A new type of bubble solutions for a Schrödinger equation with critical growth
aa r X i v : . [ m a t h . A P ] J a n A NEW TYPE OF BUBBLE SOLUTIONS FOR A SCHR ¨ODINGEREQUATION WITH CRITICAL GROWTH
QIHAN HE, CHUNHUA WANG † AND QINGFANG WANG
Abstract.
In this paper, we investigate the following critical elliptic equation − ∆ u + V ( y ) u = u N +2 N − , u > , in R N , u ∈ H ( R N ) , where V ( y ) is a bounded non-negative function in R N . Assuming that V ( y ) = V ( | ˆ y | , y ∗ ) , y =(ˆ y, y ∗ ) ∈ R × R N − and gluing together bubbles with different concentration rates, we ob-tain new solutions provided that N ≥ , whose concentrating points are close to the point( r , y ∗ ) which is a stable critical point of the function r V ( r, y ∗ ) satisfying r > V ( r , y ∗ ) > . In order to construct such new bubble solutions for the above problem, wefirst prove a non-degenerate result for the positive multi-bubbling solutions constructed in[21] by some local Pohozaev identities, which is of great interest independently. Moreover,we give an example which satisfies the assumptions we impose.Key words : critical; new bubble solutions; non-degeneracy; local Pohozaev identities.AMS Subject Classification :35B05; 35B45. Introduction and the main result
Standing waves for the following nonlinear Schr¨odinger equation in R N , i ∂ψ∂t = ∆ ψ − ˜ V ( y ) ψ + | ψ | p − ψ, (1.1)are solutions of the form ψ ( t, y ) = e iλt u ( y ) , where i denotes the imaginary part and λ ∈ R , p > . Assuming that u ( y ) is positive and vanishes at infinity, we see ψ satisfies (1.1)if and only if u satisfies the following nonlinear elliptic problem − ∆ u + V ( y ) u = u p , u > , lim | y |→∞ u ( y ) = 0 , (1.2)where V ( y ) = ˜ V ( y ) − λ . Hereafter, we assume that V ( y ) is bounded and V ( y ) ≥ . When1 < p < N +2 N − in (1.2) i.e. the subcritical exponent case, in [23] Wei and Yan showed theequation has infinitely many non-radial positive solutions when V ( y ) is a radially positivefunction. There are various existence results for the subcritical case, such as [5, 6, 8].In this paper, we will investigate the critical case i.e. p = N +2 N − : − ∆ u + V ( y ) u = u N +2 N − , u > , u ∈ H ( R N ) , (1.3) † Corresponding author: Chunhua Wang. † AND QINGFANG WANG where V ( y ) ≥ V
0. It corresponds to the following well-known Brezis-Nirenbergproblem in S N − ∆ S N u = u N +2 N − + µu, u > , on S N . (1.4)Indeed, after using the stereographic projection, problem (1.4) can be reduced to (1.3) with V ( y ) = − µ − N ( N − | y | ) , and V ( y ) > µ < − N ( N − . Problem (1.4) has been studied extensively. In [3], Brezisand Li proved if µ > − N ( N − , then the only solutions to (1.4) is the constant u = ( − µ ) N − . When µ = − N ( N − , in [10] Druet (see also Druet and Hebey [11, 12]) proved that the setof positive solutions to (1.4) is compact provided that the energy is bounded. Furthermore,in [1, 4] it has been proved that there are more and more non-radial solutions as µ → −∞ . In [7], Chen, Wei and Yan proved that µ < − N ( N − and N ≥ , there are infinitely manynon-radial solutions to (1.4) whose energy can be made arbitrarily large. This impliesthat the boundedness of energy in [10, 11] is necessary. When µ = − N ( N − and u N +2 N − istaken place of K ( y ) u N +2 N − in (1.4) where K ( y ) being a fixed smooth function, in [24] Weiand Yan showed that it has infinitely many non-radial positive solutions. More results onthe existence, multiplicity and qualitative properties of solutions for non-compact ellipticproblems can also be found in [3, 9, 13, 16, 19, 20, 25] and the references therein.It is not difficult to see that if V ≥ V
0, then the mountain pass value for problem(1.3) is not a critical value of the corresponding functional. Hence all the arguments basedon the concentration compactness arguments [17, 18] can not be used to obtain an existenceresult of solutions for (1.3). To our best knowledge, the first existence result for (1.3) isdue to Benci and Cerami [2]. They proved that if k V k L N ( R N ) is suitably small, (1.3) has asolution whose energy is in the interval (cid:0) N S N , N S N (cid:1) , where S is the best Sobolev constantin the embedding D , ( R N ) ֒ → L NN − ( R N ). For the Brezis–Nirenberg problem in S N , Benciand Cerimi’s result only yields an existence result if − µ − N ( N − > N ≥ V ( y ) is radially symmetric and r V ( r ) has a local maximum point, or a local minimumpoint r > V ( r ) >
0. Note that this condition is necessary for the existence ofsolutions since by the following Pohozaev identity Z R N (cid:16) V ( | y | ) + 12 | y | V ′ ( | y | ) (cid:17) u = 0 , (1.5)(1.3) has no solution if r V ( r ) is always non-decreasing, or non-increasing. Recently, in [21]Peng, Wang and Yan showed problem (1.3) has infinitely many solutions by introducingsome local Pohozaev type identities in the finitely dimensional reduction method, where V ( y ) satisfies the following condition: UBBLE SOLUTIONS FOR A SCHR ¨ODINGER EQUATION WITH CRITICAL GROWTH 3 ( V ′ ) suppose that V ( y ) = V ( | y ′ | , y ′′ ) = V ( r, y ′′ ) , ( y ′ , y ′′ ) ∈ R × R N − and r V ( r, y ′′ ) has acritical point ( r , y ′′ ) satisfying r > V ( r , y ′′ ) > deg ( ∇ ( r V ( r, y ′′ )) , ( r , y ′′ )) =0 . It is well known that the functions U x,λ ( y ) = [ N ( N − N − (cid:16) λ λ | y − x | (cid:17) N − , λ > , x ∈ R N , are the only solutions to the problem − ∆ u = u N +2 N − , u > R N . (1.6)Define H s = n u : u ∈ D , ( R N ) , u ( y , − y , y ′′ ) = u ( y , y , y ′′ ) ,u ( r cos θ, r sin θ, y ′′ ) = u (cid:16) r cos (cid:16) θ + 2 πjm (cid:17) , r sin (cid:16) θ + 2 πjm (cid:17) , y ′′ (cid:17)o . Let x j = (cid:16) ¯ r cos 2( j − πm , ¯ r sin 2( j − πm , ¯ y ′′ (cid:17) , j = 1 , , · · · , m, where ¯ y ′′ is a vector in R N − . Let δ > r V ( r, y ′′ ) > | ( r, y ′′ ) − ( r , y ′′ ) | ≤ δ . Let ζ ( y ) = ζ ( | y ′ | , y ′′ ) be a smooth function satisfying ζ = 1 if | ( r, y ′′ ) − ( r , y ′′ ) | ≤ δ , ζ = 0 if | ( r, y ′′ ) − ( r , y ′′ ) | ≥ δ , and 0 ≤ ζ ≤
1. Denote Z x j ,λ ( y ) = ζ U x j ,λ , Z ∗ ¯ r, ¯ y ′′ ,λ = m X j =1 U x j ,λ , Z ¯ r, ¯ y ′′ ,λ ( y ) = m X j =1 Z x j ,λ ( y ) . Set x j = (cid:16) ¯ r cos 2( j − πm , ¯ r sin 2( j − πm , ¯ y ′′ (cid:17) , j = 1 , · · · , m, ¯ y ′′ ∈ R N − . We recall that the result obtained in [21] is as follows.
Theorem A.
Suppose that V ≥ is bounded and belongs to C . If V ( | y ′ | , y ′′ ) satisfies ( V ′ ) and N ≥ , then there exists a positive integer m > , such that for any integer m ≥ m , (1.3) has a solution u m of the form u m = Z ¯ r m , ¯ y ′′ m ,λ m , + ϕ m = m X j =1 ζ U x j ,λ m + ϕ m , (1.7) where ϕ m ∈ H s . Moreover, as m → + ∞ , λ m ∈ [ L m N − N − , L m N − N − ] , (¯ r m , ¯ y ′′ m ) → ( r , y ′′ ) , and λ − N − m k ϕ m k L ∞ → . To construct new bubble solutions for problem (1.3), we first want to apply some localPohozaev identities to prove the multi-bubbling solutions in
Theorem A above is non-degenerate.In order to state our main result, we give some assumptions of the function V ( y ) : QIHAN HE, CHUNHUA WANG † AND QINGFANG WANG ( V ) suppose that V ( y ) = V ( | ˆ y | , y ∗ ) = V ( r, y ∗ ) , (ˆ y, y ∗ ) ∈ R × R N − and r V ( r, y ∗ ) has acritical point ( r , y ∗ ) satisfying r > V ( r , y ∗ ) > deg ( ∇ ( r V ( r, y ∗ )) , ( r , y ∗ )) =0 . ( ˜ V ) det ( A i,l ) ( N − × ( N − = 0 , i, l = 1 , , · · · , N − , where A i,l = h ∂ V∂r − (cid:0) ∂ ∆ V∂y V + ν h ν,x i (cid:1)(cid:0) r ∂ V∂r + N P j =5 y j ∂ V∂r∂y j (cid:1)i ( r , y ∗ ) , when i = l = 1; h ∂ V∂r∂y l +3 − (cid:0) ∂ ∆ V∂y V + ν h ν,x i (cid:1)(cid:0) r ∂ V∂r∂y l +3 + N P j =5 y j ∂ V∂y j ∂y l +3 (cid:1)i ( r , y ∗ ) , when i = 1 , l = 2 , , ..., N − iπm h ∂ V∂r∂y i +3 − (cid:0) ∂ ∆ V∂yi +3 V + ν i +3 h ν,x i (cid:1)(cid:0) r ∂ V∂r + N P j =5 y j ∂ V∂r∂y j (cid:1)i ( r , y ∗ ) , when i = 2 , , ..., N − , l = 1; h ∂ V∂y i +3 ∂y l +3 − (cid:0) ∂ ∆ V∂yi +3 V + ν i +3 h ν,x i (cid:1)(cid:0) r ∂ V∂r∂y l +3 + N P j =5 y j ∂ V∂y j ∂y l +3 (cid:1)i ( r , y ∗ ) , when i, l = 2 , , ..., N − ,ν i and ν are the i -th unit outward normal and unit outward normal respectively on Ω defined in (2.3).Assume that δ > r V ( r, y ∗ ) > | ( r, y ∗ ) − ( r , y ∗ ) | ≤ δ .We also define a cut-off function ˆ ζ ( y ) = ˆ ζ ( | ˆ y | , y ∗ ) be a smooth function satisfying ˆ ζ = 1 if | ( r, y ∗ ) − ( r , y ∗ ) | ≤ δ , ˆ ζ = 0 if | ( r, y ∗ ) − ( r , y ∗ ) | ≥ δ , and 0 ≤ ˆ ζ ≤ Remark 1.1.
From the proof of
Theorem A in [21], if we substitute the assumption ( V )for the assumption ( V ′ ) , then only by making some minor modifications we can also provethat the result of Theorem A is still true. For simplicity of notations, we still denote thesolution as u m , and u m = m P j =1 ˆ ζ ( y ) U ˆ x j ,λ + ϕ m , whereˆ x j = (cid:16) ¯ r cos 2( j − πm , ¯ r sin 2( j − πm , , , ˜ y ∗ (cid:17) , j = 1 , · · · , m. Then, like Remark 1.1 we can also find a solution with n -bubbles, whose centers lie nearthe surface ( r , y ∗ ) satisfying | ˆ y | = | ( y , y , y , y ) | = r . The question we want to discussin this paper is whether these two solutions can be glued together to generate a new typeof solutions. In other words, we are concerned with looking for a new solution to (1.3),whose shape is, at main order u ≈ m X j =1 ˆ ζ ( y ) U ˆ x j ,λ + n X j =1 ˆ ζ ( y ) U p j ,µ := m X j =1 Z ˆ x j ,λ + n X j =1 Z p j ,µ , (1.8)for m and n big integers, where we take UBBLE SOLUTIONS FOR A SCHR ¨ODINGER EQUATION WITH CRITICAL GROWTH 5 p j = (cid:16) , , t cos 2( j − πn , t sin 2( j − πn , ˜ y ∗ (cid:17) , j = 1 , · · · , n, ˜ y ∗ ∈ R N − . Here ¯ r and t are close to r and ˜ y ∗ → y ∗ = ( y , , y , , ..., y ,N ) . The energy functional corresponding to equation (1.3) is I ( u ) = 12 Z R N (cid:0) |∇ u | + V u (cid:1) − ∗ Z R N | u | ∗ , u ∈ H ( R N ) . Therefore, generally speaking, a function of the form (1.8) is an approximate solution to(1.3) provided that ¯ r, t, ˜ y ∗ and the parameters µ and λ are such that I ′ (cid:16) m X j =1 Z ˆ x j ,λ + n X j =1 Z p j ,µ (cid:17) ∼ . Letting that λ, µ → ∞ , ¯ r, t → r and ˜ y ∗ → y ∗ , we can easily obtain that I (cid:16) m X j =1 Z ˆ x j ,λ + n X j =1 Z p j ,µ (cid:17) = ( m + n ) A + m (cid:16) B V (¯ r, ˜ y ∗ ) λ − m X j =2 B λ N − | ˆ x − ˆ x j | N − + O (cid:0) λ ǫ (cid:1)(cid:17) + n (cid:16) C V ( t, ˜ y ∗ ) µ − m X j =2 C µ N − | p − p j | N − + O (cid:0) µ ǫ (cid:1)(cid:17) , (1.9)where A = (cid:0) − ∗ (cid:1) Z R N |∇ U , | and B , B , C , C are some positive constants and ǫ > n ≫ m, then the two terms in (1.9) are of different orders,which causes it not easy to find a critical point of I. Hence it is very difficult to apply areduction argument to construct solutions of the form (1.8).In this paper, we use a new method which was first introduced by Guo, Musso, Pengand Yan recently in [15] where they studied the prescribed scalar curvature equation witha radial potential function. Recall that we intend to glue n -bubbles, whose centers lie onthe surface ( r , y ∗ ) to the m -bubbling solution u m described in Remark 1.1. The linearoperator for such a problem is Q n η = − ∆ η + V ( y ) η − (2 ∗ − (cid:16) u m + n X j =1 Z p j ,µ (cid:17) ∗ − η. Away from the points p j , the operator Q n can be approximated by the linearized operatoraround u m , defined by L m η = − ∆ η + V ( y ) η − (2 ∗ − u ∗ − m η. (1.10) QIHAN HE, CHUNHUA WANG † AND QINGFANG WANG
The approach we use here is to construct the solution with m -bubbles whose centeris close to ( r , y ∗ ) and n -bubbles whose center lies near ( r , y ∗ ) as a perturbation of thesolution with the m -bubbles whose center lie near ( r , y ∗ ) . The main result of this paper is as follows:
Theorem 1.2.
Suppose V ( y ) satisfies the assumptions ( V ) , ( ˜ V ) and N ≥ . Let u m be asolution in Remark 1.1 and m > is a large even number. Then there is an integer n > , depending on m , such that for any even number n ≥ n , (1.3) has a solution whosemain order is of the form (1.8) for some t n → r , ˜ y ∗ → y ∗ and µ n ∼ n N − N − . Remark 1.3.
Like [21], in section 3 to deal with the slow decay of the function U p j ,µ ( y )when the dimension N is not big, we introduce the cut-off function ˆ ζ ( y ) . Remark 1.4.
We want to point out that if we assume that the small constants δ in thedefinition of the cut-off function ˆ ζ ( y ) and ϑ in (3.10) which are less or equal to cµ − , theresult in Theorem 1.2 can hold for N ≥ . It is just technical. In this case, we have thefollowing relation 1 µ ≤ C µ | y − p j | , which can help us to deal with some estimates such as (3.36). Remark 1.5.
In Theorem 1.2, in order to obtain the solution u ( y ) satisfying that it iseven about y h , h = 1 , , , , we assume that m, n are both even integers. Otherwise, onlyto obtain the existence of the solution u ( y ) , we do not need this requirement.To prove Theorem 1.2, we mainly argue as [21] and [22]. Since the concentration pointsof the bump solutions include a saddle point of V ( r, y ∗ ) , we apply some local Pohozaevidentities to locate the concentration points of the bump solutions, instead of estimatingdirectly the derivatives of the reduced functional. Compared with [21] and [22], in theprocess of doing the finite-dimensional reduction we need to compute more accurately,such as the estimate of J in Lemma 3.2, where we follow some ideas from [14].Our paper is organized as follows. In section 2, we will prove a non-degenerate resultby some local Pohozaev identities, which is very crucial in constructing a new type ofbubbling solutions by applying the finitely dimensional reduction method. Applying thenon-degenerate result, we construct new solutions and prove Theorem 1.2 in section 3. Insection A, we give some Pohozaev identities. We give an example of the potential V ( r, y ∗ )which satisfies the assumptions ( V ) and ( ˜ V ) in appendix C.2. the non-degeneracy of the solutions In this section, we mainly prove the non-degeneracy of the multi-bubbling solutionsobtained by Peng, Wang and Yan in [21].Define k u k ∗ = sup y ∈ R N (cid:16) m X j =1 λ m | y − x m,j | ) N − + τ (cid:17) − λ − N − m | u ( y ) | (2.1) UBBLE SOLUTIONS FOR A SCHR ¨ODINGER EQUATION WITH CRITICAL GROWTH 7 and k f k ∗∗ = sup y ∈ R N (cid:16) m X j =1 λ m | y − x m,j | ) N +22 + τ (cid:17) − λ − N +22 m | f ( y ) | , (2.2)where x m,j = ( r m cos j − πm , r m sin j − πm , ¯ x ′′ m ), τ = N − N − .Let Ω j = n y = ( y ′ , y ′′ ) ∈ R × R N − : h y ′ | y ′ | , x ′ m,j | x ′ m,j | i ≥ cos πm o . (2.3)Define the linear operator L m η = − ∆ η + V η − (2 ∗ − u ∗ − m η. Lemma 2.1.
There is a positive constant C such that | u m ( y ) | ≤ C m X j =1 λ N − m λ m | y − x m,j | ) N − for all y ∈ R N . Proof.
Since the proof is just the same as Lemma 2.2 of [15], here we omit it. (cid:3)
Now we prove Theorem 2.1 by an indirect method. Assume that there are m k → + ∞ ,satisfying k η k k ∗ = 1 and L m k η k = 0 . (2.4)Denote e η k ( y ) = λ − N − m k η k ( λ − m k y + x m k , ) . (2.5) Lemma 2.2.
It holds e η k → b ψ + N X i =1 ,i =2 b i ψ i , (2.6) uniformly in C ( B R (0)) for any R > , where b and b i ( i = 1 , , · · · , N ) are some constants, ψ = ∂U ,λ ∂λ (cid:12)(cid:12)(cid:12) λ =1 , ψ i = ∂U , ∂y i , i = 1 , , · · · , N. Proof.
Observing that | e η k | ≤ C , we may suppose that e η k → η in C loc ( R N ). Then η satisfies − ∆ η = (2 ∗ − U ∗ − η, x ∈ R N , which implies η = N X i =0 b i ψ i . Since η k is even in y , there holds b = 0. (cid:3) QIHAN HE, CHUNHUA WANG † AND QINGFANG WANG
We decompose η k ( y ) = b ,m λ m k m k X j =1 ∂Z x mk,j ,λ mk ∂λ m k + b ,m λ − m k m k X j =1 ∂Z x mk,j ,λ mk ∂r + N X i =3 b i,m λ − m k ∂Z x mk,j ,λ mk ∂ ¯ y i + η ∗ k , where η ∗ k satisfies Z R N Z ∗ − x mk,j ,λ mk ∂Z x mk,j ,λ mk ∂λ m k η ∗ k = Z R N Z ∗ − x mk,j ,λ mk ∂Z x mk,j ,λ mk ∂r η ∗ k = Z R N Z ∗ − x mk,j ,λ mk ∂Z x mk,j ,λ mk ∂ ¯ y i η ∗ k = 0 , ( i = 3 , · · · , N ) . It follows from Lemma 2.2 that b ,m , b ,m and b i,m ( i = 3 , · · · , N ) are bounded. Lemma 2.3.
There holds k η ∗ k k ∗ ≤ Cλ − − ǫm k , where ǫ > is a small constant.Proof. One can see easily that L m k η ∗ k = − ∆ η ∗ k + V η ∗ k − (2 ∗ − u ∗ − m k η ∗ k = − V (cid:16) b ,m λ m k m k X j =1 ∂Z x mk,j ,λ mk ∂λ m k + b ,m λ m k ∂Z x mk,j ,λ mk ∂r + N X i =3 b i,m λ m k m k X j =1 ∂Z x mk,j ,λ mk ∂ ¯ y i (cid:17) + (2 ∗ − m k X j =1 ( u ∗ − m k − U ∗ − x mk,j ,λ mk ) (cid:16) b ,m λ m k ∂Z x mk,j ,λ mk ∂λ m k + b ,m λ m k ∂Z x mk,j ,λ mk ∂r + N X i =3 b i,m λ m k ∂Z x mk,j ,λ mk ∂ ¯ y i (cid:17) + 2 ∇ ξ (cid:16) b ,m λ m k m k X j =1 ∇ ∂U x mk,j ,λ mk ∂λ m k + b ,m λ m k m k X j =1 ∇ ∂U x mk,j ,λ mk ∂r + N X i =3 b i,m λ m k m k X j =1 ∇ ∂U x mk,j ,λ mk ∂ ¯ y i (cid:17) + ∆ ξ (cid:16) b ,m λ m k m k X j =1 ∂U x mk,j ,λ mk ∂λ m k + b ,m λ m k m k X j =1 ∂U x mk,j ,λ mk ∂r + N X i =3 b i,m λ m k m k X j =1 ∂U x mk,j ,λ mk ∂ ¯ y i (cid:17) : = L + L + L + L . Similar to the proof of J in Lemma 2.4 in [21], we can prove (cid:13)(cid:13) L (cid:13)(cid:13) ∗∗ ≤ Cλ − − ǫm k . (2.7) UBBLE SOLUTIONS FOR A SCHR ¨ODINGER EQUATION WITH CRITICAL GROWTH 9
By the same argument as that of [15] in Lemma 2.4, we can estimate (cid:13)(cid:13) L (cid:13)(cid:13) ∗∗ ≤ Cλ − − ǫm k . (2.8)Similar to J of Lemma 2.4 in [21], we can check | L | ≤ C (cid:16) λ m k (cid:17) ǫ m k X j =1 λ N +22 m k (1 + λ m k | y − x m k ,j | ) N +22 + τ . Hence, we obtain k L k ∗∗ ≤ C (cid:16) λ m k (cid:17) ǫ . (2.9)Also, similar to J of Lemma 2.4 in [21], we can prove | L | ≤ C ( 1 λ m k ) ǫ m k X j =1 λ N +22 m k (1 + λ m k | y − x m k ,j | ) N +22 + τ , which yields that k L k ∗∗ ≤ C (cid:16) λ m k (cid:17) ǫ . (2.10)It follows from (2.7) to (2.10) that k L m k η ∗ k k ∗∗ ≤ Cλ − − ǫm k . (2.11)Furthermore, from Z R N Z ∗ − x mk,j ,λ mk ∂Z x mk,j ,λ mk ∂λ m k η ∗ k = Z R N Z ∗ − x mk,j ,λ mk ∂Z x mk,j ,λ mk ∂r η ∗ k = Z R N Z ∗ − x mk,j ,µ mk ∂Z x mk,j ,λ mk ∂ ¯ y i η ∗ k = 0 , ( i = 3 , · · · , N )and Lemma 2.1, we can prove that there exists ρ > k L m k η ∗ k k ∗∗ ≥ ρ k η ∗ k k ∗ . (2.12)Combining (2.11) and (2.12), the result is true. (cid:3) Now we give another assumption of V ( y ) :( ˜ V ′ ) det ( A i,l ) ( N − × ( N − = 0 , i, l = 1 , , ..., N − , † AND QINGFANG WANG where A i,l = h ∂ V∂r − (cid:0) ∂ ∆ V∂y V + ν h ν,x i (cid:1)(cid:0) r ∂ V∂r + N P j =3 y j ∂ V∂r∂y j (cid:1)i ( r , y ′′ ) , when i = l = 1; h ∂ V∂r∂y l +1 − (cid:0) ∂ ∆ V∂y V + ν h ν,x i (cid:1)(cid:0) r ∂ V∂r∂y l +1 + N P j =3 y j ∂ V∂y j ∂y l +1 (cid:1)i ( r , y ′′ ) , when i = 1 , l = 2 , , ..., N − iπm h ∂ V∂r∂y i +1 − (cid:0) ∂ ∆ V∂yi +1 V + ν i +1 h ν,x i (cid:1)(cid:0) r ∂ V∂r + N P j =3 y j ∂ V∂r∂y j (cid:1)i ( r , y ′′ ) , when i = 2 , , ..., N − , l = 1; h ∂ V∂y i +1 ∂y l +1 − (cid:0) ∂ ∆ V∂yi +1 V + ν i +1 h ν,x i (cid:1)(cid:0) r ∂ V∂r∂y l +1 + N P j =3 y j ∂ V∂y j ∂y l +1 (cid:1)i ( r , y ′′ ) , when i, l = 2 , , ..., N − , (2.13) ν i and ν are the i -th unit outward normal and unit outward normal respectively on Ω defined in (2.3). Lemma 2.4. If ( ˜ V ′ ) holds, then ˜ η k → uniformly in C ( B R (0)) for any R > .Proof. Step1. Recall thatΩ j = n y = ( y ′ , y ′′ ) ∈ R × R N − : h y ′ | y ′ | , x ′ m,j | x ′ m,j | i ≥ cos πm o . In order to prove b i,k → i = 1 , , · · · , N ), we apply the identities in Lemma A.1 in thedomain Ω , − Z ∂ Ω ∂u m k ∂ν ∂η k ∂y i − Z ∂ Ω ∂η k ∂ν ∂u m k ∂y i + Z ∂ Ω h∇ u m k , ∇ η k i ν i + Z ∂ Ω V u m k η k ν i − Z ∂ Ω u ∗ − m k η k ν i = Z Ω ∂V∂y i u m k η k , i = 1 , , · · · , N. (2.14)By the symmetry, we have ∂u mk ∂ν = 0 and ∂η k ∂ν = 0 on ∂ Ω . Hencethe left hand side of (2.14)= ν i (cid:16) Z ∂ Ω h∇ u m k , ∇ η k i + Z ∂ Ω V u m k η k − Z ∂ Ω u ∗ − m k η k (cid:17) . (2.15)Combining (2.14) and (2.15), we obtain ν i (cid:16) Z ∂ Ω h∇ u m k , ∇ η k i + Z ∂ Ω V u m k η k − Z ∂ Ω u ∗ − m k η k (cid:17) = Z Ω ∂V∂y i u m k η k . (2.16) UBBLE SOLUTIONS FOR A SCHR ¨ODINGER EQUATION WITH CRITICAL GROWTH 11
To estimate the left hand side in (2.16), we use (A.4) in Ω . Applying the symmetry, wehave Z Ω u m k η k h∇ V, y − x m k , i + 2 Z Ω V η k u m k = − Z ∂ Ω u ∗ − m k η k h ν, y − x m k , i + Z ∂ Ω h∇ u m k , ∇ η k ih ν, y − x m k , i + Z ∂ Ω V u m k η k h ν, y − x m k , i . (2.17)On ∂ Ω , it holds h ν, y i = 0 . Then, (2.17) becomes Z Ω u m k η k h∇ V, y − x m k , i + 2 Z Ω V η k u m k = − h ν, x m k , i (cid:16) − Z ∂ Ω u ∗ − m k η k + Z ∂ Ω h∇ u m k , ∇ η k i + Z ∂ Ω V u m k η k (cid:17) . (2.18)It follows from (2.16) and (2.18) that Z Ω ∂V∂y i u m k η k = − ν i h ν, x m k , i (cid:16) Z Ω u m k η k h∇ V, y − x m k , i + 2 Z Ω V η k u m k (cid:17) . (2.19)Since ∇ V ( x m k , ) = O ( | x m k , − y | ) and Z Ω u m k η k = Z (Ω ) xmk, ,λmk (cid:0) λ − N − m k u m k ( λ − m k y + x m k , ) (cid:1) ˜ η k = 1 λ m k Z R N U (cid:2) b ,m ψ + b ,m ψ + N X i =3 b i,m ψ i + λ − N − m k η ∗ k ( λ − m k y + x m k , ) (cid:3) + O (cid:0) λ − − ǫm k (cid:1) = O (cid:0) λ − − ǫm k (cid:1) , (2.20)where (Ω ) x km , = { y, λ − k m y + x k m , ∈ Ω } , we have Z Ω ∂V∂y i u m k η k = Z Ω u m k η k (cid:16) ∂V∂y i − ∂V ( x m k , ) ∂y i (cid:17) + Z Ω ∂V ( x m k , ) ∂y i u m k η k = Z Ω u m k η k h h∇ ∂V ( x m k , ) ∂y i , y − x m k , i + 12 h∇ ∂V ( x m k , ) ∂y i ( y − x m k , ) , y − x m k , i + O ( | y − x m k , | ) i + O ( λ − − ǫm k )= 1 λ m k Z R N U (cid:16) b ,k ψ + b ,k ψ + N X l =3 b l,k ψ l (cid:17)(cid:16) h∇ ∂V ( x m k , ) ∂y i , yλ m k i + 12 h∇ ∂V ( x m k , ) ∂y i yλ m k , yλ m k i (cid:17) + O ( λ − − ǫm k )= ∂ V∂y ∂y i ( x m k , ) λ m k b ,k Z R N U ψ y + N X l =3 b l,k ∂ V∂y l ∂y i ( x m k , ) λ m k Z R N U ψ l y l † AND QINGFANG WANG + ∂ △ V ( x mk, ) ∂y i b ,k N λ m k Z R N U ψ | y | + O (cid:0) λ − − ǫm k (cid:1) . (2.21)Moreover, we can estimate Z Ω u m k η k h∇ V, y − x m k , i = Z Ω u m k η k h∇ V ( y ) − ∇ V ( x m k , ) , y − x m k , i + Z Ω u m k η k h∇ V ( x m k , ) , y − x m k , i = Z Ω u m k η k h∇ V ( y ) − ∇ V ( k m k , ) , y − x m k , i + O ( λ − − ǫm k )= Z Ω u m k η k h∇ V ( x m k , )( y − x m k , ) , y − x m k , i + O ( λ − − ǫm k )= 1 λ m k Z R N U (cid:16) b ,k ψ + b ,k ψ + N X l =3 b l,k ψ l (cid:17)(cid:10) ∇ V ( x m k , ) λ − m k y, λ − m k y (cid:11) + O ( λ − − ǫm k )= b ,k ∆ V ( x m k , ) N λ m k Z R N U ψ | y | + O (cid:0) λ − − ǫm k (cid:1) . (2.22)Hence, (2.21) and (2.22) give b ,k λ m k (cid:16) ∂ ∆ V∂y i ( x m k , )2 N + ν i h ν, x m k , i ∆ V ( x m k , ) N (cid:17) Z R N U ψ | y | + b ,k ∂ V∂y ∂y i ( x m k , ) Z R N U ψ y + N X l =3 b l,k ∂ V∂y l ∂y i ( x m k , ) Z R N U ψ l y l = O (cid:0) λ − − ǫm k (cid:1) . (2.23)Step 2. Next, we apply (A.4) to get Z R N u m k η k h∇ V ( y ) , y i = 0 , which implies Z Ω i u m k η k h∇ V ( y ) , y i = 0 . (2.24)On the other hand, proceeding as in the proof of (2.20), we have Z Ω u m k η k h∇ V ( x m k , ) , y i = Z Ω u m k η k h∇ V ( x m k , ) , y − x m k , i + Z Ω u m k η k h∇ V ( x m k , ) , x m k , i = O (cid:0) λ − − ǫm k (cid:1) . UBBLE SOLUTIONS FOR A SCHR ¨ODINGER EQUATION WITH CRITICAL GROWTH 13
Therefore, from (2.23), we have Z Ω u m k η k h∇ V ( y ) , y i = Z Ω u m k η k h∇ V ( y ) − ∇ V ( x m k , ) , y i + O (cid:0) λ − − ǫm k (cid:1) = Z Ω u m k η k h∇ V ( x m k , )( y − x m k , ) , y i + O (cid:0) λ − − ǫm k (cid:1) = 1 λ m k Z R N U (cid:16) b ,k ψ + b ,k ψ + N X l =3 b l,k ψ l (cid:17) h∇ V ( x m k , ) λ − m k y, λ − m k y + x m k , i + O (cid:0) λ − − ǫm k (cid:1) = b ,k ∆ V ( x m k , ) N λ m k Z R N U ψ | y | + b ,k λ m k N X j =1 ( x m k , ) j ∂ V∂y j ∂y ( x m k , ) Z R N U ψ y + N P l =3 b l,k N X j =1 ( x m k , ) j ∂ V∂y j ∂y l ( x m k , ) λ m k Z R N U ψ l y l + O (cid:0) λ − − ǫm k (cid:1) , which combining with (2.24) implies that b ,k ∆ V ( x m k , ) N λ m k Z R N U ψ | y | + b ,k N X j =1 ( x m k , ) j ∂ V∂y j ∂y ( x m k , ) Z R N U ψ y + N X l =3 b l,k N X j =1 ( x m k , ) j ∂ V∂y j ∂y l ( x m k , ) Z R N U ψ l y l = O (cid:0) λ − − ǫm k (cid:1) . (2.25)It follows from (2.23) and (2.25) that b ,k h ∂ V∂y ∂y i ( x m k , ) − (cid:0) ∂ ∆ V∂y i ( x m k , )2∆ V ( x m k , ) + ν i h ν, x m k , i (cid:1) N X j =1 ( x m k , ) j ∂ V∂y j ∂y ( x m k , ) i Z R N U ψ y + N X l =3 b l,k h ∂ V∂y l ∂y i ( x m k , ) − (cid:0) ∂ ∆ V∂y i ( x m k , )2∆ V ( x m k , ) + ν i h ν, x m k , i (cid:1) N X j =1 ( x m k , ) j ∂ V∂y j ∂y l ( x m k , ) i Z R N U ψ l y l = O (cid:0) λ − − ǫm k (cid:1) , i = 1 , , , ..., N, † AND QINGFANG WANG which implies that h ∂ V∂r − (cid:0) ∂ ∆ V∂y V + ν h ν,x i (cid:1)(cid:0) r ∂ V∂r + N P j =3 y j ∂ V∂y r ∂y j (cid:1)i ( r , y ′′ ) b ,k + N − P l =2 h ∂ V∂r∂y l +1 − (cid:0) ∂ ∆ V∂y V + ν h ν,x i (cid:1)(cid:0) r ∂ V∂r∂y l +1 + N P j =3 y j ∂ V∂y j ∂y l +1 (cid:1)i ( r , y ′′ ) b l +1 ,k = O (cid:0) λ − − ǫm k (cid:1) cos πm h ∂ V∂r∂y − (cid:0) ∂ ∆ V∂y V + ν h ν,x i (cid:1)(cid:0) r ∂ V∂r + N P j =3 y j ∂ V∂y r ∂y j (cid:1)i ( r , y ′′ ) b ,k + N − P l =2 h ∂ V∂y ∂y l +1 − (cid:0) ∂ ∆ V∂y V + ν h ν,x i (cid:1)(cid:0) r ∂ V∂r∂y l +1 + N P j =3 y j ∂ V∂y j ∂y l +1 (cid:1)i ( r , y ′′ ) b l +1 ,k = O (cid:0) λ − − ǫm k (cid:1) ................................ cos iπm h ∂ V∂r∂y i +1 − (cid:0) ∂ ∆ V∂yi +1 V + ν i +1 h ν,x i (cid:1)(cid:0) r ∂ V∂r + N P j =3 y j ∂ V∂r∂y j (cid:1)i ( r , y ′′ ) b ,k + N − P l =2 h ∂ V∂y i +1 ∂y l +1 − (cid:0) ∂ ∆ V∂yi +1 V + ν i +1 h ν,x i (cid:1)(cid:0) r ∂ V∂r∂y l +1 + N P j =3 y j ∂ V∂y j ∂y l +1 (cid:1)i ( r , y ′′ ) b l +1 ,k = O (cid:0) λ − − ǫm k (cid:1) ................................. cos N − πm h ∂ V∂r∂y N − (cid:0) ∂ ∆ V∂yN V + ν N h ν,x i (cid:1)(cid:0) r ∂ V∂r + N P j =3 y j ∂ V∂r∂y j (cid:1)i ( r , y ′′ ) b ,k + N − P l =2 h ∂ V∂y N ∂y l +1 − (cid:0) ∂ ∆ V∂yN V + ν N h ν,x i (cid:1)(cid:0) r ∂ V∂r∂y l +1 + N P j =3 y j ∂ V∂y j ∂y l +1 (cid:1)i ( r , y ′′ ) b l +1 ,k = O (cid:0) λ − − ǫm k (cid:1) . (2.26)Obviously, the coefficient matrix of the system (2.26) is just the matrix ( A i,l ) ( N − × ( N − , where A i,l , i, l = 1 , , ..., N − V ′ ) and the theory of solutions for homogenous linear equations inlinear Algebra, we know that the only solution of the system (2.26) is b ,k = o (1) , b l,k = o (1)( l = 3 , · · · , N ) . (cid:3) Now we come to the main result of this section.
Proposition 2.1.
Suppose N ≥ . Assume that V ( y ) satisfies ( V ′ ) and ( ˜ V ′ ). Let η ∈ H s be a solution of L m η = 0 . Then η = 0 . Proof.
First we have | η k ( y ) | ≤ C Z R N | y − z | N − | u ∗ − m k ( z ) || η k ( z ) | dz ≤ C k η k k ∗ Z | y − z | N − | u ∗ − m k ( z ) | m k X j =1 λ N − m k (1 + λ m k | z − x m k ,j | ) N − + τ ≤ C k η k k ∗ m k X j =1 λ N − m k (1 + λ m k | y − x m k ,j | ) N − + τ + θ , (2.27) UBBLE SOLUTIONS FOR A SCHR ¨ODINGER EQUATION WITH CRITICAL GROWTH 15 for some θ > . Hence we have | η k ( y ) | m k X j =1 λ N − m k (1 + λ m k | y − x m k ,j | ) N − + τ ≤ C k η k k ∗ m k X j =1 λ N − m k (1 + λ m k | y − x m k ,j | ) N − + τ + θm k X j =1 λ N − m k (1 + λ m k | y − x m k ,j | ) N − + τ . Since η k → B Rλ − mk ( x m k ,j ) and k η k k ∗ = 1 , we know that | η k ( y ) | m k X j =1 λ N − m k (1 + λ m k | y − x m k ,j | ) N − + τ attains its maximum in R N \ ∪ m k j =1 B Rλ − mk ( x m k ,j ) . Therefore k η k k ∗ ≤ o (1) k η k k ∗ . Hence k η k k ∗ → k → ∞ . This contradicts with k η k k ∗ = 1 . (cid:3) Remark 2.5. If V ( y ) is radial, then the assumption ( ˜ V ′ ) is just∆ V − (cid:16) ∆ V + 12 (∆ V ) ′ (cid:17) r = 0 at r = r . We would like to point out that the local Pohozaev identities play a crucial role in theinvestigation of the non-degeneracy of the multi-bubbling solutions. This novel idea firstcomes from [15]. Also, the non-degeneracy of the solution and the uniqueness of such asolution are two very closely related problems which are both of great interest.
Remark 2.6.
From the proof of Proposition 2.1, if we we substitue the assumption ( ˜ V ′ )for the assumption ( ˜ V ) , then only by making some minor modifications we can also provethe bubbling solution u m in Remark 1.1 is non-degenerate.It follows from Remark 1.1 and Remark 2.6 that if the assumptions ( V ) and ( ˜ V ) hold,then problem (1.3) has a non-degenerate m -bubbling solution of the form u m = Z ¯ r m , ˆ y ∗ m ,λ m , + ϕ m = m P j =1 ˆ ζU ˆ x j ,λ m + ϕ m , where ϕ m ∈ H s . Moreover, as m → + ∞ , λ m ∈ [ L m N − N − , L m N − N − ],(¯ r m , ¯ y ∗ m ) → ( r , y ∗ ) , and λ − N − m k ϕ m k L ∞ → Construction of a new bubble solution
With the non-degenerate result obtained in section 2 at hand, we can construct a newmulti-bubbling solution for (1.3) as in [21, 22].Set n ≥ m be a large even integer. Recall thatˆ x j = (cid:16) ¯ r cos 2( j − πm , ¯ r sin 2( j − πm , , , ˜ y ∗ (cid:17) , j = 1 , · · · , m, ˜ y ∗ = (¯ y , ¯ y , · · · , ¯ y N ) , † AND QINGFANG WANG and p j = (cid:16) , , t cos 2( j − πn , t sin 2( j − πn , ˜ y ∗ ) , where t is close to r and ˜ y ∗ is close to y ∗ = ( y , , y , , ..., y ,N ) ∈ R N − . Define k u k ˜ ∗ = sup y ∈ R N (cid:16) n X j =1 µ n | y − p n,j | ) N − + τ (cid:17) − µ − N − n | u ( y ) | (3.1)and k f k ˜ ∗ ˜ ∗ = sup y ∈ R N (cid:16) n X j =1 µ n | y − p n,j | ) N +22 + τ (cid:17) − µ − N +22 n | f ( y ) | , (3.2)where p n,j = ( t n cos j − πn , t n sin j − πn , x ∗ n ), τ = N − N − . Let u m be the m -bubbling solutions in Remark 2.6, where m > m is even, u m is even in y j , j = 1 , , , . We define X s = n u : u ∈ H s , u is even in y h , h = 1 , , , ,u ( y , y , t cos θ, t sin θ, y ∗ ) = u (cid:0) y , y , t cos( θ + 2 πjn ) , t sin( θ + 2 πjn ) , y ∗ (cid:1)o , where y ∗ = ( y , y , · · · , y N ).Denote M j = n y = ( y ′ , y , y , y ∗ ) ∈ R × R × R N − : h ( y , y ) | ( y , y ) | , ( p j , p j ) | ( p j , p j ) | i ≥ cos πn o . Assume that | ( t, ˜ y ∗ ) − ( r , y ∗ ) | ≤ ϑ, (3.3)where ϑ > u m and n P j =1 U p j ,µ belong to X s , while u m and n P j =1 U p j ,µ are separatedfrom each other. We intend to construct a solution for (1.3) of the form u = u m + n X j =1 ˆ ζ ( y ) U p j ,µ + ψ := u m + ˆ ζ ( y ) Z ∗ t, ˜ y ∗ ,µ ( y ) + ψ := u m + Z t, ˜ y ∗ ,µ ( y ) + ψ, where ψ ∈ X s is a small perturbed term. Recall that Z p j ,µ = ˆ ζ ( y ) U p j ,µ . Define the linear operator Q n ψ = − ∆ ψ + V ( y ) ψ − (2 ∗ − (cid:16) u m + n X j =1 Z p j ,µ (cid:17) ∗ − ψ, ψ ∈ X s . (3.4)Denote D j, = ∂Z p j ,µ ∂µ , D j, = ∂Z p j ,µ ∂t , D j,k = ∂Z p j ,µ ∂ ˜ y ∗ k , k = 5 , , · · · , N. UBBLE SOLUTIONS FOR A SCHR ¨ODINGER EQUATION WITH CRITICAL GROWTH 17
Let g n ∈ X s . Now we consider Q n ψ n = g n + N − P i =1 a n,i n P j =1 Z ∗ − p j ,µ D j,i ,ψ n ∈ X s , Z R N Z ∗ − p j ,µ D j,i ψ n = 0 , i = 1 , · · · , N − , j = 1 , , · · · , n (3.5)for some constants a n,i , depending on ψ n . Lemma 3.1.
Suppose that ψ n solves (3.5) . If k g n k ˜ ∗ ˜ ∗ → , then k ψ n k ˜ ∗ → .Proof. We argue by contradiction. Suppose that there exist n → + ∞ , ¯ t n → r , ¯ y ∗ n → y ∗ , µ n ∈ [ L n N − N − , L n N − N − ] and ψ n solving (3.5) for g = g n , µ = µ n , ¯ t = ¯ t n , ¯ y ∗ = ¯ y ∗ n with k g n k ˜ ∗ ˜ ∗ → k ψ n k ˜ ∗ ≥ c > . We may assume that k ψ n k ˜ ∗ = 1. For simplicity, we dropthe subscript n. We have | ψ ( y ) | ≤ C Z R N | z − y | N − (cid:0) u m ( z ) + Z t, ˜ y ∗ ,µ ( z ) (cid:1) ∗ − | ψ ( z ) | dz + C Z R N | z − y | N − (cid:16) | g | + (cid:12)(cid:12)(cid:12) N − X l =1 a l m X j =1 D ∗ − p j ,µ D j,l (cid:12)(cid:12)(cid:12)(cid:17) dz. (3.6)Applying the following inequality that for β, γ being any two complex numbers, thereholds | β + γ | p ≤ max { , p − } (cid:0) | β | p + | γ | p (cid:1) , (0 < p < + ∞ ) (3.7)we have (cid:0) u m + Z t, ˜ y ∗ ,µ (cid:1) ∗ − ≤ C (cid:0) u ∗ − m + | Z t, ˜ y ∗ ,µ | ∗ − (cid:1) ≤ C (cid:0) Z t, ˜ y ∗ ,µ (cid:1) ∗ − , in every M j , (3.8)where we use the fact that u m is bounded in M j . As in [24], using Lemma B.2 and B.3, from (3.8) we can prove Z R N | z − y | N − (cid:0) u m + Z t, ˜ y ∗ ,µ (cid:1) ∗ − | ψ | dz ≤ Z S nj =1 M j | z − y | N − (cid:0) u m + Z t, ˜ y ∗ ,µ (cid:1) ∗ − | ψ | dz ≤ C m X j =1 Z M j | z − y | N − (cid:0) u m + Z t, ˜ y ∗ ,µ (cid:1) ∗ − | ψ | dz ≤ C m X j =1 Z M j | z − y | N − (cid:0) Z t, ˜ y ∗ ,µ (cid:1) ∗ − | ψ | dz ≤ C Z R N | z − y | N − (cid:0) Z t, ˜ y ∗ ,µ (cid:1) ∗ − | ψ | dz ≤ C k ψ k ∗ n X j =1 µ N − (1 + µ | y − p j | ) N − + τ + ι , (3.9) † AND QINGFANG WANG
Assume that | ( t, ˜ y ∗ ) − ( r , y ∗ ) | ≤ ϑ, (3.10)where ϑ > Z R N | z − y | N − | g ( z ) | dz ≤ C k g k ˜ ∗ ˜ ∗ µ N − n X j =1 µ | y − x j | ) N − + τ (3.11)and Z R N | z − y | N − (cid:12)(cid:12)(cid:12) n X j =1 D ∗ − x j ,µ Z j,l (cid:12)(cid:12)(cid:12) dz ≤ Cµ N − + n l n X j =1 µ | y − p j | ) N − + τ , (3.12)where n j = 1, j = 2 , · · · , N − , and n = − . To estimate a l , l = 1 , , · · · , N − , multiplying (3.5) by D ,l ( l = 1 , , · · · , N −
2) andintegrating, we see that a l satisfies N − X h =1 a h n X j =1 Z R N Z ∗ − p j ,µ D j,h D ,l = D − ∆ ψ + V ( r, y ∗ ) ψ − (2 ∗ − Z ∗ − t,y ∗ ,µ ψ, D ,l E − h g, D ,l i . (3.13)It follows from Lemma B.1 that (cid:12)(cid:12)(cid:10) g, D ,l (cid:11)(cid:12)(cid:12) ≤ C k g k ˜ ∗ ˜ ∗ Z R N µ N − + n l (1 + µ | y − p | ) N − n X j =1 µ N +22 (1 + µ | y − p j | ) N +22 + τ ≤ Cµ n l k g k ˜ ∗ ˜ ∗ (cid:16) C + C n X j =2 µ | p j − p | ) τ (cid:17) ≤ Cµ n l k g k ˜ ∗ ˜ ∗ . (3.14)Similar to (2.10) in [21], we can estimate | (cid:10) V ( r, y ∗ ) ψ, D ,l (cid:11) | = O (cid:16) µ n l k ψ k ˜ ∗ µ ǫ (cid:17) , (3.15)where we use the fact that for any | ( r, y ∗ ) − ( r , y ∗ ) | ≤ δ, µ ≤ C µ | y − p j | . (3.16)On the other hand, direct calculation gives (cid:12)(cid:12)(cid:12)D − ∆ ψ − (2 ∗ − u m + Z t, ˜ y ∗ ,µ ) ∗ − ψ, D ,l E(cid:12)(cid:12)(cid:12) = O (cid:16) µ n l k ψ k ˜ ∗ µ ǫ (cid:17) . (3.17)Combining (3.14), (3.15), (3.17), we have D − ∆ ψ + V ( t, y ∗ ) ψ − (2 ∗ − u m + Z t, ˜ y ∗ ,µ ) ∗ − ψ, D ,l E − h g, D ,l i = O (cid:16) µ n l (cid:0) k ψ k ˜ ∗ µ ǫ + k g k ˜ ∗ ˜ ∗ (cid:1)(cid:17) . (3.18) UBBLE SOLUTIONS FOR A SCHR ¨ODINGER EQUATION WITH CRITICAL GROWTH 19
It is easy to check that n X j =1 h D ∗ − p j ,µ D j,h , D ,l i = (¯ c + o (1)) δ hl µ n l (3.19)for some constant ¯ c > a l = 1 µ n l (cid:0) o ( k ψ k ˜ ∗ ) + O ( k g k ˜ ∗ ˜ ∗ ) (cid:1) . (3.20)So, k ψ k ˜ ∗ ≤ o (1) + k g n k ˜ ∗ ˜ ∗ + P nj =1 1(1+ µ | y − p j | ) N −
22 + τ + ι P nj =1 1(1+ µ | y − p j | ) N −
22 + τ ! . (3.21)Since k ψ k ˜ ∗ = 1, we obtain from (3.21) that there is R > k µ − N − ψ k L ∞ ( B R/µ ( p j )) ≥ a > , (3.22)for some j . But ˜ ψ ( y ) = µ − N − ψ ( µ ( y − p j )) converges uniformly in any compact set to asolution u of − ∆ u + V u − (2 ∗ − u ∗ − m u = 0 , in R N , (3.23)for some Λ ∈ [Λ , Λ ]. So it follows from Proposition 2.1 that u = 0. This is a contradictionto (3.22). (cid:3) We want to construct a solution u for (1.3) with u = u m + n X j =1 ˆ ζU p j ,µ + ψ, where ψ ∈ X s is a small perturbed term, satisfying Z R N Z ∗ − p j ,µ D j,l ψ = 0 , j = 1 , · · · , n, l = 1 , , · · · , N − . Then ψ satisfies Q n ψ = l n + R n ( ψ ) , where Q n ψ = − ∆ ψ + V ( y ) ψ − (2 ∗ − (cid:16) u m + n X j =1 Z p j ,µ (cid:17) ∗ − ψ,l n = (cid:16) u m + n X j =1 Z p j ,µ (cid:17) ∗ − − u ∗ − m − n X j =1 ˆ ζU ∗ − p j ,µ − V ( y ) n X j =1 Z p j ,µ + Z ∗ t, ˜ y ∗ ,µ ∆ ˆ ζ + 2 ∇ ˆ ζ ∇ Z ∗ t, ˜ y ∗ ,µ , † AND QINGFANG WANG and R n ( ψ ) = (cid:16) u m + n X j =1 Z p j ,µ + ψ (cid:17) ∗ − − (cid:16) u m + n X j =1 Z p j ,µ (cid:17) ∗ − − (2 ∗ − (cid:16) u m + n X j =1 Z p j ,µ (cid:17) ∗ − ψ. We have the following estimate for k l n k ˜ ∗ ˜ ∗ . Lemma 3.2.
There exists a small ǫ > , such that k l n k ˜ ∗ ˜ ∗ ≤ Cµ ǫ . (3.24) Proof.
First, we write l n = h(cid:0) n X j =1 ˆ ζU p j ,µ (cid:1) ∗ − − n X j =1 ˆ ζU ∗ − p j ,µ i − V ( y ) n X j =1 ˆ ζU p j ,µ + Z ∗ t, ˜ y ∗ ,µ ∆ ˆ ζ + 2 ∇ ˆ ζ ∇ Z ∗ t, ˜ y ∗ ,µ + h(cid:0) u m + n X j =1 ˆ ζU p j ,µ (cid:1) ∗ − − u ∗ − m − (cid:0) n X j =1 ˆ ζU p j ,µ (cid:1) ∗ − i := J + J + J + J + J . (3.25)Just by the same argument as that of Lemma 2.5 in [21], we can estimate J + J + J + J ≤ Cµ ǫ n X j =1 µ N +22 (1 + µ | y − p j | ) N +22 + τ . (3.26)Now we estimate J . Using the assumed symmetry, we just need to estimate J in M . Denote B = M T B µ − ( p ) . Note that, it holds U p ,µ ≥ c in B . When y ∈ B , applying the following formula(1 + t ) p = 1 + pt + O ( t p ) , for t ≥ p ∈ (1 , , we have | J | ≤ CU ∗ − p ,µ (cid:16) u m + n X j =2 U p j ,µ (cid:17) + (cid:16) n X j =2 U p j ,µ (cid:17) ∗ − + ˜ J := J , + J , + ˜ J , (3.27)where ˜ J ≤ C in B . For y ∈ M , by Lemma B.1 we obtain J , ≤ Cµ N +22 (cid:16) n X j =2 µ | y − p j | ) N − (cid:17) N +2 N − ≤ Cµ N +22 (cid:16) n X j =2 µ | y − p | ) N +22 µ | y − p j | ) N − (cid:17) N +2 N − UBBLE SOLUTIONS FOR A SCHR ¨ODINGER EQUATION WITH CRITICAL GROWTH 21 ≤ Cµ N +22 h n X j =2 µ | p j − p | ) N − − N − N +2 τ (cid:0) µ | y − p | ) N − + N − N +2 τ + 1(1 + µ | y − p j | ) N − + N +2 N − τ (cid:1)i N +2 N − ≤ Cµ N +22 (cid:0) nµ (cid:1) N +22 − τ µ | y − p | ) N +22 + τ ≤ Cµ N +2 N − − N − N − µ N +22 (1 + µ | y − p | ) N +22 + τ ≤ Cµ ε µ N +22 (1 + µ | y − p | ) N +22 + τ , y ∈ M . (3.28)Noting that µ N +22 µ | y − p | N +22 + τ ≥ µ N +22 (1 + µ ) N +22 + τ ≥ c µ N +24 − τ , y ∈ B , we have c ≤ µ N +22 (1 + µ | y − p | ) N +22 + τ µ − ( N +24 − τ ) ≤ Cµ ε µ N +22 (1 + µ | y − p | ) N +22 + τ , (3.29)which implies that | ˜ J | ≤ Cµ ε µ N +22 (1 + µ | y − p | ) N +22 + τ , y ∈ B . (3.30)On the other hand, when y ∈ B , there holds J , ≤ (cid:12)(cid:12)(cid:12) U ∗ − p ,µ ( u m + n X j =2 U p j ,µ ) (cid:12)(cid:12)(cid:12) ≤ CU ∗ − p ,µ + CU ∗ − p ,µ n X j =2 U p j ,µ := J , , + J , , . Similar to Lemma 2.5 in [21], we can prove | J , , | ≤ Cµ ε µ N +22 (1 + µ | y − p | ) N +22 + τ , y ∈ B . (3.31)Moreover, if N ≥ y ∈ B , (1 + µ | y − p | ) N +22 + τ − ≤ Cµ ( N +22 + τ − , noting that N +24 − τ > , then we have J , , ≤ µ N +22 (1 + µ | y − p | ) N +22 + τ µ − N +22 (1 + µ | y − p | ) − ( N +22 + τ ) † AND QINGFANG WANG ≤ Cµ − ( N +24 − τ ) µ N +22 (1 + µ | y − p | ) N +22 + τ ≤ Cµ ε µ N +22 (1 + µ | y − p | ) N +22 + τ . (3.32)Therefore, it follows from (3.27) to (3.32) that J ≤ Cµ ǫ µ N +22 (1 + µ | y − p | ) N +22 + τ , y ∈ B . (3.33)On the other hand, noting that in M \ B , it holds U p ,µ ≤ C , therefore, | J | ≤ C n X j =1 Z p j ,µ + C (cid:16) n X j =1 U p j ,µ (cid:17) ∗ − = ˆ J , + ˆ J , . (3.34)Observe that ˆ J , = n X j =1 ˆ ζ ( y ) U p j ,µ ≤ ˆ ζ ( y ) U p ,µ + n X j =2 ˆ ζ ( y ) U p j ,µ . We have ˆ ζU p ,µ ≤ ˆ ζ ( y ) µ N − (1 + µ | y − p | ) N − ≤ ˆ ζ ( y ) µ N +22 µ (1 + µ | y − p | ) N − ≤ Cµ ǫ ˆ ζ ( y ) µ − ǫ (1 + µ | y − p | ) N − ≤ Cµ ǫ µ N +22 (1 + µ | y − p | ) N − − ǫ ≤ Cµ ǫ µ N +22 (1 + µ | y − p | ) N +22 + τ , (3.35)since N − − ǫ > N +22 + τ. Also for y ∈ M \ B we can estimate n X j =2 ˆ ζ ( y ) U p j ,µ ≤ C n X j =2 ˆ ζ ( y ) µ N − (1 + µ | y − p j | ) N − ≤ Cµ ǫ n X j =2 ˆ ζ ( y ) µ N +22 µ − ǫ (1 + µ | y − p j | ) N − ≤ Cµ ǫ nµ N +22 µ − ǫ (1 + µ | y − p | ) N − ≤ Cµ ǫ µ N +22 µ − τ − ǫ (1 + µ | y − p | ) N − ≤ Cµ ǫ µ N +22 (1 + µ | y − p | ) N − − τ − ǫ ≤ Cµ ǫ µ N +22 (1 + µ | y − p | ) N +22 + τ , (3.36)since N − − τ − ǫ ≥ N +22 + τ. UBBLE SOLUTIONS FOR A SCHR ¨ODINGER EQUATION WITH CRITICAL GROWTH 23
Finally, we have | ˆ J , | ≤ CU ∗ − p ,µ + C (cid:16) n X j =2 U p j ,µ (cid:17) ∗ − = ˆ J , , + ˆ J , , . And from (3.28), we have | ˆ J , , | ≤ C | J , | ≤ Cµ ǫ µ N +22 (1 + µ | y − p | ) N +22 + τ y ∈ M \ B . (3.37)For y ∈ M \ B , | y − p | ≥ µ − and µ | y − p | ≥ µ , then from ( N +22 − τ ) = N +24 − N − N − > , we haveˆ J , , ≤ C µ N +22 (1 + µ | y − p | ) N +2 ≤ C µ N +22 (1 + µ | y − p | ) N +22 + τ µ | y − p | ) N +22 − τ ≤ µ N +22 (1 + µ | y − p | ) N +22 + τ µ ( N +22 − τ ) (3.38) ≤ Cµ N +22 µ ǫ (1 + µ | y − p | ) N +22 + τ , y ∈ M \ B . (3.39)From (3.34) to (3.38), we obtain J ≤ Cµ ǫ µ N +22 (1 + µ | y − p | ) N +22 + τ , y ∈ M \ B . (3.40)Combining (3.33) and (3.40), applying the symmetry we have J ≤ Cµ ǫ n X j =1 µ N +22 (1 + µ | y − p j | ) N +22 + τ . (3.41)By (3.25), (3.26) and (3.41), we obtain k l n k ˜ ∗ ˜ ∗ ≤ Cµ ǫ . (cid:3) We also need the following estimates.
Lemma 3.3. If N ≥ , then there holds k R n ( ψ ) k ˜ ∗ ˜ ∗ ≤ C k ψ k min { ∗ − , } ˜ ∗ . Proof.
Since it can be proved by the same argument as that of Lemma 2.4 in [21], here weomit its proof. (cid:3) † AND QINGFANG WANG
We define I ( u ) = 12 Z R N (cid:0) |∇ u | + V ( y ) u (cid:1) dy − ∗ Z R N | u | ∗ dy. Set F ( t, ˜ y ∗ , µ ) = I (cid:16) u m + n X j =1 Z p j ,µ + ψ n (cid:17) . To get a solution of the form u m + n P j =1 Z p j ,µ + ψ n , we only need to find a critical pointfor F ( t, ˜ y ∗ , µ ) in B ϑ ( r , y ∗ ) × [ C n N − N − , C n N − N − ] , where ϑ > C , C are differentconstants.Now we will prove Theorem 1.2. Proof of Theorem 1.2 . By direct computation, we have F ( t, ˜ y ∗ , µ ) = I (cid:16) u m + n X j =1 Z p j ,µ (cid:17) + nO (cid:16) µ ǫ (cid:17) . (3.42)On the other hand, we get I (cid:16) u m + n X j =1 Z p j ,µ (cid:17) = I (cid:16) n X j =1 Z p j ,µ (cid:17) + I ( u m ) + 12 Z R N n X j =1 u ∗ − m Z p j ,µ − ∗ Z R N (cid:16)(cid:0) u m + n X j =1 Z p j ,µ (cid:1) ∗ − u ∗ m − (cid:0) n X j =1 Z p j ,µ (cid:1) ∗ (cid:17) . (3.43)It is not difficult to check Z R N u ∗ − m Z p j ,µ = O (cid:16) µ N − (cid:17) . For y ∈ R N \ S nj =1 B j , we have (cid:12)(cid:12)(cid:12)(cid:0) u m + n X j =1 Z p j ,µ (cid:1) ∗ − u ∗ m − (cid:0) n X j =1 Z p j ,µ (cid:1) ∗ (cid:12)(cid:12)(cid:12) ≤ Cu ∗ − m n X j =1 Z p j ,µ + (cid:16) n X j =1 Z p j ,µ (cid:17) ∗ ≤ Cu ∗ − m n X j =1 Z p j ,µ . Hence we obtain Z R N \ S nj =1 B j (cid:12)(cid:12)(cid:12)(cid:0) u m + n X j =1 Z p j ,µ (cid:1) ∗ − u ∗ m − (cid:0) n X j =1 Z p j ,µ (cid:1) ∗ (cid:12)(cid:12)(cid:12) ≤ C Z R N u ∗ − m n X j =1 Z p j ,µ = O (cid:16) nµ N − (cid:17) . UBBLE SOLUTIONS FOR A SCHR ¨ODINGER EQUATION WITH CRITICAL GROWTH 25
By symmetry, we have Z S nj =1 B j (cid:12)(cid:12)(cid:12)(cid:0) u m + n X j =1 Z p j ,µ (cid:1) ∗ − u ∗ m − (cid:0) n X j =1 Z p j ,µ (cid:1) ∗ (cid:12)(cid:12)(cid:12) = n Z B (cid:12)(cid:12)(cid:12) ( u m + n X j =1 Z p j ,µ ) ∗ − u ∗ m − ( n X j =1 Z p j ,µ ) ∗ (cid:12)(cid:12)(cid:12) . There holds Z B u ∗ m = O (cid:16) µ N (cid:17) , and Z B (cid:12)(cid:12)(cid:12)(cid:0) u m + n X j =1 Z p j ,µ (cid:1) ∗ − (cid:0) n X j =1 Z p j ,µ (cid:1) ∗ (cid:12)(cid:12)(cid:12) ≤ C Z B (cid:16) n X j =1 Z p j ,µ (cid:17) ∗ − ≤ C Z B (cid:16) U ∗ − p ,µ + µ N +22 (1 + µ | y − p | ) (2 ∗ − N − − τ ) (cid:17) ≤ Cµ N − , where τ = N − N − . Hence we have proved I (cid:16) u m + n X j =1 Z p j ,µ (cid:17) = I (cid:16) n X j =1 Z p j ,µ (cid:17) + I ( u m ) + O (cid:16) nµ N − (cid:17) . (3.44)Moreover, by direct computation we can obtain I (cid:16) n X j =1 Z p j ,µ (cid:17) = n (cid:16) A + A V ( t, ˜ y ∗ ) µ − n X j =2 A µ N − | p j − p | N − + O (cid:0) µ ǫ (cid:1)(cid:17) , (3.45)where A = (cid:0) − ∗ (cid:1) Z R N U ∗ , and A i ( i = 2 ,
3) are some positive constants.It follows from (3.42), (3.44) and (3.45) that F ( t, ˜ y ∗ , µ ) = I (cid:16) n X j =1 Z p j ,µ (cid:17) + I ( u m ) + nO (cid:16) µ ǫ (cid:17) = I ( u m ) + nA + n (cid:16) A µ V ( t, ˜ y ∗ ) − n X j =2 A µ N − | p − p j | N − (cid:17) + O (cid:16) nµ ǫ (cid:17) , (3.46)where A i ( i = 1 , ,
3) are the same as those of (3.45). † AND QINGFANG WANG
Now in order to find a critical point for F ( t, ˜ y ∗ , µ ), we only need to continue exactly assection 3 in [21]. One can also see section 3 in [22]. Here we omit the detailed process ofits proof. (cid:3) Appendix A. Some Pohozaev identities
Set − ∆ u + V ( | y ′ | , y ′′ ) u = u ∗ − , (A.1)and − ∆ η + V ( | y ′ | , y ′′ ) η = (2 ∗ − u ∗ − η. (A.2)Suppose that Ω is a smooth domain in R N .We have the following identities which are used in section 2 by proving the non-degeneracyof the multi-bubbling solutions obtained in [21]. Lemma A.1.
There holds − Z ∂ Ω ∂u∂ν ∂η∂y i − Z ∂ Ω ∂η∂ν ∂u∂y i + Z ∂ Ω h∇ u, ∇ η i ν i + Z ∂ Ω V uην i − Z ∂ Ω u ∗ − ην i = Z Ω ∂V∂y i uη, (A.3) and Z Ω uη h∇ V, y − x i + 2 Z Ω V ηu = − Z ∂ Ω u ∗ − η h ν, y − x i − Z ∂ Ω ∂u∂ν h∇ η, y − x i− Z ∂ Ω ∂η∂ν h∇ u, y − x i + Z ∂ Ω h∇ u, ∇ η ih ν, y − x i + Z ∂ Ω V uη h ν, y − x i + 2 − N Z ∂ Ω η ∂u∂ν + 2 − N Z ∂ Ω u ∂η∂ν . (A.4) Proof.
Proof of (A.3) . First we have Z Ω ( − ∆ u + V u ) ∂η∂y i = Z Ω u ∗ − ∂η∂y i , and Z Ω ( − ∆ η + V η ) ∂u∂y i = Z Ω (2 ∗ − u ∗ − η ∂u∂y i , which implies that Z Ω (cid:16) − ∆ u ∂η∂y i + ( − ∆ η ) ∂u∂y i + V u ∂η∂y i + V η ∂u∂y i (cid:17) = Z Ω (cid:16) u ∗ − ∂η∂y i + (2 ∗ − u ∗ − η ∂u∂y i (cid:17) . (A.5) UBBLE SOLUTIONS FOR A SCHR ¨ODINGER EQUATION WITH CRITICAL GROWTH 27
It is easy to check that Z Ω (cid:16) u ∗ − ∂η∂y i + (2 ∗ − u ∗ − η ∂u∂y i (cid:17) = Z Ω ∂ ( u ∗ − η ) ∂y i = Z ∂ Ω u ∗ − ην i . (A.6)Moreover, similar to (2.7) in [15], we have Z Ω (cid:16) − ∆ u ∂η∂y i + ( − ∆ η ) ∂u∂y i (cid:17) = − Z ∂ Ω ∂u∂ν ∂η∂y i − Z ∂ Ω ∂η∂ν ∂u∂y i + Z ∂ Ω h∇ u, ∇ η i ν i , (A.7)and Z Ω (cid:16) V u ∂η∂y i + V η ∂u∂y i (cid:17) = Z Ω V ∂∂y i ( uη ) = Z ∂ Ω V uην i − Z Ω uη ∂V∂y i . (A.8)It follows from (A.5) to (A.8) that (A.3) holds. Proof of (A.4) . It is easy to check that Z Ω (cid:0) ( − ∆ u + V u ) h∇ η, y − x i + ( − ∆ η + V η ) h∇ u, y − x i (cid:1) = Z Ω (cid:0) u ∗ − h∇ η, y − x i + (2 ∗ − u ∗ − η h∇ u, y − x i (cid:1) . (A.9)We find that Z Ω (cid:0) u ∗ − h∇ η, y − x i + (2 ∗ − u ∗ − η h∇ u, y − x i (cid:1) = Z Ω h∇ ( u ∗ − η ) , y − x i = Z ∂ Ω u ∗ − η h ν, y − x i − N Z Ω u ∗ − η. (A.10)Also similar to (2.10) in [15], we have Z Ω (cid:0) − ∆ u h∇ η, y − x i + ( − ∆ η ) h∇ u, y − x i (cid:1) = − Z ∂ Ω ∂u∂ν h∇ η, y − x i − Z ∂ Ω ∂η∂ν h∇ u, y − x i + Z ∂ Ω h∇ u, ∇ η ih ν, y − x i + (2 − N ) Z Ω h∇ u, ∇ η i . (A.11)On the other hand, there holds2 ∗ Z Ω u ∗ − η = Z Ω (cid:0) ( − ∆ uη + u ( − ∆ η ) + V uη + V ηu (cid:1) =2 Z Ω h∇ u, ∇ η i − Z ∂ Ω η ∂u∂ν − Z ∂ Ω u ∂η∂ν + 2 Z Ω V uη, (A.12)which yields Z Ω h∇ u, ∇ η i = 2 ∗ Z Ω u ∗ − η + 12 Z ∂ Ω η ∂u∂ν + 12 Z ∂ Ω u ∂η∂ν − Z Ω V uη. (A.13) † AND QINGFANG WANG
Moreover, we obtain Z Ω (cid:0) V u h∇ η, y − x i + V η h∇ u, y − x i (cid:1) = Z Ω V h∇ ( uη ) , y − x i = Z ∂ Ω V uη h ν, y − x i − Z Ω uη h∇ V, y − x i − N Z Ω V uη. (A.14)Therefore, from (A.9) to (A.14) we know that (A.4) holds. (cid:3)
Appendix B. Basic estimates
For each fixed k and j , k = j , we consider the following function g k,j ( y ) = 1(1 + | y − x j | ) α | y − x k | ) β , (B.1)where α ≥ β ≥ Lemma B.1. (Lemma B.1, [24] ) For any constants < δ ≤ min { α, β } , there is a constant C > , such that g k,j ( y ) ≤ C | x k − x j | δ (cid:16) | y − x k | ) α + β − δ + 1(1 + | y − x j | ) α + β − δ (cid:17) . Lemma B.2. (Lemma B.2, [24] ) For any constant < δ < N − , there is a constant C > , such that Z | y − z | N − | z | ) δ dz ≤ C (1 + | y | ) δ . Let us recall that Z t, ˜ y ∗ ,µ ( y ) = n X j =1 ˆ ζ ( y ) U p j ,µ = [ N ( N − N − n X j =1 ˆ ζ (cid:16) µ µ | y − p j | (cid:17) N − . Just by the same argument as that of Lemma B.3 in [21], we can prove
Lemma B.3.
Suppose that N ≥ . Then there is a small constant ι > , such that Z | y − z | N − Z N − t, ˜ y ∗ ,µ ( z ) n X j =1 µ | z − p j | ) N − + ι dz ≤ n X j =1 C (1 + µ | y − p j | ) N − + τ + ι . Appendix C. An example of the potential V ( r, y ∗ )Here we give an example of V (ˆ y, y ∗ ) which satisfies the assumptions ( V ) and ( ˜ V ) . Wedefine V ( r, y ∗ ) = r − r (cid:16) N P j =5 y j (cid:17) + (cid:16) N P j =5 y j (cid:17) + 1 , B ρ ( r , y ∗ ) , ≥ , R N \ B ρ ( r , y ∗ ) . UBBLE SOLUTIONS FOR A SCHR ¨ODINGER EQUATION WITH CRITICAL GROWTH 29 where ρ is the same as that of [21] and ( r , y ∗ ) is defined below. By some direct computa-tions, we can check that f ( r, y ∗ ) := r V ( r, y ∗ ) = r − r (cid:16) N X j =5 y j (cid:17) + r (cid:16) N X j =5 y j (cid:17) + r , We have ∂f∂r = 4 r − r (cid:16) N X j =5 y j (cid:17) + 2 r (cid:16) N X j =5 y j (cid:17) + 2 r, and ∂f∂y i = − r + 2 r y i , i = 5 , · · · , N. Suppose that ∂f∂r = 0 , ∂f∂y i = 0, we obtain y i = 2 r, for i = 5 , · · · , N, r = r N − , y ,i = 2 r N −
34 ( i = 5 , ..., N ) . Therefore, ( r , y ∗ ) is a critical point of the function f ( r, y ∗ ) and V ( r , y ∗ ) = >
0. Also ∂ f∂r = 12 r − r (cid:16) N X j =5 y j (cid:17) + 2 (cid:16) N X j =5 y j (cid:17) + 2 ,∂ f∂r∂y i = − r + 4 ry i , i = 5 , · · · , N, and ∂ f∂y i = 2 r ( i = 5 , · · · , N ) , ∂ f∂y i ∂y j = 0( i, j = 5 , ..., N, i = j ) . By direct computation, we obtain B = ∂ f∂r ∂ f∂r∂y · · · ∂ f∂r∂y N ∂ f∂r∂y ∂ f∂y ∂y · · · · · · · · · · · · · · · ∂ f∂r∂y N · · · ∂ f∂y N ∂y N ( N − × ( N − . Also by some tedious computation, we can obtain the eigenvalues of the matrix B are asfollows: | λI − B | ( r ,y ∗ ) = h(cid:0) λ − ∂ f∂r (cid:1) ( λ − r ) − N X j =5 ( − r + 4 ry j ) i ( λ − r ) N − (cid:12)(cid:12)(cid:12) ( r ,y ∗ ) = 0 , † AND QINGFANG WANG which implies that (cid:16) λ − ∂ f∂r | ( r ,y ∗ ) (cid:17) ( λ − r ) = N X j =5 ( − r + 4 r y j ) , or ( λ − r ) N − (cid:12)(cid:12)(cid:12) ( r ,y ∗ ) = 0 . By further computation, we can check that min { λ , λ } < λ = · · · = λ N − = 2 r > . Hence the assumption ( V ) holds.On the other hand, we recall V ( r, y ) = r − r (cid:16) N X j =5 y j (cid:17) + (cid:16) N X j =5 y j (cid:17) + 1 , in B ρ ( r , y ∗ ) . We obtain in B ρ ( r , y ∗ ) ∂V∂r = 2 r − N X j =5 y j , ∂V∂y i = 2 y i − y i r (cid:16) N X j =5 y j (cid:17) , i = 1 , , , ,∂V∂y k = − r + 2 y k , k = 5 , · · · , N,∂ V∂r = 2 , ∂ V∂r∂y i = 2 y ,i r ( i = 1 , , , , ∂ V∂r∂y k = − k = 5 , ..., N ) , (C.1) ∂ V∂y i ∂y i = 2 − r − y i r (cid:16) N X j =5 y j (cid:17) ( i = 1 , , , , ∂V∂y k = 2( k = 5 , · · · , N ) . (C.2)and ∂ V∂y k ∂y j = 0( j = k, k, j = 5 , · · · , N ) . (C.3)Then from (C.1) to (C.3), we obtain in B ρ ( r , y ∗ )∆ V = N X j =1 ∂ V∂y i ∂y i = X i =1 (cid:16) − r − y i r ( y + · · · + y N ) (cid:17) + 2( N − N − r ( y + · · · + y N ) . Hence ∆ V (cid:12)(cid:12)(cid:12) ( r ,y ∗ ) = 96 − N. For y ∗ = ( y , , y , · · · , y ,N ) , one have ∂ ∆ V∂y (cid:12)(cid:12)(cid:12) ( r ,y ∗ ) = 12 y r ( y + · · · + y N ) (cid:12)(cid:12)(cid:12) ( r ,y ∗ ) = 24 y , r ( N − , UBBLE SOLUTIONS FOR A SCHR ¨ODINGER EQUATION WITH CRITICAL GROWTH 31 ∂ (∆ V ) ∂y k (cid:12)(cid:12)(cid:12) ( r ,y ∗ ) = − r , k = 5 , · · · , N. Since y ,i = 2 r ( i = 5 , · · · , N ) , we obtain A i,l = h − ( y ,i ( N − r (96 − N ) + ν h ν,x i )(34 − N ) r ] , when i = l = 1; − , when i = 1 , l = 2 , , ..., N − iπm ( − − ( − r (96 − N ) + ν i +1 h ν,x i )(34 − N ) r ) , when i = 2 , , ..., N − , l = 1;2 , when i = l, i, l = 2 , , ..., N − , , when i = l, i, l = 2 , , ..., N − , (C.4)Therefore, from (C.4) we havedet( A i,l ) ( N − × ( N − = N − Y i =2 A i,i (cid:0) A , − N − X k =2 A k, A ,k A k,k (cid:1) = N − Y i =2 (cid:16) − ( 24 y ,i ( N − r (8 − N − ν h ν, x i )(34 − N ) r (cid:1) + N − X k =2 iπm ( − − ( − r (8 − N − ν i +3 h ν, x i )(34 − N ) r ) (cid:17) = 0 . Hence, the assumption ( ˜ V ) also holds. Acknowledgements
Q. He was partially supported by the fund from NSF of China (No.11701107 and No. 11701108) and the NSF of Guangxi Province (2017GXNSFBA198190,2017GXNSFBA198088). C. Wang was partially supported by NSFC (No.12071169) andthe Fundamental Research Funds for the Central Universities(No. KJ02072020-0319). Q.Wang was partially supported by NSFC (No.11701439).
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