Local Uniqueness and Refined Spike Profiles of Ground States for Two-Dimensional Attractive Bose-Einstein Condensates
aa r X i v : . [ m a t h . A P ] A p r Local Uniqueness and Refined Spike Profiles of GroundStates for Two-Dimensional Attractive Bose-EinsteinCondensates
Yujin Guo ∗ , Changshou Lin † , and Juncheng Wei ‡ July 4, 2018
Abstract
We consider ground states of two-dimensional Bose-Einstein condensates in atrap with attractive interactions, which can be described equivalently by positiveminimizers of the L − critical constraint Gross-Pitaevskii energy functional. It isknown that ground states exist if and only if a < a ∗ := k w k , where a denotes theinteraction strength and w is the unique positive solution of ∆ w − w + w = 0 in R . In this paper, we prove the local uniqueness and refined spike profiles of groundstates as a ր a ∗ , provided that the trapping potential h ( x ) is homogeneous and H ( y ) = R R h ( x + y ) w ( x ) dx admits a unique and non-degenerate critical point. Keywords:
Bose-Einstein condensation; spike profiles; local uniqueness; Pohozaev iden-tity.
The phenomenon of Bose-Einstein condensation (BEC) has been investigated intensivelysince its first realization in cold atomic gases, see [1, 5] and references therein. In theseexperiments, a large number of (bosonic) atoms are confined to a trap and cooled to verylow temperatures. Condensation of a large fraction of particles into the same one-particlestate is observed below a critical temperature. These Bose-Einstein condensates displayvarious interesting quantum phenomena, such as the critical-mass collapse, the superflu-idity and the appearance of quantized vortices in rotating traps (e.g.[5]). Specially, if theforce between the atoms in the condensates is attractive, the system collapses as soonas the particle number increases beyond a critical value, see, e.g., [23] or [5, Sec. III.B].Bose-Einstein condensates (BEC) of a dilute gas with attractive interactions in R can be described ([2, 5, 10]) by the following Gross-Pitaevskii (GP) energy functional E a ( u ) := Z R (cid:16) |∇ u | + V ( x ) | u | (cid:17) dx − a Z R | u | dx, (1.1) ∗ Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, P.O. Box 71010, Wuhan430071, P. R. China. Email: [email protected] . Y. J. Guo is partially supported by NSFC grants11322104 and 11671394. † Taida Institute of Mathematical Sciences, National Taiwan University, Taipei 10617, Taiwan. Email: [email protected] . ‡ Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2, Canada.Email: [email protected] . J.C. Wei is partially supported by NSERC of Canada. a > V ( x ) ≥ | x |→∞ V ( x ) = ∞ . As addressed recently in [10, 11],ground states of attractive BEC in R can be described by the constraint minimizers ofthe GP energy e ( a ) := inf { u ∈H , k u k =1 } E a ( u ) , (1.2)where the space H is defined by H := n u ∈ H ( R ) : Z R V ( x ) | u ( x ) | dx < ∞ o . (1.3)The minimization problem e ( a ) was analyzed recently in [2, 10, 11, 12, 26] and referencestherein. Existing results show that e ( a ) is an L − critical constraint variational problem.Actually, it was shown in [2, 10] that e ( a ) admits minimizers if and only if a < a ∗ := k w k ,where w = w ( | x | ) is the unique (up to translations) radial positive solution (cf. [7, 19, 14])of the following nonlinear scalar field equation∆ w − w + w = 0 in R , where w ∈ H ( R ) . (1.4)It turns out that the existence and nonexistence of minimizers for e ( a ) are well connectedwith the following Gagliardo-Nirenberg inequality Z R | u ( x ) | dx ≤ k w k Z R |∇ u ( x ) | dx Z R | u ( x ) | dx, ∀ u ∈ H ( R ) , (1.5)where the equality is attained at w (cf. [25]).Since E a ( u ) ≥ E a ( | u | ) for any u ∈ H , any minimizer u a of e ( a ) must be eithernon-negative or non-positive, and it satisfies the Euler-Lagrange equation − ∆ u a + V ( x ) u a = µ a u a + au a in R , (1.6)where µ a ∈ R is a suitable Lagrange multiplier. Thus, by applying the maximum princi-ple to the equation (1.6), any minimizer u a of e ( a ) is further either negative or positive.Therefore, without loss of generality one can restrict the minimizations of e ( a ) to positivefunctions. In this paper positive minimizers of e ( a ) are called ground states of attractiveBEC. Applying energy estimates and blow-up analysis, the spike profiles of positive min-imizers for e ( a ) as a ր a ∗ were recently discussed in [10, 11, 12] under different types ofpotentials V ( x ), see our Proposition 2.1 for some related results. In spite of these facts,it remains open to discuss the refined spike profiles of positive minimizers. On the otherhand, the local uniqueness of positive minimizers for e ( a ) as a.e. a ր a ∗ was also proved[11] by the ODE argument, for the case where V ( r ) = V ( | x | ) is radially symmetric andsatisfies V ′ ( r ) ≥
0, see Corollary 1.1 in [11] for details. Here the locality of uniquenessmeans that a is near a ∗ . It is therefore natural to ask whether such local uniquenessstill holds for the case where V ( x ) is not radially symmetric. We should remark that allthese results mentioned above were obtained mainly by analyzing the variational struc-tures of the minimization problem e ( a ), instead of discussing the PDE properties of theassociated elliptic equation (1.6).By investigating thoroughly the associated equation (1.6), the main purpose of thispaper is to derive the refined spike profiles of positive minimizers for e ( a ) as a ր a ∗ , andextend the above local uniqueness to the cases of non-symmetric potentials V ( x ) as well.Throughout the whole paper, we shall consider the trapping potential V ( x ) satisfyinglim | x |→∞ V ( x ) = ∞ in the class of homogeneous functions, for which we define2 efinition 1.1. h ( x ) ≥ R is homogeneous of degree p ∈ R + (about the origin), ifthere exists some p > h ( tx ) = t p h ( x ) in R for any t > . (1.7)Following [9, Remark 3.2], the above definition implies that the homogeneous function h ( x ) ∈ C ( R ) of degree p > ≤ h ( x ) ≤ C | x | p in R , (1.8)where C > h ( x ) on ∂B (0). Moreover, since we assume thatlim | x |→∞ h ( x ) = ∞ , x = 0 is the unique minimum point of h ( x ). Additionally, we oftenneed to assume that V ( x ) = h ( x ) ∈ C ( R ) satisfies y is the unique critical point of H ( y ) = Z R h ( x + y ) w ( x ) dx. (1.9)The following example shows that for some non-symmetric potentials h ( x ), H ( y ) admitsa unique critical point y , where y satisfies y = 0 and is non-degenerate in the sensethat det (cid:16) ∂ H ( y ) ∂x i ∂x j (cid:17) = 0 , where i, j = 1 , . (1.10) Example 1.1.
Suppose that the potential h ( x ) satisfies h ( x ) = | x | p (cid:2) δh ( θ ) (cid:3) ≥ , where p ≥ δ ∈ R , (1.11)where h ( θ ) ∈ C ([0 , π ]) satisfies (cid:16) Z π h ( θ ) cos θdθ (cid:17) + (cid:16) Z π h ( θ ) sin θdθ (cid:17) > . (1.12)One can check from (1.12) that if | δ | ≥ H ( y ) admits a uniquecritical point y = − δ ˆ y ∈ R , where ˆ y satisfiesˆ y ∼ (cid:16) C Z π h ( θ ) cos θdθ, C Z π h ( θ ) sin θdθ (cid:17) = (0 ,
0) as δ → C and C depending only on w and p . Furthermore, if | δ | ≥ (cid:16) ∂ H ( y ) ∂x i ∂x j (cid:17) >
0, which implies that the unique criticalpoint y of H ( y ) is non-degenerate.Our first main result is concerned with the following local uniqueness as a ր a ∗ ,which holds for some non-symmetric homogeneous potentials h ( x ) in view of Example1.1. Theorem 1.1.
Suppose V ( x ) = h ( x ) ∈ C ( R ) is homogeneous of degree p ≥ , where lim | x |→∞ h ( x ) = ∞ , and satisfies y is the unique and non-degenerate critical point of H ( y ) = Z R h ( x + y ) w ( x ) dx. (1.14) Then there exists a unique positive minimizer for e ( a ) as a ր a ∗ . e ( a ) must beunique as a is near a ∗ . It is possible to extend Theorem 1.1 to more general potentials V ( x ) = g ( x ) h ( x ) for a class of functions g ( x ), which is however beyond the discussionranges of the present paper. We also remark that the proof of Theorem 1.1 is moreinvolved for the case where y = 0 occurs in (1.14). Our proof of such local uniquenessis motivated by [3, 6, 9]. Roughly speaking, as derived in Proposition 2.1 we shallfirst obtain some fundamental estimates on the spike behavior of positive minimizers.Under the non-degeneracy assumption of (1.14), the local uniqueness is then proved inSubsection 2.1 by establishing various types of local Pohozaev identities.The proof of Theorem 1.1 shows that if one considers the local uniqueness of Theorem1.1 in other dimensional cases, where R is replaced by R d and u is replaced by u d for d = 2, the fundamental estimates of Proposition 2.1 are not enough. Therefore,in the following we address the refined spike behavior of positive minimizers under theassumption (1.14). To introduce our second main result, for convenience we next denote λ = (cid:18) p Z R h ( x + y ) w ( x ) dx (cid:19) p , (1.15)where y ∈ R is given by (1.14), and ψ ( x ) = ϕ ( x ) − C ∗ (cid:2) w ( x ) + x · ∇ w ( x ) (cid:3) , where ϕ ( x ) ∈ C ( R ) ∩ L ∞ ( R ) is the unique solution of ∇ ϕ (0) = 0 and (cid:2) − ∆ + (1 − w ) (cid:3) ϕ ( x ) = − w R R w − h ( x + y ) wp R R h ( x + y ) w in R , (1.16)and the nonzero constant C ∗ is given by C ∗ = 22 + p (cid:16) Z R wψ + Z R ϕ (cid:17) with ψ ∈ C ( R ) ∩ L ∞ ( R ) being the unique solution of (3.29). Using above notations,we shall derive the following theorem. Theorem 1.2.
Suppose V ( x ) = h ( x ) ∈ C ( R ) is homogeneous of degree p ≥ , where lim | x |→∞ h ( x ) = ∞ , and satisfies (1.14) for some y ∈ R . If u a is a positive minimizerof e ( a ) as a ր a ∗ , then we have u a ( x ) = λ k w k n a ∗ − a ) p w (cid:16) λ ( x − x a )( a ∗ − a ) p (cid:17) + ( a ∗ − a ) p p ψ (cid:16) λ ( x − x a )( a ∗ − a ) p (cid:17) +( a ∗ − a ) p p φ (cid:16) λ ( x − x a )( a ∗ − a ) p (cid:17)o + o (cid:0) ( a ∗ − a ) p p (cid:1) as a ր a ∗ (1.17) uniformly in R for some function φ ∈ C ( R ) ∩ L ∞ ( R ) , where x a is the uniquemaximum point of u a satisfying (cid:12)(cid:12)(cid:12) λx a ( a ∗ − a ) p − y (cid:12)(cid:12)(cid:12) = ( a ∗ − a ) O ( | y | ) as a ր a ∗ (1.18) for some y ∈ R . φ ∈ C ( R ) ∩ L ∞ ( R ) is given explicitly. In Section 4 we shall extendthe refined spike behavior of Theorem 1.2 to more general potentials V ( x ) = g ( x ) h ( x ),where h ( − x ) = h ( x ) is homogeneous and satisfies (1.14) and 0 ≤ C ≤ g ( x ) ≤ C holdsin R , see Theorem 4.4 for details. To establish Theorem 1.2 and Theorem 4.4, ourProposition 2.1 shows that the arguments of [10, 11, 12] give the leading expansionterms of the minimizer u a and the associated Lagrange multiplier µ a satisfying (1.6) aswell. In order to get (1.17) for the rest terms of u a , the difficulty is to obtain the moreprecise estimate of µ a , which is overcome by the very delicate analysis of the associatedequation (1.6), together with the constraint condition of u a .This paper is organized as follows: In Section 2 we shall prove Theorem 1.1 on thelocal uniqueness of positive minimizers. Section 3 is concerned with proving Theorem1.2 on the refined spike profiles of positive minimizers for e ( a ) as a ր a ∗ . The mainaim of Section 4 is to derive Theorem 4.4, which extends the refined spike behavior ofTheorem 1.2 to more general potentials V ( x ) = g ( x ) h ( x ). We shall leave the proof ofLemma 3.4 to Appendix A. This section is devoted to the proof of Theorem 1.1 on the local uniqueness of positiveminimizers. Towards this purpose, we need some estimates of positive minimizers for e ( a )as a ր a ∗ , which hold essentially for more general potential V ( x ) ∈ C ( R ) satisfying V ( x ) = g ( x ) h ( x ) , where 0 < C ≤ g ( x ) ≤ C in R and h ( x ) is homogeneous ofdegree p ≥
2. (2.1)For convenience, we always denote { u k } to be a positive minimizer sequence of e ( a k )with a k ր a ∗ as k → ∞ , and define λ = (cid:18) pg (0)2 Z R h ( x + y ) w ( x ) dx (cid:19) p , (2.2)where V ( x ) = g ( x ) h ( x ) is assumed to satisfy (2.1) with p ≥ y ∈ R is given by(1.9). Recall from (1.4) that w ( | x | ) satisfies Z R |∇ w | dx = Z R | w | dx = 12 Z R | w | dx, (2.3)see also Lemma 8.1.2 in [4]. Moreover, it follows from [7, Prop. 4.1] that w admits thefollowing exponential decay w ( x ) , |∇ w ( x ) | = O ( | x | − e −| x | ) as | x | → ∞ . (2.4) Proposition 2.1.
Suppose V ( x ) = g ( x ) h ( x ) ∈ C ( R ) satisfies lim | x |→∞ V ( x ) = ∞ and(2.1), and assume (1.9) holds for some y ∈ R . Then there exist a subsequence, stilldenoted by { a k } , of { a k } and { x k } ⊂ R such that(I). The subsequence { u k } satisfies ( a ∗ − a k ) p u k (cid:16) x k + x ( a ∗ − a k ) p (cid:17) → λw ( λx ) k w k as k → ∞ (2.5)5 niformly in R , and x k is the unique maximum point of u k satisfying lim k →∞ λx k ( a ∗ − a k ) p = y , (2.6) where y ∈ R is the same as that of (1.9). Moreover, u k satisfies ( a ∗ − a k ) p u k (cid:16) x k + x ( a ∗ − a k ) p (cid:17) ≤ Ce − λ | x | in R , (2.7) where the constant C > is independent of k .(II). The energy e ( a k ) satisfies lim k →∞ e ( a k )( a ∗ − a k ) p/ (2+ p ) = λ a ∗ p + 2 p . (2.8) Proof.
Since the proof of Proposition 2.1 is similar to those in [10, 11, 12], which handle(1.1) with different potentials V ( x ), we shall briefly sketch the structure of the proof.If V ( x ) ∈ C ( R ) satisfies (2.1) with p ≥
2, we note that h ( x ) ≥ u τ ( x ) = A τ τ k w k ϕ ( x ) w ( τ x ) , where the nonnegative cut-off function ϕ ∈ C ∞ ( R ) satisfies 0 ≤ ϕ ( x ) ≤ R , and A τ > R R u τ ( x ) dx = 1. The same proof of Lemma 3 in [10] thenyields that e ( a ) ≤ C ( a ∗ − a ) pp +2 for 0 ≤ a < a ∗ , (2.9)where the constant C > a . By (2.9), we can follow Lemma 4 in [10]to derive that there exists a positive constant K , independent of a , such that Z R | u a ( x ) | dx ≤ K ( a ∗ − a ) − p +2 for 0 ≤ a < a ∗ , (2.10)where u a > e ( a ). Applying (2.9) and (2.10), a proof similar tothat of Theorem 2.1 in [12] then gives that there exist two positive constants m < M ,independent of a , such that m ( a ∗ − a ) pp +2 ≤ e ( a ) ≤ M ( a ∗ − a ) pp +2 for 0 ≤ a < a ∗ . (2.11)Based on (2.11), similar to Theorems 1.2 and 1.3 in [12], one can further deduce thatthere exist a subsequence (still denoted by { a k } ) of { a k } and { x k } ⊂ R , where a k ր a ∗ as k → ∞ , such that (2.7) and (2.8) hold, and( a ∗ − a k ) p u k (cid:16) x k + x ( a ∗ − a k ) p (cid:17) → λw ( λx ) k w k strongly in H ( R ) (2.12)as k → ∞ , where x k is the unique maximum point of u k . Finally, since w decaysexponentially, the standard elliptic regularity theory applied to (2.12) yields that (2.5)holds uniformly in R (e.g. Lemma 4.9 in [18] for similar arguments).We finally follow (1.9) and (2.5) to derive the estimate (2.6). Following (2.5), wedefine ¯ u k ( x ) := √ a ∗ ε k λ u k (cid:16) ε k λ x + x k (cid:17) , where ε k := ( a ∗ − a k ) p > ,
6o that ¯ u k ( x ) → w ( x ) uniformly in R as k → ∞ . We then derive from (1.5) that e ( a k ) = E a k ( u k ) = λ a ∗ ε k h Z R |∇ ¯ u k ( x ) | dx − Z R ¯ u k ( x ) dx i + λ ε pk a ∗ ) Z R ¯ u k ( x ) dx + 1 a ∗ Z R V (cid:0) ε k λ x + x k )¯ u k ( x ) dx (2.13) ≥ λ ε pk a ∗ ) Z R ¯ u k ( x ) dx + 1 a ∗ (cid:16) ε k λ (cid:17) p Z R g (cid:0) ε k λ x + x k ) h (cid:0) x + λx k ε k )¯ u k ( x ) dx, which then implies from (2.5) that | λx k ε k | is bounded uniformly in k . Therefore, thereexist a subsequence (still denoted by { λx k ε k } ) of { λx k ε k } and y ∈ R such that λx k ε k → y as k → ∞ . Note that lim inf k →∞ Z R g (cid:0) ε k λ x + x k ) h (cid:0) x + λx k ε k )¯ u k ( x ) dx ≥ lim inf k →∞ Z B √ εk (0) g (cid:0) ε k λ x + x k ) h (cid:0) x + λx k ε k )¯ u k ( x ) dx = g (0) Z R h ( x + y ) w ( x ) dx. (2.14)Since u k gives the least energy of e ( a k ) and the assumption (1.9) implies that y isessentially the unique global minimum point of H ( y ) = R R h ( x + y ) w ( x ) dx , we concludefrom (2.13) and (2.14) that y = y , which thus implies that (2.6) holds, and the proofis therefore complete. Following Proposition 2.1, this subsection is focussed on the proof of Theorem 1.1, andin the whole subsection we always assume that V ( x ) = h ( x ) ∈ C ( R ) is homogeneousof degree p ≥ | x |→∞ h ( x ) = ∞ . Our proof is stimulated by[3, 6, 9]. We first define the linearized operator L by L := − ∆ + (1 − w ) in R , where w = w ( | x | ) > w satisfies the expo-nential decay (2.4). Recall from [14, 20] that ker ( L ) = span n ∂w∂x , ∂w∂x o . (2.15)For any positive minimizer u k of e ( a k ), where a k ր a ∗ as k → ∞ , one can note that u k solves the Euler-Lagrange equation − ∆ u k ( x ) + V ( x ) u k ( x ) = µ k u k ( x ) + a k u k ( x ) in R , (2.16)where µ k ∈ R is a suitable Lagrange multiplier and satisfies µ k = e ( a k ) − a k Z R u k ( x ) dx. (2.17)7oreover, under the more general assumption (2.1), one can derive from (2.3) and (2.5)that u k satisfies Z R u k ( x ) dx = ( a ∗ − a k ) − p h λ a ∗ + o (1) i as k → ∞ . (2.18)It then follows from (2.3), (2.17) and (2.18) that µ k satisfies µ k ε k λ → − k → + ∞ , (2.19)where we denote ε k := ( a ∗ − a k ) p > . Set ¯ u k ( x ) := √ a ∗ ε k λ u k (cid:16) ε k λ x + x k (cid:17) , so that Proposition 2.1 gives ¯ u k ( x ) → w ( x ) uniformly in R as k → ∞ . Note from (2.16)that ¯ u k satisfies − ∆¯ u k ( x ) + (cid:16) ε k λ (cid:17) V (cid:16) ε k λ x + x k (cid:17) ¯ u k ( x ) = µ k ε k λ ¯ u k ( x ) + a k a ∗ ¯ u k ( x ) in R . (2.20)Moreover, by the exponential decay (2.7), there exist C > R > | ¯ u k ( x ) | ≤ C e − | x | for | x | > R, (2.21)which then implies that (cid:12)(cid:12)(cid:12)(cid:16) ε k λ (cid:17) V (cid:16) ε k λ x + x k (cid:17) ¯ u k ( x ) (cid:12)(cid:12)(cid:12) ≤ CC e − | x | for | x | > R, if V ( x ) satisfies (2.1) with p ≥
2. Therefore, under the assumption (2.1), applying thelocal elliptic estimates (see (3.15) in [8]) to (2.20) yields that |∇ ¯ u k ( x ) | ≤ Ce − | x | as | x | → ∞ , (2.22)where the estimates (2.19) and (2.21) are also used. In the following, we shall followProposition 2.1 and (2.22) to derive Theorem 1.1 on the local uniqueness of positiveminimizers as a ր a ∗ . Proof of Theorem 1.1.
Suppose that there exist two different positive minimizers u ,k and u ,k of e ( a k ) with a k ր a ∗ as k → ∞ . Let x ,k and x ,k be the unique local maximumpoint of u ,k and u ,k , respectively. Following (2.16), u i,k then solves the Euler-Lagrangeequation − ∆ u i,k ( x ) + h ( x ) u i,k ( x ) = µ i,k u i,k ( x ) + a k u i,k ( x ) in R , i = 1 , , (2.23)where V ( x ) = h ( x ) and µ i,k ∈ R is a suitable Lagrange multiplier. Define¯ u i,k ( x ) := √ a ∗ ε k λ u i,k ( x ) , where i = 1 , . (2.24)8roposition 2.1 then implies that ¯ u i,k (cid:0) ε k λ x + x ,k (cid:1) → w ( x ) uniformly in R , and ¯ u i,k satisfies the equation − ε k ∆¯ u i,k ( x ) + ε k h ( x )¯ u i,k ( x ) = µ i,k ε k ¯ u i,k ( x ) + λ a k a ∗ ¯ u i,k ( x ) in R , i = 1 , . (2.25)Because u ,k u ,k , we consider¯ ξ k ( x ) = u ,k ( x ) − u ,k ( x ) k u ,k − u ,k k L ∞ ( R ) = ¯ u ,k ( x ) − ¯ u ,k ( x ) k ¯ u ,k − ¯ u ,k k L ∞ ( R ) . Then ¯ ξ k satisfies the equation − ε k ∆ ¯ ξ k + ¯ C k ( x ) ¯ ξ k = ¯ g k ( x ) in R , (2.26)where the coefficient ¯ C k ( x ) satisfies¯ C k ( x ) := − µ ,k ε k − λ a k a ∗ (cid:0) ¯ u ,k + ¯ u ,k ¯ u ,k + ¯ u ,k (cid:1) + ε k h ( x ) , (2.27)and the nonhomogeneous term ¯ g k ( x ) satisfies¯ g k ( x ) := ε k ¯ u ,k ( µ ,k − µ ,k ) k ¯ u ,k − ¯ u ,k k L ∞ ( R ) = − λ a k ¯ u ,k a ∗ ) ε k Z R ¯ u ,k − ¯ u ,k k ¯ u ,k − ¯ u ,k k L ∞ ( R ) dx = − λ a k ¯ u ,k a ∗ ) ε k Z R ¯ ξ k (cid:0) ¯ u ,k + ¯ u ,k (cid:1)(cid:0) ¯ u ,k + ¯ u ,k (cid:1) dx, (2.28)due to the relation (2.17).Motivated by [3], we first claim that for any x ∈ R , there exists a small constant δ > Z ∂B δ ( x ) h ε k |∇ ¯ ξ k | + λ | ¯ ξ k | + ε k h ( x ) | ¯ ξ k | i dS = O ( ε k ) as k → ∞ . (2.29)To prove the above claim, multiplying (2.26) by ¯ ξ k and integrating over R , we obtainthat ε k Z R |∇ ¯ ξ k | − µ i,k ε k Z R | ¯ ξ k | + ε k Z R h ( x ) | ¯ ξ k | = λ a k a ∗ Z R (cid:0) ¯ u ,k + ¯ u ,k ¯ u ,k + ¯ u ,k (cid:1) | ¯ ξ k | − λ a k a ∗ ) ε k Z R ¯ u ,k ¯ ξ k Z R ¯ ξ k (cid:0) ¯ u ,k + ¯ u ,k (cid:1)(cid:0) ¯ u ,k + ¯ u ,k (cid:1) ≤ λ a k a ∗ Z R (cid:0) ¯ u ,k + ¯ u ,k ¯ u ,k + ¯ u ,k (cid:1) + λ a k a ∗ ) ε k Z R ¯ u ,k Z R (cid:0) ¯ u ,k + ¯ u ,k (cid:1)(cid:0) ¯ u ,k + ¯ u ,k (cid:1) ≤ Cε k as k → ∞ , since | ¯ ξ k | and ¯ u i,k (cid:0) ε k λ x + x ,k (cid:1) are bounded uniformly in k , and ¯ u i,k (cid:0) ε k λ x + x ,k (cid:1) decaysexponentially as | x | → ∞ , i = 1 ,
2. This implies that there exists a constant C > I := ε k Z R |∇ ¯ ξ k | + λ Z R | ¯ ξ k | + ε k Z R h ( x ) | ¯ ξ k | < C ε k as k → ∞ . (2.30)9pplying Lemma 4.5 in [3], we then conclude that for any x ∈ R , there exist a smallconstant δ > C > Z ∂B δ ( x ) h ε k |∇ ¯ ξ k | + λ | ¯ ξ k | + ε k h ( x ) | ¯ ξ k | i dS ≤ C I ≤ C C ε k as k → ∞ , which therefore implies the claim (2.29).We next define ξ k ( x ) = ¯ ξ k (cid:0) ε k λ x + x ,k (cid:1) , k = 1 , , · · · , (2.31)and ˜ u i,k ( x ) := √ a ∗ ε k λ u i,k (cid:0) ε k λ x + x ,k (cid:1) , where i = 1 , , so that ˜ u i,k ( x ) → w ( x ) uniformly in R as k → ∞ in view of Proposition 2.1. Underthe non-degeneracy assumption (1.14), we shall carry out the proof of Theorem 1.1 byderiving a contradiction through the following three steps. Step 1.
There exist a subsequence { a k } and some constants b , b and b such that ξ k ( x ) → ξ ( x ) in C loc ( R ) as k → ∞ , where ξ ( x ) = b (cid:0) w + x · ∇ w (cid:1) + X i =1 b i ∂w∂x i . (2.32)Note that ξ k satisfies − ∆ ξ k + C k ( x ) ξ k = g k ( x ) in R , (2.33)where the coefficient C k ( x ) satisfies C k ( x ) := − (cid:16) − ε pk a ∗ (cid:17)h ˜ u ,k ( x ) + ˜ u ,k ( x )˜ u ,k ( x ) + ˜ u ,k ( x ) i − ε k λ µ ,k + ε k λ h (cid:0) ε k xλ + x ,k (cid:1) , (2.34)and the nonhomogeneous term g k ( x ) satisfies g k ( x ) := ˜ u ,k λ ε k ( µ ,k − µ ,k ) k ˜ u ,k − ˜ u ,k k L ∞ = − ˜ u ,k λ a k ε k Z R u ,k − u ,k k ˜ u ,k − ˜ u ,k k L ∞ dx = − a k ˜ u ,k a ∗ ) Z R ξ k (cid:0) ˜ u ,k + ˜ u ,k (cid:1)(cid:0) ˜ u ,k + ˜ u ,k (cid:1) dx. (2.35)Here we have used (2.17) and (2.25). Since k ξ k k L ∞ ( R ) ≤
1, the standard elliptic regu-larity then implies (cf. [8]) that k ξ k k C ,αloc ( R ) ≤ C for some α ∈ (0 , C > k . Therefore, there exist a subsequence { a k } and a function ξ = ξ ( x ) such that ξ k ( x ) → ξ ( x ) in C loc ( R ) as k → ∞ . Applying Proposition 2.1,direct calculations yield from (2.17) and (2.18) that C k ( x ) → − w ( x ) uniformly on R as k → ∞ , and g k ( x ) → − w ( x ) a ∗ Z R w ξ uniformly on R as k → ∞ . ξ solves L ξ = − ∆ ξ + (1 − w ) ξ = (cid:16) − a ∗ Z R w ξ (cid:17) w in R . (2.36)Since L ( w + x · ∇ w ) = − w , we then conclude from (2.15) and (2.36) that (2.32) holdsfor some constants b , b and b . Step 2.
The constants b = b = b = 0 in (2.32).We first derive the following Pohozaev-type identity b Z R ∂h ( x + y ) ∂x j (cid:0) x · ∇ w (cid:1) − X i =1 b i Z R ∂ h ( x + y ) ∂x j ∂x i w = 0 , j = 1 , . (2.37)Multiplying (2.25) by ∂ ¯ u i,k ∂x j , where i, j = 1 ,
2, and integrating over B δ ( x ,k ), where δ > − ε k Z B δ ( x ,k ) ∂ ¯ u i,k ∂x j ∆¯ u i,k + ε k Z B δ ( x ,k ) h ( x ) ∂ ¯ u i,k ∂x j ¯ u i,k = µ i,k ε k Z B δ ( x ,k ) ∂ ¯ u i,k ∂x j ¯ u i,k + λ a k a ∗ Z B δ ( x ,k ) ∂ ¯ u i,k ∂x j ¯ u i,k = 12 µ i,k ε k Z ∂B δ ( x ,k ) ¯ u i,k ν j dS + λ a k a ∗ Z ∂B δ ( x ,k ) ¯ u i,k ν j dS, (2.38)where ν = ( ν , ν ) denotes the outward unit normal of ∂B δ ( x ,k ). Note that − ε k Z B δ ( x ,k ) ∂ ¯ u i,k ∂x j ∆¯ u i,k = − ε k Z ∂B δ ( x ,k ) ∂ ¯ u i,k ∂x j ∂ ¯ u i,k ∂ν dS + ε k Z B δ ( x ,k ) ∇ ¯ u i,k · ∇ ∂ ¯ u i,k ∂x j = − ε k Z ∂B δ ( x ,k ) ∂ ¯ u i,k ∂x j ∂ ¯ u i,k ∂ν dS + 12 ε k Z ∂B δ ( x ,k ) |∇ ¯ u i,k | ν j dS, and ε k Z B δ ( x ,k ) h ( x ) ∂ ¯ u i,k ∂x j ¯ u i,k = ε k Z ∂B δ ( x ,k ) h ( x )¯ u i,k ν j dS − ε k Z B δ ( x ,k ) ∂h ( x ) ∂x j ¯ u i,k . We then derive from (2.38) that ε k Z B δ ( x ,k ) ∂h ( x ) ∂x j ¯ u i,k = − ε k Z ∂B δ ( x ,k ) ∂ ¯ u i,k ∂x j ∂ ¯ u i,k ∂ν dS + ε k Z ∂B δ ( x ,k ) |∇ ¯ u i,k | ν j dS + ε k Z ∂B δ ( x ,k ) h ( x )¯ u i,k ν j dS − µ i,k ε k Z ∂B δ ( x ,k ) ¯ u i,k ν j dS − λ a k a ∗ Z ∂B δ ( x ,k ) ¯ u i,k ν j dS. (2.39)11ollowing (2.39), we thus have ε k Z B δ ( x ,k ) ∂h ( x ) ∂x j (cid:0) ¯ u ,k + ¯ u ,k (cid:1) ¯ ξ k dx = − ε k Z ∂B δ ( x ,k ) h ∂ ¯ u ,k ∂x j ∂ ¯ ξ k ∂ν + ∂ ¯ ξ k ∂x j ∂ ¯ u ,k ∂ν i dS + ε k Z ∂B δ ( x ,k ) ∇ ¯ ξ k · ∇ (cid:0) ¯ u ,k + ¯ u ,k (cid:1) ν j dS + ε k Z ∂B δ ( x ,k ) h ( x ) (cid:0) ¯ u ,k + ¯ u ,k (cid:1) ¯ ξ k ν j dS − µ ,k ε k Z ∂B δ ( x ,k ) (cid:0) ¯ u ,k + ¯ u ,k (cid:1) ¯ ξ k ν j dS − λ a k a ∗ Z ∂B δ ( x ,k ) (cid:0) ¯ u ,k + ¯ u ,k (cid:1)(cid:0) ¯ u ,k + ¯ u ,k (cid:1) ¯ ξ k ν j dS − (cid:0) µ ,k − µ ,k (cid:1) ε k k ¯ u ,k − ¯ u ,k k L ∞ Z ∂B δ ( x ,k ) ¯ u ,k ν j dS. (2.40)We now estimate the right hand side of (2.40) as follows. Applying (2.29), if δ > ε k Z ∂B δ ( x ,k ) (cid:12)(cid:12)(cid:12) ∂ ¯ u ,k ∂x j ∂ ¯ ξ k ∂ν (cid:12)(cid:12)(cid:12) dS ≤ ε k (cid:16) Z ∂B δ ( x ,k ) (cid:12)(cid:12)(cid:12) ∂ ¯ u ,k ∂x j (cid:12)(cid:12)(cid:12) dS (cid:17) (cid:16) ε k Z ∂B δ ( x ,k ) (cid:12)(cid:12)(cid:12) ∂ ¯ ξ k ∂ν (cid:12)(cid:12)(cid:12) dS (cid:17) ≤ Cε k e − Cδεk as k → ∞ , (2.41)due to the fact that ∇ ¯ u ,k (cid:0) ε k λ x + x ,k (cid:1) satisfies the exponential decay (2.22), where C > k . Similarly, we have ε k Z ∂B δ ( x ,k ) (cid:12)(cid:12)(cid:12) ∂ ¯ ξ k ∂x j ∂ ¯ u ,k ∂ν (cid:12)(cid:12)(cid:12) dS ≤ Cε k e − Cδεk as k → ∞ , and ε k (cid:12)(cid:12)(cid:12) Z ∂B δ ( x ,k ) ∇ ¯ ξ k · ∇ (cid:0) ¯ u ,k + ¯ u ,k (cid:1) ν j dS (cid:12)(cid:12)(cid:12) ≤ Cε k e − Cδεk as k → ∞ , On the other hand, since both | ¯ ξ k | and | (cid:0) µ ,k − µ ,k (cid:1) ε k | are bounded uniformly in k , wealso get from (2.22) that (cid:12)(cid:12)(cid:12) ε k Z ∂B δ ( x ,k ) h ( x ) (cid:0) ¯ u ,k + ¯ u ,k (cid:1) ¯ ξ k ν j dS − µ ,k ε k Z ∂B δ ( x ,k ) (cid:0) ¯ u ,k + ¯ u ,k (cid:1) ¯ ξ k ν j dS − λ a k a ∗ Z ∂B δ ( x ,k ) (cid:0) ¯ u ,k + ¯ u ,k (cid:1)(cid:0) ¯ u ,k + ¯ u ,k (cid:1) ¯ ξ k ν j dS − (cid:0) µ ,k − µ ,k (cid:1) ε k k ¯ u ,k − ¯ u ,k k L ∞ Z ∂B δ ( x ,k ) ¯ u ,k ν j dS (cid:12)(cid:12)(cid:12) = o ( e − Cδεk ) as k → ∞ , (2.42)due to the fact that (2.28) gives (cid:12)(cid:12) µ ,k − µ ,k (cid:12)(cid:12) ε k k ¯ u ,k − ¯ u ,k k L ∞ ≤ λ a k a ∗ ) ε k Z R (cid:0) ¯ u ,k + ¯ u ,k (cid:1)(cid:0) ¯ u ,k + ¯ u ,k (cid:1) | ¯ ξ k | ≤ M, (2.43)12here the constants M > k . Because h ( x ) is homogeneous of degree p , it then follows from (2.40) that for small δ > o ( e − Cδεk ) = ε k Z B δ ( x ,k ) ∂h ( x ) ∂x j (cid:2) ¯ u ,k ( x ) + ¯ u ,k ( x ) (cid:3) ¯ ξ k ( x ) dx = ε k λ Z B λδεk (0) ∂∂y j h (cid:0) ε k λ y + x ,k (cid:1) ¯ ξ k (cid:0) ε k λ y + x ,k (cid:1) · h ¯ u ,k (cid:0) ε k λ y + x ,k (cid:1) + ¯ u ,k (cid:0) ε k λ y + x ,k (cid:1)i dy = ε p +3 k λ p +1 h Z B λδεk (0) ∂∂y j h (cid:0) y + λx ,k ε k (cid:1) ¯ ξ k (cid:0) ε k λ y + x ,k (cid:1) · h ¯ u ,k (cid:0) ε k λ y + x ,k (cid:1) + ¯ u ,k (cid:0) ε k λ y + x ,k (cid:1)i dy + o (1) i (2.44)as k → ∞ . Applying (1.14), we thus derive from (2.6), (2.32) and (2.44) that0 = 2 Z R ∂h ( x + y ) ∂x j w ξ = 2 Z R ∂h ( x + y ) ∂x j w h b (cid:0) w + x · ∇ w (cid:1) + X i =1 b i ∂w∂x i i = b Z R ∂h ( x + y ) ∂x j (cid:0) x · ∇ w (cid:1) − X i =1 b i Z R ∂ h ( x + y ) ∂x j ∂x i w , where j = 1 ,
2, which thus implies (2.37).We next derive b = 0. Using the integration by parts, we note that − ε k Z B δ ( x ,k ) (cid:2) ( x − x ,k ) · ∇ ¯ u i,k (cid:3) ∆¯ u i,k = − ε k Z ∂B δ ( x ,k ) ∂ ¯ u i,k ∂ν (cid:2) ( x − x ,k ) · ∇ ¯ u i,k (cid:3) + ε k Z B δ ( x ,k ) ∇ ¯ u i,k ∇ (cid:2) ( x − x ,k ) · ∇ ¯ u i,k (cid:3) = − ε k Z ∂B δ ( x ,k ) ∂ ¯ u i,k ∂ν (cid:2) ( x − x ,k ) · ∇ ¯ u i,k (cid:3) + ε k Z ∂B δ ( x ,k ) (cid:2) ( x − x ,k ) · ν (cid:3) |∇ ¯ u i,k | . (2.45)Multiplying (2.25) by ( x − x ,k ) · ∇ ¯ u i,k , where i = 1 ,
2, and integrating over B δ ( x ,k ),13here δ > i = 1 , , − ε k Z B δ ( x ,k ) (cid:2) ( x − x ,k ) · ∇ ¯ u i,k (cid:3) ∆¯ u i,k = ε k Z B δ ( x ,k ) (cid:2) µ i,k − h ( x ) (cid:3) ¯ u i,k (cid:2) ( x − x ,k ) · ∇ ¯ u i,k (cid:3) + λ a k a ∗ Z B δ ( x ,k ) ¯ u i,k (cid:2) ( x − x ,k ) · ∇ ¯ u i,k (cid:3) = − ε k Z B δ ( x ,k ) ¯ u i,k n (cid:2) µ i,k − h ( x ) (cid:3) − ( x − x ,k ) · ∇ h ( x ) o + ε k Z ∂B δ ( x ,k ) ¯ u i,k (cid:2) µ i,k − h ( x ) (cid:3) ( x − x ,k ) νdS − λ a k a ∗ Z B δ ( x ,k ) ¯ u i,k + λ a k a ∗ Z ∂B δ ( x ,k ) ¯ u i,k ( x − x ,k ) νdS = − µ i,k ε k Z R ¯ u i,k + 2 + p ε k Z R h ( x )¯ u i,k − λ a k a ∗ Z R ¯ u i,k + I i , (2.46)where the lower order term I i satisfies I i = µ i,k ε k Z R \ B δ ( x ,k ) ¯ u i,k − p ε k Z R \ B δ ( x ,k ) h ( x )¯ u i,k + λ a k a ∗ Z R \ B δ ( x ,k ) ¯ u i,k − ε k Z B δ ( x ,k ) ¯ u i,k (cid:2) x ,k · ∇ h ( x ) (cid:3) + ε k Z ∂B δ ( x ,k ) ¯ u i,k (cid:2) µ i,k − h ( x ) (cid:3) ( x − x ,k ) νdS + λ a k a ∗ Z ∂B δ ( x ,k ) ¯ u i,k ( x − x ,k ) νdS, i = 1 , . (2.47)Since it follows from (2.17) that − µ i,k ε k Z R ¯ u i,k − λ a k a ∗ Z R ¯ u i,k = − a ∗ ε k λ h µ i,k + a k Z R u i,k i = − a ∗ ε k λ e ( a k ) , we reduce from (2.45)–(2.47) that a ∗ ε k λ e ( a k ) − p ε k Z R h ( x )¯ u i,k = I i + ε k Z ∂B δ ( x ,k ) ∂ ¯ u i,k ∂ν (cid:2) ( x − x ,k ) · ∇ ¯ u i,k (cid:3) − ε k Z ∂B δ ( x ,k ) (cid:2) ( x − x ,k ) · ν (cid:3) |∇ ¯ u i,k | , i = 1 , , which implies that − p ε k Z R h ( x ) (cid:2) ¯ u ,k + ¯ u ,k (cid:3) ¯ ξ k = T k . (2.48)14ere the term T k satisfies that for small δ > T k = I − I k ¯ u ,k − ¯ u ,k k L ∞ − ε k Z ∂B δ ( x ,k ) (cid:2) ( x − x ,k ) · ν (cid:3)(cid:0) ∇ ¯ u ,k + ∇ ¯ u ,k (cid:1) ∇ ¯ ξ k + ε k Z ∂B δ ( x ,k ) n(cid:2) ( x − x ,k ) · ∇ ¯ u ,k (cid:3)(cid:0) ν · ∇ ¯ ξ k (cid:1) + (cid:0) ν · ∇ ¯ u ,k (cid:1)(cid:2) ( x − x ,k ) · ∇ ¯ ξ k (cid:3)o = I − I k ¯ u ,k − ¯ u ,k k L ∞ + o ( e − Cδεk ) as k → ∞ , (2.49)due to (2.22) and (2.29), where the second equality follows by applying the argument ofestimating (2.41).Using the arguments of estimating (2.41) and (2.42), along with the exponentialdecay of ¯ u i,k , we also derive that for small δ > I − I k ¯ u ,k − ¯ u ,k k L ∞ = µ ,k ε k Z R \ B δ ( x ,k ) (cid:0) ¯ u ,k + ¯ u ,k (cid:1) ¯ ξ k − p ε k Z R \ B δ ( x ,k ) h ( x ) (cid:0) ¯ u ,k + ¯ u ,k (cid:1) ¯ ξ k + λ a k a ∗ Z R \ B δ ( x ,k ) (cid:0) ¯ u ,k + ¯ u ,k (cid:1)(cid:0) ¯ u ,k + ¯ u ,k (cid:1) ¯ ξ k + (cid:0) µ ,k − µ ,k (cid:1) ε k k ¯ u ,k − ¯ u ,k k L ∞ Z R \ B δ ( x ,k ) ¯ u ,k − ε k Z B δ ( x ,k ) (cid:2) x ,k · ∇ h ( x ) (cid:3)(cid:0) ¯ u ,k + ¯ u ,k (cid:1) ¯ ξ k + λ a k a ∗ Z ∂B δ ( x ,k ) (cid:0) ¯ u ,k + ¯ u ,k (cid:1)(cid:0) ¯ u ,k + ¯ u ,k (cid:1) ¯ ξ k ( x − x ,k ) νdS − ε k Z ∂B δ ( x ,k ) (cid:0) ¯ u ,k + ¯ u ,k (cid:1) ¯ ξ k h ( x )( x − x ,k ) νdS + µ ,k ε k Z ∂B δ ( x ,k ) (cid:0) ¯ u ,k + ¯ u ,k (cid:1) ¯ ξ k ( x − x ,k ) νdS + (cid:0) µ ,k − µ ,k (cid:1) ε k k ¯ u ,k − ¯ u ,k k L ∞ Z ∂B δ ( x ,k ) ¯ u ,k ( x − x ,k ) νdS = (cid:0) µ ,k − µ ,k (cid:1) ε k k ¯ u ,k − ¯ u ,k k L ∞ h Z R \ B δ ( x ,k ) ¯ u ,k + 12 Z ∂B δ ( x ,k ) ¯ u ,k ( x − x ,k ) νdS i − ε k Z B δ ( x ,k ) (cid:2) x ,k · ∇ h ( x ) (cid:3)(cid:0) ¯ u ,k + ¯ u ,k (cid:1) ¯ ξ k + o ( e − Cδεk ) as k → ∞ . (2.50)Note from (2.43) that (cid:0) µ ,k − µ ,k (cid:1) ε k k ¯ u ,k − ¯ u ,k k L ∞ h Z R \ B δ ( x ,k ) ¯ u ,k + 12 Z ∂B δ ( x ,k ) ¯ u ,k ( x − x ,k ) νdS i = O ( e − Cδεk ) (2.51)as k → ∞ , where the constant C > k . Moreover, we follow from the15rst identity of (2.44) that12 ε k Z B δ ( x ,k ) (cid:2) x ,k · ∇ h ( x ) (cid:3)(cid:0) ¯ u ,k + ¯ u ,k (cid:1) ¯ ξ k = 12 ε k X i =1 x i ,k Z B δ ( x ,k ) ∂h ( x ) ∂x i (cid:2) ¯ u ,k ( x ) + ¯ u ,k ( x ) (cid:3) ¯ ξ k ( x ) dx = o ( e − Cδεk ) as k → ∞ , (2.52)where we denote x ,k = ( x ,k , x ,k ). Therefore, we deduce from (2.49)–(2.52) that T k = o ( ε pk ) as k → ∞ . Further, we obtain from (2.48) that o ( ε pk ) = − p ε k Z R h ( x ) h ¯ u ,k + ¯ u ,k i ¯ ξ k = − p λ ε k Z R h (cid:0) ε k λ x + x ,k (cid:1)h ¯ u ,k (cid:0) ε k λ x + x ,k (cid:1) + ¯ u ,k (cid:0) ε k λ x + x ,k (cid:1)i ξ k ( x ) dx − p λ ε k Z R h (cid:0) ε k λ x + x ,k (cid:1)h ¯ u ,k (cid:0) ε k λ x + x ,k (cid:1) − ¯ u ,k (cid:0) ε k λ x + x ,k (cid:1)i ξ k ( x ) dx = − p λ p ε pk Z R h (cid:0) x + λx ,k ε k (cid:1)h ¯ u ,k (cid:0) ε k λ x + x ,k (cid:1) + ¯ u ,k (cid:0) ε k λ x + x ,k (cid:1)i ξ k ( x ) dx + O ( ε pk | x ,k − x ,k | ) as k → ∞ . Since ( x + y ) · ∇ h ( x + y ) = ph ( x + y ), by Proposition 2.1 and Step 1, we thus obtainfrom (1.14) and above that0 = 2 Z R h ( x + y ) wξ = 2 b Z R h ( x + y ) w (cid:0) w + x · ∇ w (cid:1) + X i =1 b i Z R h ( x + y ) ∂w ∂x i = 2 b h Z R h ( x + y ) w + 12 Z R h ( x + y ) (cid:0) x · ∇ w (cid:1)i = 2 b n Z R h ( x + y ) w − Z R w (cid:2) h ( x + y ) + x · ∇ h ( x + y ) (cid:3)o = − pb Z R h ( x + y ) w + b Z R w (cid:2) y · ∇ h ( x + y ) (cid:3) = − pb Z R h ( x + y ) w , which therefore implies that b = 0.By the non-degeneracy assumption (1.14), setting b = 0 into (2.37) then yields that b = b = 0, and Step 2 is therefore proved. Step 3. ξ ≡ y k be a point satisfying | ξ k ( y k ) | = k ξ k k L ∞ ( R ) = 1. By the same argumentas employed in proving Lemma 3.1 in next section, applying the maximum principle to(2.33) yields that | y k | ≤ C uniformly in k . Therefore, we conclude that ξ k → ξ R , which however contradicts to the fact that ξ ≡ R . This completesthe proof of Theorem 1.1. 16 Refined Spike Profiles
In the following two sections, we shall derive the refined spike profiles of positive minimiz-ers u k = u a k for e ( a k ) as a k ր a ∗ . The purpose of this section is to prove Theorem 1.2.Recall first that u k satisfies the Euler-Lagrange equation (2.16). Under the assumptionsof Proposition 2.1, for convenience, we denote ε k = ( a ∗ − a k ) p > , α k := ε pk > β k := 1 + µ k ε k λ , (3.1)where µ k ∈ R is the Lagrange multiplier of the equation (2.16), so that α k → β k → k → ∞ , where (2.19) is used. In order to discuss the refined spike profiles of u k as k → ∞ , thekey is thus to obtain the refined estimate of µ k (equivalently β k ) in terms of ε k .We next define w k ( x ) := ¯ u k ( x ) − w ( x ) := √ a ∗ ε k λ u k (cid:16) ε k λ x + x k (cid:17) − w ( x ) , (3.2)where x k is the unique maximum point of u k , so that w k ( x ) → R byProposition 2.1. By applying (2.16), direct calculations then give that ¯ u k satisfies − ∆¯ u k ( x ) + ε k λ V (cid:0) ε k λ x + x k (cid:1) ¯ u k ( x ) = µ k ε k λ ¯ u k ( x ) + a k a ∗ ¯ u k ( x ) in R . Relating to the operator L := − ∆ + (1 − w ) in R , we also denote the linearizedoperator L k := − ∆ + (cid:2) − (cid:0) ¯ u k + ¯ u k w + w (cid:1)(cid:3) in R , so that w k satisfies L k w k ( x ) = − α k h a ∗ ¯ u k ( x ) + 1 λ p g (cid:0) ε k xλ + x k (cid:1) h (cid:0) x + λx k ε k (cid:1) ¯ u k ( x ) i + β k ¯ u k ( x ) in R , ∇ w k (0) = 0 , (3.3)where V ( x ) = g ( x ) h ( x ) satisfies the assumptions of Proposition 2.1 and the coefficients α k > β k > L k ψ ,k ( x ) = − α k h λ p g (cid:0) ε k xλ + x k (cid:1) h (cid:0) x + λx k ε k (cid:1) ¯ u k ( x )+ 1 a ∗ ¯ u k ( x ) i in R , ∇ ψ ,k (0) = 0 , L k ψ ,k ( x ) = β k ¯ u k ( x ) in R , ∇ ψ ,k (0) = 0 , (3.4)so that the solution w k ( x ) of (3.3) satisfies w k ( x ) := ψ ,k ( x ) + ψ ,k ( x ) in R . (3.5)We first employ Proposition 2.1 to address the following estimates of w k as k → ∞ . Lemma 3.1.
Under the assumptions of Proposition 2.1, where V ( x ) = g ( x ) h ( x ) , wehave . ψ ,k ( x ) satisfies ψ ,k ( x ) = α k ψ ( x ) + o ( α k ) as k → ∞ , (3.6) where ψ ( x ) ∈ C ( R ) ∩ L ∞ ( R ) solves uniquely ∇ ψ (0) = 0 , L ψ ( x ) = − a ∗ w ( x ) − g (0) λ p h ( x + y ) w ( x ) in R , (3.7) where y ∈ R is given by (1.9).2. ψ ,k ( x ) satisfies ψ ,k ( x ) = β k ψ ( x ) + o ( β k ) as k → ∞ , (3.8) where ψ ( x ) solves uniquely ∇ ψ (0) = 0 , L ψ ( x ) = w ( x ) in R , (3.9) i.e., ψ ( x ) ∈ C ( R ) ∩ L ∞ ( R ) satisfies ψ = − (cid:0) w + x · ∇ w (cid:1) . (3.10) w k satisfies w k ( x ) := α k ψ ( x ) + β k ψ ( x ) + o ( α k + β k ) as k → ∞ . (3.11) Proof.
1. We first derive | ψ ,k | ≤ Cα k in R by contradiction. On the contrary, weassume that lim k →∞ k ψ ,k k L ∞ α k = ∞ . (3.12)Set ¯ ψ ,k = ψ ,k k ψ ,k k L ∞ so that k ¯ ψ ,k k L ∞ = 1. Following (3.4), ¯ ψ ,k then satisfies − ∆ ¯ ψ ,k + (cid:2) − (cid:0) ¯ u k + ¯ u k w + w (cid:1)(cid:3) ¯ ψ ,k = − α k k ψ ,k k ∞ h λ p g (cid:0) ε k xλ + x k (cid:1) h (cid:0) x + λx k ε k (cid:1) ¯ u k ( x ) + 1 a ∗ ¯ u k ( x ) i in R . (3.13)Let y k be the global maximum point of ¯ ψ ,k so that ¯ ψ ,k ( y k ) = max y ∈ R ψ ,k ( y ) k ψ ,k k L ∞ = 1.Since both ¯ u k and w decay exponentially in view of (2.7), using the maximum principlewe derive from (3.13) that | y k | ≤ C uniformly in k .On the other hand, applying the usual elliptic regularity theory, there exists a sub-sequence, still denoted by { ¯ ψ ,k } , of { ¯ ψ ,k } such that ¯ ψ ,k → ¯ ψ weakly in H ( R ) andstrongly in L qloc ( R ) for all q ∈ [2 , ∞ ). Here ¯ ψ satisfies ∇ ¯ ψ (0) = 0 , L ¯ ψ ( x ) = 0 in R , which implies that ¯ ψ = P i =1 c i ∂w∂y i . Since ∇ ¯ ψ (0) = 0, we obtain that c = c = 0.Thus, we have ¯ ψ ( y ) ≡ R , which however contradicts to the fact that 1 = ¯ ψ ,k ( y k ) → ¯ ψ (¯ y ) for some ¯ y ∈ R by passing to a subsequence if necessary. Therefore, we have | ψ ,k | ≤ Cα k in R .We next set φ ,k ( x ) = ψ ,k ( x ) − α k ψ ( x ), where ψ ( x ) ∈ C ( R ) ∩ L ∞ ( R ) is asolution of (3.7). Then either φ ,k ( x ) = O ( α k ) or φ ,k ( x ) = o ( α k ) as k → ∞ , and φ ,k satisfies ∇ φ ,k (0) = 0 , − ∆ φ ,k + (cid:2) − (cid:0) ¯ u k + ¯ u k w + w (cid:1)(cid:3) φ ,k = − α k f k ( x ) in R , f k ( x ) satisfies f k ( x ) = (cid:0) w − ¯ u k − ¯ u k w (cid:1) ψ ( x ) + 1 a ∗ (cid:0) ¯ u k ( x ) − w ( x ) (cid:1) + 1 λ p h g (cid:0) ε k xλ + x k (cid:1) h (cid:0) x + λx k ε k (cid:1) ¯ u k ( x ) − g (0) h ( x + y ) w ( x ) i . One can note that f k ( x ) → k → ∞ . Therefore, applying the previousargument yields necessarily that φ ,k ( x ) = o ( α k ) as k → ∞ , and the proof of (3.6) isthen complete. Also, the property (2.15) gives the uniqueness of solutions for (3.7).2. Since the proof of (3.8) is very similar to that of (3.6), we omit the details.Further, the property (2.15) gives the uniqueness of ψ . Also, one can check directlythat − ( w + x · ∇ w ) / The main aim of this subsection is to prove Theorem 1.2 on the refined spike behaviorof positive minimizers. In this whole subsection, we assume that the potential V ( x ) = h ( x ) ∈ C ( R ) satisfies lim | x |→∞ h ( x ) = ∞ and (1.14), where h ( x ) is homogeneous ofdegree p ≥
2. Following (3.1), from now on we denote for simplicity that o (cid:0) [ α k + β k ] (cid:1) = o (cid:0) α k (cid:1) + o (cid:0) α k β k (cid:1) + o (cid:0) β k (cid:1) as k → ∞ , (3.14)where α k and β k are defined in (3.1). We first use Lemma 3.1 to establish the followinglemmas. Lemma 3.2.
Suppose that V ( x ) = h ( x ) ∈ C ( R ) satisfies lim | x |→∞ h ( x ) = ∞ and(1.14) for some y ∈ R , where h ( x ) is homogeneous of degree p ≥ . Then there existsan x ∈ R such that the unique maximum point x k of u k satisfies (cid:12)(cid:12)(cid:12) α k (cid:16) λx k ε k − y (cid:17) − α k β k y (cid:12)(cid:12)(cid:12) = α k O ( | x | ) + o (cid:0) [ α k + β k ] (cid:1) as k → ∞ . (3.15) Proof.
Multiplying (3.7) and (3.9) by ∂w∂x and then integrating over R , respectively,we obtain from (1.14) and (2.15) that Z R ∂w∂x L w k = Z R ∂w∂x w = Z R ∂w∂x h ( x + y ) w = 0 , (3.16)where y is given by the assumption (1.14). Similarly, we derive from (3.3) and (3.11)that Z R ∂w∂x L k w k = β k Z R ∂w∂x ¯ u k − α k Z R ∂w∂x h a ∗ ¯ u k + ¯ u k λ p h (cid:0) x + λx k ε k (cid:1)i = α k β k Z R ∂w∂x ψ + o ( α k β k + β k ) − I , (3.17)where the identity R R ∂w∂x ψ = 0 is used, since ∂w∂x ψ is odd in x by the radial symmetry19f ψ . We obtain from (1.14) and (3.16) that I = α k Z R ∂w∂x h a ∗ ¯ u k + ¯ u k λ p h (cid:0) x + λx k ε k (cid:1)i = α k Z R ∂w∂x n a ∗ (cid:0) ¯ u k − w (cid:1) + 1 λ p h h (cid:0) x + λx k ε k (cid:1) ¯ u k − h ( x + y ) w io = α k a ∗ Z R ∂w∂x w k (cid:0) w + 3 ww k + w k (cid:1) + α k λ p Z R ∂w∂x h h (cid:0) x + λx k ε k (cid:1) ¯ u k − h ( x + y ) w i = 3 α k a ∗ Z R ∂w∂x w ψ + o ( α k + α k β k ) + I , (3.18)where we have used the identity R R ∂w∂x w ψ = 0, since ∂w∂x w ψ is odd in x by theradial symmetry of ψ . Further, applying (3.11) and (3.16) yields that λ p α k I = Z R ∂w∂x n h (cid:0) x + λx k ε k (cid:1)(cid:2) ¯ u k − w (cid:3) + (cid:2) h (cid:0) x + λx k ε k (cid:1) − h ( x + y ) (cid:3) w o = Z R ∂w∂x h ( x + y ) w k + o ( α k + β k )+ Z R ∂w∂x h(cid:16) λx k ε k − y (cid:17) · ∇ h ( x + y ) i w + o (cid:16)(cid:12)(cid:12)(cid:12) λx k ε k − y (cid:12)(cid:12)(cid:12)(cid:17) = α k Z R ∂w∂x h ( x + y ) ψ + β k Z R ∂w∂x h ( x + y ) ψ + Z R ∂w∂x h(cid:16) λx k ε k − y (cid:17) · ∇ h ( x + y ) i w + o (cid:16) α k + (cid:12)(cid:12) λx k ε k − y (cid:12)(cid:12) + β k (cid:17) , (3.19)where (2.6) is used for the second identity. We thus get that I = α k h a ∗ Z R ∂w∂x w ψ + 1 λ p Z R ∂w∂x h ( x + y ) ψ i + α k β k λ p Z R ∂w∂x h ( x + y ) ψ + α k λ p Z R ∂w∂x h(cid:16) λx k ε k − y (cid:17) · ∇ h ( x + y ) i w + o (cid:16) α k (cid:12)(cid:12) λx k ε k − y (cid:12)(cid:12) + [ α k + β k ] (cid:17) . (3.20)On the other hand, we obtain from (3.16) that Z R ∂w∂x L k w k = Z R ∂w∂x L w k + Z R ∂w∂x (cid:0) L k − L (cid:1) w k = − Z R ∂w∂x w k (3 w + w k )= − α k Z R ∂w∂x wψ − α k β k Z R ∂w∂x wψ ψ + o ( α k + α k β k ) . (3.21)20ombining (3.17), (3.21) and (3.20), we now conclude from (1.14) and (3.11) that α k λ p Z R ∂w∂x h(cid:16) λx k ε k − y (cid:17) · ∇ h ( x + y ) i w = α k β k h Z R ∂w∂x ψ + 6 Z R ∂w∂x wψ ψ − λ p Z R ∂w∂x h ( x + y ) ψ i − α k h a ∗ Z R ∂w∂x w ψ + 1 λ p Z R ∂w∂x h ( x + y ) ψ − Z R ∂w∂x wψ i + o ([ α k + β k ] ) . (3.22)We claim that the coefficient I of the term α k β k in (3.22) satisfies I : = Z R ∂w∂x ψ + 6 Z R ∂w∂x wψ ψ − λ p Z R ∂w∂x h ( x + y ) ψ = 12 λ p Z R w h y · ∇ h ( x + y ) i ∂w∂x . (3.23)If (3.23) holds, we then derive from (3.22) that there exists some x = ( x , x ) ∈ R such that 12 λ p Z R ∂w ∂x j h α k (cid:16) λx k ε k − y (cid:17) − α k β k y i · ∇ h ( x + y )= α k O ( | x j | ) + o ([ α k + β k ] ) , j = 1 , . (3.24)By the non-degeneracy assumption of (1.14), we further conclude from (3.24) that (3.15)holds for some x ∈ R , and the lemma is therefore proved.To complete the proof of the lemma, the rest is to prove the claim (3.23). Indeed,using the integration by parts, we derive from (3.10) that A : = Z R ∂w∂x ψ + 6 Z R ∂w∂x wψ ψ = Z R ∂w∂x ψ − Z R ∂w∂x w ψ − Z R ∂w∂x ψ ( x · ∇ w )= Z R ∂w∂x ψ − Z R ∂w∂x w ψ + 32 Z R w h ∂w∂x ψ + x · ∇ (cid:16) ∂w∂x ψ (cid:17)i = Z R ∂w∂x ψ + 32 Z R w h ∂w∂x ( x · ∇ ψ ) + ψ x · ∇ (cid:16) ∂w∂x (cid:17)i . Since ( x + y ) · ∇ h ( x + y ) = ph ( x + y ), we obtain from (1.14), (3.10) and (3.16) that B : = − λ p Z R ∂w∂x h ( x + y ) ψ = 12 λ p Z R ∂w∂x h ( x + y )( w + x · ∇ w )= − λ p Z R w h h ( x + y ) ∂w∂x + x · ∇ (cid:16) ∂w∂x h ( x + y ) (cid:17)i = − λ p Z R w n(cid:2) x · ∇ h ( x + y ) (cid:3) ∂w∂x + h ( x + y ) x · ∇ (cid:16) ∂w∂x (cid:17)o = − λ p Z R wh ( x + y ) h x · ∇ (cid:16) ∂w∂x (cid:17)i + 12 λ p Z R w h y · ∇ h ( x + y ) i ∂w∂x .
21y above calculations, we then get from (3.23) that I = A + B = Z R ∂w∂x ψ + 32 Z R w ∂w∂x ( x · ∇ ψ )+ 12 Z R h w ψ − wh ( x + y ) λ p ih x · ∇ (cid:16) ∂w∂x (cid:17)i + 12 λ p Z R w h y · ∇ h ( x + y ) i ∂w∂x := I + 12 λ p Z R w h y · ∇ h ( x + y ) i ∂w∂x . (3.25)Applying the integration by parts, we derive from (3.7) that Z R ∂w∂x ψ + 12 Z R h w ψ − wh ( x + y ) λ p ih x · ∇ (cid:16) ∂w∂x (cid:17)i = Z R ∂w∂x ψ + 12 Z R h w ψ − (cid:16) w a ∗ + wh ( x + y ) λ p (cid:17)ih x · ∇ (cid:16) ∂w∂x (cid:17)i = Z R ∂w∂x ψ + 12 Z R (cid:0) − ∆ ψ + ψ (cid:1)h x · ∇ (cid:16) ∂w∂x (cid:17)i = Z R ∂w∂x ψ + 12 Z R (cid:0) − ∆ ψ (cid:1)h x · ∇ (cid:16) ∂w∂x (cid:17)i − Z R ∂w∂x (cid:2) ψ + x · ∇ ψ (cid:3) = 12 Z R (cid:0) − ∆ ψ (cid:1)h x · ∇ (cid:16) ∂w∂x (cid:17)i − Z R ∂w∂x (cid:0) x · ∇ ψ (cid:1) , which then gives from (3.25) that − I = Z R ∆ ψ h x · ∇ (cid:16) ∂w∂x (cid:17)i + Z R ∂∂x h w − w i(cid:0) x · ∇ ψ (cid:1) = Z R ∆ ψ h x · ∇ (cid:16) ∂w∂x (cid:17)i + Z R ∂ ∆ w∂x (cid:0) x · ∇ ψ (cid:1) . (3.26)To further simplify I , we next rewrite ψ as ψ ( x ) = ψ ( r, θ ), where x = r (cos θ, sin θ )22nd ( r, θ ) is the polar coordinate in R . We then follow from (3.7) and (3.26) that − I = Z ∞ Z π n(cid:2) r (cid:0) ψ (cid:1) r (cid:3) r + 1 r (cid:0) ψ (cid:1) θθ o r ∂∂r (cid:0) w ′ cos θ (cid:1) dθdr + Z ∞ Z π ∂∂r (cid:16) w ′′ + w ′ r (cid:17) cos θ r (cid:0) ψ (cid:1) r dθdr = − Z ∞ Z π r (cid:0) ψ (cid:1) r ( rw ′′ ) ′ cos θdθdr + Z ∞ Z π r (cid:0) ψ (cid:1) r h r ∂∂r (cid:16) w ′′ + w ′ r (cid:17)i cos θdθdr + Z ∞ Z π (cid:0) ψ (cid:1) θθ w ′′ cos θdθdr = − Z ∞ Z π r (cid:0) ψ (cid:1) r n ( rw ′′ ) ′ − h r ∂∂r (cid:16) w ′′ + w ′ r (cid:17)io cos θdθdr − Z ∞ Z π ψ w ′′ cos θdθdr = − Z ∞ Z π (cid:0) ψ (cid:1) r w ′ cos θdθdr − Z ∞ Z π ψ w ′′ cos θdθdr = 0 , (3.27) i.e., I = 0, which therefore implies that the claim (3.23) holds by applying (3.25). Remark 3.1.
Whether the point x ∈ R in Lemma 3.2 is the origin or not is determinedcompletely by the fact that whether the coefficient I of the term α k in (3.22) is zero ornot, where I satisfies I := 3 a ∗ Z R ∂w∂x w ψ + 1 λ p Z R ∂w∂x h ( x + y ) ψ − Z R ∂w∂x wψ . If h ( x ) is not even in x , it however seems difficult to derive that whether I = 0 or not. Lemma 3.3.
Suppose that V ( x ) = h ( x ) ∈ C ( R ) satisfies lim | x |→∞ h ( x ) = ∞ and(1.14) for some y ∈ R , where h ( x ) is homogeneous of degree p ≥ . Then we have w k := α k ψ + β k ψ + α k ψ + β k ψ + α k β k ψ + o ([ α k + β k ] ) as k → ∞ , (3.28) where ψ ( x ) , ψ ( x ) ∈ C ( R ) ∩ L ∞ ( R ) are given in Lemma 3.1 with g (0) = 1 , and ψ i ( x ) ∈ C ( R ) ∩ L ∞ ( R ) , i = 3 , , , solves uniquely ∇ ψ i (0) = 0 and L ψ i ( x ) = f i ( x ) in R , i = 3 , , , (3.29) and f i ( x ) satisfies for some y ∈ R , f i ( x ) = wψ − (cid:16) w a ∗ + h ( x + y ) λ p (cid:17) ψ − wλ p (cid:2) y · ∇ h ( x + y ) (cid:3) , if i = 3;3 wψ + ψ , if i = 4;6 wψ ψ + ψ − (cid:16) w a ∗ + h ( x + y ) λ p (cid:17) ψ − w λ p h y · ∇ h ( x + y ) i , if i = 5; (3.30) where y ∈ R is given by (1.14). Moreover, ψ is radially symmetric. roof. Following Lemma 3.1(3), set v k = w k − α k ψ − β k ψ , so that L k w k = L k ( v k + α k ψ + β k ψ )= L k v k + α k ( L k − L ) ψ + β k ( L k − L ) ψ + α k L ψ + β k L ψ = L k v k − w k ( α k ψ + β k ψ )(3 w + w k ) − α k h w a ∗ + h ( x + y ) wλ p i + β k w. (3.31)Applying (3.3), we then have L k v k = L k w k + w k ( α k ψ + β k ψ )(3 w + w k ) + α k h w a ∗ + h ( x + y ) wλ p i − β k w = w k ( α k ψ + β k ψ )(3 w + w k ) + β k (¯ u k − w ) − α k n a ∗ (¯ u k − w ) + 1 λ p h h (cid:0) x + λx k ε k (cid:1) ¯ u k − h ( x + y ) w io = w k ( α k ψ + β k ψ )(3 w + w k ) + β k w k − I , (3.32)where I satisfies I = α k a ∗ w k (3 w + 3 ww k + w k )+ α k λ p n h ( x + y )(¯ u k − w ) + h h (cid:0) x + λx k ε k (cid:1) − h ( x + y ) i ¯ u k o = α k a ∗ w k (3 w + 3 ww k + w k ) + α k λ p h ( x + y ) w k + α k λ p h(cid:16) λx k ε k − y (cid:17) · ∇ h ( x + y ) i ¯ u k + o (cid:0) [ α k + β k ] (cid:1) = α k w k (cid:16) w a ∗ + h ( x + y ) λ p (cid:17) + α k a ∗ w k (3 w + w k )+ α k λ p h(cid:16) λx k ε k − y (cid:17) · ∇ h ( x + y ) i ¯ u k + o (cid:0) [ α k + β k ] (cid:1) , where Lemma 3.2 is used in the second equality. By Lemma 3.2 again, there exists y ∈ R such that (cid:12)(cid:12)(cid:12) α k (cid:16) λx k ε k − y (cid:17) − α k β k y − α k y (cid:12)(cid:12)(cid:12) = o (cid:0) [ α k + β k ] (cid:1) as k → ∞ . We thus obtain from above that L k v k = w k ( α k ψ + β k ψ )(3 w + w k ) + β k w k − α k w k (cid:16) w a ∗ + h ( x + y ) λ p (cid:17) − α k λ p h(cid:16) λx k ε k − y (cid:17) · ∇ h ( x + y ) i ¯ u k − α k a ∗ w k (3 w + w k ) + o ([ α k + β k ] )= α k n wψ − (cid:16) w a ∗ + h ( x + y ) λ p (cid:17) ψ − wλ p h y · ∇ h ( x + y ) io + α k β k n wψ ψ + ψ − (cid:16) w a ∗ + h ( x + y ) λ p (cid:17) ψ − λ p w h y · ∇ h ( x + y ) io + β k (3 wψ + ψ ) + o ([ α k + β k ] ) in R . (3.33)24ollowing (3.33), the same argument of proving Lemma 3.1 then gives (3.28). Finally,since f ( x ) is radially symmetric, there exists a radial solution ψ . Further, the property(2.15) gives the uniqueness of ψ . Therefore, ψ must be radially symmetric, and theproof is complete. Lemma 3.4.
Suppose that V ( x ) = h ( x ) ∈ C ( R ) satisfies lim | x |→∞ h ( x ) = ∞ and(1.14) for some y ∈ R , where h ( x ) is homogeneous of degree p ≥ . Then we have Z R wψ = 0 , Z R wψ = 0 , (3.34) and I = Z R (cid:0) wψ + ψ (cid:1) = 0 . (3.35) However, we have II = 2 Z R wψ + 2 Z R ψ ψ = − p < . (3.36) Here ψ ( x ) , · · · , ψ ( x ) ∈ C ( R ) ∩ L ∞ ( R ) are given in Lemma 3.1 with g (0) = 1 andLemma 3.3. Since the proof of Lemma 3.4 is very involved, we leave it to the appendix. Followingabove lemmas, we are now ready to derive the comparison relation between β k and α k . Proposition 3.5.
Suppose that V ( x ) = h ( x ) ∈ C ( R ) satisfies lim | x |→∞ h ( x ) = ∞ and(1.14) for some y ∈ R , where h ( x ) is homogeneous of degree p ≥ . Then we have β k = C ∗ α k as k → ∞ , (3.37) where the constant C ∗ satisfies C ∗ = 22 + p (cid:16) Z R wψ + Z R ψ (cid:17) = 0 . (3.38) Moreover, w k satisfies w k := (cid:2) ψ + C ∗ ψ (cid:3) α k + (cid:2) ψ + ( C ∗ ) ψ + C ∗ ψ (cid:3) α k + o ( α k ) as k → ∞ , (3.39) Here ψ ( x ) , · · · , ψ ( x ) ∈ C ( R ) ∩ L ∞ ( R ) are given in Lemma 3.1 with g (0) = 1 andLemma 3.3. Proof.
Note from (3.2) that w k satisfies Z R w = Z R ¯ u k = Z R (cid:0) w + w k (cid:1) , i.e. Z R ww k + Z R w k = 0 . (3.40)25pplying (3.40), we then derive from Lemma 3.3 that0 = 2 Z R ww k + Z R w k = 2 Z R w ( α k ψ + β k ψ + α k ψ + β k ψ + α k β k ψ )+ Z R ( α k ψ + β k ψ + α k ψ + β k ψ + α k β k ψ ) + o ([ α k + β k ] )= α k (cid:16) Z R wψ (cid:17) + β k (cid:16) Z R wψ (cid:17) + β k (cid:16) Z R wψ + Z R ψ (cid:17) + α k β k (cid:16) Z R wψ + 2 Z R ψ ψ (cid:17) + α k (cid:16) Z R wψ + Z R ψ (cid:17) + o ([ α k + β k ] )= − p α k β k + α k (cid:16) Z R wψ + Z R ψ (cid:17) + o ([ α k + β k ] ) , (3.41)where Lemma 3.4 is used in the last equality. It then follows from (3.41) that2 Z R wψ + Z R ψ = 0 , and moreover, − p β k + α k (cid:16) Z R wψ + Z R ψ (cid:17) = 0 , i.e., β k = C ∗ α k , where C ∗ = 0 is as in (3.38). Finally, the expansion (3.39) follows directly from (3.37)and Lemma 3.3, and we are done.We remark from (3.1) and Proposition 3.5 that the Lagrange multiplier µ k ∈ R ofthe Euler-Lagrange equation (2.16) satisfies µ k = − λε k + λ C ∗ ε pk + o ( ε pk ) as k → ∞ , (3.42)where λ > g (0) = 1, and C ∗ = 0 is given by (3.38). Moreover,following above results we finally conclude the following refined spike profiles. Theorem 3.6.
Suppose that V ( x ) = h ( x ) ∈ C ( R ) satisfies lim | x |→∞ h ( x ) = ∞ and(1.14) for some y ∈ R , where h ( x ) is homogeneous of degree p ≥ . If u a is a positiveminimizer of e ( a ) for a < a ∗ . Then for any sequence { a k } with a k ր a ∗ as k → ∞ ,there exist a subsequence, still denoted by { a k } , of { a k } and { x k } ⊂ R such that thesubsequence solution u k = u a k satisfies for ε k := ( a ∗ − a k ) p , u k ( x ) = λ k w k n ε k w (cid:16) λ ( x − x k ) ε k (cid:17) + ε pk h ψ + C ∗ ψ i(cid:16) λ ( x − x k ) ε k (cid:17) + ε pk h ψ + ( C ∗ ) ψ + C ∗ ψ i(cid:16) λ ( x − x k ) ε k (cid:17)o + o ( ε pk ) as k → ∞ (3.43) uniformly in R , where the unique maximum point x k of u k satisfies (cid:12)(cid:12)(cid:12) λx k ε k − y (cid:12)(cid:12)(cid:12) = ε pk O ( | y | ) as k → ∞ (3.44) for some y ∈ R , and C ∗ = 0 is given by (3.38). Here ψ ( x ) , · · · , ψ ( x ) ∈ C ( R ) ∩ L ∞ ( R ) are given in Lemma 3.1 with g (0) = 1 and Lemma 3.3. roof. The refined spike profile (3.43) follows immediately from (3.2) and (3.39). Also,Lemma 3.2 and (3.37) yield that the estimate (3.44) holds.
Proof of Theorem 1.2.
Since the local uniqueness of Theorem 1.1 implies that Theo-rem 3.6 holds for the whole sequence { a k } , Theorem 1.2 is proved. V ( x ) = g ( x ) h ( x ) The main purpose of this section is to derive Theorem 4.4 which extends the refinedspike behavior of Theorem 1.2 to more general potentials V ( x ) = g ( x ) h ( x ) ∈ C ( R ),where V ( x ) satisfies lim | x |→∞ V ( x ) = ∞ and( V ) . h ( − x ) = h ( x ) satisfies (1.14) and is homogeneous of degree p ≥ g ( x ) ∈ C m ( R )for some 2 ≤ m ∈ N ∪ { + ∞} satisfies 0 < C ≤ g ( x ) ≤ C in R and G ( x ) := g ( x ) − g (0), D α G (0) = 0 for all | α | ≤ m − , and D α G (0) = 0 for some | α | = m. Here it takes m = + ∞ if g ( x ) ≡ Remark 4.1.
The property h ( − x ) = h ( x ) in the above assumption ( V ) implies that y = 0 must occur in (1.14). For the above type of potentials V ( x ), suppose { u k } is a positive minimizer sequenceof e ( a k ) with a k ր a ∗ as k → ∞ , and let w k be defined by (3.2), where x k is the uniquemaximum point of u k . Then Lemma 3.1 still holds in this case, where α k > β k > Lemma 4.1.
Suppose V ( x ) = g ( x ) h ( x ) ∈ C ( R ) satisfies lim | x |→∞ V ( x ) = ∞ and theassumption ( V ) for p ≥ and ≤ m ∈ N ∪ { + ∞} . Then the unique maximum point x k of u k satisfies the following estimates:1. If m is even, then we have λα k | x k | ε k = o (cid:0) [ α k + β k ] + α k ε mk (cid:1) as k → ∞ . (4.1)
2. If m is odd, then we have λα k | x k | ε k = O ( α k ε mk | x | ) + o (cid:0) [ α k + β k ] + α k ε mk (cid:1) as k → ∞ , (4.2) where x ∈ R satisfies g (0) Z R ∂w∂x (cid:2) x · ∇ h ( x ) (cid:3) w + 1 λ m X | α | = m Z R ∂w∂x h x α α ! D α g (0) i h ( x ) w = 0 . (4.3) Proof.
Recall that ψ ( x ) and ψ ( x ) are given in Lemma 3.1. Since h ( − x ) = h ( x ), wehave ψ i ( − x ) = ψ i ( x ) for i = 1 , Z R ∂w∂x ψ = Z R ∂w∂x wψ = Z R ∂w∂x wψ ψ = 0 . (4.4)27ince (1.14) holds with y = 0 as shown in Remark 4.1, the same calculations of (3.17)–(3.18) then yield that o ( α k + α k β k ) = Z R ∂w∂x L k w k = o ( α k β k + β k ) − α k Z R ∂w∂x h a ∗ ¯ u k + ¯ u k λ p g (cid:0) ε k xλ + x k (cid:1) h (cid:0) x + λx k ε k (cid:1)i = o ( α k β k + β k ) − α k a ∗ Z R ∂w∂x (cid:0) ¯ u k − w (cid:1) − α k λ p Z R ∂w∂x h g (cid:0) ε k xλ + x k (cid:1) h (cid:0) x + λx k ε k (cid:1) ¯ u k − g (0) h ( x ) w i = o ( α k β k + β k ) − α k λ p Z R ∂w∂x h g (cid:0) ε k xλ + x k (cid:1) h (cid:0) x + λx k ε k (cid:1) ¯ u k − g (0) h ( x ) w i = o ( α k β k + β k ) − I , (4.5)where the first equality follows from (3.21) and (4.4). Similar to (3.19), we deduce from(1.14) with y = 0 that λ p α k I = Z R ∂w∂x n g (0) h (cid:0) x + λx k ε k (cid:1)(cid:2) ¯ u k − w (cid:3) + g (0) (cid:2) h (cid:0) x + λx k ε k (cid:1) − h ( x ) (cid:3) w o + Z R ∂w∂x h g (cid:0) ε k xλ + x k (cid:1) − g (0) i h (cid:0) x + λx k ε k (cid:1) ¯ u k = o ( α k + (cid:12)(cid:12) λx k ε k (cid:12)(cid:12) + β k ) + g (0) Z R ∂w∂x (cid:16) λx k ε k · ∇ h ( x ) (cid:17) w + (cid:16) ε k λ (cid:17) m X | α | = m Z R ∂w∂x h α ! (cid:16) x + λx k ε k (cid:17) α D α g (0) i h (cid:0) x + λx k ε k (cid:1) ¯ u k + o ( ε mk ) , which then implies that I = α k λ p (cid:16) ε k λ (cid:17) m X | α | = m Z R ∂w∂x h α ! (cid:16) x + λx k ε k (cid:17) α D α g (0) i h (cid:0) x + λx k ε k (cid:1) ¯ u k + α k λ p g (0) Z R ∂w∂x (cid:16) λx k ε k · ∇ h ( x ) (cid:17) w + o ( α k + α k β k + (cid:12)(cid:12) λx k ε k (cid:12)(cid:12) + α k ε mk ) . (4.6)Combining (4.5) and (4.6), we then conclude from the estimate (3.11) that α k λ p g (0) Z R ∂w∂x (cid:16) λx k ε k · ∇ h ( x ) (cid:17) w = − α k λ p (cid:16) ε k λ (cid:17) m X | α | = m Z R ∂w∂x h x α α ! D α g (0) i h ( x ) w + o (cid:16) [ α k + β k ] + α k ε mk (cid:17) . (4.7)If m is even, one can note that X | α | = m Z R ∂w∂x h x α α ! D α g (0) i h ( x ) w = 0 , and it then follows from (4.7) and (1.14) with y = 0 that (4.1) holds. If m is odd, wethen derive from (4.7) that both (4.2) and (4.3) hold.28 emma 4.2. Suppose V ( x ) = g ( x ) h ( x ) ∈ C ( R ) satisfies lim | x |→∞ V ( x ) = ∞ and theassumption ( V ) for p ≥ and ≤ m ∈ N ∪ { + ∞} . Let ψ ( x ) and ψ ( x ) be given inLemma 3.1 with y = 0 . Then w k satisfies w k : = α k ψ + β k ψ + α k ψ + β k ψ + α k ε mk φ + α k β k ψ + o (cid:0) [ α k + β k ] + α k ε mk (cid:1) as k → ∞ , (4.8) where ψ i ( x ) ∈ C ( R ) ∩ L ∞ ( R ) , i = 3 , , , solves uniquely ∇ ψ i (0) = 0 and L ψ i ( x ) = g i ( x ) in R , i = 3 , , , (4.9) and g i ( x ) satisfies g i ( x ) = wψ − (cid:16) w a ∗ + g (0) h ( x ) λ p (cid:17) ψ , if i = 3;3 wψ + ψ , if i = 4;6 wψ ψ + ψ − (cid:16) w a ∗ + g (0) h ( x ) λ p (cid:17) ψ , if i = 5 . (4.10) Here φ ∈ C ( R ) ∩ L ∞ ( R ) solves uniquely L φ ( x ) = − λ p n(cid:2) x · ∇ h ( x ) (cid:3) g (0) w + 1 λ m X | α | = m h x α α ! D α g (0) i h ( x ) w o in R , and ∇ φ (0) = 0 , (4.11) where x = 0 holds for the case where m is even, and x ∈ R satisfies (4.3) for the casewhere m is odd. Proof.
Following Lemma 3.1(3), we set v k = w k − α k ψ − β k ψ . Similar to (3.32), we then have L k v k = w k ( α k ψ + β k ψ )(3 w + w k ) + β k w k − α k a ∗ (¯ u k − w ) − α k λ p h g (cid:0) ε k xλ + x k (cid:1) h (cid:0) x + λx k ε k (cid:1) ¯ u k − g (0) h ( x ) w i = w k ( α k ψ + β k ψ )(3 w + w k ) + β k w k − α k a ∗ w k (3 w + 3 ww k + w k ) − I , (4.12)29here I satisfies I = α k λ p nh g (cid:0) ε k xλ + x k (cid:1) − g (0) i h (cid:0) x + λx k ε k (cid:1) ¯ u k + g (0) h h (cid:0) x + λx k ε k (cid:1) − h ( x ) i ¯ u k + g (0) h ( x ) (cid:0) ¯ u k − w (cid:1)o = α k λ p n(cid:16) ε k λ (cid:17) m X | α | = m h α ! (cid:16) x + λx k ε k (cid:17) α D α g (0) i h (cid:0) x + λx k ε k (cid:1) ¯ u k + g (0) (cid:16) λx k ε k · ∇ h ( x ) (cid:17) ¯ u k + g (0) h ( x ) w k o + o (cid:16) α k x k ε k + α k ε mk (cid:17) = α k w k g (0) h ( x ) λ p + α k λ p (cid:16) λx k ε k · ∇ h ( x ) (cid:17) g (0)¯ u k + α k ε mk λ p + m X | α | = m h α ! (cid:16) x + λx k ε k (cid:17) α D α g (0) i h (cid:0) x + λx k ε k (cid:1) ¯ u k + o (cid:0) α k ε mk (cid:1) , (4.13)where Lemma 4.1 is used in the last equality. Applying Lemma 4.1 again, we then obtainfrom (4.12) and (4.13) that L k v k = w k ( α k ψ + β k ψ )(3 w + w k ) + β k w k − α k a ∗ w k (3 w + w k ) − α k w k h w a ∗ + g (0) h ( x ) λ p i − α k λ p (cid:16) λx k ε k · ∇ h ( x ) (cid:17) g (0)¯ u k − α k ε mk λ p + m X | α | = m h α ! (cid:16) x + λx k ε k (cid:17) α D α g (0) i h (cid:0) x + λx k ε k (cid:1) ¯ u k + o (cid:0) α k ε mk (cid:1) = α k h wψ − (cid:16) w a ∗ + g (0) h ( x ) λ p (cid:17) ψ i + α k β k h wψ ψ + ψ − (cid:16) w a ∗ + g (0) h ( x ) λ p (cid:17) ψ i − α k ε mk λ p n(cid:2) x · ∇ h ( x ) (cid:3) g (0) w + 1 λ m X | α | = m h x α α ! D α g (0) i h ( x ) w o + β k (3 wψ + ψ ) + o (cid:0) [ α k + β k ] + α k ε mk (cid:1) in R , (4.14)where x = 0 holds for the case where m is even, and x ∈ R satisfies (4.3) for the casewhere m is odd. Following (4.14), the same argument of proving Lemma 3.1 then gives(4.8), and the proof is therefore complete. Proposition 4.3.
Suppose V ( x ) = g ( x ) h ( x ) ∈ C ( R ) satisfies lim | x |→∞ V ( x ) = ∞ and the assumption ( V ) for p ≥ and ≤ m ∈ N ∪ { + ∞} . Let ψ ( x ) , · · · , ψ ( x ) ∈ C ( R ) ∩ L ∞ ( R ) be given in Lemma 3.1 with y = 0 and Lemma 4.2, and φ is given by(4.11).1. If m > p , then β k = C ∗ α k , (4.15) and w k satisfies w k := (cid:2) ψ + C ∗ ψ (cid:3) α k + (cid:2) ψ + ( C ∗ ) ψ + C ∗ ψ (cid:3) α k + o ( α k ) as k → ∞ , (4.16) where the constant C ∗ satisfies C ∗ := 22 + p (cid:16) Z R wψ + Z R ψ (cid:17) = 0 . (4.17)30 . If ≤ m ≤ p and m is odd, then β k = C ∗ α k and w k satisfies w k : = (cid:2) ψ + C ∗ ψ (cid:3) α k + φ α k ε mk + (cid:2) ψ + ( C ∗ ) ψ + C ∗ ψ (cid:3) α k + o ( α k ε mk ) as k → ∞ , (4.18) where the constant C ∗ = 0 is given by (4.17).3. If ≤ m < p and m is even, consider S = X | α | = m Z R h x α α ! D α g (0) i h ( x ) w . (4.19) Then for the case where S = 0 , we have β k = C ∗ α k and w k satisfies (4.18), wherethe constant C ∗ = 0 is given by (4.17). However, for the case where S 6 = 0 , wehave β k = C ∗ ε mk , (4.20) and w k satisfies w k := C ∗ ψ ε mk + ψ α k + ( C ∗ ) ψ ε mk + o ( ε min { p, m } k ) as k → ∞ , (4.21) where the constant C ∗ satisfies C ∗ = − m + p (2 + p ) λ p + m X | α | = m Z R h x α α ! D α g (0) i h ( x ) w = 0 . (4.22)
4. If m = 2 + p is even, then β k = C ∗ α k , (4.23) and w k satisfies w k : = (cid:2) ψ + C ∗ ψ (cid:3) α k + (cid:2) ψ + ( C ∗ ) ψ + C ∗ ψ + φ (cid:3) α k + o ( α k ) as k → ∞ , (4.24) where the constant C ∗ satisfies C ∗ = 22 + p h Z R wψ + Z R ψ + 2 Z R wφ i = 0 . (4.25) Proof.
The same argument of proving Lemma 3.4 with y = 0 yields that Z R wψ = 0 , Z R wψ = 0 and I = Z R (cid:0) wψ + ψ (cid:1) = 0 , (4.26)and II = 2 Z R wψ + 2 Z R ψ ψ = − p < . (4.27)31t thus follows from (3.40) and Lemma 4.2 that0 = 2 Z R ww k + Z R w k = 2 Z R w (cid:0) α k ψ + β k ψ + α k ψ + β k ψ + α k ε mk φ + α k β k ψ (cid:1) + Z R (cid:0) α k ψ + β k ψ + α k ψ + β k ψ + α k ε mk φ + α k β k ψ (cid:1) + o (cid:0) [ α k + β k ] + α k ε mk (cid:1) = α k (cid:16) Z R wψ (cid:17) + β k (cid:16) Z R wψ (cid:17) + β k (cid:16) Z R wψ + Z R ψ (cid:17) + α k β k (cid:16) Z R wψ + 2 Z R ψ ψ (cid:17) + α k (cid:16) Z R wψ + Z R ψ (cid:17) + α k ε mk (cid:16) Z R wφ (cid:17) + o (cid:0) [ α k + β k ] + α k ε mk (cid:1) = − p α k β k + α k (cid:16) Z R wψ + Z R ψ (cid:17) + α k ε mk (cid:16) Z R wφ (cid:17) + o (cid:0) [ α k + β k ] + α k ε mk (cid:1) , (4.28)where (4.26) and (4.27) are used in the last equality. Following (4.28), we next carry outthe proof by considering separately the following four cases: Case 1. m > p . In this case, it follows from (4.28) that the constant C ∗ defined in(4.17) is nonzero and − p β k + α k (cid:16) Z R wψ + Z R ψ (cid:17) = 0 , i.e., β k = C ∗ α k . Moreover, the expansion (4.16) follows directly from (4.15) and Lemma 4.2, and Case 1is therefore proved.
Case 2. ≤ m ≤ p and m is odd. In this case, since m is odd and h ( − x ) = h ( x ), weobtain from (3.10) and (4.11) that2 Z R wφ = 2 Z R φ L ψ = 2 Z R ψ L φ = 1 λ p Z R n(cid:2) x · ∇ h ( x ) (cid:3) g (0) w + 1 λ m X | α | = m h x α α ! D α g (0) i h ( x ) w o(cid:0) w + x · ∇ w (cid:1) = 0 . We then derive from (4.28) that (4.17) still holds and thus β k = C ∗ α k . Further, theexpansion (4.18) follows directly from (4.8) and (4.15). Case 3. ≤ m < p and m is even. Since m is even, then x = 0 holds in (4.11).32urther, since x α h ( x ) is homogeneous of degree m + p , we then obtain from (4.11) that2 Z R wφ = 2 Z R φ L ψ = 2 Z R ψ L φ = 1 λ p + m X | α | = m Z R h x α α ! D α g (0) i h ( x ) w (cid:0) w + x · ∇ w (cid:1) = 1 λ p + m X | α | = m Z R h x α α ! D α g (0) i h ( x ) w + 12 λ p + m X | α | = m Z R h x α α ! D α g (0) i h ( x ) (cid:0) x · ∇ w (cid:1) = 1 λ p + m X | α | = m Z R h x α α ! D α g (0) i h ( x ) w − λ p + m X | α | = m Z R w n h x α α ! D α g (0) h ( x ) i + x · ∇ h x α α ! D α g (0) h ( x ) io = − m + p λ p + m X | α | = m Z R h x α α ! D α g (0) i h ( x ) w := − m + p λ p + m S , (4.29)where S is as in (4.19). Therefore, if S = 0, then we are in the same situation as thatof above Case 2, which gives that β k = C ∗ α k and w k satisfies (4.18), where the constant C ∗ = 0 is given by (4.17).We next consider the case where S 6 = 0. By applying (4.29), in this case we derivefrom (4.28) that − p α k β k + α k ε mk (cid:16) Z R wφ (cid:17) = 0 , which implies that β k = C ∗ ε mk , where the constant C ∗ = 0 satisfies (4.22) in view of(4.29). Further, the expansion (4.21) follows directly from (4.20) and Lemma 4.2. Case 4. m = 2 + p is even. In this case, we derive from (4.28) that − p α k β k + α k (cid:16) Z R wψ + Z R ψ + 2 Z R wφ (cid:17) = 0 , which gives that β k = C ∗ α k , where the constant C ∗ = 0 satisfies (4.25). Further, theexpansion (4.24) follows directly from (4.23) and Lemma 4.2.Applying directly Lemmas 4.1 and 4.2 as well as Proposition 4.3, we now concludethe following main results of this section. Recall that λ > y = 0, ψ ( x ) , · · · , ψ ( x ) ∈ C ( R ) ∩ L ∞ ( R ) are given in Lemma 3.1 with y = 0 andLemma 4.2, and φ is given by (4.11). Theorem 4.4.
Suppose V ( x ) = g ( x ) h ( x ) ∈ C ( R ) satisfies lim | x |→∞ V ( x ) = ∞ andthe assumption ( V ) for p ≥ and ≤ m ∈ N ∪ { + ∞} . Let u a be a positive minimizerof (1.1) for a < a ∗ . Then for any sequence { a k } with a k ր a ∗ as k → ∞ , there existsa subsequence, still denoted by { a k } , of { a k } such that u k = u a k has a unique maximumpoint x k ∈ R and satisfies for ε k := ( a ∗ − a k ) p , . If m > p , then we have u k ( x ) = λ k w k n ε k w (cid:16) λ ( x − x k ) ε k (cid:17) + ε pk h ψ + C ∗ ψ i(cid:16) λ ( x − x k ) ε k (cid:17) + ε pk h ψ + ( C ∗ ) ψ + C ∗ ψ i(cid:16) λ ( x − x k ) ε k (cid:17)o + o ( ε pk ) as k → ∞ (4.30) uniformly in R , where x k satisfies | x k | ε k = O ( ε mk | y | ) + o ( ε pk ) as k → ∞ (4.31) for some y ∈ R , and the constant C ∗ = 0 is given by (4.17). Further, if m iseven, then x k satisfies | x k | ε pk = o (1) as k → ∞ . (4.32)
2. If ≤ m ≤ p and m is odd, then we have u k ( x ) = λ k w k n ε k w (cid:16) λ ( x − x k ) ε k (cid:17) + ε pk h ψ + C ∗ ψ i(cid:16) λ ( x − x k ) ε k (cid:17) + ε pk (cid:2) ψ + ( C ∗ ) ψ + C ∗ ψ (cid:3)(cid:16) λ ( x − x k ) ε k (cid:17) + ε m + pk φ (cid:16) λ ( x − x k ) ε k (cid:17)o + o ( ε m + pk ) as k → ∞ (4.33) uniformly in R , where x k satisfies | x k | ε m +1 k = O ( | y | ) as k → ∞ . (4.34) for some y ∈ R , and the constant C ∗ = 0 is given by (4.17).3. If m = 2 + p is even, then we have u k ( x ) = λ k w k n ε k w (cid:16) λ ( x − x k ) ε k (cid:17) + ε pk h ψ + C ∗ ψ i(cid:16) λ ( x − x k ) ε k (cid:17) + ε pk h ψ + ( C ∗ ) ψ + C ∗ ψ + φ i(cid:16) λ ( x − x k ) ε k (cid:17)o + o ( ε pk ) as k → ∞ (4.35) uniformly in R , where x k satisfies (4.32) and the constant C ∗ = 0 is defined by(4.25).4. If ≤ m < p and m is even, let the constant S be defined in (4.19). Then forthe case where S = 0 , u k satisfies (4.33) and x k satisfies | x k | ε m +1 k = o (1) as k → ∞ . (4.36) However, for the case where
S 6 = 0 , u k satisfies u k ( x ) = λ k w k n ε k w (cid:16) λ ( x − x k ) ε k (cid:17) + ε m − k C ∗ ψ (cid:16) λ ( x − x k ) ε k (cid:17) + ε m − k ( C ∗ ) ψ (cid:16) λ ( x − x k ) ε k (cid:17) + ε pk ψ (cid:16) λ ( x − x k ) ε k (cid:17)o + o ( ε min { p, m }− k ) as k → ∞ (4.37)34 niformly in R , where x k satisfies (4.36), and the constant C ∗ = 0 is defined by(4.22). Proof. (1). If m > p , then (4.30) follows directly from Proposition 4.3(1), and(4.31) follows from Lemma 4.1. Specially, if m is even, then Lemma 4.1 gives y = 0,and therefore (4.31) implies (4.32).(2). If 1 ≤ m ≤ p and m is odd, then Proposition 4.3(2) gives (4.33). Moreover,it yields from (4.2) that x k satisfies (cid:12)(cid:12)(cid:12) x k ε k (cid:12)(cid:12)(cid:12) = O ( ε mk | y | ) + o ( ε mk ) as k → ∞ , which thenimplies (4.34) for some y ∈ R .(3). If m = 2 + p is even, then Proposition 4.3(4) gives (4.35), and we reduce from(4.1) that x k satisfies (4.32).(4). If 1 ≤ m < p and m is even, it then follows from (4.1) that x k alwayssatisfies (4.36). Moreover, Proposition 4.3(3) gives that if S = 0, then u k satisfies (4.33);if S 6 = 0, then u k satisfies (4.37). A Appendix: The Proof of Lemma 3.4
In this appendix, we shall follow Lemmas 3.1 and 3.3 to address the proof of Lemma 3.4,i.e., (3.34)–(3.36).
The proof of (3.34).
Under the assumptions of Lemma 3.4, we first note that the equation(3.7) can be simplified as ∇ ψ (0) = 0 , L ψ = − w R R w − h ( x + y ) wp R R h ( x + y ) w in R , (A.1)due to the fact that a ∗ = k w k = 12 Z R w . (A.2)By (1.14), (3.10) and (A.1), we then have2 Z R wψ = 2 Z R L ψ ψ = 2 Z R ψ L ψ = Z R h w R R w + 2 h ( x + y ) wp R R h ( x + y ) w i(cid:0) w + x · ∇ w (cid:1) = 2 + 2 p + 2 R R w Z R w (cid:0) x · ∇ w (cid:1) + 2 p R R h ( x + y ) w Z R h ( x + y ) w (cid:0) x · ∇ w (cid:1) = 2 + 2 p + 12 R R w Z R (cid:0) x · ∇ w (cid:1) + 1 p R R h ( x + y ) w Z R h ( x + y ) (cid:0) x · ∇ w (cid:1) = 2 + 2 p − − p R R h ( x + y ) w Z R w h h ( x + y ) + (cid:0) x · ∇ h ( x + y ) (cid:1)i = 2 + 2 p − − ( p + 2) p = 0 , since ( x + y ) · ∇ h ( x + y ) = ph ( x + y ) and R R w (cid:2) y · ∇ h ( x + y ) (cid:3) = 0. Also, we deducefrom (3.10) that2 Z R wψ = − Z R w ( w + x · ∇ w ) = − Z R w − Z R ( x · ∇ w ) = 0 , The proof of (3.35).
By Lemmas 3.1 and 3.3, we obtain that I = Z R (cid:0) wψ + ψ (cid:1) = Z R ψ + 2 hL ψ , ψ i = Z R ψ + 2 h ψ , L ψ i = Z R ψ + 2 h ψ , (3 wψ + ψ ) i = 3 Z R ψ + 6 Z R wψ , which implies that4 I ÷ π = 4 h Z R ψ + 2 Z R wψ i ÷ π = Z ∞ r ( w + rw ′ ) − Z ∞ rw ( w + rw ′ ) := A − B. (A.3)Here we have A = Z ∞ r ( w + rw ′ ) = Z ∞ r ( w ′ ) + Z ∞ rw + Z ∞ r dw = Z ∞ r ( w ′ ) − Z ∞ rw , where (A.2) is used, and B = Z ∞ rw ( w + rw ′ ) = h Z ∞ rw + 3 Z ∞ r w w ′ i + 3 Z ∞ r w w ′ w ′ + Z ∞ r w ( w ′ ) = − Z ∞ rw + 3 Z ∞ r w w ′ w ′ + Z ∞ r w ( w ′ ) . Therefore, we get from (A.3) that4 I ÷ π = Z ∞ r ( w ′ ) − Z ∞ r w ( w ′ ) − Z ∞ r w w ′ w ′ := C + D + E. (A.4)To further simplify I , recall that rw ′′ = − w ′ + rw − rw , (A.5)by which we then have C = Z ∞ r w ′ dw = − Z ∞ w ( r w ′ ) ′ = − Z ∞ w (cid:2) r w ′ + r ( − w ′ + rw − rw ) (cid:3) = − Z ∞ w (cid:2) r w ′ + r w − r w (cid:3) = 2 Z ∞ rw − Z ∞ r w + Z ∞ r w . D = − Z ∞ r ( w ′ ) dw = 12 Z ∞ w (cid:2) r ( w ′ ) + 2 r w ′ ( − w ′ + rw − rw ) (cid:3) = Z ∞ r w ( w ′ ) + 14 Z ∞ r dw − Z ∞ r dw = Z ∞ r w ( w ′ ) − Z ∞ r w + 23 Z ∞ r w . Note from (A.5) that − Z ∞ r w ( w ′ ) = − Z ∞ r w ′ dw = 23 Z ∞ w (cid:2) r w ′ + r ( − w ′ + rw − rw ) (cid:3) = 43 Z ∞ w r w ′ + 23 Z ∞ r w − Z ∞ r w = − Z ∞ rw + 23 Z ∞ r w − Z ∞ r w . We thus derive that D + E = − Z ∞ r w ( w ′ ) − Z ∞ r w + 23 Z ∞ r w = − Z ∞ rw − Z ∞ r w , by which we conclude from (A.2) and (A.4) that4 I ÷ π = C + D + E = 13 h Z ∞ rw − Z ∞ r w + 2 Z ∞ r w i . (A.6)In the following, we note that w satisfies( rw ′ ) ′ = rw − rw , r > . (A.7)Multiplying (A.7) by r w ′ and integrating on [0 , ∞ ), we get that Z ∞ r w ′ ( rw ′ ) ′ = Z ∞ r w ′ [ rw − rw ] = 12 Z ∞ r dw − Z ∞ r dw = − Z ∞ r w + Z ∞ r w . Note also that Z ∞ r w ′ ( rw ′ ) ′ = Z ∞ r ( w ′ ) + 12 Z ∞ r d ( w ′ ) = − Z ∞ r ( w ′ ) . By combining above two identities, it yields that Z ∞ r ( w ′ ) = 2 Z ∞ r w − Z ∞ r w . (A.8)37n the other hand, multiplying (A.7) by r w and integrating on [0 , ∞ ), we obtain that Z ∞ r w − Z ∞ r w = Z ∞ r w ( rw ′ ) ′ = Z ∞ r ww ′ + Z ∞ r wdw ′ = Z ∞ r ww ′ − Z ∞ w ′ (3 r w + r w ′ )= − Z ∞ r ww ′ − Z ∞ r ( w ′ ) = 2 Z ∞ rw − Z ∞ r ( w ′ ) , which then implies that Z ∞ r ( w ′ ) = 2 Z ∞ rw − Z ∞ r w + Z ∞ r w . (A.9)We thus conclude from (A.8) and (A.9) that2 Z ∞ rw − Z ∞ r w + 2 Z ∞ r w = 0 , which therefore implies that I = 0 in view of (A.6), i.e., (3.35) holds. The proof of (3.36).
Following Lemmas 3.1 and 3.3 again, we get that II = 2 Z R ψ L ψ + 2 Z R ψ ψ = 2 Z R ψ [ L ψ + ψ ]= − Z R ( w + x · ∇ w )(6 wψ ψ + 2 ψ ) − Z R h w a ∗ + h ( x + y ) λ p i(cid:0) w + x · ∇ w (cid:1) + 12 λ p Z R ( w + x · ∇ w ) (cid:2) y · ∇ h ( x + y ) (cid:3) w := A + B. (A.10)Since ( x + y ) · ∇ h ( x + y ) = ph ( x + y ) holds in R , we derive from (1.14) and (A.1)that B = − Z R h w a ∗ + h ( x + y ) λ p i(cid:2) w + 2 w ( x · ∇ w ) + ( x · ∇ w ) (cid:3) + 12 λ p Z R ( w + x · ∇ w ) (cid:2) y · ∇ h ( x + y ) (cid:3) w = − Z R h w R R w + h ( x + y ) p R R h ( x + y ) w i(cid:2) w + 2 w ( x · ∇ w ) (cid:3) − Z R h w a ∗ + h ( x + y ) λ p i(cid:0) x · ∇ w (cid:1) + 12 λ p Z R w (cid:2) y · ∇ h ( x + y ) (cid:3) ( x · ∇ w )= − − p − R R w Z R ( x · ∇ w ) − p R R h ( x + y ) w Z R h ( x + y ) (cid:0) x · ∇ w (cid:1) − Z R h w a ∗ + h ( x + y ) λ p i(cid:0) x · ∇ w (cid:1) + 12 λ p Z R w (cid:2) y · ∇ h ( x + y ) (cid:3) ( x · ∇ w ):= − − p + 3 + 2 + pp + C = p + 1 p + C , C satisfies C = − Z R h w a ∗ + h ( x + y ) λ p i(cid:0) x · ∇ w (cid:1) + 12 λ p Z R w (cid:2) y · ∇ h ( x + y ) (cid:3) ( x · ∇ w )= − a ∗ Z R ( x · ∇ w )( x · ∇ w ) − λ p Z R h ( x + y ) (cid:0) x · ∇ w (cid:1) ( x · ∇ w )+ 12 λ p Z R w (cid:2) y · ∇ h ( x + y ) (cid:3) ( x · ∇ w )= 12 a ∗ Z R w h x · ∇ w ) + x · ∇ ( x · ∇ w ) i + 12 λ p Z R w n h ( x + y )( x · ∇ w ) + (cid:2) x · ∇ h ( x + y ) (cid:3) ( x · ∇ w )+ h ( x + y ) h x · ∇ ( x · ∇ w ) io + 12 λ p Z R w (cid:2) y · ∇ h ( x + y ) (cid:3) ( x · ∇ w )= 12 Z R h w a ∗ + wh ( x + y ) λ p i(cid:2) x · ∇ (cid:0) x · ∇ w (cid:1)(cid:3) + 1 a ∗ Z R w ( x · ∇ w ) + 2 + p λ p Z R wh ( x + y )( x · ∇ w )= 12 Z R h w a ∗ + wh ( x + y ) λ p i(cid:2) x · ∇ (cid:0) x · ∇ w (cid:1)(cid:3) + 12 R R w Z R ( x · ∇ w ) + 2 + p p R R h ( x + y ) w Z R h ( x + y ) (cid:0) x · ∇ w (cid:1) = 12 Z R h w a ∗ + wh ( x + y ) λ p i(cid:2) x · ∇ (cid:0) x · ∇ w (cid:1)(cid:3) − − (2 + p ) p , in view of (A.1). We thus have B = 12 Z R h w a ∗ + wh ( x + y ) λ p i(cid:2) x · ∇ (cid:0) x · ∇ w (cid:1)(cid:3) − p + 4 p + 22 p . (A.11)We next calculate the term A as follows. Observe that12 Z R ψ x · ∇ (cid:0) x · ∇ w (cid:1) = − Z R (cid:0) x · ∇ w (cid:1)(cid:2) ψ + (cid:0) x · ∇ ψ (cid:1)(cid:3) = − Z R ψ (cid:0) x · ∇ w (cid:1) − Z R (cid:0) x · ∇ ψ (cid:1)(cid:0) x · ∇ w (cid:1) = − Z R ψ (cid:0) x · ∇ w (cid:1) + 12 Z R w (cid:2) (cid:0) x · ∇ ψ (cid:1) + x · ∇ (cid:0) x · ∇ ψ (cid:1)(cid:3) = − Z R ψ (cid:0) x · ∇ w (cid:1) + Z R w (cid:0) x · ∇ ψ (cid:1) + 12 Z R wx · ∇ (cid:0) x · ∇ ψ (cid:1) , which implies that − Z R ψ (cid:0) x · ∇ w (cid:1) + Z R w (cid:0) x · ∇ ψ (cid:1) = 12 Z R ψ x · ∇ (cid:0) x · ∇ w (cid:1) − Z R wx · ∇ (cid:0) x · ∇ ψ (cid:1) . (A.12)39sing (A.12), we then derive that A = − Z R ( w + x · ∇ w )(6 wψ ψ + 2 ψ )= − Z R wψ − Z R ψ ( x · ∇ w ) + 3 Z R wψ ( w + x · ∇ w ) = − Z R wψ − Z R ψ ( x · ∇ w )+ Z R w (cid:2) ψ + x · ∇ ψ (cid:3) + 3 Z R wψ ( w + x · ∇ w ) = − Z R ψ ( x · ∇ w ) + Z R w ( x · ∇ ψ ) + D = 12 Z R ψ x · ∇ (cid:0) x · ∇ w (cid:1) − Z R wx · ∇ (cid:0) x · ∇ ψ (cid:1) + D, (A.13)where the term D satisfies D = 3 Z R wψ h w + 2 w ( x · ∇ w ) + ( x · ∇ w ) i = 3 Z R w ψ + 6 Z R w ψ ( x · ∇ w ) + 32 Z R ψ ( x · ∇ w )( x · ∇ w )= 3 Z R w ψ + 6 Z R w ψ ( x · ∇ w ) − Z R w n ψ ( x · ∇ w ) + ( x · ∇ w )( x · ∇ ψ ) + ψ h x · ∇ ( x · ∇ w ) io . Since − Z R w ( x · ∇ w )( x · ∇ ψ )= − Z R ( x · ∇ ψ )( x · ∇ w )= 12 Z R w h x · ∇ ( x · ∇ ψ ) + 2( x · ∇ ψ ) i = 12 Z R w x · ∇ ( x · ∇ ψ ) + Z R w ( x · ∇ ψ )= 12 Z R w x · ∇ ( x · ∇ ψ ) − Z R ψ h w + 3 w ( x · ∇ w ) i = 12 Z R w x · ∇ ( x · ∇ ψ ) − Z R w ψ − Z R w ψ ( x · ∇ w ) , the term D can be further simplified as D = Z R w ψ − Z R w ψ (cid:2) x · ∇ ( x · ∇ w ) (cid:3) + 12 Z R w x · ∇ ( x · ∇ ψ ) . (A.14)Applying (A.14), we then obtain from (A.13) that A = Z R w ψ + 12 Z R (1 − w ) ψ (cid:2) x · ∇ ( x · ∇ w ) (cid:3) − Z R ∆ w (cid:2) x · ∇ ( x · ∇ ψ ) (cid:3) , (A.15)40ince w solves the equation w − w = − ∆ w in R .Combining (A.11) and (A.15) now yields that II = A + B = Z R w ψ − p + 4 p + 22 p + 12 Z R (cid:2) x · ∇ ( x · ∇ w ) (cid:3) ∆ ψ − Z R (cid:2) x · ∇ ( x · ∇ ψ ) (cid:3) ∆ w. (A.16)We claim that Z R w ψ = p + 1 p . (A.17)Actually, multiplying (A.1) by w and integrating on R gives that Z R ∇ ψ ∇ w − Z R w ψ = − Z R h w R R w + 2 h ( x + y ) w p R R h ( x + y ) w i = − p + 1) p , due to the fact that R R wψ = 0 by (3.34). On the other hand, multiplying (1.4) by ψ and integrating on R gives that Z R ∇ ψ ∇ w = − Z R wψ + Z R w ψ = Z R w ψ . The claim (A.17) then follows directly from above two identities. We next claim that Z R (cid:2) x · ∇ ( x · ∇ w ) (cid:3) ∆ ψ = Z R (cid:2) x · ∇ ( x · ∇ ψ ) (cid:3) ∆ w. (A.18)To prove (A.18), rewrite ψ as ψ ( x ) = ψ ( r, θ ), where ( r, θ ) is the polar coordinate in R , such that∆ ψ = (cid:0) ψ (cid:1) rr + 1 r (cid:0) ψ (cid:1) r + 1 r (cid:0) ψ (cid:1) θθ , ∇ ψ = xr (cid:0) ψ (cid:1) r + x ⊥ r (cid:0) ψ (cid:1) θ , (A.19)where x ⊥ = ( − x , x ) for x = ( x , x ) ∈ R . We then derive from (3.7) that Z R (cid:2) x · ∇ ( x · ∇ w ) (cid:3) ∆ ψ = Z π Z ∞ r ( rw ′ ) ′ n(cid:2) r (cid:0) ψ (cid:1) r (cid:3) r + (cid:0) ψ (cid:1) θθ r o drdθ = Z π Z ∞ r ( rw ′ ) ′ (cid:2) r (cid:0) ψ (cid:1) r (cid:3) r drdθ + Z π Z ∞ ( rw ′ ) ′ (cid:0) ψ (cid:1) θθ drdθ = Z π Z ∞ r ( rw ′ ) ′ (cid:2) r (cid:0) ψ (cid:1) r (cid:3) r drdθ, and Z R (cid:2) x · ∇ ( x · ∇ ψ ) (cid:3) ∆ w = Z π Z ∞ r (cid:2) r (cid:0) ψ (cid:1) r (cid:3) r ( rw ′ ) ′ drdθ, which thus imply that (A.18) holds. Applying (A.17) and (A.18), we therefore concludefrom (A.16) that II = p + 1 p − p + 4 p + 22 p = − p , which gives (3.36), and the proof is complete.41 cknowledgements: The authors are very grateful to the referees for many useful sug-gestions which lead to some improvements of the present paper. The first author thanksProf. Robert Seiringer for fruitful discussions on the present work. Part of the presentwork was finished when the first author was visiting Taida Institute of Mathematical Sci-ences (TIMS) in October 2013 and Pacific Institute for Mathematical Sciences (PIMS)from March to April in 2016. He would like to thank both institutes for their warmhospitality.
References [1] M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman and E. A. Cornell,
Observation of Bose-Einstein condensation in a dilute atomic vapor , Science (1995), 198–201.[2] W. Z. Bao and Y. Y. Cai,
Mathematical theory and numerical methods for Bose-Einstein condensation , Kinetic and Related Models (2013), 1–135.[3] D. M. Cao, S. L. Li and P. Luo, Uniqueness of positive bound states with multi-bump for nonlinear Schr¨odinger equations , Calc. Var. Partial Differential Equations (2015), no. 4, 4037–4063.[4] T. Cazenave, Semilinear Schr¨odinger Equations, Courant Lecture Notes in Math. , Courant Institute of Mathematical Science/AMS, New York, (2003).[5] F. Dalfovo, S. Giorgini, L. P. Pitaevskii and S. Stringari, Theory of Bose-Einsteincondensation in trapped gases , Rev. Modern Phys. (1999), 463–512.[6] Y. B. Deng, C. S. Lin and S. Yan, On the prescribed scalar curvature problem in R N ,local uniqueness and periodicity , J. Math. Pures Appl. (2015), no. 6, 1013–1044.[7] B. Gidas, W. M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinearelliptic equations in R n , Mathematical analysis and applications Part A, Adv. inMath. Suppl. Stud. vol. (1981), 369–402.[8] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of SecondOrder, Springer, (1997).[9] M. Grossi, On the number of single-peak solutions of the nonlinear Schr¨odingerequations , Ann. Inst H. Poincar Anal. Non Linaire (2002), 261–280.[10] Y. J. Guo and R. Seiringer, On the mass concentration for Bose-Einstein conden-sates with attractive interactions , Lett. Math. Phys. (2014), 141–156.[11] Y. J. Guo, Z. Q. Wang, X. Y. Zeng and H. S. Zhou,
Properties for ground states ofattractive Gross-Pitaevskii equations with multi-well potentials , arXiv:1502.01839,submitted, (2015).[12] Y. J. Guo, X. Y. Zeng and H. S. Zhou,
Energy estimates and symmetry breakingin attractive Bose-Einstein condensates with ring-shaped potentials , Ann. Inst. H.Poincar´e Anal. Non Lin´eaire (2016), 809-828.4213] Y. Kagan, A. E. Muryshev and G. V. Shlyapnikov, Collapse and Bose-Einsteincondensation in a trapped Bose gas with nagative scattering length , Phys. Rev. Lett. (1998), 933–937.[14] M. K. Kwong, Uniqueness of positive solutions of ∆ u − u + u p = 0 in R N , Arch.Rational Mech. Anal. (1989), 243–266.[15] E. H. Lieb and M. Loss, Analysis, Graduate Studies in Math. , Amer. Math. Soc.,Providence, RI, second edition (2001).[16] T. C. Lin and J. C. Wei, Orbital stability of bound states of semiclassical nonlinearSchr¨odinger equations with critical nonlinearity , SIAM J. Math. Anal. (2008),no. 1, 365–381.[17] T. C. Lin, J. C. Wei and W. Yao, Orbital stability of bound states of nonlinearSchr¨odinger equations with linear and nonlinear optical lattices , J. Differential Equa-tions (2010), no. 9, 2111–2146.[18] M. Maeda,
On the symmetry of the ground states of nonlinear Schr¨odinger equationwith potential , Adv. Nonlinear Stud. (2010), 895–925.[19] K. McLeod and J. Serrin, Uniqueness of positive radial solutions of ∆ u + f ( u ) = 0 in R n , Arch. Rational Mech. Anal. (1987), 115–145.[20] W.-M. Ni and I. Takagi, On the shape of least-energy solutions to a semilinearNeumann problem , Comm. Pure Appl. Math. (1991), 819–851.[21] W.-M. Ni and J. C. Wei, On the location and profile of spike-layer solutions tosingularly perturbed semilinear Dirichlet problems , Comm. Pure Appl. Math. (1995), 731–768.[22] M. Reed and B. Simon, Methods of Modern Mathematical Physics. IV. Analysis ofOperators, Academic Press, New York-London, 1978.[23] C. A. Sackett, H. T. C. Stoof and R. G. Hulet, Growth and collapse of a Bose-Einstein condensate with attractive interactions , Phys. Rev. Lett. (1998), 2031.[24] X. F. Wang, On concentration of positive bound states of nonlinear Schr¨odingerequations , Comm. Math. Phys. (1993), 229–244.[25] M. I. Weinstein,
Nonlinear Schr¨odinger equations and sharp interpolations esti-mates , Comm. Math. Phys. (1983), 567–576.[26] J. Zhang, Stability of attractive Bose-Einstein condensates , J. Stat. Phys.101