Mass Concentration and Local Uniqueness of Ground States for L 2 -subcritical Nonlinear Schrödinger Equations
aa r X i v : . [ m a t h . A P ] J un Mass Concentration and Local Uniqueness of Ground Statesfor L -subcritical Nonlinear Schr¨odinger Equations ∗ Shuai Li a b and Xincai Zhu a b † a Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences,Wuhan 430071, P. R. China b University of Chinese Academy of Sciences, Beijing 100049, P. R. China
October 15, 2018
Abstract
We consider ground states of L -subcritical nonlinear Schr¨odinger equation (1.1),which can be described equivalently by minimizers of the following constraint mini-mization problem e ( ρ ) := inf (cid:8) E ρ ( u ) : u ∈ H ( R d ) , k u k = 1 (cid:9) . The energy functional E ρ ( u ) is defined by E ρ ( u ) := 12 Z R d |∇ u | d x + 12 Z R d V ( x ) | u | d x − ρ p − p + 1 Z R d | u | p +1 d x, where d ≥ ρ > p ∈ (cid:0) , d (cid:1) and 0 ≤ V ( x ) → ∞ as | x | → ∞ . We present adetailed analysis on the concentration behavior of ground states as ρ → ∞ , whichextends the concentration results shown in [22]. Moreover, the uniqueness of non-negative ground states is also proved when ρ is large enough. Keywords: L -subcritical, ground states, minimizers, mass concentration, local unique-ness MSC(2010):
In this paper, we study the following time-independent nonlinear Schr¨odinger equation − ∆ u + V ( x ) u = µu + ρ p − u p , u ∈ H ( R d ) , (1.1)where d ≥ µ ∈ R , p ∈ (cid:0) , d (cid:1) , and ρ > ∗ This work is partially supported by NSFC under Grant No. 11671394. † Corresponding author. E-mail: lishuai [email protected] (S. Li); [email protected] (X. Zhu). H ( R d ) is defined as H ( R d ) := n u ( x ) ∈ H ( R d ) (cid:12)(cid:12)(cid:12) Z R d V ( x ) | u ( x ) | d x < ∞ o , (1.2)and the associated norm is given by k u k H = (cid:8) R R d (cid:0) |∇ u ( x ) | + [1 + V ( x )] | u ( x ) | (cid:1) d x (cid:9) .Equation (1.1) arises in Bose-Einstein condensates (BEC) and nonlinear optics. Espe-cially, when p = 3 and d = 1, it is the well known time-independent Gross-Pitaevskii(GP) equation which describes the one-dimensional BEC problem, see, e.g., [7, 8, 25] andthe references therein. From the physical point of view, we assume that the trappingpotential V ( x ) ≥ V ( x ) ∈ L ∞ loc ( R d ) ∩ C α ( R d ) with α ∈ (0 , x ∈ R d V ( x ) = 0 and lim | x |→∞ V ( x ) = ∞ . (1.3)It is well known that, a minimizer of the following minimization problem solvesequation (1.1) for some suitable Lagrange multiplier µ , e ( ρ ) := inf (cid:8) E ρ ( u ) : u ∈ H ( R d ) , k u k = 1 (cid:9) , (1.4)where E ρ ( u ) is the Gross-Pitaevskii (GP) energy functional defined by E ρ ( u ) := 12 Z R d |∇ u | d x + 12 Z R d V ( x ) | u | d x − ρ p − p + 1 Z R d | u | p +1 d x. (1.5)Equivalence between ground states of equation (1.1) and constraint minimizers of (1.4)is proved in Theorem 1.1. To discuss equivalently ground states of (1.1), in this paper,we shall therefore focus on investigating (1.4), instead of (1.1). On the other hand, asfor the general constraint k u k = N ∈ (0 , ∞ ), one can check that this latter case can beeasily reduced to (1.4), by minimizing (1.5) under the constraint k u k = 1 but simplyreplacing ρ by ρN .When p > d , (1.4) is the so called L -supcritical problem (also known as masssupcritical problem). Taking a suitable trial function and substituting it into (1.5), onecan check that problem (1.4) admits no minimizer for any ρ ∈ (0 , ∞ ) under this case.For the case p = 1 + d , (1.4) is known as the L -critical problem (also called mass criticalproblem). Recently, some interesting results on this L -critical problem were obtainedby Guo and his co-authors (cf. [11]-[15]). Roughly speaking, the authors proved in [12]that there exists a finite value ρ ∗ such that (1.4) admits minimizers if and only if ρ < ρ ∗ (see also [1] for similar results). The threshold value ρ ∗ is determined by k w k , where w is the unique (up to translations) positive radially symmetric solution of the followingnonlinear scalar field equation (cf. [5, 17, 18, 23])∆ w − w + w p = 0 , w ∈ H ( R d ) . (1.6)The concentration behavior of minimizers as ρ ր ρ ∗ was also analyzed in [12, 13, 15] un-der different types of trapping potentials. Furthermore, the local uniqueness of minimiz-ers as ρ ր ρ ∗ was proved in [11], where the trapping potential is a class of homogeneousfunctions. 2s for the L -subcritical case, i.e. , 1 < p < d , problem (1.4) admits minimizersfor any ρ ∈ (0 , ∞ ), see, e.g., [3, 14, 22, 26, 31]. Some qualitative properties of minimizersfor (1.4), such as uniqueness, concentration behavior and symmetry, were also studied in[14, 22, 31] and the references therein. In detail, M. Maeda showed in [22] that minimizersof (1.4) are unique when ρ is small enough and the minimizers must concentrate at aglobal minimum of V ( x ) as ρ → ∞ . Further, Guo, Zeng and Zhou presented in [14]a detailed analysis on the concentration behavior of minimizers for e ( ρ ) with d = 2 as q ր
3, and more recently, Zeng has generalized these results in [31].Motivated by the works mentioned above, in this paper we focus on proving the localuniqueness of minimizers for the L -subcritical problem (1.4) as ρ → ∞ . Towards thispurpose, it is necessary to analyze the concentration behavior of minimizers as ρ → ∞ .However, the concentration results shown in [22] are not enough, and we therefore needto give a more detailed analysis on the limit behavior of minimizers for e ( ρ ) as ρ → ∞ .Besides, the equivalence between ground states of (1.1) and constraint minimizers of(1.4) is also addressed.Before stating our results, we need to introduce the following classical Gagliardo-Nirenberg type inequality (cf. [29]) C GN ≤ k∇ u k d ( p − k u k p +1 − d ( p − k u k p +1 p +1 for any u ∈ H ( R d ) \ { } , (1.7)where C GN := k w k p − (cid:16) − p − p + 1 d (cid:17)h d ( p − p + 1) − d ( p − i d ( p − , (1.8)and w is the unique positive solution of (1.6). The equality in (1.7) is attained at w .Applying the following Pohozaev identity of (1.6) (cf. [3, Lemma 8.1.2])( d − Z R d |∇ w | d x + d Z R d w d x = 2 dp + 1 Z R d w p +1 d x, (1.9)one can deduce from (1.6) that w satisfies Z R d |∇ w | d x = d p − p + 1 Z R d w p +1 d x = d ( p − p + 1) − d ( p − Z R d w d x. (1.10)Note also from [5, Proposition 4.1] that w ( x ) decays exponentially in the sense that w ( x ) , |∇ w ( x ) | = O ( | x | − d − e −| x | ) as | x | → ∞ . (1.11)Our first result is concerned with the equivalence between minimizers of (1.4) andground states of (1.1). For convenience, we introduce some notations in advance. Forany given ρ ∈ (0 , ∞ ), the set of nontrivial weak solutions for (1.1) is defined by S µ,ρ := n u ∈ H \ { (0) } : h F ′ µ,ρ ( u ) , ϕ i = 0 , ∀ ϕ ∈ H o , where the energy functional F µ,ρ ( u ) is defined as F µ,ρ ( u ) := 12 Z R d |∇ u | d x + 12 Z R d (cid:0) V ( x ) − µ (cid:1) | u | d x − ρ p − p + 1 Z R d | u | p +1 d x. (1.12)3urther, the set of ground states for (1.1) is given by G µ,ρ := n u ∈ S µ,ρ : F µ,ρ ( u ) ≤ F µ,ρ ( v ) for all v ∈ S µ,ρ o . (1.13)Moreover, the set of minimizers for e ( ρ ) is defined as M ρ := n u ρ ∈ H ( R d ) : u ρ is a minimizer of e ( ρ ) o . (1.14)Our first result is stated as the following theorem. Theorem 1.1.
Suppose V ( x ) satisfies (1.3) . Then we have follows.(i). For a.e. ρ ∈ (0 , ∞ ) , all minimizers of e ( ρ ) satisfy equation (1.1) with a fixedLagrange multiplier µ = µ ρ .(ii). For a.e. ρ ∈ (0 , ∞ ) , G µ ρ ,ρ = M ρ . Theorem 1.1 indicates that, for a.e. ρ ∈ (0 , ∞ ), there exists a unique µ = µ ρ such thatequation (1.1) admits ground states, which are equivalent to minimizers of e ( ρ ) in thesense that G µ ρ ,ρ = M ρ . Theorem 1.1 is largely inspired by some similar conclusions ondifferent types of problems, such as [3, Chapter 8], [13, Theorem 1.1] and [19, Theorem1.2]. The proof of Theorem 1.1 is left to Appendix A.1.We next focus on analyzing the limit behavior of minimizers as ρ → ∞ . Since |∇| u || ≤ |∇ u | holds for a.e. x ∈ R d , without loss of generality, we always supposeminimizers of e ( ρ ) are nonnegative. Motivated by [12]-[15], in order to analyze theblow-up behavior of minimizers as ρ → ∞ , some additional assumptions on V ( x ) arerequired. Definition 1.1. h ( x ) in R d is homogeneous of degree q ∈ R + (about the origin), if thereexists some q > h ( tx ) = t q h ( x ) in R d for any t > . This definition indicates that if h ( x ) ∈ C ( R d ) is homogeneous of degree q >
0, then0 ≤ h ( x ) ≤ C | x | q in R d , where C := max x ∈ ∂B (0) h ( x ),because h ( x | x | ) ≤ C for any x ∈ R d \ { } . Moreover, if h ( x ) → ∞ as | x | → ∞ , then 0 isthe unique minimum point of h ( x ).Define the set of global minimum points of V ( x ) by Z := (cid:8) x ∈ R d : V ( x ) = 0 (cid:9) = (cid:8) x , x , · · · , x m (cid:9) , where m ≥ . (1.15)We then assume that, V ( x ) is almost homogeneous of degree r i > x i .Specifically, there exists some V i ( x ) ∈ C ( R d ), which is homogeneous of degree r i > | x |→∞ V i ( x ) = + ∞ , such thatlim x → V ( x + x i ) V i ( x ) = 1 , i = 1 , , · · · , m. (1.16)4dditionally, inspired by [9], we define Q i ( y ) by Q i ( y ) := Z R d V i ( x + y ) w d x, i = 1 , , · · · , m. (1.17)Set r := max ≤ i ≤ m r i , ¯ Z := (cid:8) x i ∈ Z : r i = r (cid:9) ⊂ Z, (1.18)and ¯ λ := min i ∈ Γ ¯ λ i , where ¯ λ i := min y ∈ R d Q i ( y ) and Γ := (cid:8) i : x i ∈ ¯ Z (cid:9) . (1.19)Besides, we also introduce some useful notations, λ := 12 4 − d ( p − p + 1) − d ( p − , (1.20) Q ( y ) := Z R d V ( x + y ) w d x, where V ( x ) := V i ( x ) and i satisfies ¯ λ i = ¯ λ , (1.21)and Z := (cid:8) x i ∈ ¯ Z : ¯ λ i = ¯ λ (cid:9) , K := (cid:8) y : Q ( y ) = ¯ λ i = ¯ λ } , (1.22)where Z denotes the set of the flattest global minimum points of V ( x ). Stimulated by[10, 12, 22, 28], we now give the following theorem on the blow-up behavior of nonnegativeminimizers as ρ → ∞ . Theorem 1.2.
Suppose V ( x ) ∈ C ( R d ) satisfies (1.3) and (1.16) , and there exists aconstant κ > such that V ( x ) ≤ Ce κ | x | if | x | is large . (1.23) Set a ∗ := k w k , where w is the unique positive solution of (1.6) . Let u k be a nonnegativeminimizer of e ( ρ k ) , where ρ k → ∞ as k → ∞ . Then there exists a subsequence, stilldenoted by { u k } , such that u k satisfies ¯ u k ( x ) := √ a ∗ ε d k u k ( ε k x + x k ) → w ( x ) uniformly in R d as k → ∞ , (1.24) where ε k := (cid:16) ρ k √ a ∗ (cid:17) − p − − d ( p − , (1.25) and x k is the unique local maximum point of u k satisfying x k − x ε k → y for some x ∈ Z and y ∈ K as k → ∞ . (1.26) Furthermore, u k decays exponentially in the sense that ¯ u k ( x ) ≤ Ce − | x | and |∇ ¯ u k | ≤ Ce − | x | as | x | → ∞ , (1.27) where C > is a constant independent of k . e ( ρ ) must concentrate at one of the flattestglobal minimum points of V ( x ) as ρ → ∞ . Some results similar to (1.24) were alsoobtained in [22]. In Section 2, we shall present a different proof for (1.24) by employingthe refined energy estimates in Lemma 2.1 and blow-up analysis in Lemma 2.4. Herewe point out that, the L -subcritical nonlinearity term will lead to some difficultieson analyzing the limit behavior of minimizers, due to that the Gagliardo-Nirenberginequality cannot be used directly. This is quite different from those obtained in [13, 15],where the L -critical problem is considered. The key to solve these problems is toestablish a refined energy estimate of e ( ρ ) firstly, by which one can deduce the blow-uprates of u ρ as ρ → ∞ . Towards this aim, we have to employ the fact that e ( ρ ) ≥ ˜ e ( ρ ),where ˜ e ( ρ ) is a new minimization problem defined in (A.8). Moreover, (1.26) gives theconvergence rate of the unique maximum point of each minimizer as ρ → ∞ , which isbased on a more precise energy estimate of e ( ρ ). In fact, we shall show that e ( ρ ) = − λ (cid:16) ρ √ a ∗ (cid:17) p − − d ( p − + ¯ λ + o (1) a ∗ (cid:16) ρ √ a ∗ (cid:17) − r ( p − − d ( p − as ρ → ∞ , (1.28)where a ∗ := k w k , r , ¯ λ and λ are respectively defined by (1.18), (1.19) and (1.20).Motivated by the uniqueness results addressed in [2, 4, 9, 11], we finally investigatethe uniqueness of nonnegative minimizers for e ( ρ ) as ρ → ∞ . Towards this purpose,we require some additional conditions on V ( x ). Suppose V ( x ) admits a unique flattestminimum point x , i.e. , Z contains only one element x , where Z is defined in (1.22). (1.29)Further, we suppose that V ( x ) is homogeneous of degree r ≥ x . (1.30)Moreover, we also assume that there exists a constant R small enough such that ∂V ( x + x ) ∂x i = ∂V ( x ) ∂x i + W i ( x ) and | W i ( x ) | ≤ C | x | s i in B R (0) , (1.31)where x ∈ Z , V is given in (1.21), and s i > r − i = 1 , , · · · , m . Under theseassumptions, our uniqueness results can be stated as the following theorem. Theorem 1.3.
Suppose V ( x ) ∈ C ( R d ) satisfies (1.3) , (1.16) , (1.23) and (1.29) - (1.31) .Moreover, we also assume that y is the unique and non-degenerate critical point of Q ( y ) , (1.32) where Q ( y ) is defined by (1.21) . Then there exists a unique nonnegative minimizer for e ( ρ ) as ρ → ∞ . Theorem 1.3 indicates that problem (1.4) admits only one nonnegative minimizerwhen ρ is large enough. Together with Theorem 1.1 and the uniqueness results given in[22, Theorem 1.2], one can conclude that, for any given ρ where ρ is small enough orlarge enough, there exists a unique µ = µ ρ such that(1.1) admits one and only one nonnegative ground state.6heorem 1.3 is proved by establishing various types of local Pohozaev identities,which is inspired by the [2, 4, 9, 11]. However, the proof of Theorem 1.3 requires moreinvolved and intricate calculations, because of the general assumptions on dimensionand trapping potentials. Moreover, comparing with discussing L -critical problem, theappearance of L -subcritical term also leads to some essential differences on deriving thesecond Pohozaev identity.This paper is organized as follows. Section 2 is concerned with proving Theorem 1.2on the limit behavior of minimizers for e ( ρ ) as ρ → ∞ . The main purpose of Section3 is to prove the local uniqueness of nonnegative minimizers by deriving local Pohozaevidentities. The proof of Theorem 1.1 is left to Appendix A.1, and we also give someuseful results on ˜ e ( ρ ) in Appendix A.2. In this section, we shall prove Theorem 1.2 on the limit behavior of minimizers for e ( ρ )as ρ → ∞ . We shall firstly establish the optimal energy estimates for e ( ρ ), and thenpresent a detailed analysis on the limit behavior of minimizers as ρ → ∞ . The main purpose of this section is to establish the refined estimates of e ( ρ ) by thefollowing lemma. Lemma 2.1.
Suppose V ( x ) satisfies (1.3) , and then we have lim ρ →∞ e ( ρ ) (cid:16) ρ √ a ∗ (cid:17) p − − d ( p − = − λ, (2.1) where λ is given in (1.20) , a ∗ := k w k and w is the unique positive solution of (1.6) . Proof.
We start with the upper bound estimate on the energy e ( ρ ) as ρ → ∞ . Suppose χ ( x ) ∈ C ∞ ( R d ) is a cut-off function satisfying χ ( x ) = 1 as | x | ≤ χ ( x ) = 0 as | x | ≥
2. Choose a trial function u τ ( x ) := A τ k w k τ d w ( τ x ) χ ( x ) , (2.2)where τ = (cid:16) ρ √ a ∗ (cid:17) p − − d ( p − , and A τ is chosen such that k u τ k = 1. Applying the identity(1.10) and the exponential decay of w in (1.11), some calculations yield that e ( ρ ) ≤ E ρ ( u τ ) ≤ d ( p − p + 1) − d ( p − τ − (cid:0) ρ √ a ∗ (cid:1) p − τ d ( p − p + 1) − d ( p −
1) + O (1)= − (1 + o (1)) λ (cid:16) ρ √ a ∗ (cid:17) p − − d ( p − as ρ → ∞ , where λ is given in (1.20). This gives the upper bound of e ( ρ ) as ρ → ∞ .7ext, we shall establish the lower bound estimate of e ( ρ ) as ρ → ∞ by employing theestimate of ˜ e ( ρ ) given in (A.10), where ˜ e ( ρ ) is a new minimization problem defined by(A.8). Let u ρ be a nonnegative minimizer of e ( ρ ) with ρ → ∞ . Since R R d V ( x ) u ρ d x ≥ e ( ρ ) ≥ ˜ e ( ρ ) = − λ (cid:16) ρ √ a ∗ (cid:17) p − − d ( p − as ρ → ∞ . Combining the upper and lower bound estimates then yields (2.1), and this lemmais then proved.
In this section, we shall complete the proof of Theorem 1.2. Let u k be a nonnegativeminimizer of e ( ρ k ) with ρ k → ∞ as k → ∞ , and then u k satisfies (1.1) for some suitable µ k . We firstly give the following lemma. Lemma 2.2.
Suppose V ( x ) satisfies (1.3) . Let u k be a nonnegative minimizer of e ( ρ k ) with ρ k → ∞ as k → ∞ . We then have ≤ e ( ρ k ) − ˜ e ( ρ k ) → as k → ∞ , (2.3) and Z R d V ( x ) u k d x → as k → ∞ . (2.4)As for the proof of this lemma, one can refer to [22, Lemma 4.2] and we omit ithere.Define ˆ ε k := ρ − p − − d ( p − k and ˆ w k := ˆ ε d k u k (ˆ ε k x ) . (2.5)Some calculations yield that Z R d |∇ u k | d x = ˆ ε − k Z R d |∇ ˆ w k | d x and ρ p − k Z R d u p +1 k d x = ˆ ε − k Z R d ˆ w p +1 k d x. We now give the following lemma on the boundedness of R R d |∇ ˆ w k | d x and R R d ˆ w p +1 k d x . Lemma 2.3.
Suppose V ( x ) satisfies (1.3) . Let u k be a nonnegative minimizer of e ( ρ k ) with ρ k → ∞ as k → ∞ . Then one has C ≤ Z R d |∇ ˆ w k | d x ≤ C and C ′ ≤ Z R d ˆ w p +1 k d x ≤ C ′ , (2.6) where ˆ w k is defined by (2.5) , C , C , C ′ and C ′ are positive constants independent of k . roof. It follows from (1.5) and (2.1) that( √ a ∗ ) p − − d ( p − ˆ ε k e ( ρ k ) = Z R d |∇ ˆ w k | d x − p + 1 Z R d ˆ w p +1 k d x → − λ < k → ∞ , (2.7)where λ is given in (1.20). Using the Gagliardo-Nirenberg inequality (1.7), one can thenderive from (2.7) that R R d |∇ ˆ w k | d x ≤ C and R R d ˆ w p +1 k d x ≤ C ′ . As for the lowerbounds, from (2.7) one can deduce that R R d ˆ w p +1 k d x ≥ C ′ , and using the Gagliardo-Nirenberg inequality (1.7) then yields R R d |∇ ˆ w k | d x ≥ C (cid:0) R R d ˆ w p +1 k d x (cid:1) d ( p − ≥ C .Hence, we complete the proof of this lemma.Motivated by [13, 15, 28, 30, 31], we then give the following lemma, which is a weakversion of Theorem 1.2. Lemma 2.4.
Suppose V ( x ) satisfies (1.3) . Let u k be a nonnegative minimizer of e ( ρ k ) with ρ k → ∞ as k → ∞ . We then have follows.(i). There exist a sequence { y k } ⊂ R d and positive constants ι and R such that lim inf k →∞ Z B R (0) ˆ w p +1 k d x ≥ ι > . (2.8) (ii). The sequence { y k } satisfies that, passing to a subsequence if necessary, ˆ ε k y k → z for some z ∈ R d satisfying V ( z ) = 0 .(iii). Defined w k := ˆ w k ( x + y k ) = ˆ ε d k u k (ˆ ε k x + ˆ ε k y k ) , (2.9) and then passing to a subsequence if necessary, there holds that lim k →∞ w k = ( √ a ∗ ) − d ( p − − d ( p − w (cid:0) ( √ a ∗ ) − p − − d ( p − x + x ′ (cid:1) , (2.10) strongly in H ( R d ) for some x ′ ∈ R d , where a ∗ := k w k and w is the unique (upto translations) positive solution of (1.6) . Proof. (i). As for (2.8), if it is false, then for any
R >
0, there exists a subsequence ofˆ w k (still denoted by ˆ w k ) such that lim k →∞ sup y ∈ R d R B R ( y ) ˆ w p +1 k d x = 0. Applying [21, Lemma1.1] then yields that ˆ w k → L p +1 ( R d ) as k → ∞ , which however contradicts (2.6).(ii). Employing (2.4), this conclusion can be obtained by using the proof by contra-diction. Since the proof is similar to that of [15, Lemma 2.3], we omit it here.(iii). It follows from (1.1) and (2.5) that w k solves − ∆ w k + ˆ ε k V (ˆ ε k x + ˆ ε k y k ) w k = ˆ ε k µ k w k + w pk . (2.11)Following (1.5) and (2.1), one can deduce thatˆ ε k µ k = 2ˆ ε k e ( ρ k ) − p − p + 1 Z R d w p +1 k d x < . (2.12)9sing (2.6) and (2.7), one can then obtain the uniform boundedness of { ˆ ε k µ k } as k → ∞ ,which indicates that passing to a subsequence if necessary, ˆ ε k µ k → − β for some β ∈ R + as k → ∞ . From (2.6), one can deduce that w k is bounded uniformly in H ( R d ). Taking k → ∞ , passing to a subsequence if necessary, one then has w k ⇀ w ≥ H ( R d ) forsome w ∈ H ( R d ) satisfying − ∆ w + βw = w p . (2.13)Applying the maximum principle, one can then conclude from (2.8) that w >
0, whichimplies from (1.6) that w = β p − w ( β x + x ′ ) for some x ′ ∈ R d , because of the unique-ness (up to translations) of positive solution of (1.6).Here we claim that k w k = 1 and β = k w k p − d ( p − − . (2.14)From the following Pohozaev identity of (2.13) (cf. [3, Lemma 8.1.2]),( d − Z R d |∇ w | d x + dβ Z R d w d x = 2 dp + 1 Z R d w p +10 d x, one can derive that Z R d |∇ w | d x = p − p + 1 d Z R d w p +10 d x = d ( p − p + 1) − d ( p − β Z R d w d x. (2.15)Applying the Gagliardo-Nirenberg inequality (1.7) and (2.15), some calculations yieldthat C GN ≤ k∇ w k d ( p − k w k p +1 − d ( p − k w k p +1 p +1 = (cid:16) p − p + 1 d (cid:17) d ( p − (cid:16) p + 1) β p + 1) − d ( p − (cid:17) d ( p − − k w k p − ≤ C GN k w k p − β d ( p − − , where C GN is defined in (1.8). This gives that β ≥ k w k p − d ( p − − , and further implies that β = k w k p − d ( p − − , because k w k = β − d ( p − p − k w k ≤
1. Moreover, one can deduce that k w k = 1.Since k w k k = k w k = 1, passing to a subsequence if necessary, one has w k → w strongly in L ( R d ) as k → ∞ . Using the interpolation inequality, one can further derivethat w k → w strongly in L q ( R d ) for any q ∈ [2 , ∗ ) as k → ∞ . Moreover, one canconclude from (2.11) and (2.13) that (2.10) holds. Lemma 2.5.
Suppose V ( x ) satisfies (1.3) . Let u k be a nonnegative minimizer of e ( ρ k ) with ρ k → ∞ as k → ∞ . Then u k admits only one local maximum point x k , and passingto a subsequence if necessary, there holds that ¯ u k ( x ) := √ a ∗ ε d k u k ( ε k x + x k ) → w ( x ) strongly in H ( R d ) as k → ∞ , (2.16) where ε k := (cid:16) ρ k √ a ∗ (cid:17) − p − − d ( p − is defined in (1.25) . roof. Applying the De-Giorgi-Nash-Moser theory (cf. [16, Theorem 4.1]), one canderive from (2.10) and (2.11) that w k ( x ) → | x | → ∞ uniformly for large k, (2.17)which indicates that u k ( x ) admits at least one global maximum point. Let x k is a globalmaximum point of u k ( x ) and set z k := ε k y k → x as k → ∞ . Since z ′ k := x k − z k ǫ k is aglobal maximum point of w k ( x ), one can thus derive from (2.8) and (2.17) that n x k − z k ǫ k o is bounded uniformly in R d . (2.18)Define ¯ u k ( x ) := √ a ∗ ε d k u k ( ε k x + x k ) (2.19)where ε k := (cid:16) ρ k √ a ∗ (cid:17) − p − − d ( p − is given in (1.25). It then follows from (2.10) that, passingto a subsequence if necessary, ¯ u k ( x ) → w ( x + y ′ ) for some y ′ ∈ R d strongly in H ( R d )as k → ∞ . Since V ( x ) ∈ C α ( R d ), using the standard elliptic regular theory, we have¯ u k ( x ) → w ( x + y ′ ) in C loc ( R d ) as k → ∞ , (2.20)and one can see [15, Lemma 3.1] for a detailed proof. Note that the origin is a localmaximum point of ¯ u k for all k >
0, and it follows from (2.20) that it is also a localmaximum point of w . Since w ( x ) is radially symmetric about the origin and decreasesstrictly in | x | (see, e.g., [5, 18, 29]), we know that x = 0 is the unique local maximumpoint of w ( x ), which thus implies from (2.20) that y ′ = 0. Hence, it follows that¯ u k ( x ) → w ( x ) strongly in H ( R d ) as k → ∞ . (2.21)We finally prove the uniqueness of the local maximum points of u k when k is suffi-ciently large. Suppose x k is any local maximum point of u k . It is easy to know that ¯ u k satisfies − ∆¯ u k + ε k V ( ε k x + x k )¯ u k = ε k µ k ¯ u k + ¯ u pk in R d . (2.22)From this, one can deduce that ¯ u k ( x k ) ≥ C > k > u k must stay in a finite ball B R (0) as k → ∞ ,where R > k . Employing the uniqueness of local maximum points of w , one can deduce from (2.20) that the origin is the unique maximum point of ¯ u k , i.e. , u k admits only one local maximum point x k when k large enough. Proof of Theorem 1.2.
As for the exponential decay of u k in (1.27), one can obtainit by using the comparison principle. Similar to (2.12), one can check that ε k µ k → − k → ∞ , and then we can derive that, there exists a constant R > − ∆¯ u k + 12 ¯ u k ≤ u k ≤ Ce − R for | x | ≥ R. Comparing ¯ u k with Ce − | x | then yields ¯ u k ( x ) ≤ Ce − | x | when | x | ≥ R . Further more,applying the local elliptic estimate (cf. (3.15) in [6]) then yields |∇ ¯ u k | ≤ Ce − | x | when11 x | > R . We thus give the proof of (1.27). Moreover, by (1.27) and (2.21), applyingthe standard elliptic regularity theory then yields (1.24), (see, e.g., [22, Lemma 4.9] forsimilar arguments).Finally, we aim at proving (1.26). Suppose ˜ u k is a nonnegative minimizer of ˜ e ( ρ k ),and then ˜ u k ( x − ǫ k y − x ) is also a nonnegative minimizer of ˜ e ( ρ k ), where x ∈ Z , y ∈ K , and Z , K are defined by (1.22). We then derive from (1.11), (1.16), (1.23)and (A.11) that e ( ρ k ) − ˜ e ( ρ k ) ≤ Z R d V ( x )˜ u k ( x − ε k y − x ) d x ≤ a ∗ (cid:0) o (1) (cid:1) Z B √ εk (0) V ( ε k x + ε k y + x ) V ( ε k x + ε k y ) V ( ε k x + ε k y ) w d x ≤ a ∗ ε rk (cid:0) o (1) (cid:1) Z R d V ( x + y ) w d x = 1 a ∗ ε rk (cid:0) o (1) (cid:1) ¯ λ , (2.23)where ¯ λ is given by (1.19). Suppose u k is a nonnegative minimizer of e ( ρ k ), and thenone can deduce from (1.16) and (1.27) that e ( ρ k ) − ˜ e ( ρ k ) ≥ Z R d V ( x ) u k d x = 1 a ∗ Z R d V ( ε k x + x k )¯ u k d x = 1 a ∗ Z R d V ( ε k x + x k − x i + x i ) V i ( ε k x + x k − x i ) V i ( ε k x + x k − x i )¯ u k d x ≥ o (1) a ∗ ε r i k Z B √ εk ( x i ) V i (cid:16) x + x k − x i ε k (cid:17) ¯ u k d x, (2.24)where x i ∈ Z . Comparing with the upper estimate (2.23), one can directly check that r i = r and x i = x ∈ ¯ Z , where r and ¯ Z is given by (1.18). Since V ( x ) → ∞ as | x | → ∞ ,one can further check that, { x k − x ε k } is bounded uniformly in k . More precisely, one canalso verify that, passing to a subsequence if necessary, x k − x ε k → y , which implies that x ∈ Z and Z is defined in (1.22), i.e. , (1.26) holds. Moreover, wealso have lim k →∞ e ( ρ k ) − ˜ e ( ρ k ) ε rk = 1 a ∗ ¯ λ , (2.25)where ¯ λ is defined by (1.19). This gives (1.28), and the proof of Theorem 1.2 is thuscompleted. In this section, we focus on the proof of local uniqueness of minimizers as ρ → ∞ .Argue by contradiction. Suppose it is not true, and there exist two different nonnegative12inimizers u k and u k for e ( ρ k ) with ρ k → ∞ as k → ∞ . Let x k and x k denote theunique local maximum point of u k and u k , respectively. Following (1.1), we have − ∆ u ik + V ( x ) u ik = µ ik u ik + ρ p − k u pik in R d , i = 1 , . (3.1)Define ˆ u ik ( x ) := √ a ∗ ε d k u ik ( x ) and ¯ u ik ( x ) := ˆ u ik ( ε k x + x k ) , i = 1 , , (3.2)where ε k is given by (1.25). Since lim k →∞ x k − x k ε k = 0, by Theorem 1.2, one then has¯ u ik → w ( x ) uniformly in R d as k → ∞ . One can check that ¯ u ik satisfies − ∆¯ u ik + ε k V ( ε k x + x k )¯ u ik = ε k µ ik ¯ u ik + ¯ u pik in R d , i = 1 , . (3.3)Since u k u k , define η k := u k − u k k u k − u k k L ∞ ( R d ) and ˆ η k := ˆ u k − ˆ u k k ˆ u k − ˆ u k k L ∞ ( R d ) , (3.4)and then we have η k = ˆ η k . Further we define¯ η k ( x ) := ˆ η k ( ε k x + x k ) , (3.5)and thus one can deduce from (3.3) that ¯ η k satisfies − ∆¯ η k + ε k V ( ε k x + x k )¯ η k = ε k µ k ¯ η k + ¯ g k ( x ) + ¯ f k ( x ) , (3.6)where ¯ g k ( x ) := ε k µ k − µ k k ¯ u k − ¯ u k k L ∞ ( R d ) ¯ u k and ¯ f k ( x ) := ¯ u p k − ¯ u p k k ¯ u k − ¯ u k k L ∞ ( R d ) . (3.7)Now we give the following lemma on the limit of ¯ η k . Lemma 3.1.
Suppose all the assumptions of Theorem 1.3 hold. Then passing to asubsequence if necessary, ¯ η k → ¯ η in C loc ( R d ) as k → ∞ , where ¯ η satisfies ¯ η ( x ) = b (cid:0) w + p − x · ∇ w (cid:1) + d X i =1 b i ∂w∂x i , (3.8) and b , b , ...b d are some constants. Proof.
Since k ¯ η k k ∞ ≤
1, the standard elliptic regularity then implies that k ¯ η k k C ,αloc ( R d ) ≤ C , where C is a constant independent of k . Therefore, passing to a subsequence ifnecessary, one can deduce that¯ η k → ¯ η in C loc ( R d ) as k → ∞ for some function ¯ η ∈ C loc ( R d ). (3.9)Similar to (2.12), from (1.5) and (3.2), one can derive that ε k µ ik = 2 ε k e ( ρ k ) − p − a ∗ ( p + 1) Z R d | ¯ u ik | p +1 d x. (3.10)13efine ¯ u p +11 k − ¯ u p +12 k = Z d d t (cid:2) t ¯ u k + (1 − t )¯ u k (cid:3) p +1 d t =( p + 1)(¯ u k − ¯ u k ) Z (cid:2) t ¯ u k + (1 − t )¯ u k (cid:3) p d t :=( p + 1) ¯ C pk ( x )(¯ u k − ¯ u k ) , (3.11)which implies from (1.24) that ¯ C pk ( x ) → w p ( x ) uniformly in R d as k → ∞ . Further onecan derive that¯ g k ( x ) = ε k µ k − µ k k ¯ u k − ¯ u k k L ∞ ( R d ) ¯ u k = − p − a ∗ ( p + 1) R R d (cid:0) | ¯ u k | p +1 − | ¯ u k | p +1 (cid:1) d x k ¯ u k − ¯ u k k L ∞ ( R d ) ¯ u k = − p − a ∗ Z R d ¯ C pk ( x )¯ η k d x ¯ u k , (3.12)which implies from (1.24) that¯ g k ( x ) → − p − a ∗ Z R d w p ¯ η d x w uniformly in R d as k → ∞ . (3.13)On the other hand, similar to (3.11), one can also define ¯ D p − k ( x ) satisfying p ¯ D p − k ( x )(¯ u k − ¯ u k ) := ¯ u p k − ¯ u p k , (3.14)and then ¯ D p − k ( x ) → w p − ( x ) uniformly in R d as k → ∞ . Further, one has¯ f k ( x ) = ¯ u p k − ¯ u p k k ¯ u k − ¯ u k k L ∞ ( R d ) = p ¯ D p − k ( x )¯ η k , (3.15)and ¯ f k ( x ) → pw p − ¯ η uniformly in R d as k → ∞ . (3.16)By the above results, taking k → ∞ , it follows from (3.6) that ¯ η solves − ∆¯ η + (1 − pw p − )¯ η = − p − a ∗ Z R d w p ¯ η d x w. (3.17)Set L := − ∆ + (1 − pw p − ) and one can check that L (cid:0) w + p − x · ∇ w (cid:1) = − ( p − w .Recall from (cf. [17, 24]) thatker L = n ∂w∂x , ∂w∂x , ... ∂w∂x d o , and then one can derive that¯ η ( x ) = b (cid:16) w + p − x · ∇ w (cid:17) + d X i =1 b i ∂w∂x i , (3.18)where b , b , b ,... and b d are some constants.14 emma 3.2. Under the assumptions of Theorem 1.3, there holds that, b p − Z R d ∂V ( x + y ) ∂x j (cid:0) x · ∇ w (cid:1) − d X i =1 b i Z R d ∂ V ( x + y ) ∂x i ∂x j w = 0 , (3.19) where j = 1 , , · · · , d and V is given by (1.21) . Proof.
At first, we claim that for any ¯ x , there exists a small δ > C > ε k Z ∂B δ (¯ x ) |∇ ˆ η k | d S + ε k Z ∂B δ (¯ x ) V ( x )ˆ η k d S + Z ∂B δ (¯ x ) ˆ η k d S ≤ Cε dk , (3.20)where ˆ η k is given by (3.4).Following from (3.4), (3.5) and (3.6), one can deduce that ˆ η k satisfies − ε k ∆ˆ η k + ε k V ( x )ˆ η k = ε k µ k ˆ η k + ˆ g k ( x ) + ˆ f k ( x ) , (3.21)where ˆ g k ( x ) := ε k µ k − µ k k ˆ u k − ˆ u k k L ∞ ( R d ) ˆ u k and ˆ f k ( x ) := ˆ u p k − ˆ u p k k ˆ u k − ˆ u k k L ∞ ( R d ) . (3.22)Similar to (3.12) and (3.15), one hasˆ g k ( x ) = − p − a ∗ ε − dk Z R d ˆ C pk ( x )ˆ η k d x ˆ u k and ˆ f k ( x ) = p ˆ D p − k ( x )ˆ η k , (3.23)where ˆ C pk ( ε k x + x k ) := ¯ C pk ( x ) and ˆ D p − k ( ε k x + x k ) := ¯ D p − k ( x ).Multiplying (3.21) by ˆ η k and integrating over R d yield that ε k Z R d |∇ ˆ η k | d x + ε k Z R d V ( x )ˆ η k d x − ε k µ k Z R d ˆ η k d x = p Z R d ˆ D p − k ( x )ˆ η k d x − p − a ∗ ε − dk Z R d ˆ C pk ( x )ˆ η k d x Z R d ˆ u k ˆ η k d x = pε dk Z R d ¯ D p − k ( x )¯ η k d x − p − a ∗ ε dk Z R d ¯ C pk ( x )¯ η k d x Z R d ¯ u k ¯ η k d x = O ( ε dk ) as k → ∞ , where the last equality holds because ¯ η k → ¯ η , ¯ u k → w , ¯ C pk ( x ) → w p and ¯ D p − k ( x ) → w p − uniformly in R d as k → ∞ . Applying Lemma 4.5 in [2] then yields (3.20), and thiscompletes the proof of this claim.Following from (3.1), one can deduce that ˆ u ik solves − ε k ∆ˆ u ik + ε k V ( x )ˆ u ik = ε k µ ik ˆ u ik + ˆ u pik in R d , i = 1 , . (3.24)Multiplying (3.24) by ∂ ˆ u ik ∂x j and integrating over B δ ( x k ), where i = 1 , j = 1 , , · · · , d and δ is given by (3.20), one can obtain the following equality, − ε k Z B δ ( x k ) ∆ˆ u ik ∂ ˆ u ik ∂x j + ε k Z B δ ( x k ) V ( x ) ∂ ˆ u ik ∂x j = ε k µ ik Z B δ ( x k ) ∂ ˆ u ik ∂x j + 1 p + 1 Z B δ ( x k ) ∂ ˆ u p +1 ik ∂x j . (3.25)15ome calculations yield that − ε k Z B δ ( x k ) ∆ˆ u ik ∂ ˆ u ik ∂x j = − ε k d X l =1 Z B δ ( x k ) ∂ ˆ u ik ∂x l ∂ ˆ u ik ∂x j = − ε k d X l =1 h Z ∂B δ ( x k ) ∂ ˆ u ik ∂x l ∂ ˆ u ik ∂x j ν l d S − Z B δ ( x k ) ∂ ˆ u ik ∂x l ∂∂x j ∂ ˆ u ik ∂x l i = − ε k d X l =1 h Z ∂B δ ( x k ) ∂ ˆ u ik ∂x l ∂ ˆ u ik ∂x j ν l d S − Z B δ ( x k ) ∂∂x j (cid:16) ∂ ˆ u ik ∂x l (cid:17) i = − ε k h Z ∂B δ ( x k ) ∂ ˆ u ik ∂ν ∂ ˆ u ik ∂x j d S − Z ∂B δ ( x k ) |∇ ˆ u ik | ν j d S i , (3.26)and ε k Z B δ ( x k ) V ( x ) ∂ ˆ u ik ∂x j = ε k h Z ∂B δ ( x k ) V ( x )ˆ u ik ν j d S − Z B δ ( x k ) ∂V ( x ) ∂x j ˆ u ik i . (3.27)It then follows from (3.25)-(3.27) that ε k Z B δ ( x k ) ∂V ( x ) ∂x j ˆ u ik = ε k Z ∂B δ ( x k ) V ( x )ˆ u ik ν j d S − ε k µ ik Z ∂B δ ( x k ) ˆ u ik ν j d S − p + 1 Z ∂B δ ( x k ) ˆ u p +1 ik ν j d S − ε k h Z ∂B δ ( x k ) ∂ ˆ u ik ∂ν ∂ ˆ u ik ∂x j d S − Z ∂B δ ( x k ) |∇ ˆ u ik | ν j d S i . Further, we have ε k Z B δ ( x k ) ∂V ( x ) ∂x j (ˆ u k + ˆ u k )ˆ η k = ε k Z ∂B δ ( x k ) V ( x )(ˆ u k + ˆ u k )ˆ η k ν j d S + ε k Z ∂B δ ( x k ) ( ∇ ˆ u k + ∇ ˆ u k ) ∇ ˆ η k ν j d S − ε k µ k Z ∂B δ ( x k ) (ˆ u k + ˆ u k )ˆ η k ν j d S − Z ∂B δ ( x k ) ˆ g k ˆ u k ν j d S − p + 1 Z ∂B δ ( x k ) ˆ u p +11 k − ˆ u p +12 k k ˆ u k − ˆ u k k L ∞ ( R d ) ν j d S − ε k h Z ∂B δ ( x k ) ∂ ˆ u k ∂ν ∂ ˆ η k ∂x j d S + Z ∂B δ ( x k ) ∂ ˆ η k ∂ν ∂ ˆ u k ∂x j d S i , (3.28)where ˆ g k is defined in (3.22).Using the H¨older inequality, one can derive from (3.20) and (1.27) that ε k (cid:12)(cid:12)(cid:12) Z ∂B δ ( x k ) ∂ ˆ u k ∂ν ∂ ˆ η k ∂x j d S + Z ∂B δ ( x k ) ∂ ˆ η k ∂ν ∂ ˆ u k ∂x j d S (cid:12)(cid:12)(cid:12) ≤ ε k (cid:16) Z ∂B δ ( x k ) |∇ ˆ η k | (cid:17) h(cid:16) Z ∂B δ ( x k ) |∇ ˆ u k | (cid:17) + (cid:16) Z ∂B δ ( x k ) |∇ ˆ u k | (cid:17) i ≤ Cε k ε d k ε d − k e − cδεk = Cε d − k e − cδεk = o (1) e − cδεk , (3.29)16here C is a suitable positive constant. Similarly, we also have (cid:12)(cid:12)(cid:12) ε k Z ∂B δ ( x k ) ( ∇ ˆ u k + ∇ ˆ u k ) ∇ ˆ η k ν j d S (cid:12)(cid:12)(cid:12) = o (1) e − cδεk , (3.30) ε k (cid:12)(cid:12)(cid:12) Z ∂B δ ( x k ) V ( x )(ˆ u k + ˆ u k )ˆ η k ν j d S (cid:12)(cid:12)(cid:12) ≤ ε k (cid:16) Z ∂B δ ( x k ) V ( x )ˆ η k d S (cid:17) (cid:16) Z ∂B δ ( x k ) V ( x )(ˆ u k + ˆ u k ) d S (cid:17) ≤ o (1) e − cδεk , (3.31) (cid:12)(cid:12)(cid:12) ε k µ ik Z ∂B δ ( x k ) (ˆ u k + ˆ u k )ˆ η k ν j d S (cid:12)(cid:12)(cid:12) + 12 (cid:12)(cid:12)(cid:12) Z ∂B δ ( x k ) ˆ g k ˆ u k ν j d S (cid:12)(cid:12)(cid:12) ≤ C (cid:16) Z ∂B δ ( x k ) (ˆ u k + ˆ u k ) d S (cid:17) (cid:16) Z ∂B δ ( x k ) ˆ η k d S (cid:17) + C Z R d | ¯ C pk ( x ) | ¯ η k d x Z ∂B δ ( x k ) ˆ u k d S = o (1) e − cδεk , (3.32)and (cid:12)(cid:12)(cid:12) p + 1 Z ∂B δ ( x k ) ˆ u p +11 k − ˆ u p +12 k k ˆ u k − ˆ u k k L ∞ ( R d ) ν j (cid:12)(cid:12)(cid:12) ≤ C Z ∂B δ ( x k ) | ˆ C pk ( x ) | ˆ η k = o (1) e − cδεk . (3.33)On the other hand, let x be the unique point of Z , where Z satisfies (1.22) and(1.29). Employing (3.29)-(3.33), and applying (1.16), (1.31) and (3.9), one can derivefrom (3.28) that o (1) e − cδεk = ε k Z B δ ( x k ) ∂V ( x ) ∂x j (ˆ u k + ˆ u k )ˆ η k = ε dk ε k Z B δεk (0) ∂V ( ε k x + x k − x + x ) ∂ε k x j (¯ u k + ¯ u k )¯ η k = ε d + r +1 k Z B δεk (0) ∂V ( x + x k − x ε k ) ∂x j (¯ u k + ¯ u k )¯ η k + ε d +2 k Z B δεk (0) W j ( ε k x + x k − x )(¯ u k + ¯ u k )¯ η k =(1 + o (1)) ε d + r +1 k Z R d ∂V ( x + y ) ∂x j w ¯ η . (3.34)17urthermore, one can deduce from (3.18) and (3.34) that0 = Z R d ∂V ( x + y ) ∂x j w ¯ η = Z R d ∂V ( x + y ) ∂x j w h b (cid:0) w + p − x · ∇ w (cid:1) + d X i =1 b i ∂w∂x i i = b Z R d ∂V ( x + y ) ∂x j w + b p − Z R d ∂V ( x + y ) ∂x j x · ∇ w + d X i =1 b i Z R d ∂V ( x + y ) ∂x j ∂w ∂x i = b p − Z R d ∂V ( x + y ) ∂x j (cid:0) x · ∇ w (cid:1) − d X i =1 b i Z R d ∂ V ( x + y ) ∂x i ∂x j w , which gives (3.19).In the following, we shall follow the above two lemma to complete the proof ofTheorem 1.3. Proof of Theorem 1.3.
At first, we claim that the coefficient b given in (3.18) satisfies b ≡ . (3.35)Multiplying (3.24) by ( x − x k ) · ∇ ˆ u ik and integrating over B δ ( x k ), where i = 1 , δ is given in (3.20), one has − ε k Z B δ ( x k ) ∆ˆ u ik (cid:2) ( x − x k ) · ∇ ˆ u ik (cid:3) + ε k Z B δ ( x k ) V ( x ) (cid:2) ( x − x k ) · ∇ ˆ u ik (cid:3) = ε k µ ik Z B δ ( x k ) (cid:2) ( x − x k ) · ∇ ˆ u ik (cid:3) + 1 p + 1 Z B δ ( x k ) (cid:2) ( x − x k ) · ∇ ˆ u p +1 ik (cid:3) . (3.36)Some calculations yield that − ε k Z B δ ( x k ) ∆ˆ u ik [( x − x k ) · ∇ ˆ u ik ]= − ε k Z ∂B δ ( x k ) ∂ ˆ u ik ∂ν [( x − x k ) · ∇ ˆ u ik ] + ε k Z B δ ( x k ) ∇ ˆ u ik · ∇ [( x − x k ) · ∇ ˆ u ik ]= − ε k Z ∂B δ ( x k ) ∂ ˆ u ik ∂ν [( x − x k ) · ∇ ˆ u ik ] + ε k Z ∂B δ ( x k ) [( x − x k ) · ν ] |∇ ˆ u ik | + 2 − d ε k Z ∂B δ ( x k ) ∇ ˆ u ik · ν − − d h ε k Z B δ ( x k ) V ( x )ˆ u ik − ε k µ ik Z B δ ( x k ) ˆ u ik − Z B δ ( x k ) ˆ u p +1 ik i , (3.37)18here the last “ = ” holds due to that ε k Z B δ ( x k ) ∇ ˆ u ik · ∇ [( x − x k ) · ∇ ˆ u ik ]= ε k d X j =1 Z B δ ( x k ) ∂ ˆ u ik ∂x j h ∂ ˆ u ik ∂x j + ( x − x k ) · ∇ ∂ ˆ u ik ∂x j i = ε k d X j =1 Z B δ ( x k ) (cid:16) ∂ ˆ u ik ∂x j (cid:17) + ε k d X j =1 Z B δ ( x k ) ( x − x k ) · ∇ (cid:16) ∂ ˆ u ik ∂x j (cid:17) = ε k Z B δ ( x k ) |∇ ˆ u ik | + ε k Z B δ ( x k ) ( x − x k ) · ∇|∇ ˆ u ik | = ε k Z ∂B δ ( x k ) |∇ ˆ u ik | ( x − x k ) · ν + 2 − d ε k Z B δ ( x k ) |∇ ˆ u ik | , and 2 − d ε k Z B δ ( x k ) |∇ ˆ u ik | = 2 − d ε k Z ∂B δ ( x k ) ∇ ˆ u ik · ν − − d ε k Z B δ ( x k ) ˆ u ik ∆ˆ u ik = 2 − d ε k Z ∂B δ ( x k ) ∇ ˆ u ik · ν − − d h ε k Z B δ ( x k ) V ( x )ˆ u ik − ε k µ ik Z B δ ( x k ) ˆ u ik − Z B δ ( x k ) ˆ u p +1 ik i . Moreover, one can also deduce that Z B δ ( x k ) [( x − x k ) · ∇ ˆ u ik ] = Z ∂B δ ( x k ) ˆ u ik [( x − x k ) · ν ] − d Z B δ ( x k ) ˆ u ik , Z B δ ( x k ) [( x − x k ) · ∇ ˆ u p +1 ik ] = Z ∂B δ ( x k ) ˆ u p +1 ik [( x − x k ) · ν ] − d Z B δ ( x k ) ˆ u p +1 ik , and Z B δ ( x k ) V ( x )[( x − x k ) · ∇ ˆ u ik ]= Z ∂B δ ( x k ) V ( x )ˆ u ik [( x − x k ) · ν ] − Z B δ ( x k ) (cid:2) ∇ V ( x ) · ( x − x k )ˆ u ik + dV ( x )ˆ u ik (cid:3) . − ε k Z ∂B δ ( x k ) ∂ ˆ u ik ∂ν [( x − x k ) · ∇ ˆ u ik ] + ε k Z ∂B δ ( x k ) [( x − x k ) · ν ] |∇ ˆ u ik | + 2 − d ε k Z ∂B δ ( x k ) ∇ ˆ u ik · ν − − d h ε k Z B δ ( x k ) V ( x )ˆ u ik − ε k µ ik Z B δ ( x k ) ˆ u ik − Z B δ ( x k ) ˆ u p +1 ik i − dε k Z B δ ( x k ) V ( x )ˆ u ik − ε k Z B δ ( x k ) ∇ V ( x ) · ( x − x k )ˆ u ik + ε k Z ∂B δ ( x k ) V ( x )ˆ u ik [( x − x k ) · ν ]= ε k µ ik Z ∂B δ ( x k ) ˆ u ik [( x − x k ) · ν ] − dε k µ ik Z B δ ( x k ) ˆ u ik + 1 p + 1 Z ∂B δ ( x k ) ˆ u p +1 ik [( x − x k ) · ν ] − dp + 1 Z B δ ( x k ) ˆ u p +1 ik . Following from (3.10), one has ε k µ ik Z R d ˆ u ik = 2 a ∗ ε dk ε k e ( ρ k ) − p − p + 1 Z R d | ˆ u ik | p +1 d x, and it then follows that − ε k Z ∂B δ ( x k ) ∂ ˆ u ik ∂ν [( x − x k ) · ∇ ˆ u ik ] + ε k Z ∂B δ ( x k ) [( x − x k ) · ν ] |∇ ˆ u ik | + 2 − d ε k Z ∂B δ ( x k ) ∇ ˆ u ik · ν + ε k Z ∂B δ ( x k ) V ( x )ˆ u ik [( x − x k ) · ν ] − ε k Z B δ ( x k ) [ ∇ V ( x ) · ( x − x k )]ˆ u ik − ε k µ ik Z ∂B δ ( x k ) ˆ u ik [( x − x k ) · ν ] − p + 1 Z ∂B δ ( x k ) ˆ u p +1 ik [( x − x k ) · ν ]= ε k Z B δ ( x k ) V ( x )ˆ u ik − ε k µ ik Z B δ ( x k ) ˆ u ik − p + 1) − d ( p − p + 1) Z B δ ( x k ) ˆ u p +1 ik = ε k Z B δ ( x k ) V ( x )ˆ u ik + d ( p − − p + 1) Z R d ˆ u p +1 ik − a ∗ ε dk ε k e ( ρ k )+ ε k µ ik Z R d \ B δ ( x k ) ˆ u ik + 2( p + 1) − d ( p − p + 1) Z R d \ B δ ( x k ) ˆ u p +1 ik . (3.38)By (3.4), one can deduce from (3.38) that d ( p − − p + 1) Z R d ˆ u p +11 k − ˆ u p +12 k k ˆ u k − ˆ u k k L ∞ ( R d ) = T + T + T + T + T , (3.39)20here T := − ε k Z ∂B δ ( x k ) ∂ ˆ u k ∂ν [( x − x k ) · ∇ ˆ η k ] − ε k Z ∂B δ ( x k ) ∂ ˆ η k ∂ν [( x − x k ) · ∇ ˆ u k ]+ ε k Z ∂B δ ( x k ) [( x − x k ) · ν ] ∇ ˆ η k · ∇ (ˆ u k + ˆ u k )+ 2 − d ε k h Z ∂B δ ( x k ) ˆ η k (cid:0) ∇ (ˆ u k + ˆ u k ) · ν (cid:1) + Z ∂B δ ( x k ) (ˆ u k + ˆ u k ) (cid:0) ∇ ˆ η k · ν (cid:1)i ,T := ε k Z ∂B δ ( x k ) V ( x )[( x − x k ) · ν ](ˆ u k + ˆ u k )ˆ η k ,T := − ε k µ k Z ∂B δ ( x k ) [( x − x k ) · ν ](ˆ u k + ˆ u k )ˆ η k − Z ∂B δ ( x k ) [( x − x k ) · ν ]ˆ g k ˆ u k − ε k µ k Z R d \ B δ ( x k ) (ˆ u k + ˆ u k )ˆ η k − Z R d \ B δ ( x k ) ˆ g k ˆ u k ,T := − p + 1 Z ∂B δ ( x k ) [( x − x k ) · ν ] ˆ u p +11 k − ˆ u p +12 k k ˆ u k − ˆ u k k L ∞ ( R d ) − p + 1) − d ( p − p + 1) Z R d \ B δ ( x k ) ˆ u p +11 k − ˆ u p +12 k k ˆ u k − ˆ u k k L ∞ ( R d ) , and T := − ε k Z B δ ( x k ) V ( x )(ˆ u k + ˆ u k )ˆ η k − ε k Z B δ ( x k ) [ ∇ V ( x ) · ( x − x k )](ˆ u k + ˆ u k )ˆ η k . Similar to the estimates (3.29)-(3.33), one can deduce that | T | , | T | , | T | , | T | = o (1) e − cδεk . (3.40)As for T , the estimate (3.34) gives that ε k (cid:12)(cid:12)(cid:12) Z B δ ( x k ) [ ∇ V ( x ) · x k ](ˆ u k + ˆ u k )ˆ η k (cid:12)(cid:12)(cid:12) = o (1) e − cδεk , and by (1.16), one has ε k Z B δ ( x k ) V ( x )(ˆ u k + ˆ u k )ˆ η k = ε dk Z B δεk (0) V ( ε k x + x k − x + x ) V ( ε k x + x k − x ) V ( ε k x + x k − x )(¯ u k + ¯ u k )¯ η k =(1 + o (1)) ε d + rk Z B δεk (0) V (cid:16) x + x k − x ε k (cid:17) (¯ u k + ¯ u k )¯ η k =(2 + o (1)) ε r + dk Z R d V ( x + y ) w ¯ η , V is given by (1.21) and x ∈ Z with Z defined by (1.22). Moreover, since ∇ V ( x ) · x = rV ( x ), one can derive from (1.31) and (3.34) that ε k Z B δ ( x k ) (cid:2) ∇ V ( x ) · x (cid:3) (ˆ u k + ˆ u k )ˆ η k = ε k Z B δ ( x k ) (cid:2) ∇ V ( x − x + x ) · ( x − x + x ) (cid:3) (ˆ u k + ˆ u k )ˆ η k = ε k Z B δ ( x k ) (cid:2) ∇ V ( x − x ) + W ( x − x ) (cid:3) · ( x − x )(ˆ u k + ˆ u k )ˆ η k + ε k Z B δ ( x k ) (cid:2) ∇ V ( x ) · x (cid:3) (ˆ u k + ˆ u k )ˆ η k = ε k Z B δ ( x k ) (cid:2) rV ( x − x ) + W ( x − x ) (cid:0) x − x (cid:1)(cid:3) (ˆ u k + ˆ u k )ˆ η k + o (1) e − cδεk = r (1 + o (1))2 ε d + r +1 k Z B δεk (0) V (cid:16) x + x k − x ε k (cid:17) (¯ u k + ¯ u k )¯ η k + o (1) e − cδεk =(1 + o (1)) rε r + dk Z R d V ( x + y ) w ¯ η , where W := ( W , W , · · · , W d ). It then follows from these estimates that T = O (1) ε r + dk . (3.41)As for the left hand of (3.39), one can deduce from (3.40) and (3.41) that O (1) ε r + dk = d ( p − − p + 1) Z R d ˆ u p +11 k − ˆ u p +12 k k ˆ u k − ˆ u k k L ∞ ( R d ) = d ( p − − ε dk Z R d ¯ C pk ( x )¯ η k = d ( p − −
42 (1 + o (1)) ε dk Z R d w p ¯ η , (3.42)where ¯ C pk ( x ) is defined in (3.11) and the last “ = ” holds because ¯ C pk ( x ) → w p uniformlyin R d as k → ∞ . Using (3.18), then (3.42) gives that0 = Z R d w p ¯ η = Z R d w p h b (cid:0) w + p − x · ∇ w (cid:1) + d X i =1 b i ∂w∂x i i = b Z R d w p +1 + b p − p + 1 Z R d x · ∇ w p +1 + d X i =1 b i p + 1 Z R d ∂w p +1 ∂x i = b Z R d w p +1 − db p − p + 1 Z R d w p +1 = h − d p − p + 1 i b Z R d w p +1 . − d p − p +1 = 0 when 1 < p < d , one then has b = 0. Hence, we complete theproof of (3.35).Further, it follows from (3.19) that d X i =1 b i Z R d ∂ V ( x + y ) ∂x i ∂x j w = d X i =1 b i ∂ Q ( y ) ∂x i ∂x j = 0 , which implies from the non-degeneracy assumption of Q ( y ) that b i = 0 for i = 1 , , ...d .From (3.18) we thus have ¯ η ≡ R d .At last, we claim that ¯ η = 0 cannot occur. Suppose ¯ y k ∈ R d is a maximum point of¯ η k , and then | ¯ η k (¯ y k ) | = k ¯ η k k L ∞ ( R d ) = 1. It thus follows from (3.6) that ¯ g k (¯ y k ) + ¯ f k (¯ y k ) ≥ . One can further deduce from (3.13) and (3.16) that w (¯ y k ) ≥ C >
0, which impliesthat y k is bounded uniformly in k , due to the fact that w ( x ) decays exponentially as | x | → ∞ . Therefore, one can conclude from (3.9) that ¯ η R d , which howevercontradicts to the fact that ¯ η ≡ R d . Therefore, the proof of Theorem 1.3 iscomplete. A Appendix
A.1 Equivalence between ground states and constraint minimizers
This section is devoted to the proof of Theorem 1.1 on the equivalence between groundstates of equation (1.1) and constraint minimizers of problem (1.4). At first, we give thefollowing lemma.
Lemma A.1.
Suppose V ( x ) satisfies (1.3) , and u ρ ( x ) ≥ is a nonnegative minimizerof e ( ρ ) . For any ρ , ρ k ∈ (0 , ∞ ) satisfying ρ k → ρ as k → ∞ , passing to a subsequenceif necessary, there exists ¯ u ∈ M ρ such that u ρ k → ¯ u in H ( R d ) as k → ∞ . (A.1) Proof.
For any ρ , ρ k ∈ (0 , ∞ ), we have E ρ ( u ρ ) − E ρ k ( u ρ ) ≤ e ( ρ ) − e ( ρ k ) ≤ E ρ ( u ρ k ) − E ρ k ( u ρ k ) . One can thus derive that − ρ p − k − ρ p − p + 1 Z R d | u ρ k | p +1 d x ≤ e ( ρ k ) − e ( ρ ) ≤ − ρ p − k − ρ p − p + 1 Z R d | u ρ | p +1 d x, (A.2)which implies that e ( ρ ) is a decreasing function of ρ ∈ (0 , ∞ ) and lim k →∞ e ( ρ k ) = e ( ρ ),because k u ρ k k p +1 p +1 is bounded uniformly in k .From (1.5), one can further deduce that E ρ ( u ρ k ) = e ( ρ k ) + ρ p − k − ρ p − p + 1 Z R d | u ρ k | p +1 dx → e ( ρ ) as k → ∞ . (A.3)Let { u ρ k } be a minimizing sequence for e ( ρ ). Employing the Gagliardo-Nirenberg in-equality (1.7), one can derive from (1.5) that { u ρ k } is bounded uniformly in H ( R d ) with23espect to k . Applying the well-known compact embedding theorem (cf. [27, TheoremXIII.67]), one can deduce that passing to subsequence if necessary, u ρ k → ¯ u stronglyin L q ( R d ) with q ∈ [2 , ∗ ) for some ¯ u ∈ H . This gives the weak lower-semicontinuity of E ρ ( u ρ k ), and implies from (A.3) that e ( ρ ) = lim k →∞ E ρ ( u ρ k ) ≥ E ρ (¯ u ) ≥ e ( ρ ) ,i.e. , (A.1) holds. Hence, the proof of this lemma is completed.Next, we giving the following lemma on the differentiability of e ( ρ ). Lemma A.2.
Suppose V ( x ) satisfies (1.3) , and let u ρ ( x ) ≥ be a nonnegative minimizerof e ( ρ ) . Then e ( ρ ) is differentiable for a.e. ρ ∈ (0 , ∞ ) and e ′ ( ρ ) = − p − p + 1 ρ p − Z R d | u | p +1 d x, ∀ u ∈ M ρ . (A.4)Since the proof of this lemma is similar to that of [13, Lemma 2.2], we omit it here.Based on the proof of lemma A.2, we remark that for any given ρ ∈ (0 , ∞ ), if e ( ρ )admits a unique nonnegative or nonpositive minimizer, then e ′ ( ρ ) exists and satisfies(A.4). Proof of Theorem 1.1:
For any ρ ∈ (0 , ∞ ) and 0 ≤ u ρ ∈ M ρ , u ρ satisfies (1.1) forsome Lagrange multiplier µ ρ ∈ R . It then follows from (1.1), (1.4) and (A.4) that, for a.e. ρ ∈ (0 , ∞ ), µ ρ = 2 e ( ρ ) − p − p + 1 ρ p − Z R d | u | p +1 d x = 2 e ( ρ ) + ρe ′ ( ρ ) , (A.5)which implies that µ ρ depends only on the value of ρ and is independent of the choice of u ρ . This further indicates that, for a.e. ρ ∈ (0 , ∞ ), all minimizers of e ( ρ ) satisfy equation(1.1) with the same Lagrange multiplier µ ρ .Taking any u g ∈ G ρ,µ and setting ˜ u g = k u g k u g , one then has F ρ,µ (˜ u g ) ≥ F ρ,µ ( u g ).Since u g solves (1.1), one can derive from (1.12) that F ρ,µ ( u g ) = (cid:16) − p + 1 (cid:17) ρ p +1 Z R d | u g | p +1 d x and F ρ,µ (˜ u g ) = (cid:16) k u g k − p + 1) k u g k p +12 (cid:17) ρ p +1 Z R d | u g | p +1 d x. Therefore, we have 12 k u g k − p + 1) k u g k p +12 ≥ − p + 1 . (A.6)One can check that (A.6) holds if and only if k u g k = 1, i.e. , F ρ,µ (˜ u g ) = F ρ,µ ( u g ). Thisimplies that all ground states of equation (1.1) share the same L -norm, i.e. ,for any u g ∈ G ρ,µ , u g satisfies k u g k = 1.24or any u g ∈ G ρ,µ and u ρ ∈ M ρ , one has E ρ ( u g ) ≥ E ρ ( u ρ ) and F ρ,µ ( u ρ ) ≥ F ρ,µ ( u g ).Following from (1.5) and (1.12), one has F ρ,µ ( u ) = E ρ ( u ) − µ , (A.7)which indicates that E ρ ( u ρ ) ≥ E ρ ( u g ), i.e. , u g ∈ M ρ . One can further deduce from (A.5)that for a.e. ρ ∈ (0 , ∞ ), there holds that µ = µ ρ , which implies F ρ,µ ρ ( u g ) ≥ F ρ,µ ρ ( u ρ ), i.e. , u ρ ∈ G ρ,µ ρ . Hence we complete the proof of Theorem 1.1. A.2 Some results on the problem ˜ e ρ In this section, we focus on studying the following minimization problem˜ e ( ρ ) := inf (cid:8) ˜ E ρ ( u ) : u ∈ H ( R d ) , k u k = 1 (cid:9) , (A.8)where ˜ E ρ ( u ) is defined by˜ E ρ ( u ) := 12 Z R d |∇ u | d x − ρ p − p + 1 Z R d | u | p +1 d x, < p < d . (A.9)Employing the concentration-compactness principle, one can derive that ˜ e ( ρ ) admitsminimizers for any ρ ∈ (0 , ∞ ), see, e.g., [3, 20, 21]. Similar to problem (1.4), withoutloss of generality, we restrict the minimizers of problem (A.8) to nonnegative functions.We then give our results by the following lemma. Lemma A.3.
Suppose ˜ u ρ is a nonnegative minimizer of ˜ e ( ρ ) . Set ˜ α ρ := (cid:16) ρ √ a ∗ (cid:17) p − − d ( p − and a ∗ := k w k , where w is the unique positive solution of (1.6) . We then have ˜ e ( ρ ) = − λ (cid:16) ρ √ a ∗ (cid:17) p − − d ( p − , (A.10) and, up to translations, ˜ u ρ satisfies ˜ u ρ := 1 √ a ∗ ˜ α d ρ w ( α ρ x ) , (A.11) where λ is defined in (1.20) . Proof.
Suppose ˜ u ρ is a nonnegative minimizer of ˜ e ( ρ ) and ˜ u is a nonnegative minimizerof ˜ e (1). At first, we claim that˜ e ( ρ ) = ρ p − − d ( p − ˜ e (1) and ˜ u ρ = α d ρ ˜ u ( α ρ x ) , (A.12)where α ρ := ρ p − − d ( p − . In fact, setting ˜ v := α − d ρ ˜ u ρ ( α − ρ x ), one can check that˜ e ( ρ ) = ˜ E ρ (˜ u ρ ) = ρ p − − d ( p − h Z R d |∇ ˜ v | d x − p + 1 Z R d ˜ v p +11 d x i ≥ ρ p − − d ( p − ˜ e (1) . v ρ := α d ρ ˜ u ( α ρ x ), one can check that˜ e ( ρ ) ≤ ˜ E ρ (˜ v ρ ) = ρ p − − d ( p − ˜ e (1) . (A.13)The above two inequalities then give the first equality in (A.12). Furthermore, we knowthat ˜ v is a minimizer of ˜ e (1) and ˜ v ρ is a minimizer of ˜ e ( ρ ), which gives the secondequality in (A.12).Next, we claim that ˜ e (1) = − λ ( √ a ∗ ) − p − − d ( p − , (A.14)where λ is given by (1.20), and ˜ u ρ (up to translations) satisfies˜ u ( x ) = ( √ a ∗ ) − − d ( p − w (cid:0) ( √ a ∗ ) − p − − d ( p − x (cid:1) . (A.15)Take a test function ˜ v ǫ = ǫ d ˜ v ( ǫx ), where 0 < ˜ v ∈ H ( R d ) satisfies k ˜ v k = 1 and ǫ > e (1) ≤ ˜ E (˜ v ǫ ) = ǫ Z R d |∇ ˜ v | d x − ǫ d ( p − p + 1 Z R d | ˜ v | p +1 d x < , when ǫ is small enough.Let ˜ u be a nonnegative minimizer of ˜ e (1), and then ˜ u solves − ∆˜ u = ˜ µ ˜ u + ˜ u p , (A.16)where ˜ µ is a suitable Lagrange parameter. It follows from (A.9) and (A.16) that˜ µ = 2˜ e (1) − p − p + 1 Z R d ˜ u p +11 d x < . (A.17)Applying the maximum principle (cf. [6]) then yields that ˜ u >
0, which implies that,up to translations, ˜ u = ( − ˜ µ ) p − w (cid:0) ( − ˜ µ ) x (cid:1) , due to the fact that w is the unique positive solution of (1.6). Furthermore, since k ˜ u k = 1, one can then deduce that µ satisfies µ = −k w k − p − − d ( p − = − ( √ a ∗ ) − p − − d ( p − , which implies (A.15). Further, substituting (A.15) into (A.9) then yields (A.14).Combining the above two claims then yields (A.10) and (A.11), and this completesthe proof of Lemma A.3. Acknowledgements:
The authors are grateful to Yujin Guo for his fruitful discussionson the present paper. 26 eferences [1] W. Z. Bao and Y. Y. Cai,
Mathematical theory and numerical methods for Bose-Einstein condensation , Kinetic and Related Models (2013), 1–135.[2] 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.[3] T. Cazenave, Semilinear Schr¨odinger Equations, Courant Lecture Notes in Math. , Courant Institute of Mathematical Science/AMS, New York, (2003).[4] 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.[5] 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.[6] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of SecondOrder, Springer, (1997).[7] E. P. Gross, Structure of a quantized vortex in boson systems , Nuovo Cimento (1961), 454–466.[8] E. P. Gross, Hydrodynamics of a superfluid condensate , J. Math. Phys. (1963),195–207.[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] M. Grossi, and A. Pistoia, Locating the peak of ground states of nonlinearSchr¨odinger equations , Houston J. Math. (2005), 621–635.[11] Y. J. Guo, C. S. Lin and J. C. Wei, Local uniqueness and refined spike profilesof ground states for two-dimensional attractive Bose-Einstein condensates , SIAM J.Math. Anal.
49 (5) (2017), 3671–3715.[12] Y. J. Guo and R. Seiringer,
On the mass concentration for Bose-Einstein conden-sates with attractive interactions , Lett. Math. Phys. (2014), 141–156.[13] Y. J. Guo, Z. Q. Wang, X. Y. Zeng and H. S. Zhou,
Properties for ground statesof attractive Gross-Pitaevskii equations with multi-well potentials , Nonlinearity (2018), 957–979.[14] Y. J. Guo, X. Y. Zeng and H. S. Zhou, Concentration behavior of standing wavesfor almost mass critical nonlinear Schr¨odinger equations , J. Differential Equations,
256 (7) (2014), 2079–2100.[15] 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), 809–828.2716] Q. Han and F. H. Lin, Elliptic Partial Differential Equations, second edition,Courant Lect. Notes Math., vol. 1, Courant Institute of Mathematical Science/AMS,New York, (2011).[17] M. K. Kwong, Uniqueness of positive solutions of ∆ u − u + u p = 0 in R N , Arch.Rational Mech. Anal. (1989), 243–266.[18] Y. Li and W. M. Ni, Radial symmetry of positive solutions of nonlinear ellipticequations in R n , Comm. Partial Differential Equations (1993), 1043–1054.[19] S. Li, J. L. Xiang and X. Y. Zeng, Ground states of nonlinear Choquard equationswith multi-well potentials , J. Math. Phys. (2016), 081515.[20] P. L. Lions, The concentration-compactness principle in the caclulus of variations.The locally compact case. I , Ann. Inst H. Poincar´e. Anal. Non Lin´eaire (1984),109–145.[21] P. L. Lions, The concentration-compactness principle in the caclulus of variations.The locally compact case. II , Ann. Inst H. Poincar´e. Anal. Non Lin´eaire (1984),223–283.[22] M. Maeda, On the symmetry of the ground states of nonlinear Schr¨odinger equationwith potential , Adv. Nonlinear Stud. (2010), 895–925.[23] 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.[24] W. M. Ni and I. Takagi, On the shape of least-energy solutions to a semilinearNeumann problem , Comm. Pure Appl. Math. (1991), 819–851.[25] L. P. Pitaevskii, Vortex lines in an imperfect Bose gas , Sov. Phys. JETP (1961),451-454.[26] P. H. Rabinowitz, On a class of nonlinear Schr¨odinger equations , Z. Angew. Math.Phys. (1992), 270-291.[27] M. Reed and B. Simon, Methods of modern mathematical physics. IV. analysis ofoperators, Academic Press, New York-London, (1978).[28] X. F. Wang, On concentration of positive bound states of nonlinear Schr¨odingerequations , Comm. Math. Phys. (1993), 229–244.[29] M. I. Weinstein,
Nonlinear Schr¨odinger equations and sharp interpolations estimates ,Comm. Math. Phys. (1983), 567–576.[30] X. Y. Zeng and L. Zhang, Normalized solutions for Schr¨odinger-Poisson-Slater equa-tions with unbounded potentials , J. Math. Anal. Appl. (2017), 47–61.[31] X. Y. Zeng,
Asymptotic properties of standing waves for mass subcritical nonlinearSchr¨odinger equations , Discrete Contin. Dyn. Syst. A,
37 (3)37 (3)