Properties of minimizers for L2-subcritical Kirchhoff energy functionals

Helin Guo; Lingling Zhao

doi:10.1515/anona-2025-0081

Article Open Access

Properties of minimizers for L²-subcritical Kirchhoff energy functionals

Helin Guo and Lingling Zhao

Published/Copyright: April 25, 2025

Published by

Become an author with De Gruyter Brill

Submit Manuscript Author Information Explore this Subject

From the journal Advances in Nonlinear Analysis Volume 14 Issue 1

Abstract

We consider the properties of minimizers for the following constraint minimization problem:

i ( c ) ≔ inf u ∈ S 1 I c ( u ) ,

where the L 2 -unite sphere S 1 = { u ∈ H 1 ( R N ) ∣ ∫ R N V ( x ) u 2 d x < + ∞ , ‖ u ‖ L 2 = 1 } , and the energy functional I c ( u ) is defined by

I c ( u ) = a 2 ∫ R N ∣ ∇ u ∣ 2 d x + b 4 ∫ R N ∣ ∇ u ∣ 2 d x 2 + 1 2 ∫ R N V ( x ) u 2 d x − c p + 2 ∫ R N ∣ u ∣ p + 2 d x .

If V ( x ) ≡ 0 , we shall show the uniqueness of minimizers of i ( c ) for c > c ˜ p p ∈ 0 , 4 N , or c ≥ c ˜ p p ∈ 4 N , 8 N ; if V ( x ) ≢ 0 , the concentration behavior of minimizers for i ( c ) is discussed as c → + ∞ . Moreover, the uniqueness of minimizers for i ( c ) is also proved as c large enough, which extends the related results.

Keywords: subcritical; Kirchhoff energy functional; constrained variational method; energy estimates; local uniqueness

MSC 2010: 35J20; 35J60; 47J30

1 Introduction

In this article, we are concerned with the properties of minimizers for the following constrained minimization problem:

(1.1) i ( c ) = inf u ∈ S 1 I c ( u ) ,

where the Kirchhoff energy functional is defined by

(1.2) I c ( u ) = a 2 ∫ R N ∣ ∇ u ∣ 2 d x + b 4 ∫ R N ∣ ∇ u ∣ 2 d x 2 + 1 2 ∫ R N V ( x ) u 2 d x − c p + 2 ∫ R N ∣ u ∣ p + 2 d x ,

(1.3) S 1 = { u ∈ ℋ : ‖ u ‖ 2 = 1 } with ℋ ≜ u ∈ H 1 ( R N ) ∣ ∫ R N V ( x ) u 2 d x < + ∞ ,

where a , b > 0 are the positive constants, N ≤ 4 , c > 0 , p ∈ 0 , 8 N , and V ( x ) ≥ 0 is some type of trapping potential.

It is known that a minimizer of (1.1) corresponds to a normalized L 2 -norm solution for the following elliptic equation with μ ∈ R being a suitable Lagrange multiplier:

(1.4) − a + b ∫ R N ∣ ∇ u ∣ 2 d x Δ u + V ( x ) u = c ∣ u ∣ p u + μ u , x ∈ R N .

(1.4) is a steady-state equation of certain type generalized Kirchhoff equation; we call it as a Kirchhoff-type elliptic equation. Classical Kirchhoff equation was proposed in [18], which is essentially a modified one-dimensional wave equation and can be used to make a more accurate description on the transversal oscillations of a stretched string. More backgrounds and related results on the Kirchhoff equations can be seen in [1,4,14,17,18].

It is shown in [10] that (1.1) is the so-called L 2 -subcritical problem when p ∈ 0 , 8 N in the sense that for each c > 0 , i ( c ) > − ∞ if 0 < p < 8 N and i ( c ) = − ∞ if p > 8 N . Note that when a = 1 and b = 0 , (1.4) is related to the so-called Gross-Pitaevskii equation, which arises in the study of Bose-Einstein condensates (see, e.g., [3,5]). In recent years, problem (1.1) or (1.4) with a = 1 and b = 0 has received a lot of interest in mathematics (see, e.g., [7–9,12,21,26]). Particularly, Guo and Seiringer [7] established the existence of minimizers for problem (1.1) with a = 1 if V ( x ) is some type of coercive potential and showed some detailed analysis on the concentration behavior of minimizers for i ( c ) as c approaches a critical value. Under the basis of [7], the local uniqueness of minimizers for i ( c ) is discussed in [8] as c approaches to the critical value. Lately, Li and Zhu [21] showed the concentration and local uniqueness of minimizers for i ( c p ) as c large enough. For the case of b > 0 , there still have some related results (see e.g., [10,11,16,22,27]). Particularly, for any c > 0 , Guo et al. [10] proved that problem (1.1) has always a minimizer if p ∈ 0 , 8 N , a > 0 and V ( x ) is some trapping potential. Under the basis of [10], Guo and Zhou [11] established the asymptotic behavior and local uniqueness of minimizers of i ( c ) as b → 0 + for p = 4 N and a = 1 . If c = 1 and the sphere is S β instead of S 1 in problem (1.1), Li and Ye [22] showed the concentration behavior of minimizers for i ( 1 ) as β → + ∞ under a general coercive potential and the local uniqueness of minimizers for i ( 1 ) is also discussed in [16] as a → 0 + .

Motivated by [8,11,21,22], in this article, our aim is to establish some detailed analysis on the concentration behavior of minimizers for i ( c ) as c → + ∞ under a polynomial form potential instead of a general coercive potential in [22], and the local uniqueness of minimizers for i ( c ) is also discussed as c large enough. However, the methods in [8,21] cannot be used directly to deal with our case since the existence of the nonlocal term ∫ R N ∣ ∇ u ∣ 2 d x in (1.1). Particularly, we have to encounter much more complicated and technical calculation than in [11] when we discuss the uniqueness of minimizers for i ( c ) as c large enough. To overcome these difficulties, we shall use some new ideas to obtain the energy estimates and prove the uniqueness of minimizers for i ( c ) .

Before giving the main results of this article, we need to introduce the following Gagliardo-Nirenberg inequality with the best constant [25]:

(1.5) ∫ R N ∣ u ∣ p + 2 d x ≤ p + 2 2 ‖ Q p ‖ 2 p ∫ R N ∣ ∇ u ∣ 2 d x N p 4 ∫ R N ∣ u ∣ 2 d x 4 − p ( N − 2 ) 4 ,

where p ∈ [ 0 , 2 ∗ − 2 ) if N ≥ 3 with 2 ∗ = 2 N N − 2 or p ∈ [ 0 , + ∞ ) if N = 1,2 , and the equality is achieved for Q p with Q p being the unique (up to translations) positive solution of the following elliptic equation:

(1.6) − Δ u + 4 − p ( N − 2 ) N p u = 4 N p u p + 1 , x ∈ R N .

Moreover, all optimizers of Gagliardo-Nirenberg inequality belong to the following set:

(1.7) { α Q p ( β x + y ) , α , β ∈ R + , y ∈ R N } .

Applying Lemma 8.1.2 in [2], we can obtain the following Pohozaev’s identity:

(1.8) ( N − 2 ) ∫ R N ∣ ∇ Q p ∣ 2 d x + 4 − p ( N − 2 ) p ∫ R N ∣ Q p ∣ 2 d x = 8 p ( p + 2 ) ∫ R N ∣ Q p ∣ p + 2 d x .

It then follows from (1.6) and (1.8) that Q p satisfies

(1.9) ∫ R N ∣ ∇ Q p ∣ 2 d x = ∫ R N ∣ Q p ∣ 2 d x = 2 p + 2 ∫ R N ∣ Q p ∣ p + 2 d x .

Note also from [13, Proposition 4.1] that Q p has the following decay estimate:

(1.10) Q p ( x ) , ∣ ∇ Q p ( x ) ∣ = O ( ∣ x ∣ − N − 1 2 e − ∣ x ∣ ) , as ∣ x ∣ → + ∞ .

Also, we need to introduce the auxiliary minimization problem as follows when we establish the detailed analysis on the concentration behavior of minimizers for i ( c ) as c → + ∞ . If V ( x ) ≡ 0 , then (1.1) can be written as

(1.11) i ˜ ( c ) = inf u ∈ S ˜ 1 I ˜ c ( u ) ,

where S ˜ 1 = { u ∈ H 1 ( R N ) ∣ ‖ u ‖ 2 = 1 } and

(1.12) I ˜ c ( u ) = a 2 ∫ R N ∣ ∇ u ∣ 2 d x + b 4 ∫ R N ∣ ∇ u ∣ 2 d x 2 − c p + 2 ∫ R N ∣ u ∣ p + 2 d x .

When a > 0 , b > 0 , the existence of minimizers for i ˜ ( c ) can be seen in Proposition 1.1 as follows. Since ∣ ∇ u ∣ = ∣ ∇ ∣ u ∣ ∣ holds for a.e. x ∈ R N , without loss of generality, we always suppose that minimizers of i ( c ) and i ˜ ( c ) are nonnegative. Now, we introduce some known results for the case of V ( x ) ≡ 0 .

Proposition 1.1

[10, Theorem 1.1] For p ∈ 0 , 8 N , problem (1.11) has a minimizer if and only if

(1.13) c > c ˜ p , w i t h 0 < p ≤ 4 N , o r c ≥ c ˜ p , w i t h 4 N < p < 8 N ,

where

(1.14) c ˜ p = 0 , i f 0 < p < 4 N , a ‖ Q p ‖ 2 4 N , i f p = 4 N , 2 ‖ Q p ‖ 2 p 2 a 8 − N p 8 − N p 4 b N p − 4 N p − 4 4 , i f 4 N < p < 8 N ,

and Q p is the unique positive solution of (1.6).

Set

(1.15) c ∗ = ‖ Q p ‖ 2 ,

then we have the following existence results for the case of V ( x ) ≡ 0 .

Theorem 1.2

Assume that p ∈ 0 , 8 N and (1.13) hold. Then, (1.11) has a unique minimizer (up to translations), which is the form

(1.16) u ˜ c ( x ) = t c N 2 c ∗ Q p ( t c x ) ,

where t c is the unique minimum point of

(1.17) f c ( t ) = a 2 t 2 + b 4 t 4 − c 2 c ∗ p t N p 2 , t ∈ ( 0 , + ∞ ) .

Moreover,

If p = 4 N ,
(1.18) t c = c − c ˜ p b c ∗ p 1 2 and i ˜ ( c ) = f c ( t c ) = − ( c − c ˜ p ) 2 4 b c ∗ 2 p .
If p ∈ ( 0 , 4 N ) ⋃ ( 4 N , 8 N ) ,
(1.19) t c = c N p 4 b c ∗ p 2 8 − N p ( 1 + o ( 1 ) ) , as c → + ∞ ,
and
(1.20) i ˜ ( c ) = f c ( t c ) = − b ( 8 − N p ) 4 N p c N p 4 b c ∗ p 8 8 − N p ( 1 + o ( 1 ) ) a s c → + ∞ .

Besides, when V ( x ) ≢ 0 satisfies the following condition:

(1.21) 0 ≢ V ( x ) ∈ C ( R N , R + ) , lim ∣ x ∣ → ∞ V ( x ) = + ∞ , and inf x ∈ R N V ( x ) = 0 .

It was proved in [10, Theorem 1.2] that problem (1.1) has at least one minimizer for any fixed c > 0 if p ∈ 0 , 8 N and N ≤ 4 . Under the basis of existence results of minimizers for (1.1), in the following, we shall give some detailed analysis on the concentration behavior of the minimizers for (1.1) under some polynomial-type potential. First, we introduce some definitions and assumptions.

Definition 1.3

A function f ( x ) is called homogeneous of degree q ∈ R + (about the origin) if there exists some q > 0 such that

f ( t x ) = t q f ( x ) , in R N for any t > 0 .

The aforementioned definition implies that if f ( x ) ∈ C ( R N ) is homogeneous of degree q > 0 , then

0 ≤ f ( x ) ≤ C ∣ x ∣ q , in R N ,

where C denotes the maximum of f ( x ) on ∂ B 1 ( 0 ) . Moreover, if f ( x ) → + ∞ as ∣ x ∣ → + ∞ , then 0 is the unique minimum point of f ( x ) .

Inspired by [8,11], for generality, in what follows, we assume that V ( x ) has exactly m global minimum points, namely,

(1.22) Z ≔ { x ∈ R N : V ( x ) = 0 } = { x 1 , x 2 , … , x m } , where m ≥ 1 .

We then assume that V ( x ) is almost homogeneous of degree r i > 0 around x i . Particularly, there exists some V i ( x ) ∈ C loc 2 ( R N ) satisfying lim ∣ x ∣ → + ∞ V i ( x ) = + ∞ , which is homogeneous of degree r i > 0 , such that

(1.23) lim x → 0 V ( x + x i ) V i ( x ) = 1 , i = 1 , 2 , … , m .

Additionally, we define H i ( y ) by

(1.24) H i ( y ) ≔ ∫ R N V i ( x + y ) Q p 2 ( x ) d x , i = 1 , 2 , … m .

Set

(1.25) r ≔ max 1 ≤ i ≤ m r i , and Z ¯ ≔ { x i ∈ Z : r i = r } ⊂ Z .

Define

(1.26) λ 0 ≔ min i ∈ Λ λ i , where λ i ≔ min y ∈ R N H i ( y ) and Λ ≔ { i : x i ∈ Z ¯ } .

Denote

(1.27) Z 0 ≔ { x i ∈ Z ¯ : λ i = λ 0 }

the set of the flattest minimum points of V ( x ) . Under the aforementioned assumptions, we have the following blow-up behavior of minimizers for i ( c ) as c → + ∞ .

Theorem 1.4

Assume that V ( x ) satisfies (1.21) and (1.23). Let p ∈ 0 , 8 N and u k be the minimizer of i ( c k ) , where c k → + ∞ as k → + ∞ . Then, there exists a subsequence, still denoted by { u k } , such that

(1.28) lim k → + ∞ c ∗ ε k N 2 u k ( ε k x + z k ) = Q p ( x ) , i n H 1 ( R N ) ,

where

(1.29) ε k ≔ 4 b c ∗ p c k N p 2 8 − N p ,

with c ∗ being defined in (1.15), and z k is the unique maximum point of u k satisfying

(1.30) lim k → ∞ z k − x i 0 ε k = y 0 , f o r s o m e x i 0 ∈ Z 0 and y 0 ∈ R N ,

with H i 0 ( y 0 ) = min y ∈ R N H i 0 ( y ) = λ 0 . Moreover, for p = 4 N ,

(1.31) i ( c k ) = − ( c k − c ˜ p ) 2 4 b c ∗ 2 p ( 1 + o ( 1 ) ) , a s k → + ∞ ,

and for p ∈ ( 0 , 4 N ) ⋃ ( 4 N , 8 N ) ,

(1.32) i ( c k ) = − b ( 8 − N p ) 4 N p c k N p 4 b c ∗ p 8 8 − N p ( 1 + o ( 1 ) ) , as k → + ∞ .

Next, we show the local uniqueness of minimizers of i ( c ) as c → + ∞ .

Theorem 1.5

Assume that p ∈ 0 , 8 N and V ( x ) ∈ C 2 ( R N ) satisfy (1.21) and (1.23). Moreover, we assume that Z 0 = { x 1 } holds in (1.27) and

(1.33) y 0 i s ; t h e u n i q u e a n d n o n - d e g e n e r a t e c r i t i c a l p o i n t o f H 1 ( y ) ,

where H 1 ( y ) is defined as (1.24). Suppose also that there exist γ > 0 and R 0 > 0 such that

(1.34) V ( x ) ≤ C e γ ∣ x ∣ , i f ∣ x ∣ i s l a r g e ,

and

(1.35) ∂ V ( x + x 1 ) ∂ x j = ∂ V 1 ( x ) ∂ x j + W j ( x ) , w i t h ∣ W j ( x ) ∣ ≤ C ∣ x ∣ s j in B R 0 ( 0 ) ,

where s j > r − 1 for j = 1 , 2 , … , N . Then, there exists a unique minimizer of i ( c ) as c → + ∞ .

This article is organized as follows. In Section 2, we focus on the proof of Theorem 1.2 and the estimates of i ˜ ( c ) . In Section 3, we shall give the detailed energy estimates of i ( c ) to complete the proof of Theorem 1.4. In Section 4, the local uniqueness of minimizer for i ( c ) as c → + ∞ will be shown.

2 Proof of Theorem 1.2

Proof of Theorem 1.2

For any u ∈ H 1 ( R N ) with ‖ u ‖ 2 = 1 , we can obtain from (1.5) and (1.15) that

(2.1) I ˜ c ( u ) ≥ a 2 ∫ R N ∣ ∇ u ∣ 2 d x + b 4 ∫ R N ∣ ∇ u ∣ 2 d x 2 − c 2 c ∗ p ∫ R N ∣ ∇ u ∣ 2 d x N p 4 .

Define

(2.2) f c ( t ) = a 2 t 2 + b 4 t 4 − c 2 c ∗ p t N p 2 .

Then, (2.1) shows that

(2.3) i ˜ ( c ) ≥ inf t ∈ [ 0 , + ∞ ) f c ( t ) = f c ( t c ) ,

where t c is the unique minimum point of (2.2). Take u c ( x ) = t c N 2 Q p ( t c x ) c ∗ , then ‖ u c ‖ 2 = 1 , it follows from (1.9) that

I ˜ c ( u c ) = a 2 t c 2 + b 4 t c 4 − c 2 c ∗ p t c N p 2 = f c ( t c ) ,

which implies that

(2.4) i ˜ ( c ) = I ˜ c ( u c ) = f c ( t c ) ,

and u c is a minimzier of i ˜ ( c ) .

For any minimizer u 0 of i ˜ ( c ) , one can obtain from (2.1) that

(2.5) i ˜ ( c ) = I ˜ c ( u 0 ) ≥ f c ( t 0 ) , where t 0 = ∫ R N ∣ ∇ u 0 ∣ 2 d x 1 2 ,

and the equality holds if and only if u 0 is an optimizer of (1.5). Moreover, (2.4) and (2.5) imply that t 0 = t c , and the equality in (2.5) holds. Hence, u 0 is an optimizer of (1.5), and it follows from (1.7) that

u 0 = α Q p ( β x ) , for some α > 0 , β > 0 .

Then,

∫ R N ∣ ∇ u 0 ∣ 2 d x = α 2 β 2 − N ∫ R N ∣ ∇ Q p ∣ 2 d x = t c 2

and

∫ R N ∣ u 0 ∣ 2 d x = α 2 β − N ∫ R N ∣ Q p ∣ 2 d x = 1 ,

which implies from (1.9) that

α = t c N 2 c ∗ and β = t c .

Hence, u 0 ( x ) = t c N 2 Q p ( t c x ) c ∗ = u c ( x ) , and the uniqueness of minimizer is proved.

Next, if p = 4 N , one can obtain from (1.14) and (2.2) that

(2.6) f c ( t ) = b 4 t 4 − c − c ˜ p 2 c ∗ p t 2 .

Hence, the unique minimum point of f c ( ⋅ ) is the form

t c = c − c ˜ p b c ∗ p 1 2 .

It then follows from (2.4) that

i ˜ ( c ) = f c ( t c ) = − ( c − c ˜ p ) 2 4 b c ∗ 8 N ,

and (1.18) holds.

If p ∈ ( 0 , 4 N ) ⋃ ( 4 N , 8 N ) , for any u ∈ H 1 ( R N ) with ‖ u ‖ 2 = 1 , we can obtain from (2.1) that

I ˜ c ( u ) ≥ b 4 ∫ R N ∣ ∇ u ∣ 2 d x 2 − c 2 c ∗ p ∫ R N ∣ ∇ u ∣ 2 d x N p 4 .

Define

(2.7) h ( r ) ≔ b 4 r 4 − c 2 c ∗ p r N p 2 ( r > 0 ) .

Then, the function h ( ⋅ ) has a unique minimum point, denoted by r c , which is the form

(2.8) r c = c N p 4 b c ∗ p 2 8 − N p .

Hence,

(2.9) i ˜ ( c ) ≥ inf r ≥ 0 h ( r ) = h ( r c ) = − b ( 8 − N p ) 4 N p c N p 4 b c ∗ p 8 8 − N p .

On the other hand, take u r ( x ) = r N 2 Q p ( r x ) c ∗ , then ‖ u r ‖ 2 = 1 , and

I ˜ c ( u r ) = a 2 r 2 + b 4 r 4 − c 2 c ∗ p r N p 2 .

Set r 0 = r c = c N p 4 b c ∗ p 2 8 − N p , and we have

i ˜ ( c ) ≤ a 2 c N p 4 b c ∗ p 4 8 − N p − b ( 8 − N p ) 4 N p c N p 4 b c ∗ p 8 8 − N p = − b ( 8 − N p ) 4 N p c N p 4 b c ∗ p 8 8 − N p ( 1 + o ( 1 ) ) , as c → + ∞ ,

which together with (2.9) concludes that

(2.10) i ˜ ( c ) = − b ( 8 − N p ) 4 N p c N p 4 b c ∗ p 8 8 − N p ( 1 + o ( 1 ) ) , as c → + ∞ ,

and (1.20) holds.

Next, we come to prove (1.19) by contradiction:

(2.11) lim c → + ∞ t c r c → θ ,

where r c is defined in (2.8). We claim that there is always a contradiction either θ ∈ [ 0 , 1 ) or θ > 1 .

In fact, if θ ∈ [ 0 , 1 ) , then there exists ε > 0 such that δ ≜ θ + ε < 1 and t c r c ≤ δ as c → + ∞ . Since h ( r c ) < 0 by (2.9), one can obtain from (2.7) and (2.8) that

lim c → + ∞ i ˜ ( c ) h ( r c ) ≤ lim c → + ∞ h ( t c ) h ( r c ) ≤ lim c → + ∞ h ( δ r c ) h ( r c ) = 8 δ N p 2 − N p δ 4 8 − N p ∈ [ 0 , 1 ) , if δ ∈ [ 0 , 1 ) ,

which is impossible since lim c → + ∞ i ˜ ( c ) h ( r c ) = 1 by (2.9) and (2.10).

Similarly, if θ > 1 , we can also obtain the contradiction.

If the limit in (2.11) does not exist, then there exist two constants m 1 ∈ ( 0 , + ∞ ] and m 2 ∈ [ 0 , + ∞ ) such that

limsup c → + ∞ t c r c = m 1 and liminf c → + ∞ t c r c = m 2 .

Since m 1 ≠ m 2 , we have m 1 ≠ 1 , or m 2 ≠ 1 . Then, we can have a contradiction by repeating the aforementioned proof, and θ = 1 in (2.11). Hence, (1.19) holds by (2.8) and (2.11).□

3 Blow-up analysis of minimizers

Lemma 3.1

For p ∈ 0 , 8 N , suppose V ( x ) satisfies (1.21). Let u c be the minimizer of i ( c ) . Then,

(3.1) 0 ≤ i ( c ) − i ˜ ( c ) → 0 , a s c → + ∞ ,

and

(3.2) ∫ R N V ( x ) u c 2 d x → 0 , a s c → + ∞ .

Proof

By the definitions of i ( c ) and i ˜ ( c ) , one can easy to see that

(3.3) i ( c ) − i ˜ ( c ) ≥ 0 .

Take 0 ≤ ξ ( x ) ∈ C 0 ∞ ( R N ) be a cut-off function such that

(3.4) ξ ( x ) ≡ 1 , if ∣ x ∣ ≤ 1 , ξ ( x ) ≡ 0 , if ∣ x ∣ ≥ 2 , and 0 ≤ ξ ( x ) ≤ 1 , if 1 ≤ ∣ x ∣ ≤ 2 .

For any x 0 ∈ R N , set

(3.5) u ¯ c ( x ) = A c ξ ( x − x 0 ) u ˜ c ( x − x 0 ) ,

where u ˜ c ( x ) is the unique nonnegative minimizer for i ˜ ( c ) , and A c > 1 is chosen so that ∫ R N u ¯ c 2 d x = 1 . Using (1.9) and the exponential decay of Q p ( x ) in (1.10), (1.16), (1.18), and (1.19), direct calculations show that

(3.6) 0 ≤ A c 2 − 1 = ∫ R N [ 1 − ξ 2 ( t c − 1 x ) ] Q p 2 ( x ) d x ∫ R N ξ 2 ( t c − 1 x ) Q p 2 ( x ) d x ≤ C e − 2 t c , as c → + ∞ ,

(3.7) ∫ R N u ¯ c p + 2 d x = A c p + 2 t c N p 2 ( c ∗ ) p + 2 ∫ R N ξ p + 2 ( t c − 1 x ) Q p p + 2 ( x ) d x ≤ t c N p 2 ( c ∗ ) p + 2 ∫ R N Q p p + 2 ( x ) d x + C e − 4 t c = ∫ R N u ˜ c p + 2 d x + C e − 4 t c , as c → + ∞ ,

and

(3.8) ∫ R N V ( x ) u ¯ c 2 ( x ) d x = A c 2 c ∗ 2 ∫ R N V ( t c − 1 x + x 0 ) ξ 2 ( t c − 1 x ) Q p 2 ( x ) d x → V ( x 0 ) , as c → + ∞ .

Similarly, we can also show that

(3.9) ∫ R N ∣ ∇ u ¯ c ∣ 2 d x ≤ ∫ R N ∣ ∇ u ˜ c ∣ 2 d x + C e − 2 t c , as c → + ∞ .

Taking x 0 ∈ R N such that V ( x 0 ) = 0 , then we can deduce from the aforementioned estimates and (3.3) that

0 ≤ i ( c ) − i ˜ ( c ) ≤ I c ( u ¯ c ) − I ˜ c ( u ˜ c ) = I ˜ c ( u ¯ c ) − I ˜ c ( u ˜ c ) + 1 2 ∫ R N V ( x ) u ¯ c 2 ( x ) d x ≤ 1 2 V ( x 0 ) + C e − 2 t c + o ( 1 ) → 0 , as c → + ∞ ,

and hence, (3.1) holds. Moreover, since u c is a minimizer for i ( c ) , we have

∫ R N V ( x ) u c 2 d x = I c ( u c ) − I ˜ c ( u c ) ≤ i ( c ) − i ˜ ( c ) → 0 , as c → + ∞ ,

which indicates (3.2) holds and the proof of the lemma is complete.□

Lemma 3.2

For p ∈ 0 , 8 N , suppose that V ( x ) satisfies (1.21). Let u c be the minimizer of i ( c ) . Then,

(3.10) ∫ R N ∣ ∇ u c ∣ 2 d x ∫ R N ∣ ∇ u ˜ c ∣ 2 d x → 1 , a s c → + ∞ ,

where u ˜ c ( x ) is the unique minimizer for i ˜ ( c ) and defined in (1.16).

Proof

First, (1.8), (1.15), and (1.16) indicate that

(3.11) ∫ R N ∣ ∇ u ˜ c ∣ 2 d x = t c 2 c ∗ 2 ∫ R N ∣ ∇ Q p ∣ 2 d x = t c 2 .

If there exists some θ ≥ 0 such that

(3.12) ∫ R N ∣ ∇ u c ∣ 2 d x ∫ R N ∣ ∇ u ˜ c ∣ 2 d x → θ , as c → + ∞ ,

we claim that there is always a contradiction either θ ∈ [ 0 , 1 ) or θ ≥ 1 .

In fact, if θ ∈ [ 0 , 1 ) , one can obtain from (3.11) and (3.12) that there exists ε > 0 such that δ ≜ θ + ε < 1 and

(3.13) ∫ R N ∣ ∇ u c ∣ 2 d x t c 2 ≤ δ , as c → + ∞ .

For p = 4 N , it then follows from (1.18), (3.1), and (3.13) that

(3.14) lim c → + ∞ i ( c ) i ˜ ( c ) ≤ lim c → + ∞ f c ( δ t c ) f c ( t c ) = − δ 2 + 2 δ ∈ [ 0 , 1 ) , for all δ ∈ [ 0 , 1 ) ,

where f c ( ⋅ ) is defined by (2.2) and has a unique minimum point at t c .

For p ∈ ( 0 , 4 N ) ⋃ ( 4 N , 8 N ) , we can also obtain from (1.19), (1.20), (2.8), (3.1), and (3.13) that

(3.15) lim c → + ∞ i ( c ) i ˜ ( c ) ≤ lim c → + ∞ f c ( δ t c ) f c ( t c ) ≤ lim c → + ∞ h c ( δ t c ) f c ( t c ) = 8 δ N p 4 − N p δ 2 8 − N p ∈ [ 0 , 1 ) , for all δ ∈ [ 0 , 1 ) ,

where h ( ⋅ ) is defined by (2.7).

Moreover, (3.1) indicates that

lim c → + ∞ i ( c ) i ˜ ( c ) = 1 , for p ∈ 0 , 8 N ,

which, however, contradicts with (3.14) and (3.15).

Similarly, if θ > 1 , we can also obtain the contradiction.

If the limit in (3.10) does not exist, then there exist two constants m 1 ∈ ( 0 , + ∞ ] and m 2 ∈ [ 0 , + ∞ ) such that

limsup c → + ∞ ∫ R N ∣ ∇ u c ∣ 2 d x ∫ R N ∣ ∇ u ˜ c ∣ 2 d x = m 1 and liminf c → + ∞ ∫ R N ∣ ∇ u c ∣ 2 d x ∫ R N ∣ ∇ u ˜ c ∣ 2 d x = m 2 .

Since m 1 ≠ m 2 , we have m 1 ≠ 1 , or m 2 ≠ 1 . Then, we can have a contradiction by repeating the aforementioned proof, and θ = 1 in (3.12). Hence, (3.10) holds.□

Lemma 3.3

For p ∈ 0 , 8 N , suppose that V ( x ) satisfies (1.21). Let u k be the minimizer of i ( c k ) , and z k be one global maximum point of u k , where c k → + ∞ as k → + ∞ . Set

(3.16) w k ( x ) = c ∗ ε k N 2 u k ( ε k x + z k ) , w h e r e ε k i s d e f i n e d b y ( 1.29 ) .

Then,

There exists some M > 0 such that
(3.17) liminf k → + ∞ ∫ B 2 ( 0 ) w k 2 ( x ) d x ≥ M .
Passing to a subsequence if necessary, there exists a z 0 ∈ R N such that
(3.18) z k → z 0 a s k → + ∞ , w i t h V ( z 0 ) = 0 .

Proof

(i) Since u k is a minimizer for i ( c k ) , one can obtain that u k satisfies the following Euler-Lagrange equation:

(3.19) − a + b ∫ R N ∣ ∇ u k ∣ 2 d x Δ u k + V ( x ) u k = μ k u k + c k u k p + 1 , x ∈ R N ,

where μ k ∈ R is a suitable Lagrange multiplier associated with u k , and

(3.20) i ( c k ) = 1 2 ∫ R N [ a ∣ ∇ u k ∣ 2 + V ( x ) u k 2 ] d x + b 4 ∫ R N ∣ ∇ u k ∣ 2 d x 2 − c k p + 2 ∫ R N u k p + 2 d x .

Moreover, (3.19) and (3.20) indicate that

(3.21) μ k = ∫ R N [ a ∣ ∇ u k ∣ 2 + V ( x ) u k 2 ] d x + b ∫ R N ∣ ∇ u k ∣ 2 d x 2 − c k ∫ R N u k p + 2 d x = 2 i ( c k ) + b 2 ∫ R N ∣ ∇ u k ∣ 2 d x 2 − p c k p + 2 ∫ R N u k p + 2 d x .

It then follows from (1.18)–(1.20), Lemmas 3.1 and 3.2 that

(3.22) ε k N p 2 ∫ R N u k p + 2 d x → p + 2 2 c ∗ p , as k → + ∞ ,

and

(3.23) μ k ε k 4 → − b [ 4 − p ( N − 2 ) ] N p , as k → + ∞ .

Hence,

(3.24) lim k → + ∞ ∫ R N w k p + 2 d x = lim k → + ∞ c ∗ p + 2 ε k N p 2 ∫ R N u k p + 2 d x = ( p + 2 ) c ∗ 2 2 .

Using (3.16) and (3.19), we have

(3.25) − a ε k 2 + b c ∗ 2 ∫ R N ∣ ∇ w k ∣ 2 d x Δ w k + ε k 4 V ( ε k x + z k ) w k ( x ) = μ k ε k 4 w k ( x ) + 4 b N p w k p + 1 ( x ) .

Hence, as k large enough, we can deduce from (3.23) that

(3.26) − Δ w k − c ( x ) w k ≤ 0 , where c ( x ) = 8 N p w k p ( x ) .

Then, applying the De-Giorgi-Nash-Morse theory (similar to the proof of [15, Theorem 4.1]), we deduce that

(3.27) max B 1 ( ξ ) w k ( x ) ≤ C ∫ B 2 ( ξ ) w k 2 d x 1 2 ,

where ξ is an arbitrary point in R N and C is a constant depending only on the bound of ‖ w k ‖ L 5 ( B 2 ( ξ ) ) . Since z k is one global maximum point of u k , 0 is also one global maximum point of w k . Then, as k is large enough, (3.24) and the vanishing lemma (see [24, Lemma 1.21]) imply that

w k ( 0 ) ≥ M , for some M > 0 ,

which implies (3.17) since (3.27).

(ii) Using (3.2), one can obtain

0 = liminf k → + ∞ ∫ R N V ( x ) u k 2 d x ≥ liminf k → + ∞ 1 c ∗ 2 ∫ B 2 ( 0 ) V ( ε k x + z k ) w k 2 d x .

Since V ( x ) → + ∞ as ∣ x ∣ → + ∞ , one can obtain from (3.17) that z k is uniformly bounded in k , and up to a subsequence if necessary, there exists a z 0 ∈ R N such that (3.18) holds.□

Proof of Theorem 1.4

Let u k be a minimizer for i ( c k ) , where c k → + ∞ as k → + ∞ . Then, one can obtain from (1.18), (1.19), (1.29), and (3.10) that

(3.28) lim k → + ∞ ∫ R N ∣ ∇ w k ∣ 2 d x = lim k → + ∞ ∫ R N w k 2 d x = c ∗ 2 , where w k is defined by (3.16)

Equation (3.28) implies that { w k } is a bounded sequence uniformly in H 1 ( R N ) , and up to a subsequence if necessary, there exists a w 0 ∈ H 1 ( R N ) such that

(3.29) w k ⇀ w 0 ≥ 0 , in H 1 ( R N ) as k → + ∞ ,

where w 0 ( x ) ≢ 0 since (3.17), and w 0 ( x ) ≥ 0 by strong maximum principle. Since w k satisfies (3.25), using (1.29) and (3.23), passing to the weak limit of (3.25), one can obtain w 0 satisfies

(3.30) − Δ w 0 + 4 − p ( N − 2 ) N p w 0 = 4 N p w 0 p + 1 , x ∈ R N .

After a simple scaling, there exists some x 0 ∈ R N such that

(3.31) w 0 ( x ) = Q p ( x − x 0 ) ,

where Q p ( x ) is the unique (up to translations) radially symmetric positive solution of (1.6). Combining (1.15), (3.28), (3.29), and (3.31), one can obtain that

(3.32) w k → w 0 = Q p ( x − x 0 ) , in H 1 ( R N ) as k → + ∞ .

Since 0 is the global maximum point of w k , it follows that w 0 attains its maximum point at the same point. Hence, we have x 0 = 0 by the uniqueness of maximum point of Q p . Therefore, (1.28) holds since (3.32). Furthermore, (3.25) and (3.31) indicate that w k → w 0 in C loc 2 ( R N ) as k → + ∞ since (1.21). Similar to the proof of [9, Theorem 1.1], we deduce that z k is the unique maximum point of u k as k → + ∞ .

Next, take x i 0 ∈ Z 0 and y 0 ∈ R N , where Z 0 is defined as (1.27) and y 0 satisfies H i 0 ( y 0 ) = λ 0 . Set

u ¯ k ( x ) = A k ξ ( x − x i 0 − t k − 1 y 0 ) u ˜ k ( x − x i 0 − t k − 1 y 0 ) ,

where ξ and t k ≜ t c k defined by (3.4), (1.18), and (1.19) respectively, and u ˜ k be the unique minimizer of i ˜ ( c k ) . Similar to (3.6)–(3.9), we can deduce from (1.23) that

(3.33) i ( c k ) − i ˜ ( c k ) ≤ 1 2 ∫ R N V ( x ) u ¯ k 2 d x + O ( e − 2 t k ) = 1 2 c ∗ 2 ∫ R N V ( t k − 1 x + x i 0 + t k − 1 y 0 ) ξ 2 ( t k − 1 x ) Q p 2 ( x ) d x + O ( e − 2 t k ) = 1 2 c ∗ 2 ∫ R N V ( t k − 1 x + x i 0 + t k − 1 y 0 ) V i 0 ( t k − 1 x + t k − 1 y 0 ) V i 0 ( t k − 1 x + t k − 1 y 0 ) ξ 2 ( t k − 1 x ) Q p 2 ( x ) d x + O ( e − 2 t k ) = 1 2 c ∗ 2 t k r ∫ R N V ( t k − 1 x + x i 0 + t k − 1 y 0 ) V i 0 ( t k − 1 x + t k − 1 y 0 ) V i 0 ( x + y 0 ) ξ 2 ( t k − 1 x ) Q p 2 ( x ) d x + O ( e − 2 t k ) = λ 0 2 c ∗ 2 t k r ( 1 + o ( 1 ) ) , as k → + ∞ .

Moreover, using (1.18), (1.19), and (1.29), we have

(3.34) lim k → + ∞ t k ε k = 1 .

Combining (3.33) and (3.34), one can obtain

(3.35) limsup k → + ∞ i ( c k ) − i ˜ ( c k ) ε k r ≤ λ 0 2 c ∗ 2 .

On the other hand, we know from (3.18) that the unique maximum point z k of u k satisfies z k → x 0 with V ( x 0 ) = 0 as k → + ∞ . We may assume x 0 = x i 0 for some 1 ≤ i 0 ≤ m . It then follows from (3.32) that

(3.36) liminf k → + ∞ i ( c k ) − i ˜ ( c k ) ε k r ≥ liminf k → + ∞ 1 2 ε k r ∫ R N V ( x ) u k 2 d x = liminf k → + ∞ 1 2 c ∗ 2 ε k r ∫ R N V ( ε k x + z k ) w k 2 d x = liminf k → + ∞ 1 2 c ∗ 2 ε k r − r i 0 ∫ R N V ( ε k x + z k − x i 0 + x i 0 ) V i 0 ( ε k x + z k − x i 0 ) V i 0 x + z k − x i 0 ε k w k 2 d x .

Combining (3.35) and (3.36), we deduce from (1.23) that r i 0 = r and z k − x i 0 ε k is bounded uniformly in k . Up to a subsequence if necessary, there exists y 0 ∈ R N such that

(3.37) z k − x i 0 ε k → y 0 , as k → + ∞ .

It then follows from (3.36) that

(3.38) liminf k → + ∞ i ( c k ) − i ˜ ( c k ) ε k r ≥ 1 2 c ∗ 2 ∫ R N V i 0 ( x + y 0 ) Q p 2 d x ≥ λ i 0 2 c ∗ 2 ≥ λ 0 2 c ∗ 2 ,

where all equalities hold in (3.38) if and only if H i 0 ( y 0 ) = λ i 0 = λ 0 . Combining (3.35) and (3.38), we have

(3.39) lim k → + ∞ i ( c k ) − i ˜ ( c k ) ε k r = λ 0 2 c ∗ 2 .

Therefore, all equalities in (3.38) hold, and hence, (1.30) holds since (3.37). Moreover, (1.28) follows from (3.32), (1.31), and (1.32) follows from (1.18), (1.20), and (3.39).□

4 Local uniqueness of minimizers: Proof of Theorem 1.5

In this section, we devote to prove the uniqueness of minimizers for i ( c ) as c → + ∞ for p ∈ 0 , 8 N . We shall prove it by contradiction. Suppose it is not true, and there exist two different minimizers u 1 , k and u 2 , k for i ( c k ) with c k → + ∞ as k → + ∞ . Let z 1 , k and z 2 , k denote the unique maximum point of u 1 , k and u 2 , k , respectively. Note from (3.19) that u i , k satisfies the following Euler-Lagrange equation:

(4.1) − a + b ∫ R N ∣ ∇ u i , k ∣ 2 d x Δ u i , k + V ( x ) u i , k = μ i , k u i , k + c k u i , k p + 1 , i = 1 , 2 , x ∈ R N .

Define

(4.2) u ^ i , k ( x ) = c ∗ ε k N 2 u i , k ( x ) and u ¯ i , k = u ^ i , k ( ε k x + z 1 . k ) , i = 1 , 2 ,

where ε k is given by (1.29). For i = 1, 2, one can check that u ¯ i , k and u ^ i , k satisfy

(4.3) − a ε k 2 + b c ∗ 2 ∫ R N ∣ ∇ u ¯ i , k ∣ 2 d x Δ u ¯ i , k + ε k 4 V ( ε k x + z 1 , k ) u ¯ i , k = μ i , k ε k 4 u ¯ i , k + 4 b N p u ¯ i , k p + 1

and

(4.4) − a ε k 4 + b ε k 4 − N c ∗ 2 ∫ R N ∣ ∇ u ^ i , k ∣ 2 d x Δ u ^ i , k + ε k 4 V ( x ) u ^ i , k = μ i , k ε k 4 u ^ i , k + 4 b N p u ^ i , k p + 1 .

Moreover, since lim k → + ∞ z 1 , k − z 2 , k ε k = 0 by (1.30), one has u ¯ i , k ( x ) → Q p ( x ) uniformly in R N as k → + ∞ by (1.28). Since u 1 , k ≢ u 2 , k , define

(4.5) η ¯ k = u ¯ 1 , k − u ¯ 2 , k ∣ ∣ u ¯ 1 , k − u ¯ 2 , k ∣ ∣ L ∞ and η ^ k = u ^ 1 , k − u ^ 2 , k ∣ ∣ u ^ 1 , k − u ^ 2 , k ∣ ∣ L ∞ ,

where η ¯ k ( x ) = η ^ k ( ε k x + z 1 . k ) . Then, one can derive from (4.3) that η ¯ k satisfies

(4.6) − a ε k 2 + b c ∗ 2 ∫ R N ∣ ∇ u ¯ 2 , k ∣ 2 d x Δ η ¯ k − b c ∗ 2 ∫ R N ∇ ( u ¯ 1 , k + u ¯ 2 , k ) ∇ η ¯ k d x Δ u ¯ 1 , k + ε k 4 V ( ε k x + z 1 , k ) η ¯ k = μ 1 , k ε k 4 η ¯ k + f ¯ k ( x ) + g ¯ k ( x ) ,

where

(4.7) f ¯ k ( x ) = ε k 4 μ 1 , k − μ 2 , k ∣ ∣ u ¯ 1 , k − u ¯ 2 , k ∣ ∣ L ∞ u ¯ 2 , k and g ¯ k ( x ) = 4 b ( u ¯ 1 , k p + 1 − u ¯ 2 , k p + 1 ) N p ∣ ∣ u ¯ 1 , k − u ¯ 2 , k ∣ ∣ L ∞ .

Lemma 4.1

For p ∈ 0 , 8 N , let V ( x ) satisfy all assumptions of Theorem 1.5, and u i , k be the minimizer for i ( c k ) , where c k → + ∞ as k → + ∞ , i = 1 , 2 . Then,

(4.8) u ¯ i , k ( x ) ≤ C e − ∣ x ∣ 2 a n d ∣ ∇ u ¯ i , k ( x ) ∣ ≤ C e − ∣ x ∣ 4 , a s ∣ x ∣ → + ∞ , i = 1 , 2 ,

where u ¯ i , k is given by (4.2) and the constant C > 0 is independent of k .

Proof

Using (3.23), one can obtain that

(4.9) μ i , k ε k 4 → − b [ 4 − p ( N − 2 ) ] N p , as k → + ∞ .

It then follows from (1.28), (1.29), and (4.3) that there exists a constant M > 0 large enough such that

− Δ u ¯ i , k + [ 4 − p ( N − 2 ) ] 2 N p u ¯ i , k ≤ 0 and u ¯ i , k ≤ C e − 1 2 M , for ∣ x ∣ ≥ M ,

where the constant C > 0 is independent of k . Comparing u ¯ i , k with e − 1 2 ∣ x ∣ , one can obtain from the comparison principle that

u ¯ i , k ( x ) ≤ C e − ∣ x ∣ 2 , for ∣ x ∣ ≥ M .

Since V ( x ) satisfies (1.34), one can also have

∣ ε k 4 V ( ε k x + z 1 , k ) u ¯ i , k ∣ ≤ C e − ∣ x ∣ 4 , for ∣ x ∣ ≥ M .

Applying the local elliptic estimates (see (3.15) in [6]), it follows from the aforementioned estimates that

□ ∣ ∇ u ¯ i , k ( x ) ∣ ≤ C e − ∣ x ∣ 4 , for ∣ x ∣ ≥ M .

Lemma 4.2

For p ∈ 0 , 8 N , let V ( x ) satisfy all assumptions of Theorem 1.5. Then, up to a subsequence if necessary,

(4.10) η ¯ k → η ¯ 0 i n C loc 1 ( R N ) , a s k → + ∞ ,

where

(4.11) η ¯ 0 ( x ) = h 0 Q + h ¯ 0 ( x ⋅ ∇ Q ) + ∑ i = 1 N h i ∂ Q p ∂ x i ,

and h 0 , h ¯ 0 , h 1 , … , h N are some constants.

Proof

Since ‖ η ¯ k ‖ L ∞ = 1 , in view of (4.6) and (4.7), the standard elliptic regularity theory implies that { η ¯ k } is locally uniformly bounded with respect to k in C loc 1 , α ( R N ) for some α ∈ ( 0 , 1 ) . Therefore, up to a subsequence if necessary, there exists some function η ¯ 0 ∈ C loc 1 ( R N ) such that η ¯ k → η ¯ 0 in C loc 1 ( R N ) as k → + ∞ . Similar to (3.21), one can obtain from (4.2) that

(4.12) μ i , k ε k 4 = 2 ε k 4 i ( c k ) + b 2 c ∗ 4 ∫ R N ∣ ∇ u ¯ i , k ∣ 2 d x 2 − 4 b N ( p + 2 ) c ∗ 2 ∫ R N u ¯ i , k p + 2 d x , i = 1 , 2 .

Similar to (3.11) in [21], we can derive from (1.28) that

(4.13) u ¯ 1 , k p + 2 − u ¯ 2 , k p + 2 = ( p + 2 ) C ¯ k ( x ) ( u ¯ 1 , k − u ¯ 2 , k )

and

(4.14) u ¯ 1 , k p + 1 − u ¯ 2 , k p + 1 = ( p + 1 ) D ¯ k ( x ) ( u ¯ 1 , k − u ¯ 2 , k ) ,

where C ¯ k ( x ) → Q p p + 1 ( x ) and D ¯ k ( x ) → Q p p ( x ) as k → + ∞ . Combining (4.5), (4.7), and (4.12)–(4.14), one can obtain that

(4.15) f ¯ k ( x ) = ε k 4 μ 1 , k − μ 2 , k ∣ ∣ u ¯ 1 , k − u ¯ 2 , k ∣ ∣ L ∞ u ¯ 2 , k = b 2 c ∗ 4 ∫ R N ( ∣ ∇ u ¯ 1 , k ∣ 2 + ∣ ∇ u ¯ 2 , k ∣ 2 ) d x ∫ R N ∇ ( u ¯ 1 , k + u ¯ 2 , k ) ∇ η ¯ k d x u ¯ 2 , k − 4 b N c ∗ 2 ∫ R N C ¯ k ( x ) η ¯ k d x u ¯ 2 , k

and

(4.16) g ¯ k ( x ) = 4 b ( u ¯ 1 , k p + 1 − u ¯ 2 , k p + 1 ) N p ∣ ∣ u ¯ 1 , k − u ¯ 2 , k ∣ ∣ L ∞ = 4 b ( p + 1 ) N p D ¯ k ( x ) η ¯ k .

Moreover, it follows from (1.15) that

(4.17) f ¯ k ( x ) → 2 b c ∗ 2 ∫ R N ∇ Q p ⋅ ∇ η ¯ 0 d x Q p − 4 b N c ∗ 2 ∫ R N Q p p + 1 η ¯ 0 d x Q p , uniformly in R N as k → + ∞ ,

and

(4.18) g ¯ k ( x ) → 4 b ( p + 1 ) N p Q p p η ¯ 0 , uniformly in R N as k → ∞ .

It then follows from (4.6), (4.9), and the aforementioned estimates that η ¯ 0 satisfies

(4.19) − Δ η ¯ 0 + 4 − p ( N − 2 ) N p − 4 ( p + 1 ) N p Q p p η ¯ 0 = 2 c ∗ 2 ∫ R N ∇ Q p ⋅ ∇ η ¯ 0 d x − 4 N c ∗ 2 ∫ R N Q p p + 1 η ¯ 0 d x Q p + 2 c ∗ 2 ∫ R N ∇ Q p ⋅ ∇ η ¯ 0 d x Δ Q p .

Define

(4.20) Γ ≔ − Δ + 4 − p ( N − 2 ) N p − 4 ( p + 1 ) N p Q p p ,

and one can easily check that

(4.21) Γ Q p + p 2 x ⋅ ∇ Q p = − 4 − p ( N − 2 ) N Q p and Γ ( x ⋅ ∇ Q p ) = − 2 Δ Q p .

Moreover, recall from [19,23] that

ker Γ = span ∂ Q p ∂ x 1 , ∂ Q p ∂ x 2 … , ∂ Q p ∂ x N .

Then, one can deduce from (4.19)–(4.21) that

η ¯ 0 ( x ) = h 0 Q p + h ¯ 0 ( x ⋅ ∇ Q p ) + ∑ i = 1 N h i ∂ Q p ∂ x i ,

where h 0 , h ¯ 0 , h 1 , … , h N are some constants.□

Lemma 4.3

For p ∈ 0 , 8 N , let V ( x ) satisfy all assumptions of Theorem 1.5. Then,

(4.22) h ¯ 0 ∫ R N ∂ V 1 ( x + y 0 ) ∂ x j ( x ⋅ ∇ Q p 2 ) d x − ∑ i = 1 N h i ∫ R N ∂ 2 V 1 ( x + y 0 ) ∂ x i ∂ x j Q p 2 d x = 0 , j = 1,2 … , N ,

where V 1 ( x ) is given by (1.23).

Proof

First, one can derive from (4.4) and (4.5) that η ^ k satisfies

(4.23) − 2 a ε k 4 + b ε k 4 − N c ∗ 2 ∫ R N ( ∣ ∇ u ^ 1 , k ∣ 2 + ∣ ∇ u ^ 2 , k ∣ 2 ) d x Δ η ^ k + 2 ε k 4 V ( x ) η ^ k − b ε k 4 − N c ∗ 2 ∫ R N ∇ ( u ^ 1 , k + u ^ 2 , k ) ⋅ ∇ η ^ k d x Δ ( u ^ 1 , k + u ^ 2 , k ) = ( μ 1 , k + μ 2 , k ) ε k 4 η ^ k + f ^ k ( x ) + g ^ k ( x ) ,

where

(4.24) f ^ k ( x ) = ε k 4 μ 1 , k − μ 2 , k ∣ ∣ u ^ 1 , k − u ^ 2 , k ∣ ∣ L ∞ ( u ^ 1 , k + u ^ 2 , k ) and g ^ k ( x ) = 8 b ( u ^ 1 , k p + 1 − u ^ 2 , k p + 1 ) N p ∣ ∣ u ^ 1 , k − u ^ 2 , k ∣ ∣ L ∞ .

For each k > 0 and x k ∈ R N , we claim that, for any fixed small δ ¯ > 0 independence of k and x k , there exists a small constant δ k ∈ ( δ ¯ , 2 δ ¯ ) such that

(4.25) ε k 4 ∫ ∂ B δ k ( x k ) ∣ ∇ η ^ k ∣ 2 d S + ε k 4 ∫ ∂ B δ k ( x k ) V ( x ) η ^ k 2 d S + ∫ ∂ B δ k ( x k ) η ^ k 2 d S ≤ O ( ε k N ) , as k → + ∞ .

Similar to (4.15) and (4.16), it follows (4.24) that

(4.26) f ^ k ( x ) = b 2 c ∗ 4 ∫ R N ( ∣ ∇ u ¯ 1 , k ∣ 2 + ∣ u ¯ 2 , k ∣ 2 d x ) ∫ R N ∇ ( u ¯ 1 , k + u ¯ 2 , k ) ∇ η ¯ k d x ( u ^ 1 , k + u ^ 2 , k ) − 4 b N c ∗ 2 ∫ R N C ¯ k ( x ) η ¯ k d x ( u ^ 1 , k + u ^ 2 , k )

and

(4.27) g ^ k ( x ) = 8 b ( p + 1 ) N p D ^ k ( x ) η ^ k , where D ^ k ( ε k x + z 1 , k ) = D ¯ k ( x ) .

Multiplying (4.23) by η ^ k and integrating over R N yield that

(4.28) 2 a ε k 4 + b ε k 4 − N c ∗ 2 ∫ R N ( ∣ ∇ u ^ 1 , k ∣ 2 + ∣ ∇ u ^ 2 , k ∣ 2 ) d x ∫ R N ∣ ∇ η ^ k ∣ 2 d x + 2 ε k 4 ∫ R N V ( x ) η ^ k 2 d x + b ε k 4 − N c ∗ 2 ∫ R N ∇ ( u ^ 1 , k + u ^ 2 , k ) ∇ η ^ k d x 2 − ( μ 1 , k + μ 2 , k ) ε k 4 ∫ R N η ^ k 2 d x = b 2 c ∗ 4 ∫ R N ( ∣ ∇ u ¯ 1 , k ∣ 2 + ∣ u ¯ 2 , k ∣ 2 d x ) ∫ R N ∇ ( u ¯ 1 , k + u ¯ 2 , k ) ∇ η ¯ k d x ∫ R N ( u ^ 1 , k + u ^ 2 , k ) η ^ k d x − 4 b N c ∗ 2 ∫ R N C ¯ k ( x ) η ¯ k d x ∫ R N ( u ^ 1 , k + u ^ 2 , k ) η ^ k d x + 8 b ( p + 1 ) N p ∫ R N D ^ k ( x ) η ^ k 2 d x = b ε k N 2 c ∗ 4 ∫ R N ( ∣ ∇ u ¯ 1 , k ∣ 2 + ∣ u ¯ 2 , k ∣ 2 d x ) ∫ R N ∇ ( u ¯ 1 , k + u ¯ 2 , k ) ∇ η ¯ k d x ∫ R N ( u ¯ 1 , k + u ¯ 2 , k ) η ¯ k d x − 4 b ε k N N c ∗ 2 ∫ R N C ¯ k ( x ) η ¯ k d x ∫ R N ( u ¯ 1 , k + u ¯ 2 , k ) η ¯ k d x + 8 b ( p + 1 ) ε k N N p ∫ R N D ¯ k ( x ) η ¯ k 2 d x = O ( ε k N ) , as k → + ∞ .

Since ‖ η ^ k ‖ L ∞ = 1 and u ¯ i , k decay exponentially as ∣ x ∣ → + ∞ , i = 1, 2,

(4.29) ε k 4 ∫ R N ∣ ∇ η ^ k ∣ 2 d x + ε k 4 ∫ R N V ( x ) η ^ k 2 d x + ∫ R N η ^ k 2 d x ≤ O ( ε k N ) , as k → + ∞ .

Applying [20, Lemma A.4] and (4.29), we can conclude that (4.25) holds.

Multiplying (4.4) by ∂ u ^ i , k ∂ x j and integrating over B δ k ( z 1 , k ) , where i = 1, 2, j = 1 , 2 , … , N , and δ k is given by (4.25), it follows

(4.30) − a ε k 4 + b ε k 4 − N c ∗ 2 ∫ R N ∣ ∇ u ^ i , k ∣ 2 d x ∫ B δ k ( z 1 , k ) Δ u ^ i , k ∂ u ^ i , k ∂ x j d x + ε k 4 2 ∫ B δ k ( z 1 , k ) V ( x ) ∂ u ^ i , k 2 ∂ x j d x = μ i , k ε k 4 2 ∫ B δ k ( z 1 , k ) ∂ u ^ i , k 2 ∂ x j d x + 4 b N p ( p + 2 ) ∫ B δ k ( z 1 , k ) ∂ u ^ i , k p + 2 ∂ x j d x .

Some calculations yield that

(4.31) ∫ B δ k ( z 1 , k ) Δ u ^ i , k ∂ u ^ i , k ∂ x j d x = ∑ m = 1 N ∫ B δ k ( z 1 , k ) ∂ 2 u ^ i , k ∂ x m 2 ∂ u ^ i , k ∂ x j d x = ∑ m = 1 N ∫ B δ k ( z 1 , k ) ∂ ∂ x m ∂ u ^ i , k ∂ x m ∂ u ^ i , k ∂ x j d x − ∑ m = 1 N ∫ B δ k ( z 1 , k ) ∂ u ^ i , k ∂ x m ∂ 2 u ^ i , k ∂ x j ∂ x m d x = ∑ m = 1 N ∫ B δ k ( z 1 , k ) ∂ ∂ x m ∂ u ^ i , k ∂ x m ∂ u ^ i , k ∂ x j d x − 1 2 ∑ m = 1 N ∫ B δ k ( z 1 , k ) ∂ ∂ x j ∂ u ^ i , k ∂ x m 2 d x = ∫ ∂ B δ k ( z 1 , k ) ∂ u ^ i , k ∂ x j ∂ u ^ i , k ∂ ν d S − 1 2 ∫ ∂ B δ k ( z 1 , k ) ∣ ∇ u ^ i , k ∣ 2 ν j d S ,

and

(4.32) ∫ B δ k ( z 1 , k ) V ( x ) ∂ u ^ i , k 2 ∂ x j d x = ∫ ∂ B δ k ( z 1 , k ) V ( x ) u ^ i , k 2 ν j d S − ∫ B δ k ( z 1 , k ) ∂ V ( x ) ∂ x j u ^ i , k 2 d x .

It then follows from (4.30)–(4.32) that

ε k 4 2 ∫ B δ k ( z 1 , k ) ∂ V ( x ) ∂ x j u ^ i , k 2 d x = − a ε k 4 + b ε k 4 − N c ∗ 2 ∫ R N ∣ ∇ u ^ i , k ∣ 2 d x ∫ ∂ B δ k ( z 1 , k ) ∂ u ^ i , k ∂ x j ∂ u ^ i , k ∂ ν d S − 1 2 ∫ ∂ B δ k ( z 1 , k ) ∣ ∇ u ^ i , k ∣ 2 ν j d S + ε k 4 2 ∫ ∂ B δ k ( z 1 , k ) V ( x ) u ^ i , k 2 ν j d S − μ i , k ε k 4 2 ∫ ∂ B δ k ( z 1 , k ) u ^ i , k 2 ν j d S − 4 b N p ( p + 2 ) ∫ ∂ B δ k ( z 1 , k ) u ^ i , k p + 2 ν j d S .

This implies from (4.5) that

(4.33) ε k 4 2 ∫ B δ k ( z 1 , k ) ∂ V ( x ) ∂ x j ( u ^ 1 , k + u ^ 2 , k ) η ^ k d x = G 1 + G 2 + G 3 + G 4 + G 5 + G 6 ,

where

(4.34) G 1 = − a ε k 4 + b ε k 4 − N c ∗ 2 ∫ R N ∣ ∇ u ^ 1 , k ∣ 2 d x ∫ ∂ B δ k ( z 1 , k ) ∂ η ^ k ∂ x j ∂ u ^ 1 , k ∂ ν d S + ∫ ∂ B δ k ( z 1 , k ) ∂ u ^ 2 , k ∂ x j ∂ η ^ k ∂ ν d S ,

(4.35) G 2 = a 2 ε k 4 + b ε k 4 − N 2 c ∗ 2 ∫ R N ∣ ∇ u ^ 1 , k ∣ 2 d x ∫ ∂ B δ k ( z 1 , k ) ∇ ( u ^ 1 , k + u ^ 2 , k ) ⋅ ∇ η ^ k ν j d S ,

(4.36) G 3 = − b ε k 4 − N c ∗ 2 ∫ R N ∇ ( u ^ 1 , k + u ^ 2 , k ) ⋅ ∇ η ^ k d x ∫ ∂ B δ k ( z 1 , k ) ∂ u ^ 2 , k ∂ x j ∂ u ^ 2 , k ∂ ν d S − 1 2 ∫ ∂ B δ k ( z 1 , k ) ∣ ∇ u ^ 2 , k ∣ 2 ν j d S ,

(4.37) G 4 = ε k 4 2 ∫ ∂ B δ k ( z 1 , k ) V ( x ) ( u ^ 1 , k + u ^ 2 , k ) η ^ k ν j d S ,

(4.38) G 5 = − μ 1 , k ε k 4 2 ∫ ∂ B δ k ( z 1 , k ) ( u ^ 1 , k + u ^ 2 , k ) η ^ k ν j d S − ε k 4 ( μ 1 , k − μ 2 , k ) 2 ∣ ∣ u ^ 1 , k − u ^ 2 , k ∣ ∣ L ∞ ∫ ∂ B δ k ( z 1 , k ) u ^ 2 , k 2 ν j d S ,

and

(4.39) G 6 = − 4 b N p ( p + 2 ) ∫ ∂ B δ k ( z 1 , k ) u ^ 1 , k p + 2 − u ^ 2 , k p + 2 ∣ ∣ u ^ 1 , k − u ^ 2 , k ∣ ∣ L ∞ v j d S .

Similar to (4.13), one can obtain that

(4.40) u ^ 1 , k p + 2 − u ^ 2 , k p + 2 ∣ ∣ u ^ 1 , k − u ^ 2 , k ∣ ∣ L ∞ = ( p + 2 ) C ^ k ( x ) η ^ k , where C ^ k ( ε k x + z 1 , k ) = C ¯ k ( x ) .

Moreover, applying (4.15) and (4.17), it follows

(4.41) ε k 4 ( μ 1 , k − μ 2 , k ) 2 ∣ ∣ u ^ 1 , k − u ^ 2 , k ∣ ∣ L ∞ → b c ∗ 2 ∫ R N ∇ Q p ⋅ ∇ η ¯ 0 d x − 2 b N c ∗ 2 ∫ R N Q p p + 1 η ¯ 0 d x , as k → + ∞ .

Then, using the Hölder inequality, we can derive from (4.8), (4.25), and the aforementioned estimates that there exists a constant C > 0 independent of k , such that

(4.42) ∣ G 1 ∣ ≤ o ( 1 ) ∫ ∂ B δ k ( z 1 , k ) ∣ ∇ η ^ k ∣ 2 d S 1 2 ∫ ∂ B δ k ( z 1 , k ) ∣ ∇ u ^ 1 , k ∣ 2 d S 1 2 + ∫ ∂ B δ k ( z 1 , k ) ∣ ∇ u ^ 2 , k ∣ 2 d S 1 2 = o e − C δ k ε k .

Similarly, we also have

(4.43) ∣ G 2 ∣ ≤ o e − C δ k ε k ,

(4.44) ∣ G 3 ∣ ≤ o ( 1 ) ∫ ∂ B δ k ( z 1 , k ) ∣ ∇ u ^ 2 , k ∣ 2 d S = o e − C δ k ε k ,

(4.45) ∣ G 4 ∣ ≤ ε k 4 2 ∫ ∂ B δ k ( z 1 , k ) V ( x ) η ^ k 2 d S 1 2 ∫ ∂ B δ k ( z 1 , k ) V ( x ) ( u ^ 1 , k + u ^ 2 , k ) 2 d S 1 2 = o e − C δ k ε k ,

(4.46) ∣ G 5 ∣ ≤ C ∫ ∂ B δ k ( z 1 , k ) ( u ^ 1 , k + u ^ 2 , k ) 2 d S 1 2 ∫ ∂ B δ k ( z 1 , k ) ∣ η ^ k ∣ 2 d x 1 2 + C ∫ ∂ B δ k ( z 1 , k ) ∣ u ^ 2 , k ∣ 2 d S = o e − C δ k ε k ,

and

(4.47) ∣ G 6 ∣ ≤ C ∫ ∂ B δ k ( z 1 , k ) ∣ C ^ k η ^ k ∣ d S ≤ C ∫ ∂ B δ k ( z 1 , k ) C ^ k 2 d S 1 2 ∫ ∂ B δ k ( z 1 , k ) η ^ k 2 d S 1 2 = o e − C δ k ε k .

Applying the aforementioned estimates, it then follows from (1.33), (1.35), and (4.33) that

(4.48) o e − C δ k ε k = ε k 4 2 ∫ B δ k ( z 1 , k ) ∂ V ( x ) ∂ x j ( u ^ 1 , k + u ^ 2 , k ) η ^ k d x = ε k N + 4 2 ∫ B δ k ε k ( 0 ) ∂ V ( ε k [ x + ( z 1 , k − x 1 ) ⁄ ε k ] + x 1 ) ∂ ε k x j ( u ¯ 1 , k + u ¯ 2 , k ) η ¯ k d x = ε k N + r + 3 2 ∫ B δ k ε k ( 0 ) ∂ V 1 x + z 1 , k − x 1 ε k ∂ x j ( u ¯ 1 , k + u ¯ 2 , k ) η ¯ k d x + ε k N + 4 2 ∫ B δ k ε k ( 0 ) W j ( ε k x + z 1 , k − x 1 ) ( u ¯ 1 , k + u ¯ 2 , k ) η ¯ k d x = ( 1 + o ( 1 ) ) ε k N + r + 3 ∫ R N ∂ V 1 ( x + y 0 ) ∂ x j Q p η ¯ 0 d x .

Hence, we can derive from (1.33), (4.11), and (4.48) that

0 = ∫ R N ∂ V 1 ( x + y 0 ) ∂ x j Q p η ¯ 0 d x = ∫ R N ∂ V 1 ( x + y 0 ) ∂ x j Q p h 0 Q p + h ¯ 0 ( x ⋅ ∇ Q p ) + ∑ i = 1 N h i ∂ Q p ∂ x i d x = h 0 ∫ R N ∂ V 1 ( x + y 0 ) ∂ x j Q p 2 d x + h ¯ 0 2 ∫ R N ∂ V 1 ( x + y 0 ) ∂ x j ( x ⋅ ∇ Q p 2 ) d x + 1 2 ∑ i = 1 N h i ∫ R N ∂ V 1 ( x + y 0 ) ∂ x j ∂ Q p 2 ∂ x i d x = h ¯ 0 2 ∫ R N ∂ V 1 ( x + y 0 ) ∂ x j ( x ⋅ ∇ Q p 2 ) d x − 1 2 ∑ i = 1 N h i ∫ R N ∂ 2 V 1 ( x + y 0 ) ∂ x i ∂ x j Q p 2 d x ,

and hence, (4.22) holds.□

Proof of Theorem 1.5

We first prove that the coefficients h 0 and h ¯ 0 given in (4.11) satisfy

(4.49) h 0 = h ¯ 0 = 0 .

Multiplying (4.4) by ( x − z 1 , k ) ⋅ ∇ u ^ i , k and integrating over B δ k ( z 1 , k ) , where i = 1 , 2 , and δ k is given by (4.25), it follows

(4.50) − a ε k 4 + b ε k 4 − N c ∗ 2 ∫ R N ∣ ∇ u ^ i , k ∣ 2 d x ∫ B δ k ( z 1 , k ) Δ u ^ i , k [ ( x − z 1 , k ) ⋅ ∇ u ^ i , k ] d x + ε k 4 2 ∫ B δ k ( z 1 , k ) V ( x ) [ ( x − z 1 , k ) ⋅ ∇ u ^ i , k 2 ] d x = μ i , k ε k 4 2 ∫ B δ k ( z 1 , k ) ( x − z 1 , k ) ⋅ ∇ u ^ i , k 2 d x + 4 b N p ( p + 2 ) ∫ B δ k ( z 1 , k ) ( x − z 1 , k ) ⋅ ∇ u ^ i , k p + 2 d x .

Using the integration by parts, we have

(4.51) ∫ B δ k ( z 1 , k ) Δ u ^ i , k [ ( x − z 1 , k ) ⋅ ∇ u ^ i , k ] d x = ∫ ∂ B δ k ( z 1 , k ) ∂ u ^ i , k ∂ ν [ ( x − z 1 , k ) ⋅ ∇ u ^ i , k ] d S − ∫ B δ k ( z 1 , k ) ∇ u ^ i , k ⋅ ∇ [ ( x − z 1 , k ) ⋅ ∇ u ^ i , k ] d x = ∫ ∂ B δ k ( z 1 , k ) ∂ u ^ i , k ∂ ν [ ( x − z 1 , k ) ⋅ ∇ u ^ i , k ] d S − 1 2 ∫ B δ k ( z 1 , k ) ( x − z 1 , k ) ⋅ ∇ ∣ ∇ u ^ i , k ∣ 2 d x − ∫ B δ k ( z 1 , k ) ∣ ∇ u ^ i , k ∣ 2 d x = ∫ ∂ B δ k ( z 1 , k ) ∂ u ^ i , k ∂ ν [ ( x − z 1 , k ) ⋅ ∇ u ^ i , k ] d S − 1 2 ∫ ∂ B δ k ( z 1 , k ) [ ( x − z 1 , k ) ⋅ ν ] ∣ ∇ u ^ i , k ∣ 2 d x − 2 − N 2 ∫ B δ k ( z 1 , k ) ∣ ∇ u ^ i , k ∣ 2 d x = ∫ ∂ B δ k ( z 1 , k ) ∂ u ^ i , k ∂ ν [ ( x − z 1 , k ) ⋅ ∇ u ^ i , k ] d S − 1 2 ∫ ∂ B δ k ( z 1 , k ) [ ( x − z 1 , k ) ⋅ ν ] ∣ ∇ u ^ i , k ∣ 2 d x − 2 − N 4 ∫ ∂ B δ k ( z 1 , k ) ∇ u ^ i , k 2 ⋅ ν d x + 2 − N 2 ∫ B δ k ( z 1 , k ) u ^ i , k Δ u ^ i , k d x = T i + 2 − N 2 ∫ B δ k ( z 1 , k ) u ^ i , k Δ u ^ i , k d x ,

where

(4.52) T i = ∫ ∂ B δ k ( z 1 , k ) ∂ u ^ i , k ∂ ν [ ( x − z 1 , k ) ⋅ ∇ u ^ i , k ] d S − 1 2 ∫ ∂ B δ k ( z 1 , k ) [ ( x − z 1 , k ) ⋅ ν ] ∣ ∇ u ^ i , k ∣ 2 d x − 2 − N 4 ∫ ∂ B δ k ( z 1 , k ) ∇ u ^ i , k 2 ⋅ ν d x .

Combining (4.4), (4.51), and (4.52), one can obtain that

(4.53) − a ε k 4 + b ε k 4 − N c ∗ 2 ∫ R N ∣ ∇ u ^ i , k ∣ 2 d x ∫ B δ k ( z 1 , k ) Δ u ^ i , k [ ( x − z 1 , k ) ⋅ ∇ u ^ i , k ] d x = − a ε k 4 + b ε k 4 − N c ∗ 2 ∫ R N ∣ ∇ u ^ i , k ∣ 2 d x T i − 2 − N 2 a ε k 4 + b ε k 4 − N c ∗ 2 ∫ R N ∣ ∇ u ^ i , k ∣ 2 d x ∫ B δ k ( z 1 , k ) u ^ i , k Δ u ^ i , k d x = − a ε k 4 + b ε k 4 − N c ∗ 2 ∫ R N ∣ ∇ u ^ i , k ∣ 2 d x T i − 2 − N 2 ε k 4 ∫ B δ k ( z 1 , k ) V ( x ) u ^ i , k 2 d x − μ i , k ε k 4 ∫ B δ k ( z 1 , k ) u ^ i , k 2 d x − 4 b N p ∫ B δ k ( z 1 , k ) u ^ i , k p + 2 d x .

Similarly,

(4.54) ∫ B δ k ( z 1 , k ) V ( x ) [ ( x − z 1 , k ) ⋅ ∇ u ^ i , k 2 ] d x = ∫ ∂ B δ k ( z 1 , k ) V ( x ) [ ( x − z 1 , k ) ⋅ ν ] u ^ i , k 2 d S − ∫ B δ k ( z 1 , k ) [ ∇ V ( x ) ⋅ ( x − z 1 , k ) + N V ( x ) ] u ^ i , k 2 d x ,

(4.55) ∫ B δ k ( z 1 , k ) ( x − z 1 , k ) ⋅ ∇ u ^ i , k 2 d x = ∫ ∂ B δ k ( z 1 , k ) [ ( x − z 1 , k ) ⋅ ν ] u ^ i , k 2 d S − N ∫ B δ k ( z 1 , k ) u ^ i , k 2 d x ,

and

(4.56) ∫ B δ k ( z 1 , k ) ( x − z 1 , k ) ⋅ ∇ u ^ i , k p + 2 d x = ∫ ∂ B δ k ( z 1 , k ) [ ( x − z 1 , k ) ⋅ ν ] u ^ i , k p + 2 d S − N ∫ B δ k ( z 1 , k ) u ^ i , k p + 2 d x .

Combining (4.50)–(4.56), one can obtain that

(4.57) − a ε k 4 + b ε k 4 − N c ∗ 2 ∫ R N ∣ ∇ u ^ i , k ∣ 2 d x T i − 2 − N 2 ε k 4 ∫ B δ k ( z 1 , k ) V ( x ) u ^ i , k 2 d x − μ i , k ε k 4 ∫ B δ k ( z 1 , k ) u ^ i , k 2 d x − 4 b N p ∫ B δ k ( z 1 , k ) u ^ i , k p + 2 d x + ε k 4 2 ∫ ∂ B δ k ( z 1 , k ) V ( x ) [ ( x − z 1 , k ) ⋅ ν ] u ^ i , k 2 d S − ε k 4 2 ∫ B δ k ( z 1 , k ) ∇ V ( x ) ⋅ ( x − z 1 , k ) u ^ i , k 2 d x − N ε k 4 2 ∫ B δ k ( z 1 , k ) V ( x ) u ^ i , k 2 d x = μ i , k ε k 4 2 ∫ ∂ B δ k ( z 1 , k ) [ ( x − z 1 , k ) ⋅ ν ] u ^ i , k 2 d S − N μ i , k ε k 4 2 ∫ B δ k ( z 1 , k ) u ^ i , k 2 d x + 4 b N p ( p + 2 ) ∫ ∂ B δ k ( z 1 , k ) [ ( x − z 1 , k ) ⋅ ν ] u ^ i , k p + 2 d S − 4 b p ( p + 2 ) ∫ B δ k ( z 1 , k ) u ^ i , k p + 2 d x .

Moreover, using (4.4), it follows

(4.58) μ i , k ε k 4 ∫ R N u ^ i , k 2 d x = N p c ∗ 2 ε k N + 4 2 i ( c k ) + a ( 4 − N p ) ε k 4 4 ∫ R N ∣ ∇ u ^ i , k ∣ 2 d x + ( 4 − N p ) ε k 4 4 ∫ R N V ( x ) u ^ i , k 2 d x + b ( 8 − N p ) ε k 4 − N 8 c ∗ 2 ∫ R N ∣ ∇ u ^ i , k ∣ 2 d x 2 − 2 b [ 2 ( p + 2 ) − N p ] N p ( p + 2 ) ∫ R N u ^ i , k p + 2 d x .

It then follows from (4.57) and (4.58) that

(4.59) − a ε k 4 + b ε k 4 − N c ∗ 2 ∫ R N ∣ ∇ u ^ i , k ∣ 2 d x T i + ε k 4 2 ∫ ∂ B δ k ( z 1 , k ) V ( x ) [ ( x − z 1 , k ) ⋅ ν ] u ^ i , k 2 d S − ε k 4 2 ∫ B δ k ( z 1 , k ) ∇ V ( x ) ⋅ ( x − z 1 , k ) u ^ i , k 2 d x − μ i , k ε k 4 2 ∫ ∂ B δ k ( z 1 , k ) [ ( x − z 1 , k ) ⋅ ν ] u ^ i , k 2 d S − 4 b N p ( p + 2 ) ∫ ∂ B δ k ( z 1 , k ) [ ( x − z 1 , k ) ⋅ ν ] u ^ i , k p + 2 d S = ε k 4 ∫ B δ k ( z 1 , k ) V ( x ) u ^ i , k 2 d x − μ i , k ε k 4 ∫ B δ k ( z 1 , k ) u ^ i , k 2 d x − 2 b [ 2 ( p + 2 ) − N p ] N p ( p + 2 ) ∫ B δ k ( z 1 , k ) u ^ i , k p + 2 d x = ε k 4 ∫ B δ k ( z 1 , k ) V ( x ) u ^ i , k 2 d x − N p c ∗ 2 ε k N + 4 2 i ( c k ) − a ( 4 − N p ) ε k 4 4 ∫ R N ∣ ∇ u ^ i , k ∣ 2 d x − ( 4 − N p ) ε k 4 4 ∫ R N V ( x ) u ^ i , k 2 d x − b ( 8 − N p ) ε k 4 − N 8 c ∗ 2 ∫ R N ∣ ∇ u ^ i , k ∣ 2 d x 2 + μ i , k ε k 4 ∫ R N \ B δ k ( z 1 , k ) u ^ i , k 2 d x + 2 b [ 2 ( p + 2 ) − N p ] N p ( p + 2 ) ∫ R N \ B δ k ( z 1 , k ) u ^ i , k p + 2 d x .

This implies that

(4.60) b ( 8 − N p ) ε k 4 − N 8 c ∗ 2 ∫ R N ∣ ∇ u ^ 1 , k ∣ 2 d x + ∫ R N ∣ ∇ u ^ 2 , k ∣ 2 d x ∫ R N ∇ ( u ^ 1 , k + u ^ 2 , k ) ∇ η ^ k d x + a ( 4 − N p ) ε k 4 4 ∫ R N ∇ ( u ^ 1 , k + u ^ 2 , k ) ∇ η ^ k d x = K 1 + K 2 + K 3 + K 4 ,

where

(4.61) K 1 = a ε k 4 + b ε k 4 − N c ∗ 2 ∫ R N ∣ ∇ u ^ 1 , k ∣ 2 d x T 1 − a ε k 4 + b ε k 4 − N c ∗ 2 ∫ R N ∣ ∇ u ^ 2 , k ∣ 2 d x T 2 ∣ ∣ u ^ 1 , k − u ^ 2 , k ∣ ∣ L ∞ − ε k 4 2 ∫ ∂ B δ k ( z 1 , k ) V ( x ) [ ( x − z 1 , k ) ⋅ ν ] ( u ^ 1 , k + u ^ 2 , k ) η ^ k d S + μ 1 , k ε k 4 2 ∫ ∂ B δ k ( z 1 , k ) [ ( x − z 1 , k ) ⋅ ν ] ( u ^ 1 , k + u ^ 2 , k ) η ^ k d S + ε k 4 ( μ 1 , k − μ 2 , k ) 2 ∣ ∣ u ^ 1 , k − u ^ 2 , k ∣ ∣ L ∞ ∫ ∂ B δ k ( z 1 , k ) [ ( x − z 1 , k ) ⋅ ν ] u ^ 2 , k 2 d S + c k ( p + 2 ) c ∗ p ε k 8 − N p 2 ∫ ∂ B δ k ( z 1 , k ) [ ( x − z 1 , k ) ⋅ ν ] u ^ 1 , k p + 2 − u ^ 2 , k p + 2 ∣ ∣ u ^ 1 , k − u ^ 2 , k ∣ ∣ L ∞ d S ,

(4.62) K 2 = − ε k 4 ∫ R N \ B δ k ( z 1 , k ) V ( x ) ( u ^ 1 , k + u ^ 2 , k ) η ^ k d x + μ 1 , k ε k 4 ∫ R N \ B δ k ( z 1 , k ) ( u ^ 1 , k + u ^ 2 , k ) η ^ k d x + ε k 4 ( μ 1 , k − μ 2 , k ) ∣ ∣ u ^ 1 , k − u ^ 2 , k ∣ ∣ L ∞ ∫ R N \ B δ k ( z 1 , k ) u ^ 2 , k 2 d x + 2 b [ 2 ( p + 2 ) − N p ] N p ( p + 2 ) ∫ R N \ B δ k ( z 1 , k ) u ^ 1 , k p + 2 − u ^ 2 , k p + 2 ∣ ∣ u ^ 1 , k − u ^ 2 , k ∣ ∣ L ∞ d x ,

(4.63) K 3 = ε k 4 2 ∫ B δ k ( z 1 , k ) ∇ V ( x ) ⋅ ( x − z 1 , k ) ( u ^ 1 , k + u ^ 2 , k ) η ^ k d x ,

and

(4.64) K 4 = N p ε k 4 4 ∫ R N V ( x ) ( u ^ 1 , k + u ^ 2 , k ) η ^ k d x .

Similar to the estimates of (4.42)–(4.47), it follows

(4.65) ∣ K 1 ∣ , ∣ K 2 ∣ = o e − C δ k ε k .

As for K 3 , the estimate (4.48) shows that

(4.66) ε k 4 2 ∫ B δ k ( z 1 , k ) [ ∇ V ( x ) ⋅ z 1 , k ] ( u ^ 1 , k + u ^ 2 , k ) η ^ k d x = o e − C δ k ε k .

Since ∇ V 1 ( x ) ⋅ x = r V 1 ( x ) by (1.23), one can deduce from (1.35), (4.47), and (4.66) that

(4.67) K 3 = ε k 4 2 ∫ B δ k ( z 1 , k ) [ ∇ V ( x ) ⋅ x ] ( u ^ 1 , k + u ^ 2 , k ) η ^ k d x − ε k 4 2 ∫ B δ k ( z 1 , k ) [ ∇ V ( x ) ⋅ z 1 , k ] ( u ^ 1 , k + u ^ 2 , k ) η ^ k d x = ε k 4 2 ∫ B δ k ( z 1 , k ) [ ∇ V ( x − x 1 + x 1 ) ⋅ ( x − x 1 ) ] ( u ^ 1 , k + u ^ 2 , k ) η ^ k d x + ε k 4 2 ∫ B δ k ( z 1 , k ) [ ∇ V ( x ) ⋅ x 1 ] ( u ^ 1 , k + u ^ 2 , k ) η ^ k d x + o e − C δ k ε k = ε k 4 2 ∫ B δ k ( z 1 , k ) { [ ∇ V 1 ( x − x 1 ) + W ( x − x 1 ) ] ⋅ ( x − x 1 ) } ( u ^ 1 , k + u ^ 2 , k ) η ^ k d x + o e − C δ k ε k = r ( 1 + o ( 1 ) ) 2 ε k 4 + r + N ∫ B δ k ε k ( 0 ) V 1 x + z 1 , k − x 1 ε k ( u ¯ 1 , k + u ¯ 2 , k ) η ¯ k d x + o e − C δ k ε k = r ( 1 + o ( 1 ) ) ε k 4 + r + N ∫ R N V 1 ( x + y 0 ) Q p η ¯ 0 d x ,

where W ( x ) = ( W 1 ( x ) , W 2 ( x ) , … , W N ( x ) ) , and

(4.68) K 4 = N p ε k 4 + N 4 ∫ R N V ( ε k x + z 1 , k − x 1 + x 1 ) V 1 ( ε k x + z 1 , k − x 1 ) V 1 ( ε k x + z 1 , k − x 1 ) ( u ¯ 1 , k + u ¯ 2 , k ) η ¯ k d x = N p ε k 4 + r + N 4 ∫ R N V ( ε k x + z 1 , k − x 1 + x 1 ) V 1 ( ε k x + z 1 , k − x 1 ) V 1 x + z 1 , k − x 1 ε k ( u ¯ 1 , k + u ¯ 2 , k ) η ¯ k d x = N p ( 1 + o ( 1 ) ) 2 ε k 4 + r + N ∫ R N V 1 ( x + y 0 ) Q p η ¯ 0 d x .

Moreover, the left-hand side of (4.60) follows that

(4.69) b ( 8 − N p ) ε k 4 − N 8 c ∗ 2 ∫ R N ∣ ∇ u ^ 1 , k ∣ 2 d x + ∫ R N ∣ ∇ u ^ 2 , k ∣ 2 d x ∫ R N ∇ ( u ^ 1 , k + u ^ 2 , k ) ⋅ ∇ η ^ k d x + a ( 4 − N p ) ε k 4 4 ∫ R N ∇ ( u ^ 1 , k + u ^ 2 , k ) ⋅ ∇ η ^ k d x = b ( 8 − N p ) ε k N 8 c ∗ 2 ∫ R N ∣ ∇ u ¯ 1 , k ∣ 2 d x + ∫ R N ∣ ∇ u ¯ 2 , k ∣ 2 d x ∫ R N ∇ ( u ¯ 1 , k + u ¯ 2 , k ) ⋅ ∇ η ¯ k d x + a ( 4 − N p ) ε k N + 2 4 ∫ R N ∇ ( u ¯ 1 , k + u ¯ 2 , k ) ⋅ ∇ η ¯ k d x = b ( 8 − N p ) ε k N 2 ( 1 + o ( 1 ) ) ∫ R N ∇ Q p ⋅ ∇ η ¯ 0 d x + a ( 4 − N p ) ε k N + 2 2 ( 1 + o ( 1 ) ) ∫ R N ∇ Q p ⋅ ∇ η ¯ 0 d x = b ( 8 − N p ) ε k N 2 ( 1 + o ( 1 ) ) ∫ R N ∇ Q p ⋅ ∇ η ¯ 0 d x .

Applying the aforementioned estimates, it then follows from (4.60) that

(4.70) b ( 8 − N p ) ε k N 2 ( 1 + o ( 1 ) ) ∫ R N ∇ Q p ⋅ ∇ η ¯ 0 d x = O ( ε k 4 + r + N ) .

Since p ∈ 0 , 8 N , (4.70) indicates that

(4.71) ∫ R N ∇ Q p ⋅ ∇ η ¯ 0 d x = 0 .

Then, applying (4.19), we know η ¯ 0 satisfies

− Δ η ¯ 0 + 4 − p ( N − 2 ) N p − 4 ( p + 1 ) N p Q p p η ¯ 0 = − 4 N c ∗ 2 ∫ R N Q p p + 1 η ¯ 0 d x Q p .

This implies from (4.21) that

(4.72) η ¯ 0 ( x ) = d 0 Q p + p 2 x ⋅ ∇ Q p + ∑ i = 1 N d i ∂ Q p ∂ x i ,

where d 0 , d 1 , d 2 , … , d N are some constants. Comparing (4.72) with (4.11), one has

(4.73) h ¯ 0 = p 2 h 0 = p 2 d 0 .

It then follows from (4.11), (4.71), and (4.73) that

0 = ∫ R N ∇ Q p ⋅ ∇ η ¯ 0 d x = ∫ R N ∇ Q p ⋅ ∇ h 0 Q p + p 2 x ⋅ ∇ Q p + ∑ i = 1 N h i ∂ Q p ∂ x i d x = h 0 ∫ R N ∣ ∇ Q p ∣ 2 d x + p 2 h 0 ∫ R N ∇ Q p ⋅ ∇ ( x ⋅ ∇ Q p ) d x + ∑ i = 1 N h i ∫ R N ∇ Q p ⋅ ∇ ∂ Q p ∂ x i d x = 4 − p ( N − 2 ) 4 h 0 c ∗ 2 .

This implies h 0 = 0 since 4 − p ( N − 2 ) 4 ≠ 0 for p ∈ 0 , 8 N and N ≤ 4 . We then obtain from (4.73) that h ¯ 0 = p 2 h 0 = 0 , and

∑ i = 1 N h i ∫ R N ∂ 2 V 1 ( x + y 0 ) ∂ x i ∂ x j Q p 2 d x = 0 , j = 1 , 2 … , N ,

by (4.22). Applying the non-degeneracy assumption (1.21), it follows that h i = 0 , i = 1 , 2 , … , N , and hence, η ¯ 0 ≡ 0 .

At last, for p ∈ 0 , 8 N , we prove that η ¯ 0 ≡ 0 cannot occur. Let y k ∈ R N be the maximum point of η ¯ k , where η ¯ k ( y k ) = ‖ η ¯ k ‖ L ∞ = 1 . Applying the maximum principle to (4.6), we see that ∣ y k ∣ ≤ C is uniformly in k due to the exponential decay of (4.8). Therefore, we conclude from (4.10) that η ¯ 0 ≢ 0 on R N , which, however, contradicts with the fact that η ¯ 0 ≡ 0 on R N . This completes the proof of Theorem 1.5.□

Acknowledgement

The authors are very grateful to the anonymous referees to their careful reading of the manuscript and valuable comments.

Funding information: H. Guo was supported by NSFC (Grant No. 12101442) and Fundamental Research Program of Shanxi Province (Grant No. 20210302124257). L. Zhao was supported by Fundamental Research Program of Shanxi Province (Grant No. 20210302124382) and Taiyuan University of Technology Science Foundation for Youths (Grant No. 2022QN102).
Author contributions: The authors contributed equally to this manuscript.
Conflict of interest: The authors state no conflict of interest.
Data availability statement: Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

References

[1] A. Arosio and S. Panizzi, On the well-posedness of the Kirchhoff string, Trans. Amer. Math. Soc. 348 (1996), no. 1, 305–330. https://doi.org/10.1090/S0002-9947-96-01532-2.Search in Google Scholar

[2] T. Cazenave, Semilinear Schrödinger Equations, Courant Institute of Mathematical Sciences. Vol 10, Providence, RI, New York, 2003. DOI: https://doi.org/10.1090/cln/010.10.1090/cln/010Search in Google Scholar

[3] F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, Theory of Bose-Einstein condensation in trapped gases, Rev. Modern Phys. 71 (1999), no. 3, 463–512. https://doi.org/10.1103/RevModPhys.71.463.Search in Google Scholar

[4] P. D’Ancona and S. Spagnolo, Global solvability for the degenerate Kirchhoff equation with real analytic data, Invent. Math. 108 (1992), no. 2, 247–262. https://doi.org/10.1007/BF02100605.Search in Google Scholar

[5] A. L. Fetter, Rotating trapped Bose-Einstein condensates, Rev. Modern Phys. 81 (2009), no. 2, 647. DOI: https://doi.org/10.1103/RevModPhys.81.647.10.1103/RevModPhys.81.647Search in Google Scholar

[6] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Spriger, Berlin, 1997. DOI: https://doi.org/10.1007/978-3-642-61798-0.10.1007/978-3-642-61798-0Search in Google Scholar

[7] Y. J. Guo and R. Seiringer, On the mass concentration for Bose-Einstein condensates with attractive interactions, Lett. Math. Phys. 104 (2014), no. 2, 141–156. https://doi.org/10.48550/arXiv.1301.5682.Search in Google Scholar

[8] Y. J. Guo, C. S. Lin, and J. C. Wei, Local uniqueness and refined spike profiles of ground states for two-dimensional attractive Bose-Einstein condensates, SIAM J. Math. Anal. 49 (2017), no. 5, 3671–3715. https://doi.org/10.1137/16M1100290.Search in Google Scholar

[9] Y. J. Guo, X. Y. Zeng, and H. S. Zhou, Concentration behavior of standing waves for almost mass critical nonlinear Schrödinger equations, J. Differential Equations 256 (2014), no. 7, 2079–2100. https://doi.org/10.1016/j.jde.2013.12.012.Search in Google Scholar

[10] H. L. Guo, Y. M. Zhang, and H. S. Zhou, Blow-up solutions for a Kirchhoff-type elliptic equations with trapping potential, Commun. Pure Appl. Anal. 17 (2018), no. 5, 1875–1897. https://doi.org/10.3934/cpaa.2018089.Search in Google Scholar

[11] H. L. Guo and H. S. Zhou, Properties of the minimizers for a constrained minimization problem arising in Kirchhoff equation, Discrete Contin. Dyn. Syst. 41 (2021), no. 3, 1023–1050. https://doi.org/10.3934/dcds.2020308.Search in Google Scholar

[12] H. L. Guo and H. S. Zhou, A constrained variational problem arising in attractive Bose-Einstein condensate with ellipse-shaped potential, Appl. Math. Lett. 87 (2019), 35–41. https://doi.org/10.1016/j.aml.2018.07.023.Search in Google Scholar

[13] B. Gidas, W. M. Ni, and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in Rn, Mathematical analysis and applications Part A, Adv. Math. Suppl. Stud 7, (1981), 369–402. Search in Google Scholar

[14] Y. He and G. B. Li, Standing waves for a class of Kirchhoff-type problems in R3 involving critical Sobolev exponents, Calc. Var. Partial Differential Equations 54 (2015), no. 3, 3067–3106. https://doi.org/10.1007/s00526-015-0894-2.Search in Google Scholar

[15] Q. Han and F. H. Lin, Elliptic Partial Differential Equations, Courant Lect. Notes Math, Providence, RI, New York, 2011. DOI: https://doi.org/10.1090/cln/001.10.1090/cln/001Search in Google Scholar

[16] T. X. Hu and C. L. Tang, Limiting behavior and local uniqueness of normalized solutions for mass critical Kirchhoff equations, Calc. Var. Partial Differential Equations 60 (2021), no. 6, 210. https://doi.org/10.1007/s00526-021-02018-1.Search in Google Scholar

[17] X. M. He and W. M. Zou, Existence and concentration behavior of positive solutions for a Kirchhoff equation in R3, J. Differential Equations 252 (2012), no. 2, 1813–1834. https://doi.org/10.1016/j.jde.2011.08.035.Search in Google Scholar

[18] G. Kirchhoff, Mechanik, Teubner, Leipzig, 1883. Search in Google Scholar

[19] M. K. Kwong, Uniqueness of positive solutions of Δu−u+up=0 in RN, Arch. Ration. Mech. Anal. 105 (1989), no. 3, 243–266. DOI: https://doi.org/10.1007/bf00251502.10.1007/BF00251502Search in Google Scholar

[20] P. Luo, S. J. Peng, and C. H. Wang, Uniqueness of positive solutions with concentration for the Schrödinger-Newton problem, Calc. Var. Partial Differential Equations 59 (2020), no. 2, 60. https://doi.org/10.1007/s00526-020-1726-6.Search in Google Scholar

[21] S. Li and X. C. Zhu, Mass concentration and local uniqueness of ground states for L2-subcritical nonlinear Schrödinger equations, Z. Angew. Math. Phys. 70 (2019), no. 1, 34. DOI: https://doi.org/10.1007/s00033-019-1077-3.10.1007/s00033-019-1077-3Search in Google Scholar

[22] G. B. Li and H. Y. Ye, On the concentration phenomenon of L2-subcritical constrained minimizers for a class of Kirchhoff equations with potentials, J. Differential Equations 11 (2019), no. 11, 7101–7123. https://doi.org/10.1016/j.jde.2018.11.024.Search in Google Scholar

[23] W. M. Ni and I. Takagi, On the shape of least-energy solutions to a semilinear Neumann problem, Comm. Pure Appl. Math. 44 (1991), no. 7, 819–851. DOI: https://doi.org/10.1002/cpa.3160440705.10.1002/cpa.3160440705Search in Google Scholar

[24] M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996. DOI: https://doi.org/10.1007/978-1-4612-4146-1.10.1007/978-1-4612-4146-1Search in Google Scholar

[25] M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys. 87 (1983), no. 4, 567–576. https://doi.org/10.1007/BF01208265.Search in Google Scholar

[26] X. Y. Zeng, Asymptotic properties of standing waves for mass subcritical nonlinear Schrödinger equations, Discrete Contin. Dyn. Syst. 37 (2017), no. 3, 1749–1762. https://doi.org/10.3934/dcds.2017073.Search in Google Scholar

[27] X. Y. Zeng and Y. M. Zhang, Existence and uniqueness of normalized solutions for the Kirchhoff equation, Appl. Math. Lett. 74 (2017), 52–59. https://doi.org/10.1016/j.aml.2017.05.012.Search in Google Scholar

Received: 2024-07-05

Revised: 2024-11-07

Accepted: 2025-03-24

Published Online: 2025-04-25

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/anona-2025-0081

Keywords for this article

subcritical; Kirchhoff energy functional; constrained variational method; energy estimates; local uniqueness

Creative Commons

BY 4.0