K1) k is radially symmetric and there exist positive constants k and k such that k ≤ k(x) ≤ k for any 1 2 1 2 x∈RN; (K2) k ∈C1(RN,R) and there exists a constant k such that 0≤∇k(x)·x≤k for any x∈RN. 0 0 Remark 1.1 Since ∇V(x)·x=V′(|x|)|x|, it follows from (V1) that V′(|x|)≤0. Thus we can choose V to be a positive constant. Another example for V is given by V(x) = 2−arctan|x|. From (K2), k′(|x|) ≥ 0. Hence we can choose k(x)=2+arctan|x| as a simple example. The condition (V1) and (K2) were motivated by [20, 43]. Meanwhile, the nonlinearity f will satisfy: (H1) f ∈C1(R,R). For any t≤0, f(t)=0; (H2) limt→0+ f(tt) =0 and limt→+∞ t2f∗α(t−)1 =0; (H3) For any t>0, 0<2F(t)≤f(t)t; (H4) There exists T >0 such that F(T)> V2 T2. 2k1 Remark 1.2 In order to seek nonnegative solutions of (1.1), we assume that f(t) = 0 for any t ≤ 0 in (H1). Moreover, from (H2) we know that f is subcritical. Here we do not assume classical condition (1.3) or (1.4), while the weakercondition(H3) onf is employedto replace(AR) condition. A typicalexample forf is givenby 1 3 f(t)=tlog 1+t t2− t− at+a 2 2 (cid:20) (cid:18) (cid:19)(cid:21) for any t≥0 and a certain constant a>1/3 which is sufficiently close to 1/3. It is easy to see that the function f does not fulfill the monotone condition (1.3) and the (AR) condition (1.4). Now we give the definition of weak solutions for problem (1.1): Definition 1.1. We say that u is a weak solution of (1.1) if for any φ∈Hα(RN), ((−∆)α2u·(−∆)α2φ+V(x)uφ)dx= (k(x)f(u)+λ|u|2∗α−2u)φdx, ZRN ZRN where Hα(RN) is the fractional Sobolev space, see Section 2 for more details. The energy functional on Hα(RN) is defined as follows: I(u)= 1 (|(−∆)α2u|2+V(x)u2)dx− k(x)F(u)dx− λ |u|2∗αdx. 2ZRN ZRN 2∗α ZRN ItiseasytocheckthatI ∈C1(Hα(RN), R)andthecriticalpointforI istheweaksolutionofproblem(1.1). Let O(N) be the groupof orthogonallinear transformations in RN. It is immediate that I is O(N)-invariant. Then, bytheprincipleofsymmetriccriticalityofKrawcewiczandMarzantowicz[21],weknowthatu isacriticalpoint 0 of I if and only if u is a critical point of 0 I =I , Hα(RN) r where (cid:12) e (cid:12) Hα(RN)= u∈Hα(RN): u(x)=u(|x|) , r is the fractional radially symmetric Sobolev sp(cid:8)ace. Therefore, it suffices to(cid:9)prove the existence of critical points for I on Hα(RN). r Now we are in a position to state our main result as follows: e 3 Theorem 1.1. Assume that hypotheses (H1)–(H4), (V1)–(V2) and (K1)–(K2) are fulfilled. Then there exists λ > 0 such that for any λ ∈ (0,λ ), problem (1.1) has a nontrivial weak solution u ∈ Hα(RN) which is ∗ ∗ 0 nonnegative and radially symmetric. Remark 1.3 (i) In the proof of Theorem 1.1, we follow an approximation procedure to obtain a bounded (PS) sequence {u } for I, instead of starting directly from an arbitrary (PS) sequence. To show the boundedness of n (PS) sequences {u } for I, we need condition (K2)onk. It allowsus to make use of a Pohozaevtype identity to n derive the boundedeness of {u }. A key point which allows to use the identity is that {u } is a sequence of exact n n critical points. In fact, thee requirement f ∈C1(R,R) will only be used in the proof of Pohozaev identity. (ii)Tothebestofourknowledge,thereareonlyfewpapersthatstudytheexistenceandsymmetryofsolutions for problem (1.1) by using concentrationcompactness principle in the fractionalSobolev space which is different from the version used in [17]. This paper is organized as follows. In Section 2, we will give some necessary definitions and properties of fractional Sobolev spaces. In Section 3, by using the principle of concentration compactness and minimax arguments, we give the proof of Theorem 1.1. 2 The Variational Setting For the convenience of the reader, in this part we recall some definitions and basic properties of fractional Sobolev spaces Hα(RN). For a deeper treatment on these spaces and their applications to fractional Laplacian problems of elliptic type, we refer to [13, 25] and references therein. For any α∈(0,1), the fractional Sobolev space Hα(RN) is defined by Hα(RN)= u∈L2(RN):[u]Hα(RN) <∞ , (cid:8) (cid:9) where [u]Hα(RN) denotes the so-called Gagliardo semi-norm, that is |u(x)−u(y)|2 1/2 [u]Hα(RN) =(cid:18)ZZR2N |x−y|N+2α dxdy(cid:19) and Hα(RN) is endowed with the norm ||u||Hα(RN) =[u]Hα(RN)+||u||L2(RN). As it is well known, Hα(RN) turns out to be a Hilbert space with scalar product (u(x)−u(y))(v(x)−v(y)) hu, viHα(RN) =ZZR2N |x−y|N+2α dxdy+ZRN u(x)v(x)dx, for any u,v∈Hα(RN). The space H˙α(RN) is defined as the completion of C0∞(RN) under the norm [u]Hα(RN). By Proposition 3.6 in [13], we have [u]Hα(RN) =||(−∆)α2u||L2(RN) for any u∈Hα(RN), i.e. |u(x)−u(y)|2 dxdy = |(−∆)α2u(x)|2dx. (2.1) ZZR2N |x−y|N+2α ZRN Thus, (u(x)−u(y))(v(x)−v(y)) α α dxdy = (−∆)2u(x)·(−∆)2v(x)dx. (2.2) ZZR2N |x−y|N+2α ZRN Theorem 2.1. ([15, Lemma 2.1]) The embedding Hα(RN)֒→Lp(RN) is continuous for any p∈[2,2∗] and the α embedding Hα(RN)֒→֒→Lp (RN) is compact for any p∈[2,2∗). loc α 4 3 Proof of Theorem 1.1 Throughout this section, we assume that conditions (H1)–(H4), (V1)–(V2) and (K1)–(K2) are satisfied. In this part, wewill use minimax argumentsandwe denote that C are C arepositive constant,for anyi=1,2···. i A crucial step to obtain the existence of a critical point for I is to show the boundedness of (PS) sequence. But it seems difficult under our assumptions. To overcome this difficulty we use an indirect approachdeveloped in [19]. For any η ∈[1/2,1], we consider the following family ofefunctionals defined on Hα(RN): r Iη(u)= 12ZRN(|(−∆)α2u|2+V(x)u2)dx−ηZRN k(x)F(u)dx− η2∗αλZRN |u|2∗αdx. It is easy to check that I ∈ C1(Hα(RN), R) and the critical point for I is the weak solution of the following η r η equation: (−∆)αu+V(x)u=ηk(x)f(u)+ηλ|u|2∗α−2u in RN. (3.1) First, we will give the following two lemmas to show that I has a Mountain Pass geometry. η Lemma 3.1. There exists v ∈Hα(RN) and η ∈[1/2,1) such that I (v )<0 for any η ∈[η,1], where v and η 0 r η 0 0 are independent of λ. Proof. Let R>0, we define T for |x|≤R, w(x)= T(R+1−|x|) for R<|x|<R+1,  0 for |x|≥R+1, then w ∈Hα(RN). Hence, from (H4)we have r 1 k F(w)− V w2 dx 1 2 ZRN (cid:18) 2 (cid:19) 1 1 = k F(w)− V w2 dx+ k F(w)− V w2 dx 1 2 1 2 2 2 ZB(0,R)(cid:18) (cid:19) ZB(0,R+1)\B(0,R)(cid:18) (cid:19) 1 1 ≥ k F(T)− V T2 B(0,R) − B(0,R+1)\B(0,R) · max k F(t)− V t2 1 2 1 2 2 t∈[0,T] 2 (cid:18) (cid:19) ≥C1RN −C2RN−1, (cid:12)(cid:12) (cid:12)(cid:12) (cid:12)(cid:12) (cid:12)(cid:12) (cid:12)(cid:12) (cid:12)(cid:12) where |·| denotes the Lebesgue measure and C , C are positive constants. So we could choose R > 0 large 1 2 enough such that 1 k F(w)− V w2 dx>0. 1 2 ZRN (cid:18) 2 (cid:19) Define 1 V w2dx η =max , RN 2 , 2 k F(w)dx (cid:26) RRN 1 (cid:27) then we have that η ≥1/2. Thus, for any η ∈[η,1] anRd θ >0, from (K1) it follows that x 1 x x x Iη(w( ))≤ |(−∆)α2w( )|2+V(x)|w( )|2 dx−η k(x)F(w( ))dx θ 2ZRN (cid:16) θ θ (cid:17) ZRN θ 1 1 ≤ θN−2α |(−∆)α2w|2dx+ θNV2 w2dx−θNη k1F(w)dx 2 ZRN 2 ZRN ZRN 1 1 1 = θN−2α |(−∆)α2w|2dx− θNmax V2w2dx, k1F(w)dx . 2 ZRN 2 (cid:26)ZRN 2ZRN (cid:27) Then there exists θ >0 such that for any θ ≥θ, I (w(x/θ)) <0. We take v (x) =w(x/θ). Therefore the proof η 0 is complete. 5 Lemma 3.2. For any η ∈[η,1], define c = inf max I (γ(t)), η η γ∈Γηt∈[0,1] where Γ = {γ ∈ C([0,1],Hα(RN)) : γ(0) = 0,γ(1) = v }, η and v are from Lemma 3.1. Then c > η r 0 0 η max{I (0),I (v )} and there exists c >0 such that c ≤c for any η ∈[η,1], where c is independent of λ. η η 0 0 η 0 0 Proof. According to (H1) and (H2), for any ε>0, there exists a constant C(ε)>0 such that for any t∈R, f(t)≤ε|t|+C(ε)|t|2∗α−1. (3.2) By (3.2), for any ε∈(0,1), we get F(t)≤εt2+C(ε)|t|2∗α. (3.3) Taking ε=V /(4k ), for any u∈Hα(RN) and η ∈[η,1], we obtain 1 2 r Iη(u)≥12ZRN(|(−∆)α2u|2+V1u2)dx−ZRN k2F(u)dx− 2λ∗α ZRN |u|2∗αdx ≥12ZRN |(−∆)α2u|2dx+(V1/2−εk2)ZRN u2dx−C(ε)k2ZRN |u|2∗αdx− 2λ∗α ZRN |u|2∗αdx ≥1 |(−∆)α2u|2dx+ V1 u2dx−C |u|2∗αdx− λ |u|2∗αdx 2ZRN 4 ZRN ZRN 2∗α ZRN ≥min{1/2,V /4}||u||2 −C||u||2∗α . 1 Hα(RN) Hα(RN) Thanks to 2∗α > 2, there exist 0 < ρ < ||v0||Hα(RN) and σ > 0 such that Iη(u) ≥ σ for any u ∈ Hrα(RN) with ||u||Hα(RN) = ρ. For any γ ∈ Γη, we have γ(0) = 0 and γ(1) = v0. Then, there exists tη ∈ (0,1) such that ||γ(tη)||Hα(RN) =ρ, which implies c ≥ inf I (γ(t ))≥σ >max{I (0),I (v )}. η η η η η 0 γ∈Γη Take γ (t)=tv , then γ ∈Γ . For any t∈[0,1], we obtain 0 0 0 η 1 Iη(γ0(t))=Iη(tv0)≤2ZRN(|(−∆)α2v0|2+V(x)v02)dx,c0, which implies that c ≤max I (γ (t))≤c for any η ∈[η,1]. Thus we have completed the proof. η t∈[0,1] η 0 0 Theorem 3.1. ([19,Theorem1.1])Let (X,||·|| ) be a Banach space and I ⊂R+ an interval. Consider a family X {J } of C1 functionals on X with the form η η∈I J (u)=A(u)−ηB(u), ∀η ∈I, η where B(u) ≥ 0, ∀u ∈ X, and such that either A(u) → +∞ or B(u) → +∞ as ||u|| → ∞. If there are two X points v , v ∈X such that 1 2 c = inf max J (γ(t))>max{J ,J }, η ∈I, η γ∈Γηt∈[0,1] η v1 v2 where Γ ={γ ∈C([0,1],X):γ(0)=v ,γ(1)=v }, η 1 2 then, for almost every η ∈I, there exists a sequence {v }⊂X such that n (i) {v } is bounded; n (ii) J (v )→c ; η n η (iii) J′(v )→0 in the dual X′ of X. η n 6 Remark 3.1 In fact, the map η →c is nonincreasing and continuous from the left (see [19]). η By using Lemma 3.1, Lemma 3.2 and Theorem 3.1, we obtain that for any η ∈ [η,1], I possesses a bounded η (PS) sequence at the level c . η Next we will verify that each bounded (PS) sequence for the functional I contains a convergentsubsequence. η The main difficulties here are that the embedding Hrα(RN) ֒→ L2∗α(RN) is not compact and we do not have a similar radial lemma (see [5]) in Hα(RN). To get the compactness of bounded (PS) sequence in Hα(RN), we r r assume that λ in (1.1) is small. Based on the following principle of concentration compactness in Hα(RN) and r Lemma 2.4 in [12], we obtain Lemma 3.5. Theorem 3.2. ([30, Theorem 1.5]) Let Ω ⊆ RN an open subset and let {u } be a sequence in H˙α(Ω) weakly n converging to u as n→∞ and such that |(−△)α2un|2 →µ and |un|2∗α →ν weakly-∗ in M(RN). Then, either u → u in L2∗α(RN) or there exists a (at most countable) set of distinct points {x } ⊂ Ω and n loc j j∈J positive numbers {ν } such that we have j j∈J ν =|u|2∗α + νjδxj. j∈J X If, in addition, Ω is bounded, then there exist a positive measure µ ∈ M(RN) with suppµ ⊂ Ω and positive numbers {µ } such that j j∈J µ=|(−△)α2u|2+µ+ µjeδxj. e j∈J X Remark 3.2 In the case Ω = RN, the above principleeof concentration compactness does not provide any information about the possible loss of mass at infinity. The following result expresses this fact in quantitative terms, and the proof. Lemma 3.3. Let {un} ⊂ H˙α(RN) such that un → u weakly in H˙α(RN), |(−△)α2un|2 → µ and |un|2∗α → ν weakly-∗ in M(RN), as n→∞ and define µ∞ = lim limsup |(−∆)α2un|2dx, R→∞ n→∞ Z{x∈RN:|x|>R} ν∞ = lim limsup |un|2∗αdx. R→∞ n→∞ Z{x∈RN:|x|>R} The quantities µ and ν are well defined and satisfy ∞ ∞ limsup |(−∆)α2un|2dx= dµ+µ∞, n→∞ ZRN ZRN limsup |un|2∗αdx= dν+ν∞. (3.4) n→∞ ZRN ZRN Proof. The proof is similar to that of Lemma 3.5 in [48]. Thus we just give a sketch of the proof for the reader’s convenience. Take φ ∈ C∞(RN) such that 0 ≤ φ ≤ 1; φ ≡ 1 in RN \B(0,2), φ ≡ 0 in B(0,1). For any R > 0, define φ (x)=φ(x/R). Then we have R |(−∆)α2un|2dx≤ |(−∆)α2un|2φRdx≤ |(−∆)α2un|2dx, Z{x∈RN:|x|>2R} ZRN Z{x∈RN:|x|>R} thus µ∞ =limR→∞limsupn→∞ RN |(−∆)α2un|2φRdx. Note that R |(−∆)α2un|2dx= |(−∆)α2un|2φRdx+ |(−∆)α2un|2(1−φR)dx. ZRN ZRN ZRN 7 It is easy to verify that |(−∆)α2un|2(1−φR)dx→ (1−φR)dµ, ZRN ZRN as n→∞. Hence we have µ(RN)= lim lim |(−∆)α2un|2(1−φR)dx. R→∞n→∞ZRN Then limsup |(−∆)α2un|2dx= lim limsup |(−∆)α2un|2φRdx+ (1−φR)dµ n→∞ ZRN R→∞(cid:18) n→∞ ZRN ZRN (cid:19) =µ +µ(RN). ∞ Similarly, we obtain that limsupn→∞ RN |un|2∗αdx=ν(RN)+ν∞. The lemma is thus proved. In the sequel, we derive some resultRs involving ν for any i∈J and ν . i ∞ Lemma 3.4. Let {un} ⊂ H˙α(RN) such that un → u weakly in H˙α(RN), |(−△)α/2un|2 → µ and |un|2∗α → ν weakly-∗ in M(RN), as n→∞. Then, νi ≤(Sα−1µ({xi}))2∗α/2 for any i∈J and ν∞ ≤(Sα−1µ∞)2∗α/2, where xi, ν are from Theorem 3.2 and µ , ν are from Lemma 3.3, S is the best Sobolev constant of the embedding i ∞ ∞ α H˙α(RN)֒→L2∗α(RN) (see [13]), i.e. S = inf RN |(−∆)α2u|2dx. (3.5) α u∈H˙α(RN)R ||u||2L2∗α(RN) Proof. (1)Takeϕ∈C∞(RN)suchthat0≤ϕ≤1; ϕ≡1inB(0,1), ϕ≡0inRN\B(0,2). Foranyε>0,define 0 ϕ (x)=ϕ(x−xi), where i∈J. It follows from (2.1) and (3.5) that ε ε ZRN |unϕε|2∗αdx≤(cid:18)Sα−1ZZR2N |un(x)ϕε|x(x−)−y|Nun+(2yα)ϕε(y)|2 dxdy(cid:19)2∗α/2. We have |unϕε|2∗αdx→ ϕ2ε∗αdν, as n→∞, ZRN ZRN 2∗ ϕ αdν →ν({x })=ν , as ε→0. ε i i ZRN Note that |u (x)ϕ (x)−u (y)ϕ (y)|2 n ε n ε dxdy ZZR2N |x−y|N+2α |u (x)ϕ (x)−u (x)ϕ (y)+u (x)ϕ (y)−u (y)ϕ (y)|2 n ε n ε n ε n ε = dxdy ZZR2N |x−y|N+2α u2(x)(ϕ (x)−ϕ (y))2 ϕ2(y)(u (x)−u (y))2 = n ε ε dxdy+ ε n n dxdy ZZR2N |x−y|N+2α ZZR2N |x−y|N+2α 2u (x)ϕ (y)(u (x)−u (y))(ϕ (x)−ϕ (y)) n ε n n ε ε + dxdy, ZZR2N |x−y|N+2α we get ϕ2(y)(u (x)−u (y))2 ε n n dxdy → ϕ2dµ, as n→∞, ZZR2N |x−y|N+2α ZRN ε ϕ2dµ→µ({x }), as ε→0. ε i ZRN 8 Since {u } is bounded in H˙α(RN), by the Ho¨lder inequality we obtain n u (x)ϕ (y)(u (x)−u (y))(ϕ (x)−ϕ (y)) n ε n n ε ε dxdy (cid:12)ZZR2N |x−y|N+2α (cid:12) (cid:12)(cid:12) ϕ2(y)(u (x)−u (y))2 21 u2(x)(ϕ(cid:12)(cid:12) (x)−ϕ (y))2 1/2 ≤(cid:12) ε n n dxdy n (cid:12)ε ε dxdy (cid:18)ZZR2N |x−y|N+2α (cid:19) (cid:18)ZZR2N |x−y|N+2α (cid:19) u2(x)(ϕ (x)−ϕ (y))2 1/2 ≤C n ε ε dxdy . (cid:18)ZZR2N |x−y|N+2α (cid:19) In the following, we claim that u2(x)(ϕ (x)−ϕ (y))2 lim lim n ε ε dxdy =0. ε→0n→∞ZZR2N |x−y|N+2α Note that RN ×RN =((RN \B(x ,2ε))∪B(x ,2ε))×((RN \B(x ,2ε))∪B(x ,2ε)) i i i i =((RN \B(x ,2ε))×(RN \B(x ,2ε)))∪(B(x ,2ε)×RN) i i i ∪((RN \B(x ,2ε))×B(x ,2ε)). i i (i) If (x,y)∈(RN \B(x ,2ε))×(RN \B(x ,2ε)), then ϕ (x)=ϕ (y)=0. i i ε ε (ii) (x,y)∈B(x ,2ε)×RN. If |x−y|≤ε, |y−x |≤|x−y|+|x−x |≤3ε, which implies i i i u2(x)(ϕ (x)−ϕ (y))2 dx n ε ε dy ZB(xi,2ε) Z{y∈RN:|x−y|≤ε} |x−y|N+2α u2(x)|∇ϕ(ξ)|2|x−y|2 = dx n ε dy ZB(xi,2ε) Z{y∈RN:|x−y|≤ε} |x−y|N+2α u2(x) ≤Cε−2 dx n dy ZB(xi,2ε) Z{y∈RN:|x−y|≤ε} |x−y|N+2α−2 =Cε−2α u2(x)dx, n ZB(xi,2ε) where ξ =(y−x )/ε+τ(x−x )/ε and τ ∈(0,1). i i If |x−y|>ε, then we have u2(x)(ϕ (x)−ϕ (y))2 dx n ε ε dy ZB(xi,2ε) Z{y∈RN:|x−y|>ε} |x−y|N+2α u2(x) ≤C dx n dy ZB(xi,2ε) Z{y∈RN:|x−y|>ε} |x−y|N+2α =Cε−2α u2(x)dx. n ZB(xi,2ε) (iii) (x,y)∈(RN \B(x ,2ε))×B(x ,2ε). If |x−y|≤ε, |x−x |≤|x−y|+|y−x |≤3ε. Then i i i i u2(x)(ϕ (x)−ϕ (y))2 dx n ε ε dy ZRN\B(xi,2ε) Z{y∈B(xi,2ε):|x−y|≤ε} |x−y|N+2α u2(x) ≤Cε−2 dx n dy |x−y|N+2α−2 ZB(xi,3ε) Z{y∈B(xi,2ε):|x−y|≤ε} u2(x) ≤Cε−2 dx n dz ZB(xi,3ε) Z{z∈RN:|z|≤ε} |z|N+2α−2 =Cε−2α u2(x)dx. n ZB(xi,3ε) 9 Notice that there exists K > 4 such that (RN \B(x ,2ε))×B(x ,2ε) ⊂ (B(x ,Kε)×B(x ,2ε))∪((RN \ i i i i B(x ,Kε))×B(x ,2ε)). i i If |x−y|>ε, we obtain u2(x)(ϕ (x)−ϕ (y))2 dx n ε ε dy |x−y|N+2α ZB(xi,Kε) Z{y∈B(xi,2ε):|x−y|>ε} u2(x) ≤C dx n dy |x−y|N+2α ZB(xi,Kε) Z{y∈B(xi,2ε):|x−y|>ε} u2(x) ≤C dx n dz ZB(xi,Kε) Z{z∈RN:|z|>ε} |z|N+2α ≤Cε−2α u2(x)dx. n ZB(xi,Kε) If (x,y)∈(RN \B(x ,Kε))×B(x ,2ε), we get i i |x−x | |x−x | i i |x−y|≥|x−x |−|y−x |= + −|y−x | i i i 2 2 |x−x | K |x−x | i i ≥ + ε−2ε> , 2 2 2 which implies u2(x)(ϕ (x)−ϕ (y))2 dx n ε ε dy ZRN\B(xi,Kε) Z{y∈B(xi,2ε):|x−y|>ε} |x−y|N+2α u2(x) ≤C dx n dy ZRN\B(xi,Kε) Z{y∈B(xi,2ε):|x−y|>ε} |x−xi|N+2α u2(x) ≤CεN n dx ZRN\B(xi,Kε) |x−xi|N+2α 2/2∗ (2∗−2)/2∗ ≤CεN |un(x)|2∗αdx α |x−xi|−(N+2α)2∗α2∗α−2 dx α α ZRN\B(xi,Kε) ! ZRN\B(xi,Kε) ! 2/2∗ α =CK−N |un(x)|2∗αdx . ZRN\B(xi,Kε) ! In views of (i), (ii) and (iii), we have u2(x)(ϕ (x)−ϕ (y))2 n ε ε dxdy ZZR2N |x−y|N+2α u2(x)(ϕ (x)−ϕ (y))2 u2(x)(ϕ (x)−ϕ (y))2 = n ε ε dxdy+ n ε ε dxdy ZZB(xi,2ε)×RN |x−y|N+2α ZZ(RN\B(xi,2ε))×B(xi,2ε) |x−y|N+2α 2/2∗ (3.6) α ≤Cε−2α u2n(x)dx+CK−N |un(x)|2∗αdx ZB(xi,Kε) ZRN\B(xi,Kε) ! ≤Cε−2α u2(x)dx+CK−N. n ZB(xi,Kε) Note that u →u weakly in Hα(RN), by Theorem 2.1 we obtain u →u in L2 (RN), which implies n n loc Cε−2α u2(x)dx+CK−N →Cε−2α u2(x)dx+CK−N, n ZB(xi,Kε) ZB(xi,Kε) 10