On Rank Two Toda System with Arbitrary Singularities: Local Mass and New Estimates
aa r X i v : . [ m a t h . A P ] S e p ON RANK TWO TODA SYSTEM WITH ARBITRARYSINGULARITIES: LOCAL MASS AND NEW ESTIMATES
CHANG-SHOU LIN, JUNCHENG WEI, WEN YANG, AND LEI ZHANGA
BSTRACT . For all rank two Toda systems with an arbitrary singular source,we use a unified approach to prove: (i) The pair of local masses ( s , s ) at eachblowup point has the expression s i = ( N i m + N i m + N i ) , where N i j ∈ Z , i = , , j = , , . (ii) Suppose at each vortex point p t , ( a t , a t ) are integers and r i / ∈ p N , then all the solutions of Toda systems are uniformlybounded. (iii) If the blow up point q is not a vortex point, then u k ( x ) + | x − x k | ≤ C , where x k is the local maximum point of u k near q . (iv) If the blow up point q isa vortex point p t and a t , a t and 1 are linearly independent over Q , then u k ( x ) + | x − p t | ≤ C . The Harnack type inequalities of (iii) or (iv) are important for studying the be-havior of bubbling solutions near each blowup points.
1. I
NTRODUCTION
Let ( M , g ) be a Riemann surface without boundary and K = ( k i j ) n × n be theCartan matrix of a simple Lie algebra of rank n . For example for the Lie algebra sl ( n + ) (the so called A n ) we have(1.1) K = − ... − − ... ... − − ... − . In this paper we consider solution u = ( u , ..., u n ) of the following system definedon M :(1.2) D g u i + n (cid:229) j = k i j r j ( h j e u j R M h j e u j dV g − ) = (cid:229) P t ∈ S pa it ( d P t − ) , Date : July 18, 2018.1991
Mathematics Subject Classification.
Key words and phrases.
SU(n+1)-Toda system, asymptotic analysis, a priori estimate, classifica-tion theorem, topological degree, blowup solutions, Riemann-Hurwitz Theorem. where D g is the Laplace-Beltrami operator ( − D g ≥ h , ..., h n are positive andsmooth functions on M , a it > − d P t , r = ( r , ..., r n ) is a constant vector with nonnegative components. Here for simplicity we just as-sume that the total area of M is 1.Obviously, equation (1.2) remains the same if u i is replaced by u i + c i for anyconstant c i . Thus we might assume that each component of u = ( u , ..., u n ) is in ˚ H ( M ) : = { v ∈ L ( M ) ; (cid:209) v ∈ L ( M ) , and Z M vdV g = } . Then equation (1.2) is the Euler-Lagrange equation for the following nonlinearfunctional J r ( u ) in ˚ H ( M ) : J r ( u ) = Z M n (cid:229) i , j = k i j (cid:209) g u i (cid:209) g u j dV g − n (cid:229) i = r i log Z M h i e u i dV g , where ( k i j ) n × n = K − .It is hard to overestimate the importance of system (1.2), as it covers a largenumber of equations and systems deeply rooted in geometry and Physics. Even if(1.2) is reduced to a single equation with Dirac sources, it is a mean field equa-tion that has been extensively studied for decades. The singular sources on theright hand side of the mean field equation describe conic singularities and solu-tions can be interpreted as metrics with prescribed conic singularities. This is aclassical problem in differential geometry and extensive references can be foundin [2, 3, 13, 21, 22, 34, 35, 37] etc. Recently profound relations among meanfield equation, classical Lame equation, hyper-elliptic curves, modular forms andPainleve equation have been discovered and developed (see [6] and [9]).When (1.2) has more than one equation, it has close ties with algebraic geometryand integrable system. For example, solutions of the sl ( n +
1) Toda system areclosely related to holomorphic curves in projective spaces. Let f be a holomorphiccurve from a domain D of R into CP n . Lift locally f to C n + and denote the liftby n ( z ) = [ n ( z ) , ..., n n ( z )] . The k th associated curve of f is defined by f k : D → G ( k , n + ) ⊂ CP n ( L k C n + ) , f k ( z ) = [ n ( z ) ∧ n ′ ( z ) ∧ ... ∧ n ( k − ) ( z )] , where n ( j ) is the j − th derivative of n with respect to z . Let L k ( z ) = n ( z ) ∧ ... ∧ n ( k − ) ( z ) , then the well known infinitesimal Pl¨uker formula gives(1.3) ¶ ¶ z ¶ ¯ z log k L k ( z ) k = k L k − ( z ) k k L k + ( z ) k k L k ( z ) k , for k = , , .., n , where we define the norm k · k = h· , ·i by the Fubini-Study metric in CP ( L k C n + ) and put k L ( z ) k =
1. We observe that (1.3) holds only for k L k ( z ) k >
0, i.e. all theunramification points z ∈ M . Setting k L n + ( z ) k = U k ( z ) = − log k L k ( z ) k + k ( n − k + ) log 2 , ≤ k ≤ n . ODA SYSTEM 3
Let p be a ramified point and { g p , , · · · , g p , n } be the total ramification index at p .Write: u ∗ i = n (cid:229) j = k i j U j , a p , i = n (cid:229) j = k i j g p , j , then we have(1.4) D u ∗ i + n (cid:229) j = k i j e u ∗ j − K = p (cid:229) p ∈ S a p , i d p , i = , ..., n , where K is the Gaussian curvature of the metric g .Therefore any holomorphic curve from M to CP n is associated with a solution u ∗ = ( u ∗ , ..., u ∗ n ) of (1.4). Conversely, given any solution u ∗ = ( u ∗ , ..., u ∗ n ) of (1.4)in S , we can construct a holomorphic curve of S into CP n , which has the givenramification index g p , i at p . One can see [21] for the detail of this construction.Therefore, equation (1.4) is related to the following problem: Given a set of ramifi-cated points and its ramification indexes at these points, can we find holomorphiccurves into CP n that satisfy the given ramification information?On the other hand, equation (1.2) is also related to many physical models fromgauge field theory. For example, to describe the physics of high critical temperaturesuperconductivity, a model of relative Chern-Simons model was proposed and thismodel can be reduced to a n × n system with exponential nonlinearity if the gaugepotential and the Higgs field are algebraically restricted. Then the Toda systemwith (1.1) is one of the limiting equations if the coupling constant tends to zero. Forextensive discussions on the relationship between Toda system and its backgroundin Physics we refer the readers to [4, 14, 16, 29, 38] and the reference therein.In this article we are concerned with rank 2 Today systems. There are threetypes of Cartan matrices of rank 2: A = (cid:18) − − (cid:19) B (= C ) = (cid:18) − − (cid:19) G = (cid:18) − − (cid:19) . One of our main theorems is the following estimate:
Theorem 1.1.
Let ( k i j ) × be one of the matrices above, h i be positive C functionson M, a it ∈ N ∪ { } , t ∈ { , , ..., N } and K be a compact subset of M \ S. If r i p N , there exists a constant C ( K , r , r ) such that for any solution u = ( u , u ) of (1.2) | u i ( x ) | ≤ C , ∀ x ∈ K , i = , . Our proof of Theorem 1.1 is based on the analysis of the behavior of solutions u k = ( u k , u k ) near each blowup point. A point p ∈ M is called a blowup point if wewrite ˜ u ki ( x ) = u ki ( x ) + p (cid:229) t a kt G ( x , p t ) , where G ( x , y ) is the Green’s function of theLaplacian operator on M with singularities at y ∈ M , and there exists a sequence ofpoints p k → p such that max i = , { u k ( p k ) , u k ( p k ) } → ¥ .Suppose u k is a sequence of solutions of (1.2). When n =
1, it has been provedthat if u k blows up somewhere, the mass distribution r he uk R M he uk will concentrate, that CHANG-SHOU LIN, JUNCHENG WEI, WEN YANG, AND LEI ZHANG is, r he u k R M he u k → S (cid:229) i = m i d p i , as k → ¥ , Which is equivalent to the fact that u k ( x ) → − ¥ if x is not a blowup point. This“blowup implies concentration” was first noted by Brezis-Merle [5] and was laterproved by Li [18], Li-Shafrir [19] and Bartolucci-Tarantello [2]. But for n ≥
2, thisphenomenon might fail in general. A component u ki is called not concentrating if u ki
6→ − ¥ away from blowup points, or equivalently, ˜ u ki converges to some smoothfunction w i away from blowup points. It is natural to ask whether it is possible tohave all components not concentrating. For n =
2, we prove it is impossible.
Theorem 1.2.
Suppose u k is a sequence of blowup solutions of a rank Todasystem (1.2). Then at least one component of u k satisfies u ki ( x ) → − ¥ if x is notcontained in the blowup set. The first example of such non-concentration phenomenon was first proved byLin-Tarantello [20]. The new phenomenon makes the study of systems ( n ≥ ) much more difficult than the mean field equation ( n = n ≥
3. This will be studied in a forthcoming project.As mentioned before our proofs of Theorem 1.1 and Theorem 1.2 are based onthe asymptotic behavior of local bubbling solutions. For simplicity we set up thesituation as follows:Let u k = ( u k , u k ) be a sequence of solutions of(1.5) D u ki + (cid:229) j = k i j h kj e u kj = pa i d , in B ( , ) , i = , , where a i > − B ( , ) is the unit ball in R ( we use B ( p , r ) to denote the ballwith centered p and radius r ). Throughout of the paper, h k , h k are smooth functionssatisfying h k ( ) = h k ( ) = C ≤ h ki ≤ C , k h ki k C ( B ( , )) ≤ C , in B ( , ) , i = , . For solutions u k = ( u k , u k ) we assume:(1.7) ( i ) : 0 is the only blowup point of u k , ( ii ) : | u ki ( x ) − u ki ( y ) | ≤ C , ∀ x , y ∈ ¶ B ( , ) , i = , , ( iii ) : R B ( , ) h ki e u ki ≤ C , i = , . For this sequence of blowup solutions we define the local mass by(1.8) s i = lim r → lim k → ¥ p Z B ( , r ) h ki e u ki , i = , . It is known that 0 is a blowup point if and only if ( s , s ) = ( , ) . The proofis to use ideas from [5] and has become standard now. We refer the readers to [17] ODA SYSTEM 5 for a complete proof. One important property of ( s , s ) is the so-called Pohozaevidentity (P.I. in short):(1.9) k s + k k s s + k s = k m s + k m s , where m i = + a i . Take A as an example, the P.I. is s − s s + s = m s + m s . The proof of (1.9) was given in [22]. At first sight, (1.9) seems not very usefulto determine the local mass. In [22] we initiated an algorithm to calculate all thepossible (finitely many) values of local masses. The P.I. plays one of importantroles. But the argument there seems not very efficient. In this work we developfurther our original approach to sharpen the result:
Theorem 1.3.
Suppose s and s are local masses of a sequence of blowup solu-tions of (1.4) such that (1.6) holds. Then s i can be written as s i = ( N i , m + N i , m + N i , ) , i = , , for some N i , , N i , , N i , ∈ Z (i = , ). Theorem 1.3 is proved in section 5 and section 6. In section 5, we give anexplicit procedure to calculate the local masses. Take A system as an example,we start with a = s = m . With s = m , the P.I. gives s = m + m and so on. Let G ( m , m ) be the set obtained by the above algorithm.Then G ( m , m ) is equal to the following set,(i) ( m , ) , ( m , m + m ) , ( m + m , m + m ) , ( m + m , m ) , ( , m ) for A , (ii) ( m , ) , ( m , m + m ) , ( m + m , m + m ) , ( m + m , m + m ) , ( , m ) , ( m + m , m ) , ( m + m , m + m ) , for B , (iii) ( m , ) , ( m , m + m ) , ( m + m , m + m ) , ( m + m , m + m ) , ( m + m , m + m ) , ( m + m , m + m ) , ( , m ) , ( m + m , m ) , ( m + m , m + m ) , ( m + m , m + m ) , ( m + m , m + m ) , for G . Definition 1.4.
A pair of local masses ( s , s ) ∈ G ( m , m ) is called special if ( s , s ) = ( m + m , m + m ) for A , ( m + m , m + m ) for B , ( m + m , m + m ) for G . The analysis of local solutions in [22] is a method to pick up these points G k = { , x k , · · · , x kN } (if 0 is a singular point, otherwise 0 can be deleted from G k ) suchthat a tiny ball B ( x ki , l kj ) can contribute an amount of mass (which is quantized), andthe following Harnack-type inequality holds:(1.10) u ki ( x ) + ( x , S k ) C , ∀ x ∈ B ( , ) . CHANG-SHOU LIN, JUNCHENG WEI, WEN YANG, AND LEI ZHANG
When a = a =
0, we can use Theorem 1.3 to calculate all the pairs of evenpositive integers of (1.9). It turns out the set of solution of (1.9) to be the same as G ( , ) . Corollary 1.5.
Suppose a = a = . Then ( s , s ) ∈ G ( , ) . Furthermore if ( s , s ) is not special, then S k = { x k } andu ki ( x ) + | x − x k | ≤ C , i = , . It is interesting to see whether any pair of the above is really the local massesof some sequence of blowup solutions of (1.2). For K = A the existence of sucha local solution has been obtained (see [30] and [24]). We remark that parts ofCorollary 1.5 was already proved by Jost-Lin-Wang [15], and by the first authorand the fourth author in [27].After S k is picked up, the difficulty at the next step is how to calculate the masscontributed from outside B ( x kj , l kj ) j = , , · · · , N . In section 6, we see that themass outside of this union could be very messy. However, if ( a , a ) satisfies the Q -condition: [ Q ] a , a and are linearly independent over Q . Then the result can be stated cleanly as follows.
Theorem 1.6.
Suppose ( a , a ) satisfies the Q-condition. Then ( s , s ) ∈ G ( m , m ).Furthermore, the Harnack-type inequality holds:u ki ( x ) + | x | C for x ∈ B ( , ) . For (1.2), let m , t = a t + m , t = a t + p t ∈ S , and define(1.11) G i = { p ( S t ∈ J s i , t + n ) | ( s , t , s , t ) ∈ G ( m , t , m , t ) , J ⊆ S , n ∈ N ∪ { }} . Together with Theorem 1.6, Theorem 1.1 can be extended:
Theorem 1.7.
Let h i be positive C functions on M, and K be a compact set inM. If either both a and a are integers or ( a , a ) satisfies the Q-condition and r i / ∈ G i for i = , , then there exists a constant C such that | u i ( x ) | C ∀ x ∈ K . The organization of this article is as follows. In Section 2 we establish the globalmass for the entire solutions of some singular Liouville equation defined in R .Then in Section 3 we review some fundamental tools proved in the previous work[22]. In section four we present two crucial lemmas, which play the key role in theproof of main results. Then in section 5 and section 6 we discuss the local mass oneach bubbling disk centered at 0 and not at 0 respectively, thereby we prove all theresults. ODA SYSTEM 7
2. T
OTOAL MASS FOR LIOUVILLE EQUATION
The main purpose of this section is to prove an estimate of the total mass of asolution of the following equation:(2.1) D u + e u = (cid:229) Nj = pa i d p i , in R , R R e u < ¥ , where p , ..., p N are distinct points in R and a i > − , ∀ i N . Theorem 2.1.
Suppose u is a solution of (2.1) and a , ..., a N are positive integers.Then p Z R e u is an even integer.Proof. It is known that any solution u of (2.1) has the following asymptotic behav-ior at infinity:(2.2) u ( z ) = − a ¥ log | z | + O ( ) , a ¥ > , and u satisfies(2.3) 12 p Z R e u dx = N (cid:229) i = a i + a ¥ . We shall prove that a ¥ + (cid:229) Ni = a i is an even integer. A classical Liouville theorem( see [10] ) says that, u can be written as(2.4) u = log 4 | f ′ ( z ) | ( + | f ( z ) | ) , z ∈ R , for some meromorphic function f . In general, f ( z ) is multi-valued and any vertex p i is a branch point. However if a i ∈ N ∪ { } , f ( z ) is single-valued. Furthermore(2.2) implies that f ( z ) is meromorphic at infinity. Hence for any solution u of (2.1)there is a meromorphic functon f on S = C ∪ { ¥ } such that (2.4) holds. Then4 p ( N (cid:229) j = a j + a ¥ ) = Z R e u = Z R | f ′ ( z ) | ( + | f ( z ) | ) dxdy = ( deg f ) Z R d ˜ xd ˜ y ( + | w | ) = p ( deg f ) , where deg ( f ) is the degree of f as a map from S = C ∪ { ¥ } onto S , and w = f ( z ) = ˜ x + i ˜ y . Thus we have N (cid:229) j = a j + a ¥ = deg ( f ) . Theorem 2.1 is established. (cid:3)
Theorem 2.2.
Suppose u is a solution of (2.5) (cid:26) D u + e u = pa d p + (cid:229) Ni = pa i d p i , in R , R R e u < ¥ . CHANG-SHOU LIN, JUNCHENG WEI, WEN YANG, AND LEI ZHANG where p , p , ..., p N are distinct points in R and a , ...., a N are positive integers, a > − . Then p Z R e u is equal to ( a + ) + k for some k ∈ Z or k for somek ∈ N .Proof. As in Theorem 2.1 there is a developing map f ( z ) of u such that(2.6) u ( z ) = log 4 | f ′ ( z ) | ( + | f ( z ) | ) , z ∈ C . On one hand by (2.5), u zz − u z is a meromorphic function in C ∪ { ¥ } becauseaway from the Dirac masses4 ( u zz − u z ) ¯ z = − ( e u ) z + u z e u = . By u ( z ) = a i log | z − p i | + O ( ) near p i we have u zz − u z = − { N (cid:229) j = a j ( a j + )( z − p j ) − + A j ( z − p j ) − + B } , where B ∈ C is an unknown constant. On the other hand by (2.6), a straightforwardcomputation shows that(2.7) u zz − u z = f ′′′ f ′ − ( f ′′ f ′ ) . Using the Schwarz derivative of f : { f ; z } = f ′′′ ( z ) f ′ ( z ) − ( f ′′ ( z ) f ′ ( z ) ) and letting I ( z ) = N (cid:229) j = a j ( a j + )( z − p j ) − + A j ( z − p j ) − + B , we write the equation for f as(2.8) { f , z } = − I ( z ) . A well known classic theorem (see [36]) says that for any two linearly independentsolutions y and y of(2.9) y ′′ ( z ) = I ( z ) y ( z ) , the ratio y / y always satisfies { y / y ; z } = − I ( z ) . By (2.8) and the basic result of the Schwarz derivative, f ( z ) can be written as theratio of two linearly independent solutions. This is how equation (2.1) is related tothe complex ODE (2.9). We refer the readers to [6] for the details. ODA SYSTEM 9
For the complex ODE (2.9), there is an associated monodromy representation r from p ( C \ { p , p , ..., p N } ; q ) to GL ( C ) where q is a base point. Note that atany singular point p j the local exponents are a j + − a j . So we have r j = r ( g j ) = C j (cid:18) e p i a j e − p i a j (cid:19) C − j , where C j is an invertible matrix, g j ∈ p ( C \ { p , ..., p N } , q ) encircles p j once only,0 ≤ j ≤ N . Then we have r ¥ r N ... r = I × . Note that r j = ± I for 1 ≤ j ≤ N . Hence r − ¥ = C e p (cid:229) Nj = a j e − p (cid:229) Nj = a j ! C − for some constant invertible matrix C .On the other hand, the local exponents at ¥ can be computed as follows. Recall(2.9) and let ˆ y ( z ) = y ( / z ) . Then we have(2.10) ˆ y ′′ ( z ) + z ˆ y ′ ( z ) = ˆ I ( z ) ˆ y ( z ) , where ˆ I ( z ) = I ( / z ) z − . Since I ( z ) is the Schwarz derivative of f ( z ) , by directcomputation ˆ I ( z ) is the Schwarz derivative of f ( / z ) . As before we let ˆ u ( z ) = u ( / z ) − | z | . Then f ( / z ) is the developing map of ˆ u ( z ) . Sinceˆ u ( z ) = ( a ¥ − ) log | z | + O ( ) near 0 , (because u ( z ) = − a ¥ log | z | + O ( ) at infinity), we haveˆ I ( z ) = a ¥ ( a ¥ − ) z − + higher order terms of z near 0 . By (2.10) we could prove that the local exponents of (2.9) are − a ¥ and a ¥ − e a ¥ p i equals either e i p (cid:229) Nj = a j or e − i p (cid:229) Nj = a j , which yields(2.11) a ¥ = − N (cid:229) j = a j + k or a ¥ = N (cid:229) j = a j + k for some k ∈ Z . Since 14 p Z R e u = N (cid:229) j = a j + a ¥ , we see that either p R R e u = k if the first case holds or p R R e u = ( a + ) + k ′ for k ′ = (cid:229) Ni = a i + k − (cid:3) Remark 2.1.
After Theorem 2.1 and Theorem 2.2 haven been proved, we found astronger version of both theorems in [13] . Because we only need the present form ofboth theorems, we include our proofs here to make the paper more self-contained.
3. R
EVIEW OF B UBBLING A NALYSIS F ROM A S ELECTION P ROCESS
Let u k = ( u k , u k ) be solutions of (1.5) such that (1.7) holds. In this section wereview the process to select a set S k = { , x k , ..., x kn } and balls B ( x ki , l k ) such that u k has nonzero local masses in B ( x ki , l k ) . This selection process was first carried outin [22]. We briefly review it below.The set S k is constructed by induction. If (1.5) has no singularity, we start with S k = /0. If (1.5) has a singularity, we start with S k = { } . By induction suppose S k consists of { , x k , ..., x km − } . Then we consider(3.1) max x ∈ B (cid:18) u ki ( x ) + ( x , S k ) (cid:19) . If the maximum is bounded from above independent of k , the process stops and S k is exactly equal to { , x k , ..., x km − } . However if the maximum tends to infinity, let q k be where (3.1) is achieved and we set d k =
12 dist ( q k , S k ) and S ki ( x ) = u ki ( x ) + ( d k − | x − q k | ) in B ( q k , d k ) , i = , . Suppose i is the component that attains(3.2) max i max x ∈ ¯ B ( q k , d k ) S ki at p k . Then we set ˜ l k = ( d k − | p k − q k | ) and scale u ki by(3.3) v ki ( y ) = u ki ( p k + e − u ki ( p k ) y ) − u ki ( p k ) , for | y | ≤ R k + e u ki ( p k ) ˜ l k . It can be shown that R k → ¥ and v ki is bounded from above over any fixed compactsubset of R . Thus by passing to a subsequence v ki satisfies one of the followingtwo alternatives:(a) ( v k , v k ) converges in C loc ( R ) to ( v , v ) which satisfies(3.4) D v i + (cid:229) j ∈ I k i j e v j = R , i ∈ I = { , } . (b) Either v k converges to(3.5) D v + e v = R and v k → − ¥ over any fixed compact subset of R or v k converges to D v + e v = R and v k → − ¥ over any fixed compact subset of R .Therefore in either case, we could choose l ∗ k → ¥ such that(3.6) v ki ( y ) + | y | ≤ C , for i = , | y | l ∗ k ODA SYSTEM 11 and Z B ( , l ∗ k ) h ki e v ki dy = Z R e v i ( y ) + o ( ) . By scaling back to u ki , we add p k in S k with l k = e − u ki ( p k ) l ∗ k . We can continue inthis way until the Harnack-type inequality (1.9) holds.We summarize what the selection process has done in the following proposition( a detailed proof for a more general case can be found in Proposition 2.1 of [22]): Proposition 3A.
Let u k be described as above. Then there exist a finite set S k : = { , x k , ...., x km } (if is not a singular point, then can be deleted from S k ) andpositive numbers l k , ..., l km → as k → ¥ such that the followings hold: (1) There exists C > independent of k such that (1.10) holds. (2) In B ( x kj , l kj ) ( j = , .., m), let R j , k = e u ki ( x kj ) l kj , u ki ( x kj ) = max i u ki ( x kj ) and (3.7) v ki ( y ) = u ki ( x kj + e − u ki ( x kj ) y ) − u ki ( x kj ) for | y | ≤ R j , k , then v k = ( v k , v k ) satisfies either (a) or (b). (3) B ( x kj , l kj ) ∩ B ( x ki , l ki ) = /0 . The inequality (1.10) is a Harnack type inequality, because it implies the follow-ing result
Proposition 3B.
Suppose u k satisfies (1.5) in B ( x , r k ) such thatu ki ( x ) + | x − x | ≤ C , for x ∈ B ( x , r k ) . Then (3.8) | u ki ( x ) − u ki ( x ) | ≤ C , for ≤ | x − x || x − x | ≤ and x , x ∈ B ( x , r k ) . The proof of Proposition 3B is standard (see [22, Lemma 2.4]), so we omit ithere. Let x kl ∈ S k and t kl = dist ( x kl , S k \ { x kl } ) , then (3.8) implies(3.9) u ki ( x ) = ¯ u kx kl , i ( r ) + O ( ) , x ∈ B ( x kl , t kl ) , where r = | x kl − x | and ¯ u kx kl , i is the average of u ki on ¶ B ( x kl , r ) :(3.10) ¯ u kx kl , i ( r ) = p r Z ¶ B ( x kl , r ) u ki dS , and O ( ) is independent of r and k .Next we introduce the notions of slow decay or fast decay in our bubbling anal-ysis. Definition 3.1.
We say u ki has fast decay at x ∈ B ( x , r k ) if along a subsequence,u ki ( x ) + | x − x | ≤ − N k , for x ∈ ¶ B ( x , r k ) for some N k → ¥ and u ki is called to have slow-decay if there is a constant Cindependent of k andu ki ( x ) + | x − x | ≥ − C , for x ∈ ¶ B ( x , r k ) . Fast decay is very important for evaluating Pohozaev identities. The followingproposition is a direct consequence of [22, Proposition 3.1] and it says if bothcomponents are fast-decay on the boundary, Pohozaev identity holds for the localmasses.In the following proposition, we let B = B ( x k , r k ) . If x k = , then we assume0 / ∈ B ( x k , r k ) . Proposition 3C.
Suppose both u k , u k have fast decay on ¶ B, where B is givenabove. Then ( s , s ) satisfies the P.I.(1.8), where s i = lim k → p Z B h ki e u ki , i = , . The proof of Proposition 3C requires some delicate analysis. We refer the read-ers to [22, Proposition 3.1] for the proofs. The P.I. plays an important role in ouranalysis later. 4. T WO L EMMAS
In this section, we will prove two crucial lemmas which play the key role insection 5 and 6. For Lemma 4.1, we assume(i). The Harnack inequality u ki ( x ) + | x | ≤ C , for 12 l k ≤ | x | ≤ s k , and i = , . (ii). Both u ki have fast-decay on ¶ B ( , l k ) and s ki ( B ( , l k )) = s i + o ( ) for i = , s i = lim r → lim k → ¥ s i ( B ( , rs k )) , i = , u ki has slow-decay on ¶ B ( , s k ) . Lemma 4.1. (a). Assume (i) and (ii), If u ki has slow-decay on ¶ B ( , s k ) , then m i − (cid:229) j = k i j s j > . (b). Assume (i), (ii) and (iii), then the other component has fast decay on ¶ B ( , s k ) .Proof. (a) Suppose u ki have slow decay on ¶ B ( , s k ) , then the following scaling v kj ( y ) = u kj ( s k y ) + s k , j = , , for y ∈ B gives D v kj ( y ) + (cid:229) l = k jl h kl ( s k y ) e v kl ( y ) = pa ki d , in y ∈ B . If the other component also has slow-decay on ¶ B ( , s k ) , then ( v k , v k ) coveragesto ( v , v ) which satisfies(4.1) D v j ( y ) + (cid:229) j = k jl e v j = , in B \{ } , j = , . ODA SYSTEM 13
If the other component has fast-decay on ¶ B ( , s k ) , then v ki ( y ) coverages to v i ( y ) and v j ( y ) → − ¥ , j = i . Furthermore, v i ( y ) satisfies(4.2) D v i ( y ) + e v i = B \{ } . For any r > Z ¶ B ( , r ) ¶ v i ( y ) ¶ v dS = lim k → ¥ ( pa ki − (cid:229) j = Z B ( , r ) k i j h kj e v kj dy )= pa i − p (cid:229) j = k i j s j + o ( ) + pb i + o ( ) , which implies RHS of both (4.1) and (4.2) should be replaced by 4 pb i d ( ) as anequation defined in B . It is known that if b i < −
1, either (4.1) or (4.2) has nosolutions. Hence a i − (cid:229) k i j s j > − ¶ B ( , l k ) , the pair ( s , s ) satisfiesthe P.I. (1.9). By a simple manipulation, the P.I. (1.9) can be written as(4.3) k s ( m − k s − k s ) + k s ( m − k s − k s ) = m i − (cid:229) l = k il s l > m i − (cid:229) l = k il s l > . Hence for j = i m j − (cid:229) j = k il s l < m j − (cid:229) j = k il s l < , where the last inequality is due to (4.3). By (a) again, u kj can not have slow-decayon ¶ B ( , s k ) . (cid:3) Our second lemma is about the fast-decay.
Lemma 4.2.
Suppose the Harnack-type inequality holds for both components overr ∈ [ l k , s k ] . If u ki is fast-decaying on r ∈ [ l k , s k ] , then s ki ( B ( , s k )) = s ki ( B ( , l k )) + o ( ) . Proof.
Obviously the conclusion holds easily if s k / l k C . So we assume s k / l k → + ¥ . The Harnack-type inequality implies u ki ( x ) = u ki ( r ) + o ( ) for l k | x | s k .Thus we have from (1.5) that ddr ( u ki ( r ) + r ) = m i − (cid:229) j = k i j s kj ( r ) r , l k r s k , i = , , where s kj ( r ) = s kj ( B ( , r )) and s j = lim k → + ¥ s kj ( l k ) , j = , l k , which implies at least one componentsatisfies 4 m l − (cid:229) k l j s kj ( l k ) > m l + o ( ) > . Thus,(4.4) ddr ( u ( k ) l ( r ) + r ) − m l + o ( ) r at r = l k . Suppose r k ∈ [ l k , s k ] is the largest r such that(4.5) ddr ( u ( k ) l ( r ) + r ) − m l r for r ∈ [ l k , r k ] , thus, either the identity holds at r = r k or r k = s k . For simplicity, we let e = m l . Byintegrating (4.4) from l k up to r r k , we have u ( k ) l ( r ) + r u ( k ) l ( l k ) + ( l k ) + e log ( l k r ) , that is for | x | = r , e u kl ( x ) O ( ) e u kl ( r ) e − N k l e k r − ( + e ) , where we used u ( k ) l ( l k ) + l k − N k by the assumption of fast-decay. Thus Z l k | x | r k e u kl ( x ) dx p e − N k l e k Z r k l k r − ( + e ) dr = p e − N k e → k → + ¥ . Hence(4.6) s kl ( r k ) = s kl ( l k ) + o ( ) . If both components are fast decaying on r ∈ [ l k , r k ] , then lim k → + ¥ ( s k ( r k ) , s k ( r k )) =( ˆ s , ˆ s ) also satisfies the P.I.(1.9). If ˆ s j > s j , then j = l by (4.6). We choose r ∗ k ≤ r k such that s j ( r ∗ k ) = s kj ( l k ) + e for small e , and let s ∗ j = lim k → s j ( r ∗ k ) . Then s ∗ j and s l satisfies the P.I.(1.9) and it yields a contradiction provided e issmall. Thus, we have s km ( r k ) = s km ( l k ) + o ( ) , m = ,
2. Then (4.4) holds at r = r k which implies r k = s k , and Lemma 4.2 is proved in this case.If one of the components cannot have fast decay on [ l k , r k ] . We have l = i and u kj , j = i , has slow decay on ¶ B ( , r ∗ k ) for some r ∗ k ≤ r k . If s k / r k ≤ C , then (4.6)implies the lemma. If s k / r k → + ¥ , then by the scaling of u kj at r = r ∗ k , the standardargument implies that there is a sequence of r ∗ k ≪ ˜ r k = R k r ∗ k ≪ s k such that bothcomponents have fast decay on ˜ r k and s ki ( ˜ r k ) = s i ( r ∗ k ) + o ( ) = s i ( l k ) + o ( ) , and s kj ( ˜ r k ) ≥ s kj ( l k ) + e for some j = i and e >
0. Therefore the assumption of Lemma 4.2 holds at r ∈ [ ˜ r k , s k ] . Then we repeat the argument starting from (4.4) and the lemma can beproved in a finite steps. (cid:3) Remark 4.3.
Both lemmas will be used in section 6 (and section 5) for the casewith singularity at (and without singularity at ). ODA SYSTEM 15
5. L
OCAL MASS ON THE BUBBLING DISK CENTERED AT x ki = u k near x kl where x kl = x k instead of x kl and ¯ u ki ( r ) rather than ¯ u x k , i ( r ) . Let t k =
12 dist ( x k , S k \ { x k } ) s ki ( r ) = p Z B ( x k , r ) h ki e u ki , i = , . By Proposition 3A, l k ≤ t k . Clearly u k = ( u k , u k ) satisfies D u ki + (cid:229) j k i j h kj e u kj = , in B ( x k , t k ) . For a sequence s k , we define(5.1) ˆ s i ( s k ) = lim k → + ¥ s ki ( s k ) if u ki has fast decay on ¶ B ( x k , s k ) , lim r → lim k → + ¥ s ki ( rs k ) if u ki has slow decay on ¶ B ( x k , s k ) , Recall that both u ki have fast decay on ¶ B ( x k , l k ) (see (3.4)). This is the startingpoint of the following proposition, which is a special case of Proposition 5.2 below.In Proposition 5.1, ( m , m ) will be ( , ) in both lemmas of section 4. Proposition 5.1.
Let u k = ( u k , u k ) be the solutions of (1.5) satisfying (1.7) and ˆ s i ( s k ) be defined in (5.1) , the followings hold: (1) At least one component u k has fast decay on ¶ B ( x k , t k ) , (2) ( ˆ s ( t k ) , ˆ s ( t k )) satisfies the P.I.(1.9) with m = m = , (3) ( ˆ s ( t k ) , ˆ s ( t k )) ∈ G ( , ) .Proof. If t k / l k C , (1)-(3) holds obviously for t k . So we assume t k / l k → + ¥ .First we remark that if u k is fully bubbling in B ( x k , l k ) (i,e, (a) in Proposition 3Aholds), ( ˆ s ( l k ) , ˆ s ( l k )) is special (see Definition 1.4) and satisfies2 m i − (cid:229) j = k i j ˆ s j ( l k ) < , i = , . Then by Lemma 4.1, both u ki have fast decay on ¶ B ( , t k ) and Proposition 5.1follows immediately.Now we assume v ki defined in (3.7) and satisfies case(b) in Proposition 3A. From(3.4), we already knew that both components have fast decay at r = l k . If bothcomponents remain fast decay as r increases from l k to t k , Lemma 4.2 implies s k ( t k ) = s k ( l k ) + o ( ) , s k ( t k ) = s k ( l k ) + o ( ) and we are done. So we only consider the case that at least one component changesto a slow decay component. For simplicity, we assume that u k changes to slowdecay for some r k ≫ l k . By Lemma 4.2, s k ( B ( x k , r k )) > s ( B ( x k , l k )) + c , for some c > . We might choose s k r k such that s k ( B ( x k , s k )) = s k ( B ( x k , l k )) + e , and s k ( B ( x k , r )) < s k ( B ( x k , l k )) + e ∀ r < s k , where e < c is small.Then Lemma 4.1 and Lemma 4.2 together implies u k has slow-decay on ¶ B ( x k , s k ) and u k has fast decay on ¶ B ( x k , s k ) withˆ s ( s k ) = s k ( l k ) + o ( ) and ˆ s ( s k ) = s k ( l k ) + o ( ) . Let v ki ( y ) = u ki ( x k + s k y ) + s k . If t k / s k ≤ C there is nothing to prove. So weassume t k / s k → ¥ . Then v k ( y ) converges to v ( y ) and v k ( y ) → − ¥ in any compactset of R as k → + ¥ and v ( y ) satisfies(5.2) D v + e v = − p (cid:229) ( k j ˆ s j ( l k )) d ( ) in R . Hence there is a sequence N ∗ k and N ∗ k → + ¥ as k → + ¥ such that N ∗ k s k ≤ t k and Z B ( , N ∗ k ) e v dy = Z R e v dy + o ( ) , andboth v ki ( y ) + | y | − N k for | y | = N ∗ k . Scaling back to u ki , we obtain that u ki , i = , , have fast-decay on ¶ B ( x k , N ∗ k s k ) .We could use the classification theorem of Prajapat and Tarantello [32] to calcu-late the total mass of v , but instead we use the P.I.(1.9) to compute it. We know thatboth ( ˆ s ( l k ) , ˆ s ( l k )) and ( ˆ s ( N ∗ k s k ) , ˆ s ( N ∗ k s k )) satisfy P.I. and ˆ s ( N ∗ k s k ) = ˆ s ( l k ) by Lemma 4.2. With a fixed s = ˆ s ( l k ) , the equation P.I. (1.9) is a quadratic poly-nomial in s , then ˆ s ( l k ) and ˆ s ( N ∗ k s k ) are two roots of the polynomial. From it,we could easily calculate ˆ s ( N ∗ k s k ) .By a direct computation, we have ( ˆ s ( N ∗ k s k ) , ˆ s ( N ∗ k s k )) ∈ G ( , ) if ( ˆ s ( l k ) , ˆ s ( l k )) ∈ G ( , ) . Thus (1)-(3) hold at r = N ∗ k s k . By denoting N ∗ k s k as l k , we could repeat the sameargument until t k / l k C . Hence Proposition 5.1 is proved. (cid:3) Local mass in a group that does not contain . In this subsection we collectsome x ki ∈ S k into a group S , a subset of S k satisfying the following S -condition:(1). 0 S and | S | ≥ | S | ≥ x ki , x kj , x kl are three distinct elements in S , thendist ( x ki , x kj ) ≤ C dist ( x kj , x kl ) for some constant C independent of k .(3). For any x km ∈ S k \ S , the ratio dist ( x km , S ) / dist ( x ki , x kj ) → ¥ as k → ¥ where x ki , x kj ∈ S .We write S as S = { x k , ..., x km } and let(5.3) l k ( S ) = ≤ j ≤ m dist ( x k , x kj ) . ODA SYSTEM 17
Recall t l , k = dist ( x kl , S k \ { x kl } ) , by (2) above we have l k ( S ) ∼ t ki for 1 ≤ i ≤ m .Let t kS =
12 dist ( x k , S k \ S ) then by (3) above t kS / t ki → ¥ for any x ki ∈ S .By Proposition 5.1, we know that at least one of u ki has fast decay on ¶ B ( x k , t k ) .Suppose u k has fast decay on ¶ B ( x k , t k ) . Then(5.4) u k has fast decay on ¶ B ( x k , l k ( S )) ,and Proposition 5.1 implies s k ( B ( x k , l k ( S ))) = p Z B ( x k , l k ( S )) h k e u k dx = p Z ∪ mj = B ( x kj , t kj ) h k e u k + p Z B ( x k , l k ( S )) \ ( ∪ mj = B ( x kj , t kj )) h k e u k . Since u k has fast decay outside of B ( x kj , t kj ) , we have e u k ( x ) ≤ o ( ) max j {| x − x kj | − } , for x / ∈ k [ j = B ( x kj , t kj ) and the second integral is o ( ) . Hence by Proposition 5.1,(5.5) s k ( B ( x k , l k ( S ))) = m + o ( ) for some m ∈ N ∪ { } . Similarly if u k has fast decay on ¶ B ( x k , t k ) , we have(5.6) s k ( B ( x k , l k ( S ))) = m + o ( ) for some m ∈ N ∪ { } . If u k has slow decay on ¶ B ( x k , t k ) , then it is easy to see that u k has slow decayon ¶ B ( x kj , t kj ) . By Proposition 5.1 we denote n i , j ∈ N by2 n i , j = lim r → lim k → ¥ s ki ( B ( x kj , r t kj )) , ≤ j ≤ m , i = , . Set ˆ n i , j by ˆ n i , j = − (cid:229) l = k il n l , j . Then the slow decay of u k on ¶ B ( x kj , t kj ) implies 1 + ˆ n i , j >
0. Since ˆ n i , j ∈ Z wehave ˆ n i , j ≥ u k by v ki ( y ) = u ki ( x k + l k ( S ) y ) + l k ( S ) , i = , , the sequence v k would converge to v ( y ) and v k tends to − ¥ over any compactsubset of R \ { } . Then v satisfies(5.7) D v ( y ) + e v ( y ) = p m (cid:229) j = ˆ n i , j d p j in R . where p j = lim k → ¥ ( x kj − x k ) / l k ( S ) . By Theorem 2.112 p Z R e v = N , for some N ∈ N . Thus using the argument in Proposition 5.1, we conclude that there is a sequenceof N ∗ k → ¥ such that both u ki ( i = ,
2) have fast decay on ¶ B ( x k , N ∗ k l k ( S )) and s ki ( B ( x k , N ∗ k l k ( S ))) = m i + o ( ) . Denote N ∗ k l k ( S ) by l k for simplicity, then we seethat (5.5) and (5.6) hold at l k . Then by using Lemma 4.1 and Lemma 4.2 we couldcontinue this process to obtain the following conclusion:(5.8) At least one component of u k has fast decay on ¶ B ( x k , t kS ) . Let ˆ s ki ( B ( x k , t ks )) be defined as in (5.1). Then(5.9) ˆ s ki ( B ( x k , t kS )) = m i ( S ) + o ( ) , where m i ( S ) ∈ N ∪ { } , and the pair ( m ( S ) , m ( S )) satisfies the P.I.(1.9).Denote the group S by S . Based on this procedure, we could continue to selecta new group S such that S -condition holds except we have to modify condition-(2). In (2), we consider S as an single point as long as we compare the distance ofdistinct elements in S .Set t kS =
12 dist ( x k , S k \ S ) , for x k ∈ S . Then we follow the same argument as above to obtain the same conclusion as(5.8)-(5.9).If equation (1.5) does not contain singularity, the final step is to collect all x ki intoone single biggest group and (5.8)-(5.9) hold. Then we get ( s , s ) = ( m , m ) satisfies the Pohozaev identity. By a direct computation, we could prove that allthe pairs of even integer solution of (1.9) is exactly G ( , ) . This proves Theorem1.3 if (1.5) has no singularities. Proof of Corollary 1.5.
The first part is already proved. For the last part, we wantto prove | S | =
1. We observe: for any ( s , s ) ∈ G ( , ) , if ( s , s ) is not specialthen ( s , s ) can not be written as a sum of ( s ( ) , s ( ) ) and ( s ( ) , s ( ) ) , where ( s ( i ) , s ( i ) ) ∈ G ( , ) . Now if | S | ≥
2, then ( s ( t ks ) , s ( t ks )) can be written as a sumof ( s ( ) , s ( ) ) and ( s ( ) , s ( ) ) , where ( s ( i ) , s ( i ) ) ∈ G ( , ) . But if ( s ( t ks ) , s ( t ks )) is not special, then it can not be written in this way. (cid:3) If 0 is a singularity of (1.5) then S k could be written as a disjoint union of { } and S j ( j = , .., m ). Here each S j is collected by the process described above andis maximal in the following sense:(i). 0 S , | S | ≥ x ki , x kj in S we havedist ( x ki , x kj ) ≪ t k ( S ) , where t k ( S ) = dist ( S , S k \ S ) . ODA SYSTEM 19 (ii). For any 0 = x ki ∈ S k \ S , dist ( x ki , ) ≤ C dist ( x ki , S ) for some constant C .For S j we define t kS j =
12 dist ( S j , S k \ S j ) . Then the process described above proves the main result of this section:
Proposition 5.2.
Let S j ( j = , .., m) be described as above, then (5.8)-(5.9) holdswhere B ( x k , t kS ) is replaced by B ( x ki , t kS j ) and x ki is any element in S j .
6. P
ROOF OF T HEOREM
HEOREM
HEOREM
AND T HEOREM S k = { } ∪ S ∪ · · · ∪ S N . From the construction, theratio | x k || ˜ x k | ≤ C for any x k , ˜ x k ∈ S j . Let k S j k = min x k ∈ S j | x k | and arrange S j by k S k ≤ k S k ≤ · · · ≤ k S N k . Assume l is the largest number such that k S l k ≤ C k S k . Then k S l k ≪ k S l + k .We recall the local mass contributed by x kj ∈ S j is ( ˆ s ( B ( x kj , t kj )) , ˆ s ( B ( x kj , t kj ))) = ( m , j , m , j ) , where m j , m j ∈ N ∪ { } . Let r k = k S k . By Proposition 5.1, we have u ki ( x ) + | x | ≤ C for 0 < | x | ≤ r k , i = , . Proof of Theorem 1.3.
Let˜ u ki ( x ) = u ki ( x ) + a i log | x | , i = , . Then equation (1.5) becomes D ˜ u ki ( x ) + (cid:229) j = k i j | x | a j h kj ( x ) e ˜ u kj ( x ) = , | x | ≤ r k , i = , . Let(6.1) − d k = max i ∈ I max x ∈ ¯ B ( , t k ) ˜ u ki + a i , then ˜ v ki ( y ) defined as(6.2) ˜ v ki ( y ) = ˜ u ki ( d k y ) + ( + a i ) log d k , | y | ≤ r k / d k , i = , , satisfies(6.3) D ˜ v ki ( y ) + (cid:229) j ∈ I k i j | y | a j h kj ( d k y ) e ˜ v kj ( y ) = , | y | ≤ r k / d k , i = , . We have either ( a ) lim k → ¥ r k / d k = ¥ or ( b ) r k / d k ≤ C . For (a). Our purpose is to prove a similar result as Proposition 5.1:(1).
At most one component of u k has slow decay on ¶ B ( , r k ) . As in section 5, we defineˆ s i , = (cid:26) lim k → + ¥ s ki ( B ( , r k )) if u ki has fast decay on ¶ B ( , r k ) , lim r → lim k → + ¥ s ki ( B ( , rr k )) if u ki has slow decay on ¶ B ( , r k ) , (2). ( ˆ s , , ˆ s , ) satisfies the Pohozaev identity (1.8), and(3). ˆ s i , = (cid:229) i = n i m i + n , n i ∈ Z . We carry out the proof in the discussion of the following two cases.Case 1. If both ˜ v ki ( y ) converge in any compact set of R , ( s , s ) can be obtainedby the classification theorem in [21]: ( s , s ) = ( m + m , m + m ) for A , ( m + m , m + m ) for B , ( m + m , m + m ) for G . By Lemma 4.1, both u ki have fast decay on ¶ B ( , r k ) . So this proves (1)-(3) in thiscase.Case 2. Only one ˜ v ki converges to v i ( y ) and the other tends to − ¥ uniformly in anycompact set. Then it is easy to see that there is l k ≪ r k such that both u ki have fastdecay on ¶ B ( , l k ) and ( s ( B ( , l k )) , s ( B ( , l k ))) = ( m , ) or ( s ( B ( , l k )) , s ( B ( , l k )) = ( , m )) . So this is the same situation as in the starting point for Proposition 5.1. Then thesame argument of Proposition 5.1 leads to the conclusion (1)-(3).The pair ( ˆ s , , ˆ s , ) can be calculated by the same method in Proposition 5.1.Then ( ˆ s , , ˆ s , ) ∈ G ( m , m ) , which is given in section 2.To continue our discussion for r ∈ [ r k , r k ] , where we denote k S l + k by r k . Weseparate our discussion into two cases also.Case 1. One component has slow decay on ¶ B ( , r k ) , say u k . Then we scale v ki ( y ) = u ki ( r k y ) + r k . By our assumption v k ( y ) converges to v ( y ) and v k ( y ) → − ¥ in any compact set.Let x kj ∈ S j and y kj = ( r k ) − x kj → p j . Then v ( y ) satisfies(6.4) D v + e v = p ˜ a d + p m (cid:229) j = ˜ n , j d p j , ODA SYSTEM 21 where(6.5) ˜ n , j = − (cid:229) i = k i m i , j for some m i j ∈ Z and ˜ a = a − ˆ s , +
12 ˆ s , . The finiteness of R R e v implies that˜ a > − n , j ≥ . By Theorem 2.2, we have(6.6) 12 p Z R e v d y = ( ˜ a + ) + k or 12 p Z R e v d y = k , where k , k ∈ Z . As before, we can choose l k , r k ≪ l k ≪ r k such that both u ki have fast decay on ¶ B ( , l k ) . Then the new pair ( ˆ s , , ˆ s , ) , which defined byˆ s t , = p lim k → Z B ( , l k ) h kt e v kt , t = , , becomes(6.7) ( ˆ s , , ˆ s , ) = ( ˆ s , + p Z R e v + m (cid:229) j = m , j , ˆ s , + m (cid:229) j = m , j ) for m j , m j ∈ N ∪ { } . Using (6.6), we get(6.8)ˆ s , = (cid:26) ˆ s , + k + (cid:229) mj = m , j , if p R R e v d y = k , m + ˆ s , − ˆ s , + k + (cid:229) mj = m , j , if p R R e v d y = ( ˜ a + ) + k . We note that if ( ˆ s , , ˆ s , ) ∈ G ( m , m ) and 2 m + ˆ s , − ˆ s , >
0, then ( m + ˆ s , − ˆ s , , ˆ s , ) ∈ G ( m , m ) . Let ( s ∗ , s ∗ ) = ( m + ˆ s , − ˆ s , , ˆ s , ) , we can write(6.9) ( ˆ s , , ˆ s , ) = ( s ∗ + m , s ∗ + m ) , with ( s ∗ , s ∗ ) ∈ G ( m , m ) and m , m ∈ Z . Case 2. If both u ki have fast decay on ¶ B ( , r k ) , then they have fast decay on ¶ B ( , cr k ) , where we choose c bounded such that S mj = S j ⊂ B ( , c r k ) . Then thenew pair ( ˆ s , , ˆ s , ) becomes ( ˆ s , , ˆ s , ) = ( ˆ s , + m (cid:229) j = m , j , ˆ s , + m (cid:229) j = m , j ) for m , j , m , j ∈ Z . Hence, in this case we can also write(6.10) ( ˆ s , , ˆ s , ) = ( s ∗ + m , s ∗ + m ) with ( s ∗ , s ∗ ) = ( ˆ s , , ˆ s , ) ∈ G ( m , m ) and m , m ∈ Z . Denote cr k = l k . Thenwe can continue our process starting from l k . After finite steps, we could prove thatat most one component u k has slow decay on ¶ B ( , ) and their local masses havethe expression in (3). For case (b), i.e. r k / d k ≤ C . Using ˜ v ki ≤ | y | a j h kj ( d k y ) e ˜ v kj ≤ C on B ( , r k / d k ) . Combined with the fact that ˜ v ki has bounded oscillation on ¶ B ( , t k / d k ) and ˜ v ki ≤ v ki ( x ) = ¯˜ v ki ( ¶ B ( , r k / d k )) + O ( ) for all x ∈ B ( r k / d k ) , where ¯˜ v ki ( ¶ B ( , r k / d k )) stands for the average of ˜ v ki on ¶ B ( , r k / d k ) . Direct com-putation shows that Z B ( , r k ) h ki e u ki d x = Z B ( , r k / d k ) h ki ( d k y ) e ˜ v ki ( y ) d y = O ( ) e ¯˜ v ki ( ¶ B ( , r k / d k )) . So R B ( , r k ) h ki e u ki d x = o ( ) if ¯˜ v ki ( ¶ B ( , r k / d k )) → − ¥ . On the other hand, we notethat ¯˜ v ki ( ¶ B ( , r k / d k )) → − ¥ is equivalent to u ki having fast decay on ¶ B ( , r k ) . Asa consequence, we have ˆ s i , = u ki has fast decay on ¶ B ( , r k ) . So if both twocomponents have fast decay on ¶ B ( , r k ) we have ( ˆ s , , ˆ s , ) = ( , ) .If some component of u k has slow decay on ¶ B ( , r k ) , say u k , then we choose˜ l k , r k ≪ ˜ l k ≪ r k such that s ( B ( , ˜ l k )) = s ( B ( , r k )) = u ki have fast decay on ¶ B ( , ˜ l k ) . Then ( s ( B ( , ˜ l k )) , s ( B ( , ˜ l k ))) satisfies(6.9) with ( s ∗ , s ∗ ) = ( ˆ s , , ˆ s , ) = ( , ) , which implies s ( B ( , ˜ l k )) = s ( B ( , ˜ l k )) = m . Hence in both cases, we could choose ˜ l k ≪ r k such that (1)-(3) holds on ¶ B ( , ˜ l k ) . Afterwards, we continue our discussion as the case (a). Then Theorem 1.3 isproved completely. (cid:3)
Next, we shall prove Theorem 1.6, that is S k = { } by way of contradiction.Suppose S k has points other than 0 . Using the notations from the beginning of thissection, we have S k = { } ∪ S ∪ · · · ∪ S N . Now suppose r k / d k → ¥ as k → ¥ . Let ( ˆ s , , ˆ s , ) be the local masses defined by(6.7) for one of the component u ki having slow decay on ¶ B ( , r k ) or by (6.8) forboth two components having fast decay on ¶ B ( , r k ) . Then we recall the followingresult(i). ˆ s i , = s ∗ i + m i , where ( s ∗ , s ∗ ) ∈ G ( m , m ) and m i , i = , ( s ∗ , s ∗ ) and ( ˆ s , , ˆ s , ) satisfy the Pohozaev identity.Based on the description above, we are able to prove Theorem 1.6. Proof of Theorem 1.6.
From the above discussion, we have ( ˆ s , , ˆ s , ) = ( s ∗ + m , s ∗ + m ) . We note that the conclusion of Theorem 1.5 is equivalent to show m i = , i = , . In the following, we shall prove m i = , i = , . ODA SYSTEM 23
From the above discussion, we have both ( ˆ s , , ˆ s , ) and ( s ∗ , s ∗ ) satisfy the Po-hozaev identity(6.11) k s + k k s s + k s = k m s + k m s . We can write them as(6.12) k ( s ∗ ) + k k s ∗ s ∗ + k ( s ∗ ) = k m s ∗ + k m s ∗ , and(6.13) k ( s ∗ + m ) + k k ( s ∗ + m )( s ∗ + m ) + k ( s ∗ + m ) = k m ( s ∗ + m ) + k m ( s ∗ + m ) . We use (6.13) and (6.12) to get(6.14) 2 k m ˆ s ∗ + k k m ˆ s ∗ + k k m s ∗ + k m s ∗ = k m m + k m m − ( k m + k k m m + k m ) . Since ( s ∗ , s ∗ ) ∈ G ( m , m ) , we set s ∗ = l , m + l , m , s ∗ = l , m + l , m . Then we can rewrite (6.14) as(6.15) ( k l , m + k k l , m − k m + k l , m + k k l , m ) m + ( k l , m + k k l , m + k l , m + k k l , m − k m ) m + ( k m + k k m m + k m ) = . Since m , m and 1 are linearly independent, we have the coefficients of m and m must vanish. Equivalently we have(6.16) (cid:18) k l , + k k l , − k k l , + k k l , k l , + k k l , k l , + k k l , − k (cid:19) (cid:18) m m (cid:19) = M K as M K = (cid:18) k l , + k k l , − k k l , + k k l , k l , + k k l , k l , + k k l , − k (cid:19) . Claim : M k is non-singular. We shall divide our proof into the following threecases.Case 1. K = A . Then we can write (6.16) as(6.17) (cid:18) l , − l , − l , − l , l , − l , l , − l , − (cid:19) (cid:18) m m (cid:19) = . We note that ( l , , l , , l , , l , ) ∈ { ( , , , ) , ( , , , ) , ( , , , ) , ( , , , ) , ( , , , ) } . Then it is easy to see that M K is non-singular when ( l , , l , , l , , l , ) belongs theabove set.Case 2. K = B . Then we can write (6.16) as(6.18) (cid:18) l , − l , − l , − l , l , − l , l , − l , − (cid:19) (cid:18) m m (cid:19) = . We note that ( l , , l , , l , , l , ) ∈ { ( , , , ) , ( , , , ) , ( , , , ) , ( , , , ) , ( , , , ) , ( , , , ) , ( , , , ) } From the above set, we can see that 4 | ( l , l , )( l , − l , ) . As a result, if thedeterminant of M K is 0, we have to make 4 | ( l , − l , − ) , which forces l , ≡ ( mod ) . However, this is impossible. Thus M k is non-singular in this case.Case 3. K = G . Then we can write (6.16) as(6.19) (cid:18) l , − l , − l , − l , l , − l , l , − l , − (cid:19) (cid:18) m m (cid:19) = . We note that ( l , , l , , l , , l , ) ∈ { ( , , , ) , ( , , , ) , ( , , , ) , ( , , , ) , ( , , , ) , ( , , , ) , ( , , , ) , ( , , , ) , ( , , , ) , ( , , , ) , ( , , , ) } . From the above list, we note 3 | l , , then we get 9 | ( l , − l , )( l , − l , ) . Onthe other hand, we note l , ≡ , ( mod ) and l , ≡ , ( mod ) , this implies ( l , − l , − )( l , − l , − ) is not multiple of 9, therefore wehave the determinant of M K is not zero. Thus we have M k is non-singular when K = G . From the above discussion, we have ( m , m ) = ( , ) . Therefore, Theorem 1.6is proved completely. (cid:3) At the end, we give the proof of Theorem 1.2 and Theorem 1.7.
Proof of Theorem 1.2 and Theorem 1.7.
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