Non-Degeneracy of Peak Solutions to the Schrödinger–Newton System

Qing Guo; Huafei Xie

doi:10.1515/ans-2021-2128

Open Access Published by De Gruyter April 15, 2021

Non-Degeneracy of Peak Solutions to the Schrödinger–Newton System

Qing Guo and Huafei Xie

From the journal Advanced Nonlinear Studies

https://doi.org/10.1515/ans-2021-2128

Abstract

We are concerned with the following Schrödinger–Newton problem:

- ε 2 ⁢ Δ ⁢ u + V ⁢ ( x ) ⁢ u = 1 8 ⁢ π ⁢ ε 2 ⁢ ( ∫ ℝ 3 u 2 ⁢ ( ξ ) | x - ξ | ⁢ 𝑑 ξ ) ⁢ u , x ∈ ℝ 3 .

For ε small enough, we prove the non-degeneracy of the positive solution to the above problem, that is, the corresponding linear operator

ℒ ε ⁢ ( η ) = - ε 2 ⁢ Δ ⁢ η ⁢ ( x ) + V ⁢ ( x ) ⁢ η ⁢ ( x ) - 1 8 ⁢ π ⁢ ε 2 ⁢ ( ∫ ℝ 3 u ε 2 ⁢ ( ξ ) | x - ξ | ⁢ 𝑑 ξ ) ⁢ η ⁢ ( x ) - 1 4 ⁢ π ⁢ ε 2 ⁢ ( ∫ ℝ 3 u ε ⁢ ( ξ ) ⁢ η ⁢ ( ξ ) | x - ξ | ⁢ 𝑑 ξ ) ⁢ u ε ⁢ ( x )

is non-degenerate, i.e., ℒε⁢(ηε)=0⇒ηε=0 for small ε>0. The main tools are the local Pohozaev identities and the blow-up analysis. This may be the first non-degeneracy result on the peak solutions to the Schrödinger–Newton system.

Keywords: Schrödinger–Newton System; Non-Degeneracy; Pohozaev Identity

MSC 2010: 35B40; 35B45; 35J40

1 Introduction and Main Results

The Schrödinger–Newton system describing the quantum mechanics of a polaron at rest,

(1.1) { ε 2 2 ⁢ m ⁢ Δ ⁢ u - V ⁢ ( x ) ⁢ u + ψ ⁢ u = 0 , x ∈ ℝ 3 , Δ ⁢ ψ + 4 ⁢ π ⁢ τ ⁢ | u | 2 = 0 , x ∈ ℝ 3 ,

was derived by R. Penrose [13] as a model of self-gravitating matter, in which quantum state reduction is understood as a gravitational phenomenon. Here, the interacts with a matter density is given by the square of the wave function u , which is the solution of the Schrödinger equation. In addition, an electric field is generated by a potential V⁢(x). In (1.1), ψ is the gravitational potential, ε is the Planck constant, τ=G⁢m2 and G is the Newton’s constant of gravitation.

Let

u ⁢ ( x ) ↦ u 4 ⁢ ε ⁢ π ⁢ τ ⁢ m , V ⁢ ( x ) ↦ 1 2 ⁢ m ⁢ V ⁢ ( x ) , ψ ⁢ ( x ) ↦ 1 2 ⁢ m ⁢ ψ ⁢ ( x ) .

Then system (1.1) can be written, maintaining the original notations, as

(1.2) { ε 2 ⁢ Δ ⁢ u - V ⁢ ( x ) ⁢ u + ψ ⁢ u = 0 , x ∈ ℝ 3 , ε 2 ⁢ Δ ⁢ ψ + | u | 2 2 = 0 , x ∈ ℝ 3 .

The second equation in (1.2) can be solved explicitly with respect to ψ, so that the system can be turned into the following single nonlocal equation:

(1.3) - ε 2 ⁢ Δ ⁢ u + V ⁢ ( x ) ⁢ u = 1 8 ⁢ π ⁢ ε 2 ⁢ ( ∫ ℝ 3 u 2 ⁢ ( ξ ) | x - ξ | ⁢ 𝑑 ξ ) ⁢ u , x ∈ ℝ 3 ,

which has been investigated extensively. Especially, when ε=1 and V⁢(x)=1, the existence and uniqueness of the ground states for (1.3) was obtained by variational methods [7, 9, 12], while the non-degeneracy was proved in [15, 17].

Theorem A (cf. [8, 17]).

For any fixed a∈R3 satisfying V⁢(a)>0, there exists a unique radial solution Ua of the problem

{ - Δ ⁢ u + V ⁢ ( a ) ⁢ u = 1 8 ⁢ π ⁢ ( ∫ ℝ 3 u 2 ⁢ ( ξ ) | x - ξ | ⁢ 𝑑 ξ ) ⁢ u in ⁢ ℝ 3 , u ⁢ ( x ) > 0 in ⁢ ℝ 3 , u ⁢ ( 0 ) = max x ∈ ℝ 3 ⁡ u ⁢ ( x ) .

The solution Ua is strictly decreasing and

lim | x | → ∞ ⁡ U a ⁢ ( x ) ⁢ e | x | ⁢ | x | = λ 0 > 0 , lim | x | → ∞ ⁡ U a ′ ⁢ ( x ) U a ⁢ ( x ) = - 1

for some constant λ0>0. Moreover, if ϕ⁢(x)∈H1⁢(R3) solves the linearized equation

- Δ ⁢ ϕ ⁢ ( x ) + V ⁢ ( a ) ⁢ ϕ ⁢ ( x ) = 1 8 ⁢ π ⁢ ( ∫ ℝ 3 U a 2 ⁢ ( ξ ) | x - ξ | ⁢ 𝑑 ξ ) ⁢ ϕ ⁢ ( x ) + 1 4 ⁢ π ⁢ ( ∫ ℝ 3 U a ⁢ ( ξ ) ⁢ ϕ ⁢ ( ξ ) | x - ξ | ⁢ 𝑑 ξ ) ⁢ U a ⁢ ( x ) ,

then ϕ⁢(x) is a linear combination of ∂⁡Ua/∂⁡xj, j=1,2,3.

If ε is small and V⁢(x) is not a constant, the existence of solutions with ground states for (1.3) was proved by [10]. Then Wei and Winter [17] considered the existence of multiple solutions concentrating at k points of the local minimum points of V⁢(x). We also refer to [3, 14, 16] and the references therein for the existence of solutions with concentration in other cases. The uniqueness result of concentrating solutions can be found in [11], by using local Pohozaev type of identity and blow-up analysis, which was recently developed in [1, 4, 6].

Theorem B (cf. [17]).

Suppose that {a1,…,ak}⊂R3 are non-degenerate critical points of V⁢(x). There exists a positive solution {uε}ε>0 concentrating at {a1,…,ak}⊂R3, i.e., there exist {xi,ε}ε>0⊂R3 such that

(1.4) u ε = ∑ i = 1 k U i ⁢ ( x - x i , ε ε ) + w ε ,

where |xi,ε-ai|=o⁢(1) for i=1,…,k, and ∥wε∥ε=o⁢(ε32).

It is well known that the non-degeneracy of the solutions is of fundamental importance when dealing with the orbital stability or instability result of the corresponding time-dependent equations. Especially, when V⁢(x) is not a constant, apart from the existence and uniqueness results, whether the positive solution is non-degenerate is still unknown.

We assume that V⁢(x) is a bounded C1 function satisfying infx∈ℝ3⁡V⁢(x)>0. Define the following Sobolev space Hε:

H ε := { u ⁢ ( x ) ∈ H 1 ⁢ ( ℝ 3 ) : ∫ ℝ 3 ( ε 2 ⁢ | ∇ ⁡ u ⁢ ( x ) | 2 + V ⁢ ( x ) ⁢ u 2 ⁢ ( x ) ) ⁢ 𝑑 x < ∞ } ,

and the corresponding norm

∥ u ∥ ε = ( u ⁢ ( x ) , u ⁢ ( x ) ) ε 1 2 = ( ∫ ℝ 3 ( ε 2 ⁢ | ∇ ⁡ u ⁢ ( x ) | 2 + V ⁢ ( x ) ⁢ u 2 ⁢ ( x ) ) ⁢ 𝑑 x ) 1 2 .

For any η∈H1⁢(ℝ3), we define

ℒ ε ⁢ ( η ) = - ε 2 ⁢ Δ ⁢ η ⁢ ( x ) + V ⁢ ( x ) ⁢ η ⁢ ( x ) - 1 8 ⁢ π ⁢ ε 2 ⁢ ( ∫ ℝ 3 u ε 2 ⁢ ( ξ ) | x - ξ | ⁢ 𝑑 ξ ) ⁢ η ⁢ ( x ) - 1 4 ⁢ π ⁢ ε 2 ⁢ ( ∫ ℝ 3 u ε ⁢ ( ξ ) ⁢ η ⁢ ( ξ ) | x - ξ | ⁢ 𝑑 ξ ) ⁢ u ε ⁢ ( x ) ,

where uε is obtained by Theorem B.

Our main results are as follows.

Theorem 1.1.

Suppose that {a1,…,ak}⊂R3 (k≥1) are non-degenerate critical points of V⁢(x). Let {uε}ε>0 be a positive solution to (1.3) concentrating at {a1,…,ak}⊂R3. If ηε is a solution to Lε⁢(ηε)=0, then ηε=0 for small ε>0.

Inspired by Guo, Musso, Peng and Yan [5], we apply local Pohozaev identity and the blow-up analysis to obtain ηε⁢(x)=o⁢(1) near the non-degenerate critical points. Especially, we point out that, distinct from the classical Schrödinger equations, the corresponding local Pohozaev identity of the Schrödinger–Newton system would have two terms involving volume integral due to the nonlocal term. Moreover, the asymptotic behavior of the concentrating points to Schrödinger–Newton problem is quite different from that of the classical Schrödinger equation. Hence, we should deal with the non-degeneracy of the single-peak and multi-peak solutions separately.

Organization of the Paper.

In Section 2, we obtain some estimates needed in the proof of Theorem 1.1, especially including the local Pohozaev identities. The main result on the non-degeneracy of the one-peak solutions will be proved in Section 3, while the result of multi-peak solutions will be obtained in Section 4.

2 The Basic Estimates

We first recall the following known results.

Proposition 2.1 ([11]).

Suppose that uε⁢(x) is a positive solution of equation (1.3) concentrating at different points a1,…,ak with k≥1. Then, for any fixed R≫1, there exist θ>0 and C>0 such that

u ε ⁢ ( x ) ≤ C ⁢ e - θ ⁢ | x - x l , ε | ε ⁢ for ⁢ l = 1 , … , k ⁢ and ⁢ x ∈ ℝ 3 ∖ ⋃ j = 1 k B R ⁢ ε ⁢ ( x j , ε ) ,

| ∇ ⁡ u ε ⁢ ( x ) | ≤ C ⁢ e - θ ⁢ R ε ⁢ for ⁢ x ∈ ℝ 3 ∖ ⋃ j = 1 k B R ⁢ ε ⁢ ( x j , ε ) .

Corollary 2.2.

Suppose that uε⁢(x) is a solution of equation (1.3) as in Proposition 2.1. Then the followings statements hold:

For any fixed R ≫ 1 , there exists θ 1 > 0 such that

(2.1) u ε ⁢ ( x ) , | ∇ ⁡ u ε ⁢ ( x ) | = O ⁢ ( e - θ 1 ⁢ R ) for ⁢ x ∈ ℝ 3 ∖ ⋃ j = 1 k B R ⁢ ε ⁢ ( x j , ε ) .
For any fixed d > 0 , there exists θ 2 > 0 such that

(2.2) u ε ⁢ ( x ) , | ∇ ⁡ u ε ⁢ ( x ) | = O ⁢ ( e - θ 2 ε ) for ⁢ x ∈ ℝ 3 ∖ ⋃ j = 1 k B d ⁢ ( x j , ε ) .

Proposition 2.3.

Let u⁢(x) be the solution of (1.3), Lε⁢(η)=0. Then we have the following local Pohozaev identities:

(2.3) ∫ Ω ∂ ⁡ V ⁢ ( x ) ∂ ⁡ x i ⁢ u 2 ⁢ ( x ) ⁢ 𝑑 x = - 2 ⁢ ε 2 ⁢ ∫ ∂ ⁡ Ω ∂ ⁡ u ⁢ ( x ) ∂ ⁡ ν ⁢ ∂ ⁡ u ⁢ ( x ) ∂ ⁡ x i ⁢ 𝑑 σ + ∫ ∂ ⁡ Ω ( ε 2 ⁢ | ∇ ⁡ u ⁢ ( x ) | 2 + V ⁢ ( x ) ⁢ u 2 ⁢ ( x ) ) ⁢ ν i ⁢ ( x ) ⁢ 𝑑 σ - 1 8 ⁢ π ⁢ ε 2 ⁢ ∫ ∂ ⁡ Ω ∫ ℝ 3 u 2 ⁢ ( ξ ) ⁢ u 2 ⁢ ( x ) | x - ξ | ⁢ ν i ⁢ ( x ) ⁢ 𝑑 ξ ⁢ 𝑑 σ + 1 8 ⁢ π ⁢ ε 2 ⁢ ∫ Ω ∫ ℝ 3 u 2 ⁢ ( ξ ) ⁢ u 2 ⁢ ( x ) ⁢ x i - ξ i | x - ξ | 3 ⁢ 𝑑 ξ ⁢ 𝑑 x

and

(2.4) ∫ Ω ∂ ⁡ V ⁢ ( x ) ∂ ⁡ x i ⁢ u ⁢ ( x ) ⁢ η ⁢ ( x ) ⁢ 𝑑 x = - ε 2 ⁢ ∫ ∂ ⁡ Ω ∂ ⁡ u ⁢ ( x ) ∂ ⁡ ν ⁢ ∂ ⁡ η ⁢ ( x ) ∂ ⁡ x i + ∂ ⁡ η ⁢ ( x ) ∂ ⁡ ν ⁢ ∂ ⁡ u ⁢ ( x ) ∂ ⁡ x i ⁢ d ⁢ σ + ∫ ∂ ⁡ Ω ( ε 2 ⁢ 〈 ∇ ⁡ u ⁢ ( x ) , ∇ ⁡ η ⁢ ( x ) 〉 + V ⁢ ( x ) ⁢ u ⁢ ( x ) ⁢ η ⁢ ( x ) ) ⁢ ν i ⁢ ( x ) ⁢ 𝑑 σ - 1 8 ⁢ π ⁢ ε 2 ⁢ ∫ ∂ ⁡ Ω ∫ ℝ 3 ( u 2 ⁢ ( ξ ) ⁢ u ⁢ ( x ) ⁢ η ⁢ ( x ) | x - ξ | + u 2 ⁢ ( x ) ⁢ u ⁢ ( ξ ) ⁢ η ⁢ ( ξ ) | x - ξ | ) ⁢ ν i ⁢ ( x ) ⁢ 𝑑 ξ ⁢ 𝑑 σ - 1 8 ⁢ π ⁢ ε 2 ⁢ ∫ Ω ∫ ℝ 3 ( u 2 ⁢ ( ξ ) ⁢ u ⁢ ( x ) ⁢ η ⁢ ( x ) ⁢ x i - ξ i | x - ξ | 3 + u ⁢ ( ξ ) ⁢ η ⁢ ( ξ ) ⁢ u 2 ⁢ ( x ) ⁢ x i - ξ i | x - ξ | 3 ) ⁢ 𝑑 ξ ⁢ 𝑑 x ,

where Ω is a bounded open domain of R3, i=1,2,3, ν⁢(x)=(ν1⁢(x),ν2⁢(x),ν3⁢(x)) is the outward unit normal of ∂⁡Ω and xi,ξi are the i-th components of x,ξ.

Proof.

Identity (2.3) is obtained by multiplying ∂⁡u⁢(x)∂⁡xi on both sides of (1.3) and integrating on Ω. While the identity in (2.4) is obtained by multiplying ∂⁡η⁢(x)∂⁡xi and ∂⁡u⁢(x)∂⁡xi on both sides of (1.3) and ℒε⁢(η)=0, respectively, and integrating on Ω. We omit the details. ∎

Let

(2.5) F 1 ⁢ ( x ) = 1 8 ⁢ π ⁢ ε 2 ⁢ ∫ ℝ 3 u ε 2 ⁢ ( ξ ) | x - ξ | ⁢ 𝑑 ξ , F 2 ⁢ ( x ) = 1 4 ⁢ π ⁢ ε 2 ⁢ ∫ ℝ 3 u ε ⁢ ( x ) ⁢ u ε ⁢ ( ξ ) | x - ξ | ⁢ η ε ⁢ ( ξ ) ⁢ 𝑑 ξ .

Proposition 2.4.

For ηε⁢(x) satisfying Lε⁢(η)=0, we have

(2.6) ∥ η ε ∥ ε = O ⁢ ( ε 3 2 ) .

Proof.

From

- ε 2 ⁢ Δ ⁢ η ⁢ ( x ) + V ⁢ ( x ) ⁢ η ⁢ ( x ) - F 1 ⁢ ( x ) ⁢ η ⁢ ( x ) - F 2 ⁢ ( x ) = 0 ,

we have

(2.7) ∥ η ε ∥ ε 2 = ∫ ℝ 3 F 1 ⁢ ( x ) ⁢ η ε 2 ⁢ ( x ) ⁢ 𝑑 x + ∫ ℝ 3 F 2 ⁢ ( x ) ⁢ η ε ⁢ ( x ) ⁢ 𝑑 x .

Next, by the Hardy–Littlewood–Sobolev inequality, Hölder’s inequality and the fact |ηε⁢(x)|≤1, we know

(2.8) | ∫ ℝ 3 F 1 ⁢ ( x ) ⁢ η ε 2 ⁢ ( x ) ⁢ 𝑑 x | ≤ C ⁢ ε - 2 ⁢ ( ∫ ℝ 3 | u ε ⁢ ( ξ ) | 12 5 ⁢ 𝑑 ξ ) 5 6 ⋅ ( ∫ ℝ 3 | η ε ⁢ ( x ) | 12 5 ⁢ 𝑑 x ) 5 6 ≤ C ⁢ ε - 2 ⁢ ( ∫ ℝ 3 | u ε ⁢ ( ξ ) | 12 5 ⁢ 𝑑 ξ ) 5 6 ⋅ ( ∫ ℝ 3 | η ε ⁢ ( x ) | 2 ⁢ 𝑑 x ) 5 6 ≤ C ⁢ ε 1 2 ⁢ ∥ u ε ∥ ε 2 ⁢ ∥ η ε ∥ ε 5 3 ≤ C ⁢ ε - 5 2 ⁢ ∥ η ε ∥ ε 5 3 ≤ C ⁢ ε 3 + 1 4 ⁢ ∥ η ε ∥ ε 2

and

(2.9) | ∫ ℝ 3 F 2 ⁢ ( x ) ⁢ η ε ⁢ ( x ) ⁢ 𝑑 x | ≤ C ⁢ ε 3 + 1 4 ⁢ ∥ η ε ∥ ε 2 .

Then (2.7), (2.8) and (2.9) imply (2.6). ∎

Proposition 2.5.

Suppose that uε⁢(x)=∑l=1kUal⁢(x-xl,εε)+wε⁢(x) is a positive solution of equation (1.3) and that {a1,…,ak}⊂R3 are the different non-degenerate critical points of V⁢(x) with k≥1. Then it holds

(2.10) ∥ w ε ∥ ε = O ⁢ ( ε 7 2 ) + O ⁢ ( ε 3 2 ⁢ max j = 1 , … , k ⁡ | x j , ε - a j | 2 ) .

Proof.

We postpone the proof to the Appendix. ∎

3 Proof of Theorem 1.1 with k=1

When k=1, we have the following modified estimate.

Proposition 3.1.

Let uε⁢(x) be the solution of (1.3) concentrating at a non-degenerate critical point a1∈R3 of V⁢(x). Then it holds

(3.1) | x 1 , ε - a 1 | = o ⁢ ( ε ) .

Proof.

This result can be found in [11], but we sketch the proof for being self-enclosed. First, for the small fixed constant d¯>0, taking u⁢(x)=uε⁢(x) and Ω=Bd⁢(x1,ε) in the Pohozaev identity (2.3) with any d∈(d¯,2⁢d¯), we have, for i=1,2,3,

(3.2) ∫ B d ⁢ ( x 1 , ε ) ∂ ⁡ V ⁢ ( x ) ∂ ⁡ x i ⁢ u ε 2 ⁢ ( x ) ⁢ 𝑑 x = ∫ ∂ ⁡ B d ⁢ ( x 1 , ε ) B ⁢ ( x ) ⁢ 𝑑 σ + 1 8 ⁢ π ⁢ ε 2 ⁢ ∫ B d ⁢ ( x 1 , ε ) ∫ ℝ 3 u ε 2 ⁢ ( x ) ⁢ u ε 2 ⁢ ( ξ ) ⁢ x i - ξ i | x - ξ | 3 ⁢ 𝑑 ξ ⁢ 𝑑 x ,

where

B ⁢ ( x ) = - 2 ⁢ ε 2 ⁢ ∂ ⁡ u ε ⁢ ( x ) ∂ ⁡ ν ⁢ ∂ ⁡ u ε ⁢ ( x ) ∂ ⁡ x i + ( ε 2 ⁢ | ∇ ⁡ u ε ⁢ ( x ) | 2 + u ε 2 ⁢ ( x ) ⁢ ( V ⁢ ( x ) - 1 8 ⁢ π ⁢ ε 2 ⁢ ∫ ℝ 3 u ε 2 ⁢ ( ξ ) | x - ξ | ⁢ 𝑑 ξ ) ) ⁢ ν i ⁢ ( x ) .

Next, since

∂ ⁡ V ⁢ ( x ) ∂ ⁡ x i = ∂ ⁡ V ⁢ ( x ) ∂ ⁡ x i - ∂ ⁡ V ⁢ ( a j ) ∂ ⁡ x i = ∑ l = 1 3 ( x l - a j l ) ⁢ ∂ 2 ⁡ V ⁢ ( a j ) ∂ ⁡ x i ⁢ ∂ ⁡ x l + o ⁢ ( | x - a j | ) for ⁢ i = 1 , 2 , 3 ,

for any d∈(d¯,2⁢d¯), (1.4) gives

(3.3)

∫ B d ⁢ ( x 1 , ε ) ∂ ⁡ V ⁢ ( x ) ∂ ⁡ x i ⁢ u ε 2 ⁢ ( x ) ⁢ 𝑑 x = ∫ B d ⁢ ( x 1 , ε ) ∂ ⁡ V ⁢ ( x ) ∂ ⁡ x i ⁢ U a 1 2 ⁢ ( x - x 1 , ε ε ) ⁢ 𝑑 x + o ⁢ ( ε 4 + ε 3 ⁢ | x 1 , ε - a 1 | )

= ε 3 ⁢ ( ∫ ℝ 3 U a 1 2 ⁢ ( x ) ⁢ 𝑑 x ) ⁢ ∑ l = 1 3 ∂ 2 ⁡ V ⁢ ( a 1 ) ∂ ⁡ x i ⁢ ∂ ⁡ x l ⁢ ( x 1 , ε l - a 1 l ) + o ⁢ ( ε 4 + ε 3 ⁢ | x 1 , ε - a 1 | ) ,

where x1,εl,a1l are the l-th components of x1,ε,a1. On the other hand, for any fixed d, we have

| ∫ ∂ ⁡ B d ⁢ ( x 1 , ε ) ε 2 ⁢ ∂ ⁡ u ε ⁢ ( x ) ∂ ⁡ ν ⁢ ∂ ⁡ u ε ⁢ ( x ) ∂ ⁡ x i ⁢ 𝑑 σ | ≤ C ⁢ ε 2 ⁢ ∫ ∂ ⁡ B d ⁢ ( x 1 , ε ) | ∇ ⁡ u ε ⁢ ( x ) | 2 ⁢ 𝑑 σ .

Then from (A.4), for any fixed d, we find

| ∫ ∂ ⁡ B d ⁢ ( x 1 , ε ) B ⁢ ( x ) ⁢ 𝑑 σ | ≤ C ⁢ ∫ ∂ ⁡ B d ⁢ ( x 1 , ε ) [ ε 2 ⁢ | ∇ ⁡ u ε ⁢ ( x ) | 2 + ( u ε ⁢ ( x ) ) 2 ] ⁢ 𝑑 σ .

So using (2.2), (A.1) and (2.10), there exists dε∈(d¯,2⁢d¯) such that

(3.4) ∫ ∂ ⁡ B d ε ⁢ ( x 1 , ε ) B ⁢ ( x ) ⁢ 𝑑 σ = O ⁢ ( e - η ε + ∥ w ε ∥ ε 2 ) = O ⁢ ( ε 7 + ε 3 ⁢ | x 1 , ε - a 1 | 4 ) .

Also for any d∈(d¯,2⁢d¯), by symmetry and (2.2), we deduce

(3.5) 1 8 ⁢ π ⁢ ε 2 ⁢ ∫ B d ⁢ ( x 1 , ε ) ∫ ℝ 3 u ε 2 ⁢ ( x ) ⁢ u ε 2 ⁢ ( ξ ) ⁢ x i - ξ i | x - ξ | 3 ⁢ 𝑑 ξ ⁢ 𝑑 x = 1 8 ⁢ π ⁢ ε 2 ⁢ ∫ ℝ 3 ∫ ℝ 3 u ε 2 ⁢ ( x ) ⁢ u ε 2 ⁢ ( ξ ) ⁢ x i - ξ i | x - ξ | 3 ⁢ 𝑑 ξ ⁢ 𝑑 x + O ⁢ ( e - η ε ) = O ⁢ ( e - η ε ) .

Let d=dε in (3.2). Then (3.3), (3.4) and (3.5) imply

∑ l = 1 3 ∂ 2 ⁡ V ⁢ ( a 1 ) ∂ ⁡ x i ⁢ ∂ ⁡ x l ⁢ ( x 1 , ε l - a 1 l ) = o ⁢ ( | x 1 , ε - a 1 | ) + o ⁢ ( ε ) ,

which gives (3.1). ∎

Next, we prove Theorem 1.1 by contradiction. Suppose that there exists εm→0 satisfying

∥ η ε m ∥ L ∞ = 1 , ℒ ε m ⁢ η ε m = 0 .

For simplicity, we will omit the subscript m, replacing εm by ε.

Lemma 3.2.

Let η1,ε⁢(x)=ηε⁢(ε⁢x+x1,ε). By taking a subsequence, if necessary, it holds

(3.6) η 1 , ε ⁢ ( x ) → ∑ i = 1 3 a 1 , i ⁢ ∂ ⁡ U a 1 ⁢ ( x ) ∂ ⁡ x i

uniformly in C1⁢(BR⁢(0)) for any R>0, where ηε⁢(x) is the solution to Lε⁢(η)=0, and a1,i (i=1,2,3) are some constants.

Proof.

Since ∥η1,ε∥L∞⁢(ℝ3)≤1, by the regularity theory, we know

η 1 , ε ⁢ ( x ) ∈ C loc 1 , α ⁢ ( ℝ 3 ) and ∥ η 1 , ε ∥ C loc 1 , α ⁢ ( ℝ 3 ) ≤ C for some ⁢ α ∈ ( 0 , 1 ) .

So we may assume that

η 1 , ε ⁢ ( x ) → η 1 ⁢ ( x ) in ⁢ C loc ⁢ ( ℝ 3 ) .

By direct calculations, we have

- Δ ⁢ η 1 , ε ⁢ ( x ) = - V ⁢ ( ε ⁢ x + x 1 , ε ) ⁢ η 1 , ε ⁢ ( x ) + A 1 ⁢ ( ε ⁢ x + x 1 , ε ) ⁢ η 1 , ε ⁢ ( x ) + A 2 ⁢ ( ε ⁢ x + x 1 , ε ) .

Since from (A.4) and (A.5), we have

A 1 ⁢ ( ε ⁢ x + x 1 , ε ) = 1 8 ⁢ π ⁢ ( ∫ ℝ 3 U a 1 2 ⁢ ( ξ ) | x - ξ | ⁢ 𝑑 ξ ) + o ⁢ ( 1 ) , x ∈ B d ε ⁢ ( 0 ) ,

and

A 2 ⁢ ( ε ⁢ x + x 1 , ε ) = U a 1 ⁢ ( x ) 4 ⁢ π ⁢ ( ∫ ℝ 3 U a 1 ⁢ ( ξ ) ⁢ η 1 , ε ⁢ ( ξ ) | x - ξ | ⁢ 𝑑 ξ ) + o ⁢ ( 1 ) , x ∈ B d ε ⁢ ( 0 ) .

Next, for any given Φ⁢(x)∈C0∞⁢(ℝ3), we have

(3.7)

∫ ℝ 3 ( - Δ ⁢ η 1 , ε ⁢ ( x ) + V ⁢ ( ε ⁢ x + x 1 , ε ) ⁢ η 1 , ε ⁢ ( x ) ) ⁢ Φ ⁢ ( x ) ⁢ 𝑑 x - 1 8 ⁢ π ⁢ ∫ ℝ 3 ∫ ℝ 3 U a 1 2 ⁢ ( ξ ) | x - ξ | ⁢ η 1 , ε ⁢ ( x ) ⁢ Φ ⁢ ( x ) ⁢ 𝑑 ξ ⁢ 𝑑 x

- 1 4 ⁢ π ⁢ ∫ ℝ 3 ∫ ℝ 3 U a 1 ⁢ ( ξ ) ⁢ η 1 , ε ⁢ ( ξ ) | x - ξ | ⁢ U a 1 ⁢ ( x ) ⁢ Φ ⁢ ( x ) ⁢ 𝑑 ξ ⁢ 𝑑 x = o ⁢ ( 1 ) ⁢ ∥ Φ ∥ H 1 ⁢ ( ℝ 3 ) .

Letting ε→0 in (3.7) and using the elliptic regularity theory, we find that η1⁢(x) satisfies

- Δ ⁢ η 1 ⁢ ( x ) + V ⁢ ( a 1 ) ⁢ η 1 ⁢ ( x ) = 1 8 ⁢ π ⁢ ( ∫ ℝ 3 U a 1 2 ⁢ ( ξ ) | x - ξ | ⁢ 𝑑 ξ ) ⁢ η 1 ⁢ ( x ) + 1 4 ⁢ π ⁢ ( ∫ ℝ 3 U a 1 ⁢ ( ξ ) ⁢ η 1 ⁢ ( ξ ) | x - ξ | ⁢ 𝑑 ξ ) ⁢ U a 1 ⁢ ( x ) in ⁢ ℝ 3 .

By Theorem A, it holds that η1⁢(x)=∑i=13a1,i⁢∂⁡Ua1⁢(x)∂⁡xi, which means (3.6). ∎

Lemma 3.3.

Let a1,i be as in Lemma 3.2. Then we have

a 1 , i = 0 for all ⁢ i = 1 , 2 , 3 .

Proof.

Recall the Pohozaev identity (2.4). First, by use of Corollary 2.2, for some γ>0 it holds that

- ε 2 ⁢ ∫ ∂ ⁡ B d ⁢ ( x 1 , ε ) ∂ ⁡ u ε ⁢ ( x ) ∂ ⁡ ν ⁢ ∂ ⁡ η ε ⁢ ( x ) ∂ ⁡ x i + ∂ ⁡ η ε ⁢ ( x ) ∂ ⁡ ν ⁢ ∂ ⁡ u ε ⁢ ( x ) ∂ ⁡ x i ⁢ d ⁢ σ

+ ∫ ∂ ⁡ B d ⁢ ( x 1 , ε ) ( ε 2 ⁢ 〈 ∇ ⁡ u ε ⁢ ( x ) , ∇ ⁡ η ε ⁢ ( x ) 〉 + V ⁢ ( x ) ⁢ u ε ⁢ ( x ) ⁢ η ε ⁢ ( x ) ) ⁢ ν i ⁢ ( x ) ⁢ 𝑑 σ = O ⁢ ( e - γ ε ) .

Moreover, combined with Hardy–Littlewood–Sobolev inequality, we know that

- 1 8 ⁢ π ⁢ ε 2 ⁢ ∫ ∂ ⁡ B d ⁢ ( x 1 , ε ) ∫ ℝ 3 ( u ε 2 ⁢ ( ξ ) ⁢ u ε ⁢ ( x ) ⁢ η ε ⁢ ( x ) | x - ξ | + u ε 2 ⁢ ( x ) ⁢ u ε ⁢ ( ξ ) ⁢ η ε ⁢ ( ξ ) | x - ξ | ) ⁢ ν i ⁢ ( x ) ⁢ 𝑑 ξ ⁢ 𝑑 σ = O ⁢ ( e - γ ε ) .

By symmetry,

- 1 8 ⁢ π ⁢ ε 2 ⁢ ∫ B d ⁢ ( x 1 , ε ) ∫ ℝ 3 ( u ε 2 ⁢ ( ξ ) ⁢ u ε ⁢ ( x ) ⁢ η ε ⁢ ( x ) ⁢ x i - ξ i | x - ξ | 3 + u ε ⁢ ( ξ ) ⁢ η ε ⁢ ( ξ ) ⁢ u ε 2 ⁢ ( x ) ⁢ x i - ξ i | x - ξ | 3 ) ⁢ 𝑑 ξ ⁢ 𝑑 x = 1 8 ⁢ π ⁢ ε 2 ⁢ ∫ ℝ 3 ∫ ℝ 3 ( u ε 2 ⁢ ( ξ ) ⁢ u ε ⁢ ( x ) ⁢ η ε ⁢ ( x ) ⁢ x i - ξ i | x - ξ | 3 + u ε ⁢ ( ξ ) ⁢ η ε ⁢ ( ξ ) ⁢ u ε 2 ⁢ ( x ) ⁢ x i - ξ i | x - ξ | 3 ) ⁢ 𝑑 ξ ⁢ 𝑑 x + O ⁢ ( e - γ ε ) = O ⁢ ( e - γ ε ) .

To sum up,

RHS of (2.4) = O ⁢ ( e - γ ε ) .

On the other hand,

∫ B d ⁢ ( x 1 , ε ) ∂ ⁡ V ⁢ ( x ) ∂ ⁡ x i ⁢ u ε ⁢ ( x ) ⁢ η ε ⁢ ( x ) ⁢ 𝑑 x = ∑ j = 1 3 ∫ B d ⁢ ( x 1 , ε ) ∂ 2 ⁡ V ⁢ ( a 1 ) ∂ ⁡ x i ⁢ ∂ ⁡ x j ⁢ ( x j - a 1 j ) ⁢ u ε ⁢ ( x ) ⁢ η ε ⁢ ( x ) ⁢ 𝑑 x + o ⁢ ( ∫ B d ⁢ ( x 1 , ε ) | x - a 1 | ⁢ u ε ⁢ ( x ) ⁢ η ε ⁢ ( x ) ⁢ 𝑑 x ) .

We estimate

∫ B d ⁢ ( x 1 , ε ) ( x j - a 1 j ) ⁢ u ε ⁢ ( x ) ⁢ η ε ⁢ ( x ) ⁢ 𝑑 x = ε 4 ⁢ ( ∫ B d ε ⁢ ( 0 ) ( x j + x 1 , ε j - a 1 j ε ) ⁢ U a 1 ⁢ ( x ) ⁢ ( ∑ i = 1 3 a 1 , i ⁢ ∂ ⁡ U a 1 ⁢ ( x ) ∂ ⁡ x i + o ⁢ ( 1 ) ) ⁢ 𝑑 x ) = a 1 , j ⁢ ε 4 ⁢ ∫ ℝ 3 x j ⁢ U a 1 ⁢ ( x ) ⁢ ∂ ⁡ U a 1 ⁢ ( x ) ∂ ⁡ x j ⁢ 𝑑 x + o ⁢ ( ε 4 ) = - a 1 , j 2 ⁢ ε 4 ⁢ ∫ ℝ 3 U a 1 2 ⁢ ( x ) ⁢ 𝑑 x + o ⁢ ( ε 4 ) .

Hence

∑ j = 1 3 ∫ B d ⁢ ( x 1 , ε ) ∂ 2 ⁡ V ⁢ ( a 1 ) ∂ ⁡ x i ⁢ ∂ ⁡ x j ⁢ a 1 , j ⁢ 𝑑 x = o ⁢ ( 1 ) .

Since, by assumption, a1 is a non-degenerate critical point of the potential V⁢(x), we find that a1,j=0 for j=1,2,3. ∎

Lemma 3.4.

For any fixed R>0, it holds

η ε ⁢ ( x ) = o ⁢ ( 1 ) , x ∈ B R ⁢ ε ⁢ ( x 1 , ε ) .

Proof.

Lemma 3.2 and Lemma 3.3 show that for any fixed R>0, one has η1,ε⁢(x)=o⁢(1) in BR⁢(0). Also, we know η1,ε⁢(x)=ηε⁢(ε⁢x+x1,ε). Then ηε⁢(x)=o⁢(1),x∈BR⁢ε⁢(x1,ε). ∎

Similar to Proposition 2.1 with k=1, we have the following estimate.

Lemma 3.5.

For large R>0 there exist θ>0 and C>0 such that

η ε ⁢ ( x ) ≤ C ⁢ e - θ ⁢ | x - x 1 , ε | ε for ⁢ l = 1 , … , k ⁢ and ⁢ x ∈ ℝ 3 ∖ B R ⁢ ε ⁢ ( x 1 , ε ) ,

| ∇ ⁡ η ε ⁢ ( x ) | ≤ C ⁢ e - θ ⁢ R ε for ⁢ x ∈ ℝ 3 ∖ B R ⁢ ε ⁢ ( x 1 , ε ) .

Proof of Theorem 1.1 with k=1.

By contradiction, we suppose that ∥ηε∥L∞ and ℒε⁢ηε=0. From Lemma 3.4 and Lemma 3.5 one has ηε⁢(x)=o⁢(1) for all x∈ℝ3, which contradicts with ∥ηε∥L∞⁢(ℝ3)=1. As a result, we obtain that ηε=0 for ε small enough. ∎

4 Proof of Theorem 1.1 with k>1

The case of k≥2 is distinct from that of k=1, which can be seen from the following known result.

Lemma 4.1 ([11]).

Let uε⁢(x) be the solution of (1.3) concentrating at k, k≥2, different non-degenerate critical points {a1,…,ak}⊂R3 of V⁢(x). Then it holds

| x j , ε - a j | = O ⁢ ( ε ) for ⁢ j = 1 , 2 , … , k .

Furthermore, there exist j0∈{1,…,k}, C1>0 and C2>0 such that

C 1 ⁢ ε ≤ | x j 0 , ε - a j 0 | ≤ C 2 ⁢ ε .

Lemma 4.2.

Let ηj,ε⁢(x)=ηε⁢(ε⁢x+xj,ε) for j=1,2,…,k and k≥2. Then, by taking a subsequence if necessary, it holds

η j , ε ⁢ ( x ) → ∑ i = 1 3 a j , i ⁢ ∂ ⁡ U a j ⁢ ( x ) ∂ ⁡ x i

uniformly in C1⁢(BR⁢(0)) for any R>0, where aj,i, i=1,2,3, are some constants.

Proof.

Following the proof of Lemma 3.2, we can obtain Lemma 4.2 similarly. ∎

Lemma 4.3.

Let aj,i be as in Proposition 4.2. Then we have aj,i=0 for all j=1,…,k, i=1,2,3.

Proof.

Similar as in the proof of Lemma 3.3, we also recall Pohozaev identity (2.4). By use of Corollary 2.2 and the Hardy–Littlewood–Sobolev inequality, for some γ>0 it holds that

- ε 2 ⁢ ∫ ∂ ⁡ B d ⁢ ( x j , ε ) ∂ ⁡ u ε ⁢ ( x ) ∂ ⁡ ν ⁢ ∂ ⁡ η ε ⁢ ( x ) ∂ ⁡ x i + ∂ ⁡ η ε ⁢ ( x ) ∂ ⁡ ν ⁢ ∂ ⁡ u ε ⁢ ( x ) ∂ ⁡ x i ⁢ d ⁢ σ + ∫ ∂ ⁡ B d ⁢ ( x j , ε ) ( ε 2 ⁢ 〈 ∇ ⁡ u ε ⁢ ( x ) , ∇ ⁡ η ε ⁢ ( x ) 〉 + V ⁢ ( x ) ⁢ u ε ⁢ ( x ) ⁢ η ε ⁢ ( x ) ) ⁢ ν i ⁢ ( x ) ⁢ 𝑑 σ

- 1 8 ⁢ π ⁢ ε 2 ⁢ ∫ ∂ ⁡ B d ⁢ ( x j , ε ) ∫ ℝ 3 ( u ε 2 ⁢ ( ξ ) ⁢ u ε ⁢ ( x ) ⁢ η ε ⁢ ( x ) | x - ξ | + u ε 2 ⁢ ( x ) ⁢ u ε ⁢ ( ξ ) ⁢ η ε ⁢ ( ξ ) | x - ξ | ) ⁢ ν i ⁢ ( x ) ⁢ 𝑑 ξ ⁢ 𝑑 σ = O ⁢ ( e - γ ε ) .

Next e claim that

(4.1) - 1 8 ⁢ π ⁢ ε 2 ⁢ ∫ B d ⁢ ( x j , ε ) ∫ ℝ 3 ( u ε 2 ⁢ ( ξ ) ⁢ u ε ⁢ ( x ) ⁢ η ε ⁢ ( x ) ⁢ x i - ξ i | x - ξ | 3 + u ε ⁢ ( ξ ) ⁢ η ε ⁢ ( ξ ) ⁢ u ε 2 ⁢ ( x ) ⁢ x i - ξ i | x - ξ | 3 ) ⁢ 𝑑 ξ ⁢ 𝑑 x = o ⁢ ( ε 4 ) ,

which then gives that

RHS of (2.4) = o ⁢ ( ε 4 ) .

Indeed, set

A 1 = 1 8 ⁢ π ⁢ ε 2 ⁢ ∫ B d ⁢ ( x j , ε ) ∫ ℝ 3 u ε 2 ⁢ ( ξ ) ⁢ u ε ⁢ ( x ) ⁢ η ε ⁢ ( x ) ⁢ x i - ξ i | x - ξ | 3 ⁢ 𝑑 ξ ⁢ 𝑑 x ,

A 2 = 1 8 ⁢ π ⁢ ε 2 ⁢ ∫ B d ⁢ ( x j , ε ) ∫ ℝ 3 u ε ⁢ ( ξ ) ⁢ η ε ⁢ ( ξ ) ⁢ u ε 2 ⁢ ( x ) ⁢ x i - ξ i | x - ξ | 3 ⁢ 𝑑 ξ ⁢ 𝑑 x ,

W j , ε = ( x ) = ∑ l = 1 , l ≠ j k U a l ⁢ ( x - x l , ε ε ) .

Now, A1 can be written as

A 1 = A 1 , 1 + A 1 , 2 + A 1 , 3 + A 1 , 4 ,

where

A 1 , 1 = 1 8 ⁢ π ⁢ ε 2 ⁢ ∫ B δ ⁢ ( x j , ε ) ∫ ℝ 3 U a j 2 ⁢ ( x - x j , ε ε ) ⁢ u ε ⁢ ( ξ ) ⁢ η ε ⁢ ( ξ ) ⁢ x i - ξ i | x - ξ | 3 ⁢ 𝑑 ξ ⁢ 𝑑 x ,

A 1 , 2 = 1 4 ⁢ π ⁢ ε 2 ⁢ ∫ B δ ⁢ ( x j , ε ) ∫ ℝ 3 U a j ⁢ ( x - x j , ε ε ) ⁢ w ε ⁢ ( x ) ⁢ u ε ⁢ ( ξ ) ⁢ η ε ⁢ ( ξ ) ⁢ x i - ξ i | x - ξ | 3 ⁢ 𝑑 ξ ⁢ 𝑑 x ,

A 1 , 3 = 1 8 ⁢ π ⁢ ε 2 ⁢ ∫ B δ ⁢ ( x j , ε ) ∫ ℝ 3 ( w ε ⁢ ( x ) ) 2 ⁢ u ε ⁢ ( ξ ) ⁢ η ε ⁢ ( ξ ) ⁢ x i - ξ i | x - ξ | 3 ⁢ 𝑑 ξ ⁢ 𝑑 x ,

A 1 , 4 = 1 8 ⁢ π ⁢ ε 2 ⁢ ∫ B δ ⁢ ( x j , ε ) ∫ ℝ 3 W j , ε ⁢ ( x ) ⁢ ( 2 ⁢ u ε ⁢ ( x ) - W j , ε ⁢ ( x ) ) ⁢ u ε ⁢ ( ξ ) ⁢ η ε ⁢ ( ξ ) ⁢ x i - ξ i | x - ξ | 3 ⁢ 𝑑 ξ ⁢ 𝑑 x ,

while A2 can be written as follows:

A 2 = A 2 , 1 + A 2 , 2 + A 2 , 3 ,

where

A 2 , 1 = 1 8 ⁢ π ⁢ ε 2 ⁢ ∫ B δ ⁢ ( x j , ε ) ∫ ℝ 3 U a j ⁢ ( x - x j , ε ε ) ⁢ η ε ⁢ ( x ) ⁢ ( u ε ⁢ ( ξ ) ) 2 ⁢ x i - ξ i | x - ξ | 3 ⁢ 𝑑 ξ ⁢ 𝑑 x ,

A 2 , 2 = 1 8 ⁢ π ⁢ ε 2 ⁢ ∫ B δ ⁢ ( x j , ε ) ∫ ℝ 3 W j , ε ⁢ ( x ) ⁢ η ε ⁢ ( x ) ⁢ ( u ε ⁢ ( ξ ) ) 2 ⁢ x i - ξ i | x - ξ | 3 ⁢ 𝑑 ξ ⁢ 𝑑 x ,

A 2 , 3 = 1 8 ⁢ π ⁢ ε 2 ⁢ ∫ B δ ⁢ ( x j , ε ) ∫ ℝ 3 w ε ⁢ ( x ) ⁢ η ε ⁢ ( x ) ⁢ ( u ε ⁢ ( ξ ) ) 2 ⁢ x i - ξ i | x - ξ | 3 ⁢ 𝑑 ξ ⁢ 𝑑 x .

Then in view of the fact that (by the Hardy–Littlewood–Sobolev inequalities) for any u1,u2,u3,u4∈Hε and 0<λ≤2,

(4.2) ∫ ℝ 3 ∫ ℝ 3 u 1 ⁢ ( ξ ) ⁢ u 2 ⁢ ( ξ ) ⁢ u 3 ⁢ ( x ) ⁢ u 4 ⁢ ( x ) ⋅ | x - ξ | - λ ⁢ 𝑑 ξ ⁢ 𝑑 x ≤ C ⁢ ε - λ ⁢ ∥ u 1 ∥ ε ⁢ ∥ u 2 ∥ ε ⁢ ∥ u 3 ∥ ε ⁢ ∥ u 4 ∥ ε ,

and there exist two positive constants d1 and η such that, for j=1,2,…,k,

(4.3) U a j ⁢ ( x - x j , ε ε ) = O ⁢ ( e - η ε ) for ⁢ x ∈ ℝ 3 ∖ B d ⁢ ( x j , ε ) ⁢ and ⁢ 0 < d < d 1 .

Then we know that (2.6) and (B.5) imply

A 1 , 3 = O ⁢ ( ε - 4 ⁢ ∥ w ε ∥ ε 2 ⁢ ∥ u ε + u ε ∥ ε ⁢ ∥ η ε ∥ ε ) = O ⁢ ( ε 6 ) , A 1 , 4 = O ⁢ ( e - η ε ) , A 2 , 2 = O ⁢ ( e - η ε ) .

Moreover, we could apply similar argument as in [11] to estimate that

A 1 , 1 = G 1 + 1 8 ⁢ π ⁢ ε 2 ⁢ ∫ ℝ 3 ∫ ℝ 3 U a j 2 ⁢ ( x - x j , ε ε ) ⁢ w ε ⁢ ( ξ ) ⁢ η ε ⁢ ( ξ ) ⁢ x i - ξ i | x - ξ | 3 ⁢ 𝑑 ξ ⁢ 𝑑 x + o ⁢ ( ε 4 ) ,

where

G 1 = 1 8 ⁢ π ⁢ ε 2 ⁢ ∫ ℝ 3 ∫ ℝ 3 U a j 2 ⁢ ( x - x j , ε ε ) ⁢ U a j ⁢ ( ξ - x j , ε ε ) ⁢ η ε ⁢ ( ξ ) ⁢ x i - ξ i | x - ξ | 3 ⁢ 𝑑 ξ ⁢ 𝑑 x ,

and

A 1 , 2 = 1 2 ⁢ π ⁢ ε 2 ⁢ ∫ ℝ 3 ∫ ℝ 3 U a j ⁢ ( x - x j , ε ε ) ⁢ w ε ⁢ ( x ) ⁢ U a j ⁢ ( ξ - x j , ε ε ) ⁢ η ε ⁢ ( ξ ) ⁢ x i - ξ i | x - ξ | 3 ⁢ 𝑑 ξ ⁢ 𝑑 x + o ⁢ ( ε 4 ) ,

A 2 , 1 = G 2 - 1 2 ⁢ π ⁢ ε 2 ⁢ ∫ ℝ 3 ∫ ℝ 3 U a j ⁢ ( x - x j , ε ε ) ⁢ w ε ⁢ ( x ) ⁢ U a j ⁢ ( ξ - x j , ε ε ) ⁢ η ε ⁢ ( ξ ) ⁢ x i - ξ i | x - ξ | 3 ⁢ 𝑑 ξ ⁢ 𝑑 x + o ⁢ ( ε 4 ) ,

where

G 2 = - 1 8 ⁢ π ⁢ ε 2 ⁢ ∫ ℝ 3 ∫ ℝ 3 U a j 2 ⁢ ( x - x j , ε ε ) ⁢ U a j ⁢ ( ξ - x j , ε ε ) ⁢ η ε ⁢ ( x ) ⁢ x i - ξ i | x - ξ | 3 ⁢ 𝑑 ξ ⁢ 𝑑 x ,

and

A 2 , 3 = - 1 8 ⁢ π ⁢ ε 2 ⁢ ∫ ℝ 3 ∫ ℝ 3 U a j 2 ⁢ ( x - x j , ε ε ) ⁢ w ε ⁢ ( ξ ) ⁢ η ε ⁢ ( ξ ) ⁢ x i - ξ i | x - ξ | 3 ⁢ 𝑑 ξ ⁢ 𝑑 x + o ⁢ ( ε 4 ) .

To sum up, A1+A2=o⁢(ε4), which implies the claim (4.1).

On the other hand, similar estimate to that in Lemma 3.3, we have

∫ B d ⁢ ( x j , ε ) ∂ ⁡ V ⁢ ( x ) ∂ ⁡ x i ⁢ u ε ⁢ ( x ) ⁢ η ε ⁢ ( x ) ⁢ 𝑑 x = ∑ l = 1 3 ∫ B d ⁢ ( x j , ε ) ∂ 2 ⁡ V ⁢ ( a j ) ∂ ⁡ x i ⁢ ∂ ⁡ x l ⁢ ( x l - a j l ) ⁢ u ε ⁢ ( x ) ⁢ η ε ⁢ ( x ) ⁢ 𝑑 x + o ⁢ ( ∫ B d ⁢ ( x j , ε ) | x - a j | ⁢ u ε ⁢ ( x ) ⁢ η ε ⁢ ( x ) ⁢ 𝑑 x )

= ∑ l = 1 3 ∂ 2 ⁡ V ⁢ ( a j ) ∂ ⁡ x i ⁢ ∂ ⁡ x l ⁢ ε 4 ⁢ ( ∫ B d ε ⁢ ( 0 ) ( x l + x j , ε , l - a j l ε ) ⁢ U a j ⁢ ( x ) ⁢ ( ∑ i = 1 3 a j , i ⁢ ∂ ⁡ U a j ⁢ ( x ) ∂ ⁡ x i + o ⁢ ( 1 ) ) ⁢ 𝑑 x ) + o ⁢ ( ε 4 )

= ∑ l = 1 3 ∂ 2 ⁡ V ⁢ ( a j ) ∂ ⁡ x i ⁢ ∂ ⁡ x l ⁢ a j , l ⁢ ε 4 ⁢ ∫ ℝ 3 x l ⁢ U a j ⁢ ( x ) ⁢ ∂ ⁡ U a j ⁢ ( x ) ∂ ⁡ x l ⁢ 𝑑 x + o ⁢ ( ε 4 )

= - ∑ l = 1 3 ∂ 2 ⁡ V ⁢ ( a j ) ∂ ⁡ x i ⁢ ∂ ⁡ x l ⁢ a j , l 2 ⁢ ε 4 ⁢ ∫ ℝ 3 U a j 2 ⁢ ( x ) ⁢ 𝑑 x + o ⁢ ( ε 4 ) .

Hence

∑ l = 1 3 ∂ 2 ⁡ V ⁢ ( a j ) ∂ ⁡ x i ⁢ ∂ ⁡ x l ⁢ a j , l = o ⁢ ( 1 ) .

Since, again by assumption, aj is a non-degenerate critical point of the potential V⁢(x), we find that aj,l=0 for l=1,2,3. ∎

Lemma 4.4.

For any fixed R>0, it holds

η ε ⁢ ( x ) = o ⁢ ( 1 ) , x ∈ ⋃ j = 1 k B R ⁢ ε ⁢ ( x j , ε ) .

Proof.

Similar to the proof of Lemma 3.4, Lemma 4.2 and Lemma 4.3 give the result. ∎

Similar to Proposition 2.1, we also have the following estimate.

Lemma 4.5.

For large R>0 there exist θ>0 and C>0 such that

η ε ⁢ ( x ) ≤ C ⁢ e - θ ⁢ | x - x l , ε | ε for ⁢ l = 1 , … , k ⁢ 𝑎𝑛𝑑 ⁢ x ∈ ℝ 3 ∖ ⋃ j = 1 k B R ⁢ ε ⁢ ( x j , ε ) ,

| ∇ ⁡ η ε ⁢ ( x ) | ≤ C ⁢ e - θ ⁢ R ε for ⁢ x ∈ ℝ 3 ∖ ⋃ j = 1 k B R ⁢ ε ⁢ ( x j , ε ) .

Proof of Theorem 1.1.

By contradiction, we suppose that ∥ηε∥L∞ and ℒε⁢ηε=0. From Lemmas 4.4 and 4.5, ηε⁢(x)=o⁢(1), for all x∈ℝ3, which contradicts with ∥ηε∥L∞⁢(ℝ3)=1. As a result, we obtain that ηε=0 for ε small enough. ∎

Communicated by Silvia Cingolani

Funding source: National Natural Science Foundation of China

Award Identifier / Grant number: 11771469

Funding statement: Qing Guo has been supported by NSFC grant (No. 11771469).

A Estimates on A1,A2

Lemma A.1.

Suppose fε∈L1⁢(R3)∩C⁢(R3), for any fixed small d¯>0 independent of ε and xε. Then there exists a small constant dε∈(d¯,2⁢d¯) such that

(A.1) ∫ ∂ ⁡ B d ε ⁢ ( x ε ) | f ε ⁢ ( x ) | ⁢ 𝑑 σ ≤ 1 d ¯ ⁢ ∫ ℝ 3 | f ε ⁢ ( x ) | ⁢ 𝑑 x .

Proof.

First, for any fixed small d¯>0 and xε,

(A.2) ∫ d ¯ 2 ⁢ d ¯ ∫ ∂ ⁡ B r ⁢ ( x ε ) | f ε ⁢ ( x ) | ⁢ 𝑑 σ ⁢ 𝑑 r = ∫ B 2 ⁢ d ¯ ⁢ ( x ε ) ∖ B d ¯ ⁢ ( x ε ) | f ε ⁢ ( x ) | ⁢ 𝑑 x ≤ ∫ ℝ 3 | f ε ⁢ ( x ) | ⁢ 𝑑 x .

Also ∫∂⁡Br⁢(xε)|fε⁢(x)|⁢𝑑σ is continuous with respect to r. By the mean value theorem of integrals, there exists a constant dε∈(d¯,2⁢d¯) such that

(A.3) ∫ d ¯ 2 ⁢ d ¯ ∫ ∂ ⁡ B r ⁢ ( x ε ) | f ε ⁢ ( x ) | ⁢ 𝑑 σ ⁢ 𝑑 r = d ε ⁢ ∫ ∂ ⁡ B r ⁢ ( x ε ) | f ε ⁢ ( x ) | ⁢ 𝑑 σ .

Then (A.2) and (A.3) imply (A.1). ∎

Using the notations from (2.5), we have the following estimates.

Lemma A.2.

For any fixed R>0, it holds

(A.4) A 1 ⁢ ( x ) = o ⁢ ( 1 ) ⋅ R + O ⁢ ( 1 R ) for ⁢ x ∈ ℝ 3 ∖ ⋃ j = 1 k B R ⁢ ε ⁢ ( x j , ε )

and

(A.5) A 2 ⁢ ( x ) = O ⁢ ( e - θ ′ ⁢ R ) for ⁢ x ∈ ℝ 3 ∖ ⋃ j = 1 k B R ⁢ ε ⁢ ( x j , ε ) ⁢ and some ⁢ θ ′ > 0 .

Proof.

First, we know

{ ξ : | x - ξ | ≤ R ⁢ ε 2 } ⊂ ℝ 3 ∖ ⋃ j = 1 k B R ⁢ ε 2 ⁢ ( x j , ε ) for ⁢ x ∈ ℝ 3 ∖ ⋃ j = 1 k B R ⁢ ε ⁢ ( x j , ε ) ,

and ∥uε∥ε=O⁢(ε32). Then by (2.1), for x∈ℝ3∖⋃j=1kBR⁢ε⁢(xj,ε), it holds

(A.6) A 1 ⁢ ( x ) = 1 8 ⁢ π ⁢ ε 2 ⁢ ∫ | x - ξ | ≤ R ⁢ ε 2 ( u ε ⁢ ( ξ ) ) 2 ⁢ | x - ξ | - 1 ⁢ 𝑑 ξ + 1 8 ⁢ π ⁢ ε 2 ⁢ ∫ | x - ξ | > R ⁢ ε 2 ( u ε ⁢ ( ξ ) ) 2 ⁢ | x - ξ | - 1 ⁢ 𝑑 ξ = O ⁢ ( ε - 2 ⁢ ∫ | x - ξ | ≤ R ⁢ ε 2 ( w ε ⁢ ( ξ ) ) 2 ⁢ | x - ξ | - 1 ⁢ 𝑑 ξ ) + O ⁢ ( e - 2 ⁢ θ ⁢ R ⁢ R 2 ) + O ⁢ ( 1 R ) .

Also, by Hölder’s inequality, we have

(A.7) ∫ | x - ξ | ≤ R ⁢ ε 2 ( w ε ⁢ ( ξ ) ) 2 ⁢ | x - ξ | - 1 ⁢ 𝑑 ξ = O ⁢ ( ( ∫ ℝ 3 ( w ε ⁢ ( ξ ) ) 6 ⁢ 𝑑 ξ ) 1 3 ⋅ ( ∫ | x - ξ | ≤ R ⁢ ε 2 | x - ξ | - 3 2 ⁢ 𝑑 ξ ) 2 3 ) = R ⋅ O ⁢ ( ε - 1 ⁢ ∥ w ε ⁢ ( ξ ) ∥ ε 2 ) = o ⁢ ( ε 2 ) ⋅ R .

Then (A.6) and (A.7) imply (A.4).

Next for x∈ℝ3∖⋃j=1kBR⁢ε⁢(xj,ε), we have

(A.8) A 2 ⁢ ( x ) = O ⁢ ( e - θ ⁢ R ) ⋅ ε - 2 ⁢ ∫ ℝ 3 u ε ⁢ ( ξ ) ⁢ | x - ξ | - 1 ⋅ | η ε ⁢ ( ξ ) | ⁢ 𝑑 ξ

and

(A.9) ∫ ℝ 3 u ε ⁢ ( ξ ) ⁢ | x - ξ | - 1 ⋅ | η ε ⁢ ( ξ ) | ⁢ 𝑑 ξ = ∫ | x - ξ | ≤ R ⁢ ε 2 u ε ⁢ ( ξ ) ⁢ | x - ξ | - 1 ⋅ | η ε ⁢ ( ξ ) | ⁢ 𝑑 ξ + O ⁢ ( ( R ⁢ ε ) - 1 ⁢ ∥ u ε + u ε ∥ ε ⁢ ∥ η ε ∥ ε ) = O ( ∥ ( u ε ( ⋅ ) + u ε ( ⋅ ) ∥ ε ) ⋅ ( ∫ | x - ξ | ≤ R ⁢ ε 2 | x - ξ | - 2 d ξ ) 1 2 + O ( R - 1 ε 2 ) = O ⁢ ( ( R 1 2 + R - 1 ) ⁢ ε 2 ) .

Then (A.8) and (A.9) imply (A.5). ∎

Lemma A.3.

For any fixed small d>0, it holds

(A.10) A 1 ⁢ ( x ) = 1 8 ⁢ π ⁢ ε 2 ⁢ ( ∫ ℝ 3 U a j 2 ⁢ ( ξ - x j , ε ε ) ⁢ | x - ξ | - 1 ⁢ 𝑑 ξ ) + o ⁢ ( 1 ) in ⁢ B d ⁢ ( x j , ε )

and

(A.11) A 2 ⁢ ( x ) = 1 4 ⁢ π ⋅ U a j ⁢ ( x - x j , ε ε ) ⁢ ( ∫ ℝ 3 U a j ⁢ ( ξ - x j , ε ε ) ⁢ η ε ⁢ ( ξ ) ⁢ | x - ξ | - 1 ⁢ 𝑑 ξ ) + o ⁢ ( 1 ) in ⁢ B d ⁢ ( x j , ε ) .

Proof.

For x∈Bd⁢(xj,ε), we have

(A.12) | A 1 ( x ) - 1 8 ⁢ π ⁢ ε 2 ( ∫ ℝ 3 U a j 2 ( ξ - x j , ε ε ) | x - ξ | - 1 d ξ ) | = O ⁢ ( ε - 2 ⁢ ∫ ℝ 3 | w ε ⁢ ( ξ ) | ⋅ ( u ε ⁢ ( ξ ) + U a j ⁢ ( ξ - x j , ε ε ) ) ⁢ | x - ξ | - 1 ⁢ 𝑑 ξ ) + O ⁢ ( e - η ε ) = O ⁢ ( ε - 2 ⁢ ∫ | x - ξ | ≤ C | w ε ⁢ ( ξ ) | ⋅ ( u ε ⁢ ( ξ ) + U a j ⁢ ( ξ - x j , ε ε ) ) ⁢ | x - ξ | - 1 ⁢ 𝑑 ξ ) + O ⁢ ( ε - 2 ⁢ ∥ w ε ⁢ ( ⋅ ) ∥ ε ⋅ ∥ u ε ⁢ ( ⋅ ) + U a j ⁢ ( ⋅ - x j , ε ε ) ∥ ε ) + O ⁢ ( e - η ε ) ,

where C is a fixed constant.

On the other hand, by Hölder’s inequality, we know

(A.13) ∫ | x - ξ | ≤ C | w ε ⁢ ( ξ ) | ⋅ ( u ε ⁢ ( ξ ) + U a j ⁢ ( ξ - x j , ε ε ) ) ⁢ | x - ξ | - 1 ⁢ 𝑑 ξ = O ⁢ ( ∥ w ε ⁢ ( ⋅ ) ∥ L 6 ⁢ ( ℝ 3 ) ⋅ ∥ u ε ⁢ ( ⋅ ) + U a j ⁢ ( ⋅ - x j , ε ε ) ∥ L 2 ⁢ ( ℝ 3 ) ⁢ ( ∫ | x - ξ | ≤ C | x - ξ | - 3 ⁢ 𝑑 ξ ) 1 3 ) = O ⁢ ( ε - 1 ⁢ ∥ w ε ⁢ ( ⋅ ) ∥ ε ⋅ ∥ u ε ⁢ ( ⋅ ) + U a j ⁢ ( ⋅ - x j , ε ε ) ∥ ε ) = o ⁢ ( ε 2 ) .

Then (A.12) and (A.13) imply (A.10). Similar to the estimates of (A.10), combining Proposition 3.1, we deduce (A.11). ∎

B Estimates of the Term wε

We denote

R ε ⁢ ( x ) = ∑ l = 1 k U a l ⁢ ( x - x l , ε ε ) , W j , ε ⁢ ( x ) = ∑ l = 1 , l ≠ j k U a l ⁢ ( x - x l , ε ε ) .

Let Mε⁢(x,wε⁢(x)) be as follows:

M ε ⁢ ( x , w ε ⁢ ( x ) ) := - ε 2 ⁢ Δ ⁢ w ε ⁢ ( x ) + G ⁢ ( x , w ε ⁢ ( x ) ) ,

where

G ⁢ ( x , w ε ⁢ ( x ) ) = V ⁢ ( x ) ⁢ w ε ⁢ ( x ) - 1 8 ⁢ π ⁢ ε 2 ⁢ ( ∫ ℝ 3 ( R ε ⁢ ( ξ ) ) 2 | x - ξ | ⁢ 𝑑 ξ ) ⁢ w ε ⁢ ( x ) + 1 4 ⁢ π ⁢ ε 2 ⁢ ( ∫ ℝ 3 R ε ⁢ ( ξ ) ⁢ w ε ⁢ ( ξ ) | x - ξ | ⁢ 𝑑 ξ ) ⁢ R ε ⁢ ( x ) .

Let uε⁢(x)=Rε⁢(x)+wε⁢(x) be the solution of (1.3). Then

M ε ⁢ ( x , w ε ⁢ ( x ) ) = N ⁢ ( x , w ε ⁢ ( x ) ) + l ε ⁢ ( x ) ,

where

N ⁢ ( x , w ε ⁢ ( x ) ) = 1 8 ⁢ π ⁢ ε 2 ⁢ ( ∫ ℝ 3 w ε 2 ⁢ ( ξ ) | x - ξ | ⁢ 𝑑 ξ ) ⁢ ( R ε ⁢ ( x ) + w ε ⁢ ( x ) ) + w ε ⁢ ( x ) 4 ⁢ π ⁢ ε 2 ⁢ ∫ ℝ 3 R ε ⁢ ( ξ ) ⁢ w ξ ⁢ ( ξ ) | x - ξ | ⁢ 𝑑 ξ

and

l ε ⁢ ( x ) = W j , ε ⁢ ( x ) 8 ⁢ π ⁢ ε 2 ⁢ ∫ ℝ 3 W j , ε ⁢ ( ξ ) ⁢ U a j ⁢ ( ξ - x j , ε ε ) | x - ξ | ⁢ 𝑑 ξ + ∑ j = 1 k ( V ⁢ ( a j ) - V ⁢ ( x ) ) ⁢ U a j ⁢ ( x - x j , ε ε ) .

Proposition B.1.

Let uε⁢(x)=Rε⁢(x)+wε⁢(x) be the solution of (1.3). Then there exists a constant ρ¯>0 independent of ε such that

(B.1) ∫ ℝ 3 M ε ⁢ ( x , w ε ⁢ ( x ) ) ⁢ w ε ⁢ ( x ) ⁢ 𝑑 x ≥ ρ ¯ ⁢ ∥ w ε ∥ ε 2 .

Proof.

Similar to the proof of [2, Proposition 3.1], we can prove (B.1) by the contradiction argument and blow-up analysis. For the more details, one can refer to [1, 2]. ∎

Proof of Proposition 2.5.

First, from Proposition B.1, we know

(B.2) ∥ w ε ∥ ε 2 ≤ C ⁢ ∫ ℝ 3 N ⁢ ( x , w ε ⁢ ( x ) ) ⁢ w ε ⁢ ( x ) ⁢ 𝑑 x + C ⁢ ∫ ℝ 3 l ε ⁢ ( x ) ⁢ w ε ⁢ ( x ) ⁢ 𝑑 x .

Next, using Theorem B and (4.2), we deduce

(B.3) ∫ ℝ 3 N ⁢ ( x , w ε ⁢ ( x ) ) ⁢ w ε ⁢ ( x ) ⁢ 𝑑 x = 1 8 ⁢ π ⁢ ε 2 ⁢ ∫ ℝ 3 ∫ ℝ 3 w ε 2 ⁢ ( ξ ) | x - ξ | ⁢ ( R ε ⁢ ( x ) + w ε ⁢ ( x ) ) ⁢ w ε ⁢ ( x ) ⁢ 𝑑 x ⁢ 𝑑 ξ + 1 4 ⁢ π ⁢ ε 2 ⁢ ∫ ℝ 3 ∫ ℝ 3 R ε ⁢ ( ξ ) ⁢ w ξ ⁢ ( ξ ) | x - ξ | ⁢ w ε 2 ⁢ ( x ) ⁢ 𝑑 x ⁢ 𝑑 ξ = O ⁢ ( ε - 3 ⁢ ∥ w ε ∥ ε 3 ⋅ ∥ w ε + R ε ∥ ε ) = o ⁢ ( 1 ) ⁢ ∥ w ε ∥ ε 2 .

Also from (4.3) and

V ⁢ ( a j ) - V ⁢ ( x ) = - ∑ i = 1 3 ∑ l = 1 3 ( x i - a j i ) ⁢ ( x l - a j l ) ⁢ ∂ 2 ⁡ V ⁢ ( a j ) ∂ ⁡ x i ⁢ ∂ ⁡ x l + o ⁢ ( | x - a j | 2 ) ,

we have

(B.4) ∫ ℝ 3 l ε ⁢ ( x ) ⁢ w ε ⁢ ( x ) ⁢ 𝑑 x = ∑ j = 1 k ∫ ℝ 3 ( V ⁢ ( a j ) - V ⁢ ( x ) ) ⁢ U a j ⁢ ( x - x j , ε ε ) ⁢ w ε ⁢ ( x ) ⁢ 𝑑 ξ ⁢ 𝑑 x - 1 8 ⁢ π ⁢ ε 2 ⁢ ∫ ℝ 3 ∫ ℝ 3 W j , ε ⁢ ( ξ ) ⁢ U a j ⁢ ( ξ - x j , ε ε ) ⁢ W j , ε ⁢ ( x ) ⁢ w ε ⁢ ( x ) ⁢ | x - ξ | - 1 ⁢ 𝑑 ξ = O ⁢ ( ε 3 2 ⁢ ∥ w ε ∥ ε ⁢ ( ε 2 + max j = 1 , … , k ⁡ | x j , ε - a j | 2 ) + e - η ε ) .

Then (B.2), (B.3) and (B.4) imply (2.10). ∎

Proposition B.2.

Let uε⁢(x) be a positive solution of (1.3) as in Proposition 2.5. Then it holds

(B.5) ∥ w ε ∥ ε = O ⁢ ( ε 7 2 ) .

Proof.

It follows from the results of Proposition 3.1 and Proposition 2.5 directly. ∎

Acknowledgements

The author would like to thank Peng Luo for interesting discussions on the relationship and distinction between local uniqueness and the non-degeneracy result on the present problem.

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Received: 2020-12-23

Revised: 2021-03-14

Accepted: 2021-03-15

Published Online: 2021-04-15

Published in Print: 2021-05-01

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

Non-Degeneracy of Peak Solutions to the Schrödinger–Newton System

Abstract

1 Introduction and Main Results

Theorem A (cf. [8, 17]).

Theorem B (cf. [17]).

Theorem 1.1.

Organization of the Paper.

2 The Basic Estimates

Proposition 2.1 ([11]).

Corollary 2.2.

Proposition 2.3.

Proof.

Proposition 2.4.

Proof.

Proposition 2.5.

Proof.

3 Proof of Theorem 1.1 with k=1

Proposition 3.1.

Proof.

Lemma 3.2.

Proof.

Lemma 3.3.

Proof.

Lemma 3.4.

Proof.

Lemma 3.5.

Proof of Theorem 1.1 with k=1.

4 Proof of Theorem 1.1 with k>1

Lemma 4.1 ([11]).

Lemma 4.2.

Proof.

Lemma 4.3.

Proof.

Lemma 4.4.

Proof.

Lemma 4.5.

Proof of Theorem 1.1.

A Estimates on A1,A2

Lemma A.1.

Proof.

Lemma A.2.

Proof.

Lemma A.3.

Proof.

B Estimates of the Term wε

Proposition B.1.

Proof.

Proof of Proposition 2.5.

Proposition B.2.

Proof.

Acknowledgements

References

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