Uniqueness and nondegeneracy of ground states for 
                  
                     
                     
                        −
                        Δ
                        u
                        +
                        u
                        =
                        
                           (
                           
                              
                                 
                                    I
                                 
                                 
                                    α
                                 
                              
                              ⋆
                              
                                 
                                    u
                                 
                                 
                                    2
                                 
                              
                           
                           )
                        
                        u
                     
                     
                  
                in 
                  
                     
                     
                        
                           
                              R
                           
                           
                              3
                           
                        
                     
                     
                  
                when 
                  
                     
                     
                        α
                     
                     
                  
                is close to 2

Huxiao Luo; Dingliang Zhang; Yating Xu

doi:10.1515/anona-2024-0048

Artikel Open Access

Uniqueness and nondegeneracy of ground states for − Δ u + u = ( I α ⋆ u 2 ) u in R 3 when α is close to 2

Huxiao Luo , Dingliang Zhang und Yating Xu

Veröffentlicht/Copyright: 22. November 2024

Veröffentlicht von

Veröffentlichen auch Sie bei De Gruyter Brill

Manuskript einreichen Informationen für Autor*innen Erkunden Sie dieses Fachgebiet

Aus der Zeitschrift Advances in Nonlinear Analysis Band 13 Heft 1

Abstract

In this article, we study the following Choquard equation:

− Δ u + u = ( I α ⋆ u 2 ) u , x ∈ R 3 ,

where I α is the Riesz potential and α is sufficiently close to 2. By investigating the limit profile of ground states of the equation as α → 2 , we prove the uniqueness and nondegeneracy of ground states.

Keywords: limit profile; Choquard equation; uniqueness and nondegeneracy

MSC 2010: 35A02; 35B20; 35J61

1 Introduction

In this article, we study the uniqueness and nondegeneracy of ground states for the Choquard equation

(P) − Δ u + u = ( I α ⋆ u 2 ) u , in R 3 , u ∈ H 1 ( R 3 ) ,

when α ∈ ( 1 , 3 ) sufficiently close to 2, ⋆ represents the convolution operation in R 3 . Here and in the following, a nontrivial solution is called ground state if it minimizes the energy of all the nontrivial solutions. I α is the Riesz potential, which is defined for each x ∈ R 3 \ { 0 } by

I α ( ⋅ ) = A α ∣ ⋅ ∣ 3 − α , α ∈ ( 0 , 3 ) ,

where A α = Γ 3 − α 2 Γ α 2 π 3 2 2 α is the normalization constant [11].

When α = 2 , (P) changes to be the Schrödinger-Newton equation

(1.1) − Δ u + u = ( I 2 ⋆ u 2 ) u , in R 3 .

(1.1) describes the standing waves for the Hartree equation or the coupling of Schrödinger’s equation under a classical Newtonian gravitational potential. Pekar [12] used (1.1) to describe the quantum theory of a polaron at rest. Later, Choquard [7] used it to model one-component plasmas. Existence and concentration behavior of positive solutions, radial solutions and normalized solutions for Choquard equations have been widely studied by many authors (see, e.g., [3,4,6]).

The uniqueness of ground state solutions for (1.1) has been solved by Lieb [7] (see also [7,9,15,17]). By developing Lieb’s methods [7], Wang and Yi [16] established the uniqueness of ground-state solutions to

(1.2) − Δ u + u = 1 ∣ x ∣ N − 2 ⋆ u 2 u , in R N , u ∈ H 1 ( R N ) ,

for general dimensions N = 3 , 4 , 5 .

The nondegeneracy of ground-state solutions for (1.1) has been proved by Lenzmann [5, Theorem 1.4] (see also Tod-Moroz [15] and Wei and Winter [17]). This nondegeneracy result has been generalized to (1.2) for general dimensions N = 3 , 4 , 5 [1].

However, there are few research results on the uniqueness and nondegeneracy result for the generalized Choquard equations

(1.3) − Δ u + u = ( I α ⋆ ∣ u ∣ p ) ∣ u ∣ p − 2 u , u ∈ H 1 ( R N ) .

When N = 3 and p > 2 is sufficiently close to 2, Xiang [18] obtained the uniqueness and nondegeneracy results of ground states for (1.3).

Seok [13] proved that the family of positive radial ground states to (1.3) converges to the unique positive solution for the Schrödinger equation

− Δ u + u = ∣ u ∣ 2 p − 2 u , u ∈ H 1 ( R N ) ,

as α ↘ 0 , and converges to the unique positive solution for the equation

− Δ u + u = ∫ R N ∣ u ∣ p d x ∣ u ∣ p − 2 u , u ∈ H 1 ( R N ) ,

as α ↗ N , respectively. By means of the uniqueness and nondegeneracy of ground-state solutions to the two aforementioned limit equations, the author obtained the uniqueness and nondegeneracy of the radial positive ground-state solution for (1.3) when α is sufficiently close to 0 or N .

A natural problem is, when α is sufficiently close to 2, the ground-state solution of (P) is unique and nondegenerate. Inspired by [13], we prove that the family of positive radial ground states to (P) converge to the unique positive solution for the Schrödinger-Newton equation (1.1) and then prove the uniqueness and nondegeneracy of the radial positive ground-state solution for (P) when α is sufficiently close to 2. Since the limit equation is also a nonlocal equation, this brings some new difficulties, such as the uniform estimates and asymptotics of ground states. To overcome difficulties, we need more detailed analyses.

We point out that the existence of ground states for (P) has been obtained. Indeed, the existence of a ground state for the generalized Choquard equations (1.3) has been proved by Moroz and Van Schaftingen [10].

Proposition 1

[10, Theorem 1] Let

(1.4) N ≥ 1 , α ∈ ( 0 , N ) , p > 1 , and N − 2 N + α < 1 p < N N + α .

Then, there exists a ground state u ∈ H 1 ( R N ) to equation (1.3).

Since (1.4) holds in our setting N = 3 , p = 2 , and α ∈ ( 1 , 3 ) , then by Proposition 1, we obtain a ground state u ∈ H 1 ( R 3 ) to (P).

First, we study the limit profile of ground states for (P) as α → 2 .

Theorem 1.1

Assume that { u α } is a family of the positive radial ground states to (P) for α close to 2, and let u 0 ∈ H 1 ( R 3 ) be the unique positive radial ground state of (1.1). Then, one has

lim α → 2 ‖ u α − u 0 ‖ H 1 ( R 3 ) = 0 .

Next, we give the uniqueness and nondegeneracy results.

Theorem 1.2

There exists ε 0 ∈ ( 0 , 1 ) such that for every α ∈ ( 2 − ε 0 , 2 + ε 0 ) , the positive radial ground state for (P) is unique, up to translations.

Theorem 1.3

Suppose that u α is the unique positive radial ground states of (P) obtained in Theorem 1.2. Then, for every α ∈ ( 2 − ε 0 , 2 + ε 0 ) , the linearized equation of (P) at u α , written by

(1.5) − Δ ϕ + ϕ − 2 ( I α ⋆ ( u α ϕ ) ) u α − ( I α ⋆ u α 2 ) ϕ = 0 , in R 3 ,

only admits solutions of the form

ϕ = ∑ i = 1 3 c i ∂ x i u α , c i ∈ R ,

in L 2 ( R 3 ) .

This article is organized as follows. In Section 2, we provide uniform estimates and asymptotics of ground states. In Sections 3 and 4, we prove Theorems 1.2–1.3, respectively.

For the sake of simplicity, integrals over the whole space will be often written ∫ ;
B r ( 0 ) denotes a sphere in R N with the origin as its center and r as its radius;
‖ ⋅ ‖ H 1 denotes the standard norm for the Sobolev space H 1 ( R N ) ;
‖ ⋅ ‖ L q denotes the L q ( R N ) -norm for q ∈ [ 1 , ∞ ] ;
o ( 1 ) denotes the infinitesimal as n → + ∞ ;
Unless otherwise specified, C represents a pure constant independent of any variable.

2 Uniform estimates and asymptotics

In this section, we prove Theorem 1.1. Let J α be the energy functional associated with (P),

J α ( u ) = 1 2 ∫ ∣ ∇ u ∣ 2 + u 2 d x − 1 4 ∫ ( I α ⋆ u 2 ) u 2 d x , ∀ u ∈ H 1 ( R 3 ) .

It is easy to confirm that J α is well defined from the Hardy-Littlewood-Sobolev inequality.

Lemma 2.1

(Hardy-Littlewood-Sobolev inequality [2, 8]) Let p , r > 1 and 0 < α < N be such that 1 p + 1 r = 1 + α N . Then, for any f ∈ L p ( R N ) and g ∈ L r ( R N ) , one has

∫ ∫ f ( x ) g ( y ) ∣ x − y ∣ N − α d x d y ≤ C ( N , α , r ) ‖ f ‖ L p ‖ g ‖ L r .

The sharp constant satisfies

(2.1) C ( N , α , r ) ≤ N α ∣ S ∣ N − 1 N N − α N 1 p r N − α N ( 1 − 1 ⁄ p ) N − α N + N − α N ( 1 − 1 ⁄ r ) N − α N .

Moreover, let 1 s + 1 p = 1 , then there exists a constant C ′ ( N , α , r ) < ∞ such that for any g ∈ L r ( R N ) , we have

‖ I α ⋆ g ‖ L s ≤ C ′ ( N , α , r ) ‖ g ‖ L r .

The following compact Sobolev embedding for radial functions is essential in our proof.

Lemma 2.2

[14] The space H r 1 ( R 3 ) is compactly embedded in L p ( R 3 ) if p ∈ ( 2 , 6 ) .

Now, we give the following estimates that will be used in the proof of Theorem 1.1.

Lemma 2.3

Assume s ∈ ( 0 , N ) and β ∈ ( 2 , + ∞ ) . For every f , g ∈ L 2 ( R N ) and α ∈ ( 0 , β ] satisfy I β 2 ⋆ f ∈ L 2 ( R N ) , I β 2 ⋆ g ∈ L 2 ( R N ) , ( − Δ ) s 4 f ∈ L 2 ( R N ) , and ( − Δ ) s 4 g ∈ L 2 ( R N ) . Then, we have

(2.2) ∫ ( I α ⋆ f ) g d x − ∫ ( I 2 ⋆ f ) g d x ≤ ∣ 2 − α ∣ β − 2 I β 2 ⋆ f L 2 I β 2 ⋆ g L 2 + ∣ 2 − α ∣ s ( − Δ ) s 4 f L 2 ( − Δ ) s 4 g L 2 .

Proof

For every f ∈ L 2 ( R N ) , let f ˆ be the Fourier transforms of f . By the Plancherel theorem and the formula for the Fourier transform of a Riesz potential, one has

∫ ( I α ⋆ f ) g d x − ∫ ( I 2 ⋆ f ) g d x = ∫ ( 2 π ∣ ξ ∣ ) − 2 [ ( 2 π ∣ ξ ∣ ) 2 − α − 1 ] f ˆ ( ξ ) g ˆ ( ξ ) d ξ ≔ I .

We split the integral I into two parts:

I = ∫ B 1 2 π ( 0 ) ( 2 π ∣ ξ ∣ ) − 2 [ ( 2 π ∣ ξ ∣ ) 2 − α − 1 ] f ˆ ( ξ ) g ˆ ( ξ ) d ξ + ∫ R N \ B 1 2 π ( 0 ) ( 2 π ∣ ξ ∣ ) − 2 [ ( 2 π ∣ ξ ∣ ) 2 − α − 1 ] f ˆ ( ξ ) g ˆ ( ξ ) d ξ ≔ II 1 + II 2 ,

and estimate the integration I in two cases: Case 1: α ∈ ( 0 , 2 ) ; Case 2: α ∈ ( 2 , β ) .

Case 1. We first estimate II 1 . For 2 π ∣ ξ ∣ ≤ 1 , by the Young’s inequality, one has

( 2 π ∣ ξ ∣ ) 2 − α ≤ 1 ≤ β − 2 β − α ( 2 π ∣ ξ ∣ ) 2 − α + 2 − α β − α ( 2 π ∣ ξ ∣ ) 2 − β ,

so that

∣ ( 2 π ∣ ξ ∣ ) 2 − α − 1 ∣ ≤ 2 − α β − 2 ( 2 π ∣ ξ ∣ ) 2 − β .

Therefore, by the Hölder inequality, we infer that

II 1 ≤ 2 − α β − 2 ∫ B 1 2 π ( 0 ) ( 2 π ∣ ξ ∣ ) − β ∣ f ˆ ( ξ ) ∣ ∣ g ˆ ( ξ ) ∣ d ξ ≤ 2 − α β − 2 ∫ ( 2 π ∣ ξ ∣ ) − β ∣ f ˆ ( ξ ) ∣ 2 d ξ 1 2 ∫ ( 2 π ∣ ξ ∣ ) − β ∣ g ˆ ( ξ ) ∣ 2 d ξ 1 2 = 2 − α β − 2 I β 2 ⋆ f L 2 I β 2 ⋆ g L 2 .

Next, we estimate II 2 . For 2 π ∣ ξ ∣ ≥ 1 , by the Young’s inequality, one has

1 ≤ ( 2 π ∣ ξ ∣ ) 2 − α ≤ 1 − 2 − α 2 + s + 2 − α 2 + s ( 2 π ∣ ξ ∣ ) 2 + s ,

and thus,

∣ ( 2 π ∣ ξ ∣ ) 2 − α − 1 ∣ ≤ 2 − α s ( 2 π ∣ ξ ∣ ) 2 + s .

Therefore,

II 2 ≤ 2 − α s ∫ R N \ B 1 2 π ( 0 ) ( 2 π ∣ ξ ∣ ) s ∣ f ˆ ( ξ ) ∣ ∣ g ˆ ( ξ ) ∣ d ξ ≤ 2 − α s ∫ ( 2 π ∣ ξ ∣ ) s ∣ f ˆ ( ξ ) ∣ 2 d ξ 1 2 ∫ ( 2 π ∣ ξ ∣ ) s ∣ g ˆ ( ξ ) ∣ 2 d ξ 1 2 = 2 − α s ‖ ( − Δ ) s 4 f ‖ L 2 ‖ ( − Δ ) s 4 g ‖ L 2 .

Combining the estimates of II 1 and II 2 , (2.2) holds.

Case 2. For 2 π ∣ ξ ∣ ≤ 1 , by the Young’s inequality, one has

1 ≤ ( 2 π ∣ ξ ∣ ) 2 − α ≤ 1 − α − 2 β − 2 + α − 2 β − 2 ( 2 π ∣ ξ ∣ ) 2 − β ,

i.e.,

∣ ( 2 π ∣ ξ ∣ ) 2 − α − 1 ∣ ≤ α − 2 β − 2 ( 2 π ∣ ξ ∣ ) 2 − β .

Therefore, we estimate II 1 as follows:

II 1 ≤ α − 2 β − 2 ∫ B 1 2 π ( 0 ) ( 2 π ∣ ξ ∣ ) − β ∣ f ˆ ( ξ ) ∣ ∣ g ˆ ( ξ ) ∣ d ξ ≤ α − 2 β − 2 ∫ ( 2 π ∣ ξ ∣ ) − β ∣ f ˆ ( ξ ) ∣ 2 d ξ 1 2 ∫ ( 2 π ∣ ξ ∣ ) − β ∣ g ˆ ( ξ ) ∣ 2 d ξ 1 2 = α − 2 β − 2 I β 2 ⋆ f L 2 I β 2 ⋆ g L 2 .

For 2 π ∣ ξ ∣ ≥ 1 , by the Young’s inequality, one has

( 2 π ∣ ξ ∣ ) 2 − α ≤ 1 ≤ 2 + s α + s ( 2 π ∣ ξ ∣ ) 2 − α + α − 2 α + s ( 2 π ∣ ξ ∣ ) 2 + s .

Thus,

∣ ( 2 π ∣ ξ ∣ ) 2 − α − 1 ∣ ≤ α − 2 s ( 2 π ∣ ξ ∣ ) 2 + s .

Therefore, II 2 is estimated as follows:

II 2 ≤ α − 2 s ∫ R N \ B 1 2 π ( 0 ) ( 2 π ∣ ξ ∣ ) s ∣ f ˆ ( ξ ) ∣ ∣ g ˆ ( ξ ) ∣ d ξ ≤ α − 2 s ∫ ( 2 π ∣ ξ ∣ ) s ∣ f ˆ ( ξ ) ∣ 2 d ξ 1 2 ∫ ( 2 π ∣ ξ ∣ ) s ∣ g ˆ ( ξ ) ∣ 2 d ξ 1 2 = α − 2 s ‖ ( − Δ ) s 4 f ‖ L 2 ‖ ( − Δ ) s 4 g ‖ L 2 .

(2.2) also holds. In summary, we achieve our conclusion.□

Lemma 2.4

Let 0 < q < 2 N N + 2 , max 2 N N + 2 , 1 < r < 2 , 0 < α ≤ 2 N q − N , then one has

∫ ( I α ⋆ f ) g d x − ∫ ( I 2 ⋆ f ) g d x ≤ C ( N , q , r ) ∣ 2 − α ∣ ( ‖ f ‖ L q ‖ g ‖ L q + ‖ ∇ f ‖ L r ‖ ∇ g ‖ L r ) .

Proof

Choose and fix β = 2 N q − N , s = 2 − 2 N 1 r − 1 2 . By the Hardy-Littlewood-Sobolev inequality, one has

‖ I β 2 ⋆ f ‖ L 2 ≤ C ( N , q ) ‖ f ‖ L q , ‖ I β 2 ⋆ g ‖ L 2 ≤ C ( N , q ) ‖ g ‖ L q , ‖ ( − Δ ) s 4 f ‖ L 2 = ‖ I 1 − s 2 ⋆ ∣ ∇ f ∣ ‖ L 2 ≤ C ( N , r ) ‖ ∇ f ‖ L r , ‖ ( − Δ ) s 4 g ‖ L 2 = ‖ I 1 − s 2 ⋆ ∣ ∇ g ∣ ‖ L 2 ≤ C ( N , r ) ‖ ∇ g ‖ L r .

Then, by Lemma 2.3, we obtain the conclusion.□

Remark 1

In our setting, f = g = u 2 , where u ∈ H 1 ( R 3 ) . By the Sobolev embedding theorem and the Hölder’s inequality, u 2 ∈ L q ( R 3 ) for 1 < q < 6 5 and ∇ ( u 2 ) ∈ L r ( R 3 ) for 1 ≤ r ≤ 3 2 . Hence, we can use Lemma 2.4 to obtain

∫ ( I α ⋆ u 2 ) u 2 d x − ∫ ( I 2 ⋆ u 2 ) u 2 d x ≤ C ( q , r ) ∣ 2 − α ∣ ∫ ∣ ∇ u ∣ 2 + ∣ u ∣ 2 d x 2 .

Lemma 2.5

Let α j → 2 and u j ⇀ u 0 weakly converging in H r 1 ( R 3 ) . Then, as j → ∞ , one has

(2.3) ∫ ( I α j ⋆ u j 2 ) u j 2 d x → ∫ ( I 2 ⋆ u 0 2 ) u 0 2 d x

and

(2.4) ∫ ( I α j ⋆ u j 2 ) u j ϕ d x → ∫ ( I 2 ⋆ u 0 2 ) u 0 ϕ d x , for any ϕ ∈ H 1 ( R 3 ) .

Proof

By Lemma 2.2, u j → u 0 in L p ( R 3 ) as j → ∞ for every p ∈ ( 2 , 6 ) . We decompose the integral by

(2.5) ∫ ( I α j ⋆ u j 2 ) u j 2 d x − ∫ ( I 2 ⋆ u 0 2 ) u 0 2 d x = ∫ ( ( I α j ⋆ u 0 2 ) − ( I 2 ⋆ u 0 2 ) ) u 0 2 d x + ∫ ( ( I α j ⋆ u j 2 ) − ( I α j ⋆ u 0 2 ) ) u 0 2 d x + ∫ ( I α j ⋆ u j 2 ) ( u j 2 − u 0 2 ) d x ≕ A j + B j + C j .

For A j , observe from Lemma 2.4 and Remark 1,

∣ A j ∣ ≤ o ( 1 ) , as j → ∞ .

For B j , using Lemma 2.1 (Hardy-Littlewood-Sobolev inequality) and the interpolation inequality, we infer that

∣ B j ∣ ≤ A α j C ( α j ) ‖ u j 2 − u 0 2 ‖ L 5 4 ‖ u 0 2 ‖ L 15 3 + 5 α j ≤ A α j C ( α j ) ‖ u j − u 0 ‖ L 5 2 ‖ u j + u 0 ‖ L 5 2 ‖ u 0 ‖ L 2 2 5 ‖ u 0 ‖ L 4 4 α j 3 = o ( 1 ) , as j → ∞ ,

where A α j = Γ ( 3 − α j 2 ) Γ ( α j 2 ) π 3 2 2 α j , C ( α j ) is the sharp constant for the Hardy-Littlewood-Sobolev inequality, and { A α j C ( α j ) } is bounded as α j → 2 by (2.1). For C j , similar to the aforementioned proof,

∣ C j ∣ ≤ A α j C ( α j ) ‖ u j 2 ‖ L 5 4 ‖ u j 2 − u 0 2 ‖ L 3 2 3 10 ‖ u j 2 − u 0 2 ‖ L 2 2 α j 3 ≤ A α j C ( α j ) ‖ u j ‖ L 5 2 2 ‖ u j − u 0 ‖ L 3 3 10 ‖ u j + u 0 ‖ L 4 2 α j 3 ‖ u j + u 0 ‖ L 3 3 10 ‖ u j − u 0 ‖ L 4 2 α j 3 = o ( 1 ) , as j → ∞ .

Therefore, (2.3) holds. (2.4) can be proved similar to (2.3), and we omit it. We achieve our conclusion.□

Let u α be a ground-state solution to (P) and c α = J α ( u α ) , where c α is called the ground-state energy level of J α . It is standard to verify that the ground-state energy level c α is also mountain pass level, and, satisfies

c α = min u ∈ H 1 ( R 3 ) \ { 0 } max t ≥ 0 J α ( t u ) .

Lemma 2.6

lim α → 2 c α = c 2 .

Proof

Fixed u ∈ H 1 ( R 3 ) \ { 0 } , by Lemma 2.4, one has

c α ≤ max t > 0 J α ( t u ) = max t > 0 t 2 2 ∫ ∣ ∇ u ∣ 2 + u 2 d x − t 4 4 ∫ ( I α ⋆ u 2 ) u 2 d x = t α , max 2 2 ∫ ∣ ∇ u ∣ 2 + u 2 d x − t α , max 4 4 ∫ ( I α ⋆ u 2 ) u 2 d x = t α , max 2 2 ∫ ∣ ∇ u ∣ 2 + u 2 d x − t α , max 4 4 ∫ ( I 2 ⋆ u 2 ) u 2 d x + o ( 1 ) ≤ max t > 0 t 2 2 ∫ ∣ ∇ u ∣ 2 + u 2 d x − t 4 4 ∫ ( I 2 ⋆ u 2 ) u 2 d x + o ( 1 ) = max t > 0 J 2 ( t u ) + o ( 1 ) , as α → 2 ,

where

t α , max 2 = ∫ ∣ ∇ u ∣ 2 + u 2 d x ∫ ( I α ⋆ u 2 ) u 2 d x is bounded, as α → 2 .

Take the infimum with respect to u ∈ H 1 ( R 3 ) \ { 0 } , one has

lim α → 2 sup c α ≤ c 2 .

On the other hand, from the Nehari identity of equation (P), one has

c α = J α ( u α ) = 1 2 ∫ ∣ ∇ u α ∣ 2 + u α 2 d x − 1 4 ∫ ( I α ⋆ u α 2 ) u α 2 d x = 1 2 ∫ ∣ ∇ u α ∣ 2 + u α 2 d x − 1 4 ∫ ∣ ∇ u α ∣ 2 + u α 2 d x = 1 4 ∫ ∣ ∇ u α ∣ 2 + u α 2 d x .

We can deduce that the ground-state solution sequence { u α } is uniformly bounded in H 1 ( R 3 ) . Then, by Lemma 2.4, one has

c α = J α ( u α ) = max t > 0 J α ( t u α ) = max t > 0 J 2 ( t u α ) + o ( 1 ) , as α → 2 .

Hence,

liminf α → 2 c α ≥ c 2

holds, and so the lemma is proved.□

Lemma 2.7

Suppose that α j → 2 and u α j is the positive radial ground state to (P) with α = α j . Then, there exists a subsequence of { u α j } (also written as { u α j } ) such that lim j → ∞ ‖ u α j − u 0 ‖ H 1 = 0 , where u 0 is the unique positive solution for (1.1).

Proof

On the one hand, by multiplying both sides of the equation (P) with α = α j by u α j and integrating by parts, one has

‖ u α j ‖ H 1 2 = ∫ ( I α j ⋆ u α j 2 ) u α j 2 d x and 1 4 ‖ u α j ‖ H 1 2 = J α j ( u α j ) .

Thus, we deduce from Lemma 2.6 that ‖ u α j ‖ H 1 is bounded uniformly for j . Then, { u α j } converges weakly in H 1 ( R 3 ) to some nonnegative radial function u 0 ∈ H 1 ( R 3 ) up to a subsequence. By Lemma 2.5 and the weak convergence of { u α j } , we obtain that u 0 is a weak solution of (1.1).

On the other hand, from Lemma 2.5, one has

‖ u α j ‖ H 1 2 = ∫ ( I α j ⋆ u α j 2 ) u α j 2 d x → ∫ ( I 2 ⋆ u 0 2 ) u 0 2 d x = ‖ u 0 ‖ H 1 2 , as j → ∞ .

Combining the weak convergence of { u α j } , we obtain lim j → ∞ ‖ u α j − u 0 ‖ H 1 = 0 .

Next, we prove that u 0 is nontrival. In fact, by Lemma 2.1 and the Sobolev inequality, one has

(2.6) ‖ u α j ‖ H 1 2 = ∫ ( I α j ⋆ u α j 2 ) u α j 2 d x ≤ C ( α j ) ‖ u α j ‖ L 12 3 + α j 4 ≤ C ‖ u α j ‖ H 1 4 ,

where C ( α j ) is bounded when α j → 2 by (2.1). Thus, ‖ u α j ‖ H 1 has a uniform lower bound. By the strong convergence of { u α j } , we can obtain u 0 , which is nontrival. Moreover, u 0 ∈ C 2 according to the classical theory of elliptic regularity, and u 0 > 0 by the strong maximum principle. We achieve our conclusion.□

Lemma 2.8

J 2 ( u 0 ) = c 2 .

Proof

It follows from Lemmas 2.5–2.7 that

□ c 2 = lim j → ∞ c α j = lim j → ∞ J α j ( u α j ) = lim j → ∞ 1 2 ‖ u α j ‖ H 1 ( R 3 ) 2 − 1 4 ∫ ( I α j ⋆ u α j 2 ) u α j 2 d x = 1 2 ‖ u 0 ‖ H 1 ( R 3 ) 2 − 1 4 ∫ ( I 2 ⋆ u 0 2 ) u 0 2 d x = J 2 ( u 0 ) .

The proof of Theorem 1.1. Combining Lemmas 2.7 and 2.8, Theorem 1.1 is proved. □

3 Proof of Theorem 1.2

In this section, we prove Theorem 1.2. To this, we need the following proposition.

Proposition 2

Let u 0 be the unique positive solution for (1.1). Then, the linearization of (1.1) at u 0 , ϕ ↦ − Δ ϕ + ϕ − ( I 2 ⋆ u 0 2 ) ϕ − 2 ( I 2 ⋆ u 0 ϕ ) u 0 , has a null kernel in H r 1 ( R 3 ) .

Proof

Let ϕ ∈ H r 1 ( R 3 ) be a solution for

− Δ u + u − ( I 2 ⋆ u 0 2 ) u − 2 ( I 2 ⋆ u 0 u ) u = 0 .

Note that u 0 is the unique positive solution for (1.1). Then, by [5, Theorem 4], the nondegenerate of u 0 implies that

ϕ = ∑ i = 1 3 b i ∂ u 0 ∂ x i for some suitable b i ∈ R .

On the other hand, ϕ is radial, and thus, b i = 0 , i = 1 , 2 , 3 . This implies ϕ ≡ 0 .□

Now, we give some L ∞ -estimates.

Lemma 3.1

Assume u , ϕ ∈ H 1 ( R 3 ) and α ∈ ( 3 2 , 3 ) . Then, there exists C > 0 independent of α such that

(3.1) 1 ∣ ⋅ ∣ 3 − α ⋆ ( u ϕ ) L ∞ ( R 3 ) ≤ C ‖ u ‖ H 1 ( R 3 ) ‖ ϕ ‖ H 1 ( R 3 )

and

(3.2) lim α → 2 1 ∣ ⋅ ∣ 3 − α ⋆ ( u ϕ ) − 1 ∣ ⋅ ∣ ⋆ ( u ϕ ) L ∞ ( K ) = 0 , for any compact set K ⊂ R 3 .

Proof

Note that if u , ϕ ∈ H 1 ( R 3 ) , the Hölder inequality ensures that

1 ∣ ⋅ ∣ 3 − α ⋆ ( u ϕ ) ( x ) ≤ ∫ B 1 ( x ) 1 ∣ x − y ∣ 3 − α ( u ϕ ) ( y ) d y + ∫ B 1 c ( x ) 1 ∣ x − y ∣ 3 − α ( u ϕ ) ( y ) d y ≤ ∫ B 1 ( 0 ) 1 ∣ y ∣ 2 ( 3 − α ) d y 1 2 ‖ u ‖ L 4 ‖ ϕ ‖ L 4 + ‖ u ‖ L 2 ( R 3 ) ‖ ϕ ‖ L 2 .

By 2 ( 3 − α ) < 3 , we infer that the integral

∫ B 1 ( 0 ) 1 ∣ y ∣ 2 ( 3 − α ) d y

is uniformly bounded for α (sufficiently close to 2). Thus, (3.1) holds.

To prove (3.2), we argue by contradiction. Suppose that there exist a compact set K and sequences α j → 2 , x j → x 0 ∈ K such that

(3.3) ∫ 1 ∣ x j − y ∣ 3 − α j ( u ϕ ) ( y ) d y ↛ ∫ 1 ∣ x 0 − y ∣ ( u ϕ ) ( y ) d y , as j → ∞ .

We simply write ( u ϕ ) ( y ) ∣ x j − y ∣ 3 − α j by f j ( y ) , so that f j ( y ) → ( u ϕ ) ( y ) ∣ x 0 − y ∣ almost everywhere as j → ∞ .

Claim 1

f j is uniformly integrable.

The claim can be obtained from that

∫ E ∣ f j ( y ) ∣ d y ≤ ∫ E 1 ∣ x j − y ∣ 2 ( 3 − α j ) d y 1 2 ‖ u ‖ L 4 ( E ) ‖ ϕ ‖ L 4 ( E ) = ∫ E ∩ B 1 ( x j ) + ∫ E ∩ B 1 c ( x j ) 1 ∣ x j − y ∣ 2 ( 3 − α j ) d y 1 2 ‖ u ‖ L 4 ( E ) ‖ ϕ ‖ L 4 ( E ) ≤ ∫ B 1 ( 0 ) 1 ∣ y ∣ 2 ( 3 − α ) d y + ∣ E ∣ 1 2 ‖ u ‖ L 4 ( E ) ‖ ϕ ‖ L 4 ( E ) .

Claim 2

f j is tight in R 3 , i.e., for any ε > 0 , there exists a large R > 0 such that

∫ B R c ( 0 ) ∣ f j ( y ) ∣ d y < ε .

Choose a large R > 0 such that B 2 ( x 0 ) ⊂ B R ( 0 ) . Then, thanks to

∫ B R c ( 0 ) ∣ f j ( y ) ∣ d y ≤ ∫ B R c ( 0 ) ∣ u ϕ ( y ) ∣ d y ,

we can infer that f j is tight in R 3 .

Combining Claim 1 with Claim 2, and using the Vitali convergence theorem, we obtain

∫ f j ( y ) d y → ∫ 1 ∣ x 0 − y ∣ ( u ϕ ) ( y ) d y , as j → ∞ ,

which contradicts to (3.3). Thus, (3.2) holds.□

Lemma 3.2

[13, Lemma 5.1] Let 2 N N + 2 ≤ q ≤ 2 . Then, the operator ( − Δ + I ) − 1 is bounded from L q ( R N ) into H 1 ( R N ) .

Choose and fix α 0 ∈ ( 3 2 , 2 ) and β 0 ∈ ( 2 , 3 ) , and define an operator A ( α , u ) by

A ( α , u ) ≔ u − ( − Δ + I ) − 1 [ ( I α ⋆ u 2 ) u ] , if α ∈ ( α 0 , 2 ) ∪ ( 2 , β 0 ) , u − ( − Δ + I ) − 1 [ ( I 2 ⋆ u 2 ) u ] , if α = 2 .

Lemma 3.3

The operator A : ( α 0 , 2 ] × H r 1 ( R 3 ) → H r 1 ( R 3 ) is continuous and is continuously differentiable with respect to u on ( α 0 , 2 ] × H r 1 ( R 3 ) .
The operator A : [ 2 , β 0 ) × H r 1 ( R 3 ) → H r 1 ( R 3 ) is continuous and is continuously differentiable with respect to u on [ 2 , β 0 ) × H r 1 ( R 3 ) .

Proof

We only prove the conclusion (i) and the conclusion (ii) can be proved similarly.

We prove the continuous of the operator A first. Suppose that a family of sequence ( α j , u j ) → ( α , u ) in ( α 0 , 2 ] × H r 1 ( R 3 ) . We claim that

( I α j ⋆ u j 2 ) u j → ( I 2 ⋆ u 2 ) u in L 2 ( R 3 ) .

Indeed, by Lemma 3.1, we infer that

‖ ( I α j ⋆ u j 2 ) u j − ( I 2 ⋆ u 2 ) u ‖ L 2 ≤ ‖ ( I α j ⋆ ( u j 2 − u 2 ) ) u j ‖ L 2 + ‖ ( I α j ⋆ u 2 − I 2 ⋆ u 2 ) u j ‖ L 2 + ‖ ( I 2 ⋆ ( u 2 − u j 2 ) ) u j ‖ L 2 + ‖ ( I 2 ⋆ u j 2 ) ( u j − u ) ‖ L 2 + ‖ ( I 2 ⋆ u j 2 − I 2 ⋆ u 2 ) u ‖ L 2 ≤ C ‖ I α j ⋆ ( u j 2 − u 2 ) ‖ L ∞ + ‖ ( I α j ⋆ u 2 − I 2 ⋆ u 2 ) u j ‖ L 2 ( K ) + ‖ ( I α j ⋆ u 2 − I 2 ⋆ u 2 ) u j ‖ L 2 ( R 3 \ K ) + C ‖ I 2 ⋆ ( u 2 − u j 2 ) ‖ L ∞ + ‖ ( I 2 ⋆ u j 2 ) ( u j − u ) ‖ L 2 + ‖ ( I 2 ⋆ u j 2 − I 2 ⋆ u 2 ) u ‖ L 2 ≤ C ‖ u j − u ‖ H 1 ‖ u j + u ‖ H 1 + ‖ I α j ⋆ u 2 − I 2 ⋆ u 2 ‖ L ∞ ( K ) ‖ u j ‖ L 2 + ‖ I α j ⋆ u 2 − I 2 ⋆ u 2 ‖ L ∞ ‖ u j ‖ L 2 ( R 3 \ K ) + ‖ I 2 ⋆ u j 2 ‖ L 4 ‖ u j − u ‖ L 4 + ‖ I 2 ⋆ ( u j 2 − u 2 ) ‖ L 4 ‖ u ‖ L 4 ≤ o ( 1 ) + C ‖ I α j ⋆ u 2 − I 2 ⋆ u 2 ‖ L ∞ ( K ) ‖ u ‖ L 2 + C ‖ I α j ⋆ u 2 − I 2 ⋆ u 2 ‖ L ∞ ‖ u ‖ L 2 ( R 3 \ K ) + C ‖ u j − u ‖ L 4 ‖ u j 2 ‖ L 12 11 + C ‖ u j 2 − u 2 ‖ L 12 11 ‖ u ‖ L 4 .

For any ε > 0 there exists a large enough compact set K ⊂ R 3 such that ‖ u ‖ L 2 ( R 3 \ K ) < ε ; thus,

(3.4) ‖ ( I α j ⋆ u j 2 ) u j − ( I 2 ⋆ u 2 ) u ‖ L 2 ≤ o ( 1 ) + C ( ‖ I α j ⋆ u 2 ‖ L ∞ + ‖ I 2 ⋆ u 2 ‖ L ∞ ) ‖ u ‖ L 2 ( R 3 \ K ) + C ‖ u j − u ‖ L 4 ‖ u j ‖ L 24 11 2 + C ‖ u j − u ‖ L 24 11 2 ‖ u ‖ L 4 ≤ o ( 1 ) + C ‖ u ‖ H 1 2 ‖ u ‖ L 2 ( R 3 \ K ) + C ‖ u j − u ‖ H 1 2 ‖ u ‖ H 1 2 ≤ o ( 1 ) + C ε , as j → ∞ ,

where we use the Hölder inequality, Sobolev inequality, Hardy-Littlewood-Sobolev inequality, and Lemma 3.1. Then, by the arbitrary of ε > 0 , one has

lim j → ∞ ‖ ( I α j ⋆ u j 2 ) u j − ( I 2 ⋆ u 2 ) u ‖ L 2 = 0 .

By Lemma 3.2, we obtain the continuous of the operator A .

Now, we prove the continuously differentiable of the operator A . Differentiating A with respect to u , we have

∂ A ∂ u ( α , u ) [ ϕ ] = ϕ − ( − Δ + I ) − 1 [ 2 ( I α ⋆ u ϕ ) u + ( I α ⋆ u 2 ) ϕ ] , if α ∈ ( α 0 , 2 ) ∪ ( 2 , β 0 ) , ϕ − ( − Δ + I ) − 1 [ 2 ( I 2 ⋆ u ϕ ) u + ( I 2 ⋆ u 2 ) ϕ ] , if α = 2 .

Similar to the proof of the continuous of the operator A , we obtain the continuous of the operator ∂ A ∂ u with respect to u .□

Lemma 3.4

Assume that u 0 is a unique positive radial ground state of (1.1). Then, there exists a neighborhood U ⊂ ( α 0 , β 0 ) × H r 1 ( R 3 ) of the point ( 2 , u 0 ) ∈ ( α 0 , β 0 ) × H r 1 ( R 3 ) such that the equation (P) admits a unique solution in U.

Proof

We only prove the case α ∈ ( α 0 , 2 ] , and the proof of another case α ∈ [ 2 , β 0 ) is similar. We claim that

(3.5) ∂ A ∂ u ( 2 , u 0 ) : H r 1 ( R 3 ) → H r 1 ( R 3 ) is a linear isomorphism map ,

where ∂ A ∂ u ( 2 , u 0 ) is the linearized operator of A with respect to u at ( 2 , u 0 ) ,

∂ A ∂ u ( 2 , u 0 ) [ ϕ ] = ϕ − ( − Δ + I ) − 1 [ 2 ( I 2 ⋆ u 0 ϕ ) u 0 + ( I 2 ⋆ u 0 2 ) ϕ ] , ∀ ϕ ∈ H r 1 ( R 3 ) .

Let { ϕ j } be a bounded sequence in H r 1 ( R 3 ) . Then, there exists a subsequence { ϕ j } such that ϕ j ⇀ ϕ in H r 1 ( R 3 ) . First, we show that

2 ( I 2 ⋆ u 0 ( ϕ j − ϕ ) ) u 0 + ( I 2 ⋆ u 0 2 ) ( ϕ j − ϕ ) → 0 in L 2 ( R 3 ) .

Indeed, for any ε > 0 , there exists a large enough R > 0 such that ‖ u 0 ‖ L 2 ( B R c ( 0 ) ) < ε due to the uniform decaying property of u 0 . Then, by the Hölder inequality, Sobolev inequality, Hardy-Littlewood-Sobolev inequality, and Lemma 2.2,

‖ 2 ( I 2 ⋆ u 0 ( ϕ j − ϕ ) ) u 0 + ( I 2 ⋆ u 0 2 ) ( ϕ j − ϕ ) ‖ L 2 ≤ 2 ‖ I 2 ⋆ u 0 ( ϕ j − ϕ ) ‖ L ∞ ‖ u 0 ‖ L 2 + ‖ I 2 ⋆ u 0 2 ‖ L 4 ‖ ϕ j − ϕ ‖ L 4 ≤ 2 ∫ B R ( x ) 1 ∣ x − y ∣ ( u 0 ( ϕ j − ϕ ) ) ( y ) d y + ∫ B R c ( x ) 1 ∣ x − y ∣ ( u 0 ( ϕ j − ϕ ) ) ( y ) d y ‖ u 0 ‖ L 2 + C ‖ u 0 2 ‖ L 12 11 ‖ ϕ j − ϕ ‖ L 4 ≤ C ∫ B R ( 0 ) 1 ∣ y ∣ 2 d y 1 2 ‖ u 0 ‖ L 4 ‖ ϕ j − ϕ ‖ L 4 + ‖ u 0 ‖ L 2 ( B R c ( 0 ) ) ‖ ϕ j − ϕ ‖ L 2 ‖ u 0 ‖ L 2 + C ‖ u 0 2 ‖ L 12 11 ‖ ϕ j − ϕ ‖ L 4 ≤ o ( 1 ) + C ε , as j → ∞ .

By the arbitrarily of ε > 0 , one has

lim j → ∞ ‖ 2 ( I 2 ⋆ u 0 ( ϕ j − ϕ ) ) u 0 + ( I 2 ⋆ u 0 2 ) ( ϕ j − ϕ ) ‖ L 2 = 0 .

Hence, the mapping

ϕ ↦ 2 ( I 2 ⋆ u 0 ϕ ) u 0 + ( I 2 ⋆ u 0 2 ) ϕ

is compact of H r 1 ( R 3 ) onto L 2 ( R 3 ) . By Lemma 3.2, the composite mapping

ϕ ↦ ( − Δ + I ) − 1 [ 2 ( I 2 ⋆ u 0 ϕ ) u 0 + ( I 2 ⋆ u 0 2 ) ϕ ]

is also compact of H r 1 ( R 3 ) onto H r 1 ( R 3 ) . This shows that ∂ A ∂ u ( 2 , u 0 ) is bounded.

On the other hand, we have from Proposition 2 that the kernel of ∂ A ∂ u ( 2 , u 0 ) is trivial. Then, by Fredholm alternative theorem, we obtain that ∂ A ∂ u ( 2 , u 0 ) is an onto map, so claim (3.5) holds. Therefore, the implicit function theorem implies our conclusion.□

The proof of Theorem 1.2

We claim that the equation (P) admits a unique positive radial ground state for α close to 2. We argue by contradiction. Suppose that there exist sequences { α j } converging to 2 as j → ∞ and { u α j 1 } ⊂ H r 1 ( R 3 ) , { u α j 2 } ⊂ H r 1 ( R 3 ) are the sequences of positive radial ground states of equation (P) as well as u α j 1 ≠ u α j 2 for all j . Using Theorem 1.1, we obtain that { u α j 1 } and { u α j 2 } both converge to the unique positive solution u 0 of (1.1) in H r 1 ( R 3 ) . This contradicts Lemma 3.4; thus, Theorem 1.2 is proved.

4 Proof of Theorem 1.3

In this section, we prove Theorem 1.3. We need the following convergence result.

Lemma 4.1

Suppose that { u α j } is a family of the unique positive radial ground states of (P) and u 0 ∈ H r 1 ( R 3 ) is the positive radial ground state to (1.1). Then, for any sequence α j → 2 and ψ j ⇀ ψ 0 , ϕ j ⇀ ϕ 0 in H r 1 ( R 3 ) , one has

(4.1) ∫ ( I α j ⋆ ( u α j ϕ j ) ) u α j ψ j d x → ∫ ( I 2 ⋆ ( u 0 ϕ 0 ) ) u 0 ψ 0 d x

and

(4.2) ∫ ( I α j ⋆ u α j 2 ) ϕ j ψ j d x → ∫ ( I 2 ⋆ u 0 2 ) ϕ 0 ψ 0 d x .

Proof

First, we claim that u 0 ϕ j is compact in L 5 4 ( R 3 ) , namely, ‖ u 0 ϕ j − u 0 ϕ 0 ‖ L 5 4 → 0 . In fact, assume ϕ j ⇀ ϕ 0 in H r 1 ( R 3 ) . For any ε > 0 , there exists a large enough R > 0 such that ‖ u 0 ‖ L 10 3 ( B R c ( 0 ) ) < ε by the uniform decaying property of u 0 , then

‖ u 0 ϕ j − u 0 ϕ 0 ‖ L 5 4 = ‖ u 0 ϕ j − u 0 ϕ 0 ‖ L 5 4 ( B R ( 0 ) ) + ‖ u 0 ϕ j − u 0 ϕ 0 ‖ L 5 4 ( B R c ( 0 ) ) ≤ ‖ u 0 ‖ L 10 3 ( B R ( 0 ) ) ‖ ϕ j − ϕ 0 ‖ L 2 ( B R ( 0 ) ) + ‖ u 0 ‖ L 10 3 ( B R c ( 0 ) ) ‖ ϕ j − ϕ 0 ‖ L 2 ( B R c ( 0 ) ) = C ( o ( 1 ) + ε ) , as j → ∞ .

By the arbitrary of ε > 0 , we have lim j → ∞ ‖ u 0 ϕ j − u 0 ϕ 0 ‖ L 5 4 = 0 . Using the Hölder inequality, we then infer that

‖ u α j ϕ j − u 0 ϕ 0 ‖ L 5 4 ≤ ‖ ( u α j − u 0 ) ϕ j ‖ L 5 4 + ‖ u 0 ϕ j − u 0 ϕ 0 ‖ L 5 4 ≤ ‖ u α j − u 0 ‖ L 2 ‖ ϕ j ‖ L 10 3 + o ( 1 ) = o ( 1 ) , as j → ∞ .

This implies that u α j ϕ j is compact in L 5 4 ( R 3 ) as well.

Next, we prove (4.1) using the following decomposing

∫ ( I α j ⋆ ( u α j ϕ j ) ) u α j ψ j d x − ∫ ( I 2 ⋆ ( u 0 ϕ 0 ) ) u 0 ψ 0 d x = ∫ ( I α j ⋆ ( u 0 ϕ 0 ) − I 2 ⋆ ( u 0 ϕ 0 ) ) u 0 ψ 0 d x + ∫ ( I α j ⋆ ( u α j ϕ j ) − I α j ⋆ ( u 0 ϕ 0 ) ) u 0 ψ 0 d x + ∫ ( I α j ⋆ ( u α j ϕ j ) ) ( u α j ψ j − u 0 ψ 0 ) d x ≕ E j + F j + G j .

For E j , we observe from Lemma 2.4 that ∣ E j ∣ ≤ o ( 1 ) , as j → ∞ .

For F j , using the Hölder inequality, Hardy-Littlewood-Sobolev inequality, and the L 5 4 − compactness of { u α j ϕ j } , we obtain

∣ F j ∣ ≤ ‖ I α j ⋆ ( u α j ϕ j ) − I α j ⋆ ( u 0 ϕ 0 ) ‖ L 15 12 − 5 α j ‖ u 0 ψ 0 ‖ L 15 3 + 5 α j ≤ C ‖ u α j ϕ j − u 0 ϕ 0 ‖ L 5 4 ‖ u 0 ψ 0 ‖ L 1 1 5 ‖ u 0 ψ 0 ‖ L 2 2 α j 3 = o ( 1 ) , as j → ∞ .

For G j , by the Hölder inequality, Hardy-Littlewood-Sobolev inequality, and the L 5 4 − compactness of { u α j ϕ j } and { u α j ψ j } , we deduce that

∣ G j ∣ ≤ ‖ I α j ⋆ ( u α j ϕ j ) ‖ L 15 12 − 5 α j ‖ u α j ψ j − u 0 ψ 0 ‖ L 15 3 + 5 α j ≤ C ‖ u α j ϕ j ‖ L 5 4 ‖ u α j ψ j − u 0 ψ 0 ‖ L 1 1 5 ‖ u α j ψ j − u 0 ψ 0 ‖ L 5 4 5 α j 12 = o ( 1 ) , as j → ∞ .

Combining the estimates of E j , F j , and G j , (4.1) holds.

Finally, we prove (4.2). We decompose the integral by

∫ ( I α j ⋆ u α j 2 ) ϕ j ψ j d x = ∫ ( I 2 ⋆ u 0 2 ) ϕ j ψ j d x + ∫ ( I α j ⋆ u 0 2 − I 2 ⋆ u 0 2 ) ϕ j ψ j d x + ∫ ( I α j ⋆ ( u α j 2 − u 0 2 ) ) ϕ j ψ j d x ≔ H j + K j + L j .

For H j , we have

H j = ∫ ( I 2 ⋆ u 0 2 ) ϕ j ψ j d x = ∫ ( I 2 ⋆ u 0 2 ) ϕ 0 ψ 0 d x + o ( 1 ) , as j → ∞ .

For K j , for any ε > 0 , there exists a large enough compact set K ⊂ R 3 such that ‖ ϕ j ‖ L 2 ( R 3 \ K ) ≤ 2 ‖ ϕ 0 ‖ L 2 ( R 3 \ K ) < ε , then

∣ K j ∣ = ∫ ( I α j ⋆ u 0 2 − I 2 ⋆ u 0 2 ) ϕ j ψ j d x ≤ ‖ ( I α j ⋆ u 0 2 − I 2 ⋆ u 0 2 ) ϕ j ‖ L 2 ‖ ψ j ‖ L 2 = ( ‖ ( I α j ⋆ u 0 2 − I 2 ⋆ u 0 2 ) ϕ j ‖ L 2 ( K ) + ‖ ( I α j ⋆ u 0 2 − I 2 ⋆ u 0 2 ) ϕ j ‖ L 2 ( R 3 \ K ) ) ‖ ψ j ‖ L 2 ≤ ( ‖ I α j ⋆ u 0 2 − I 2 ⋆ u 0 2 ‖ L ∞ ( K ) ‖ ϕ j ‖ L 2 + ‖ I α j ⋆ u 0 2 − I 2 ⋆ u 0 2 ‖ L ∞ ‖ ϕ j ‖ L 2 ( R 3 \ K ) ) ‖ ψ j ‖ L 2 ≤ C ( o ( 1 ) + ε ) , as j → ∞ .

Since we can choose ε > 0 arbitrary, lim j → ∞ K j = 0 .

For L j , using the Hölder inequality, Hardy-Littlewood-Sobolev inequality, and Theorem 1.1, we infer that

∣ L j ∣ = ∫ ( I α j ⋆ ( u α j 2 − u 0 2 ) ) ϕ j ψ j d x ≤ ‖ I α j ⋆ ( u α j 2 − u 0 2 ) ‖ L 15 12 − 5 α j ‖ ϕ j ψ j ‖ L 15 3 + 5 α j ≤ C ‖ u α j 2 − u 0 2 ‖ L 5 4 ‖ ϕ j ψ j ‖ L 1 1 5 ‖ ϕ j ψ j ‖ L 2 2 α j 3 ≤ C ‖ u α j − u 0 ‖ L 5 2 2 ‖ ϕ j ‖ L 2 1 5 ‖ ψ j ‖ L 2 1 5 ‖ ϕ j ‖ L 4 2 α j 3 ‖ ψ j ‖ L 4 2 α j 3 ≤ C ‖ u α j − u 0 ‖ H 1 2 = o ( 1 ) , as j → ∞ .

Combining the estimates of H j , K j , and L j , (4.2) holds.□

Now, we will prove the nondegeneracy of ground states for (P) when α is sufficiently close to 2. The idea follows from [13, Proposition 6.2].

The proof of Theorem 1.3

We argue by contradiction. Suppose that there exists a family of sequence α j → 2 such that for every j , there exists a nontrivial solution ϕ j ∈ L 2 ( R 3 ) of (1.5) but ϕ j ∉ V α j , where

V α ≔ ∑ i = 1 3 c i ∂ x i u α ∣ c i ∈ R .

We may suppose that ϕ j is L 2 orthogonal to V α j .

First, we prove that any solution ϕ ∈ L 2 ( R 3 ) of (1.5) belongs to H 1 ( R 3 ) . To this end, we define

L [ ϕ ] ≔ 2 ( I α ⋆ ( u α ϕ ) ) u α + ( I α ⋆ u α 2 ) ϕ , ∀ ϕ ∈ L 2 ( R 3 ) .

Using equation (1.5), we have

‖ ϕ ‖ H 1 2 = L [ ϕ ] ϕ .

So it is enough to verify that L [ ϕ ] ∈ H − 1 ( R 3 ) . Indeed, for any ψ ∈ H 1 ( R 3 ) ,

L [ ϕ ] ψ = 2 ∫ ( I α ⋆ ( u α ϕ ) ) u α ψ d x + ∫ ( I α ⋆ u α 2 ) ϕ ψ d x ≤ 2 ‖ I α ⋆ ( u α ϕ ) ‖ L 4 ‖ u α ‖ L 2 ‖ ψ ‖ L 4 + ‖ I α ⋆ u α 2 ‖ L 4 ‖ ϕ ‖ L 2 ‖ ψ ‖ L 4 ≤ C ‖ u α ϕ ‖ L 12 3 + 4 α ‖ u α ‖ L 2 ‖ ψ ‖ L 4 + C ‖ u α 2 ‖ L 12 3 + 4 α ‖ ϕ ‖ L 2 ‖ ψ ‖ L 4 ≤ C ‖ u α ‖ L 12 4 α − 3 ‖ ϕ ‖ L 2 ‖ u α ‖ L 2 + ‖ u α ‖ L 24 3 + 4 α 2 ‖ ϕ ‖ L 2 ‖ ψ ‖ L 4 ,

which implies L [ ϕ ] ∈ H − 1 ( R 3 ) .

Now, we may normalize ϕ j as ‖ ϕ j ‖ H 1 = 1 without loss of generality. By Lemma 4.1, it is obviously to see that ϕ j ⇀ ϕ 0 in H 1 ( R 3 ) as j → ∞ , and ϕ 0 satisfies

− Δ ϕ 0 + ϕ 0 − 2 ( I α ⋆ ( u 0 ϕ 0 ) ) u 0 − ( I 2 ⋆ u 0 2 ) ϕ 0 = 0 ,

where u 0 ∈ H 1 ( R 3 ) is the unique positive radial solution of (1.1). Using Lemma 4.1 again, we obtain that

1 = ‖ ϕ j ‖ H 1 2 = 2 ∫ ( I α j ⋆ ( u α j ϕ ) ) u α j ϕ j d x + ∫ ( I α j ⋆ u α j 2 ) ϕ j 2 d x = 2 ∫ ( I 2 ⋆ ( u 0 ϕ 0 ) ) u 0 ϕ 0 d x + ∫ ( I 2 ⋆ u 0 2 ) ϕ 0 2 d x + o ( 1 ) , as j → ∞ .

This implies that ϕ 0 is nontrivial. Moreover, for all i = 1 , 2 , 3 ,

0 = ∫ ∂ x i u α j ϕ j d x → ∫ ∂ x i u 0 ϕ 0 d x , as j → ∞ .

Then, it follows from the nondegeneracy of u 0 that ϕ 0 ≡ 0 . This contradiction implies that Theorem 1.3 holds.

Funding information: This study was partially supported by NSFC (No. 11901532).
Author contributions: Huxiao Luo: reviewing and editing, Dingliang Zhang and Yating Xu: writing.
Conflict of interest: The authors state no conflict of interest.
Data availability statement: Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.

References

[1] G. Chen, Nondegeneracy of ground states and multiple semiclassical solutions of the Hartree equation for general dimensions, Results Math. 76 (2021), 1–31. 10.1007/s00025-020-01332-ySuche in Google Scholar

[2] L. Grafakos, Modern Fourier Analysis, 2nd ed., Graduate Texts in Mathematics 250, Springer, New York, 2009. 10.1007/978-0-387-09434-2Suche in Google Scholar

[3] I. Kossowski, Radial solutions for nonlinear elliptic equation with nonlinear nonlocal boundary conditions, Opuscula Math. 43 (2023), 675–687. 10.7494/OpMath.2023.43.5.675Suche in Google Scholar

[4] J. Lan, X. He, and Y. Meng, Normalized solutions for a critical fractional Choquard equation with a nonlocal perturbation, Adv. Nonlinear Anal. 12 (2023), 20230112. 10.1515/anona-2023-0112Suche in Google Scholar

[5] E. Lenzmann, Uniqueness of ground states for pseudorelativistic Hartree equations, Anal. PDE 2 (2009), 1–27. 10.2140/apde.2009.2.1Suche in Google Scholar

[6] Y. Li, B. Zhang, and X. Han, Existence and concentration behavior of positive solutions to Schrödinger-Poisson-Slater equations, Adv. Nonlinear Anal. 12 (2023), 20220293. 10.1515/anona-2022-0293Suche in Google Scholar

[7] E. H. Lieb, Existence and uniqueness of the minimizing solution of Choquard’s nonlinear equation, Studies in Appl. Math. 57 (1976/77), 93–105. 10.1002/sapm197757293Suche in Google Scholar

[8] E. H. Lieb and M. Loss, Analysis, 2nd ed., Graduate Studies in Mathematics 14, American Mathematical Society, Providence, 2001. 10.1090/gsm/014Suche in Google Scholar

[9] L. Maaaa and L. Zhao, Classification of positive solitary solutions of the nonlinear Choquard equation, Arch. Ration. Mech. Anal. 195 (2010), 455–467. 10.1007/s00205-008-0208-3Suche in Google Scholar

[10] V. Moroz and J. Van Schaftingen, Groundstates of nonlinear Choquard equations: existence, qualitative properties and decay asymptotics, J. Funct. Anal. 265 (2013), no. 2, 153–184. 10.1016/j.jfa.2013.04.007Suche in Google Scholar

[11] V. Moroz and J. Van Schaftingen, A guide to the Choquard equation, J. Fixed Point Theory Appl. 19 (2017), 773–813. 10.1007/s11784-016-0373-1Suche in Google Scholar

[12] S. Pekar, Untersuchung über die Elekronentheorie der Kristalle, Akedemie Verlag, Berlin, 1954. 10.1515/9783112649305Suche in Google Scholar

[13] J. Seok, Limit profiles and uniqueness of ground states to the nonlinear Choquard equations, Adv. Nonlinear Anal. 8 (2019), no. 1, 1083–1098. 10.1515/anona-2017-0182Suche in Google Scholar

[14] W. A. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys. 55 (1977), 149–162. 10.1007/BF01626517Suche in Google Scholar

[15] P. Tod and I. M. Moroz, An analytical approach to the Schrödinger-Newton equations, Nonlinearity 12 (1999), 201–216. 10.1088/0951-7715/12/2/002Suche in Google Scholar

[16] T. Wang and T. Yi, Uniqueness of positive solutions of the Choquard type equations, Appl. Anal. 96 (2017), 409–417. 10.1080/00036811.2016.1138473Suche in Google Scholar

[17] J. Wei and M. Winter, Strongly interacting bumps for the Schrödinger-Newton equations, J. Math. Phys. 50 (2009), 012905, 22. 10.1063/1.3060169Suche in Google Scholar

[18] C.-L. Xiang, Uniqueness and nondegeneracy of ground states for Choquard equations in three dimensions, Calc. Var. Partial Differential Equations 55 (2016), 1–25. 10.1007/s00526-016-1068-6Suche in Google Scholar

Received: 2023-09-15

Revised: 2024-08-06

Accepted: 2024-09-21

Published Online: 2024-11-22

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

Artikel in diesem Heft

https://doi.org/10.1515/anona-2024-0048

Schlagwörter für diesen Artikel

limit profile; Choquard equation; uniqueness and nondegeneracy

Creative Commons

BY 4.0