Limit profiles and uniqueness of ground states to the nonlinear Choquard equations

Jinmyoung Seok

doi:10.1515/anona-2017-0182

Artikel Open Access

Limit profiles and uniqueness of ground states to the nonlinear Choquard equations

Jinmyoung Seok

Veröffentlicht/Copyright: 31. Mai 2018

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Abstract

Consider nonlinear Choquard equations

{ - Δ ⁢ u + u = ( I α * | u | p ) ⁢ | u | p - 2 ⁢ u in ⁢ ℝ N , lim x → ∞ ⁡ u ⁢ ( x ) = 0 ,

where I α denotes the Riesz potential and α ∈ ( 0 , N ) . In this paper, we investigate limit profiles of ground states of nonlinear Choquard equations as α → 0 or α → N . This leads to the uniqueness and nondegeneracy of ground states when α is sufficiently close to 0 or close to N.

Keywords: Semilinear elliptic; Choquard; limit profile; uniqueness

MSC 2010: 35J10; 35J20; 35J61

1 Introduction

Let N ≥ 3 , α ∈ ( 0 , N ) and p > 1 . We are concerned with the so-called nonlinear Choquard equations

(1.1) { - Δ ⁢ u + u = ( I α * | u | p ) ⁢ | u | p - 2 ⁢ u in ⁢ ℝ N , lim x → ∞ ⁡ u ⁢ ( x ) = 0 ,

where I α is Riesz potential given by

I α ⁢ ( x ) = Γ ⁢ ( N - α 2 ) Γ ⁢ ( α 2 ) ⁢ π N / 2 ⁢ 2 α ⁢ | x | N - α ,

and Γ denotes the Gamma function. Equation (1.1) finds its physical origin especially when N = 3 , α = 2 and p = 2 . In this case, a solution of the equation

(1.2) - Δ ⁢ u + u = ( I 2 * | u | 2 ) ⁢ u

gives a solitary wave of the Schrödinger-type nonlinear evolution equation

i ⁢ ∂ t ⁡ ψ + Δ ⁢ ψ + ( I 2 * | ψ | 2 ) ⁢ ψ = 0 ,

which describes, through Hartree–Fock approximation, a dynamics of condensed states to a system of nonrelativistic bosonic particles with two-body attractive interaction potential I 2 that is Newtonian potential [2, 6]. Equation (1.2) also arises as a model of a polaron by Pekar [14] or in an approximation of Hartree–Fock theory for a one-component plasma [7].

Equation (1.1) enjoys a variational structure. It is the Euler–Lagrange equation of the functional

J α ⁢ ( u ) = 1 2 ⁢ ∫ ℝ N | ∇ ⁡ u | 2 + u 2 ⁢ d ⁢ x - 1 2 ⁢ p ⁢ ∫ ℝ N ( I α * | u | p ) ⁢ | u | p ⁢ 𝑑 x .

From the Hardy–Littlewood–Sobolev inequality (Proposition 2.1 below) one can see that J α is well defined and is continuously differentiable on H 1 ⁢ ( ℝ N ) if p ∈ [ 1 + α N , N + α N - 2 ] . We say a function u ∈ H 1 ⁢ ( ℝ N ) is a ground state solution to (1.1) if J α ′ ⁢ ( u ) = 0 and

J α ⁢ ( u ) = inf ⁡ { J α ⁢ ( v ) ∣ v ∈ H 1 ⁢ ( ℝ N ) , J α ′ ⁢ ( v ) = 0 , v ≢ 0 } .

When N = 3 , α = 2 and p = 2 , the existence of a radial positive solution is proved in [7, 9, 11] by variational methods and in [1, 12, 18] by ODE approaches. In [13], Moroz and Van Schaftingen proved the existence of a ground state solution to (1.1) in the range of p ∈ ( 1 + α N , N + α N - 2 ) , and the nonexistence of a nontrivial finite energy solution of (1.1) for p outside of the above range. For qualitative properties of ground states to (1.1), we refer to [10, 13].

In this paper, we are interested in limit behaviors of ground state to (1.1) as either α → 0 or α → N . These shall play essential roles to prove the uniqueness and nondegeneracy of a positive radial ground state to (1.1) for α sufficiently close to 0 or N. From the existence results by Moroz and Van Schaftingen, we can see that a positive radial ground state of (1.1) exists for every α ∈ ( 0 , N ⁢ ( p - 1 ) ) when p ∈ ( 1 , N N - 2 ) is fixed. Also for given p ∈ ( 2 , 2 ⁢ N N - 2 ) , a positive radial ground state of (1.1) exists for every α ∈ ( ( N - 2 ) ⁢ p - N , N ) .

As α → 0 , it is possible to see that the functional J α formally approaches

J 0 ⁢ ( u ) := 1 2 ⁢ ∫ ℝ N | ∇ ⁡ u | 2 + u 2 ⁢ d ⁢ x - 1 2 ⁢ p ⁢ ∫ ℝ N | u | 2 ⁢ p ⁢ 𝑑 x on ⁢ H 1 ⁢ ( ℝ N )

because I α * f approaches f as α → 0 . It is well known that the Euler–Lagrange equation (equation (1.3) below) of J 0 admits a unique positive radial ground state solution. Thus it is reasonable to expect that the ground state of (1.3) is the limit profile of ground states of (1.1) as α → 0 . Our first result is to confirm this.

Theorem 1.1.

Fix p ∈ ( 1 , N N - 2 ) . Let { u α } be a family of positive radial ground states to (1.1) for α close to 0 and let u 0 be a unique positive radial ground state of the equation

(1.3) { - Δ ⁢ u + u = | u | 2 ⁢ p - 2 ⁢ u in ⁢ ℝ N , lim x → ∞ ⁡ u ⁢ ( x ) = 0 .

Then one has

lim α → 0 ⁡ ∥ u α - u 0 ∥ H 1 ⁢ ( ℝ N ) = 0 .

On the other hand, the functional J α blows up when α → N due to the term Γ ⁢ ( N - α 2 ) in the coefficient of I α . Thus we need to get rid of this by taking a scaling v = s ⁢ ( N , α , p ) ⁢ u where

s ⁢ ( N , α , p ) := ( Γ ⁢ ( N - α 2 ) Γ ⁢ ( α 2 ) ⁢ π N / 2 ⁢ 2 α ) 1 2 ⁢ p - 2 ∼ ( 1 N - α ) 1 2 ⁢ p - 2 as ⁢ α → N .

With this scaling, J α transforms into the following functional, which we still denote by J α for simplicity:

J α ⁢ ( v ) = 1 2 ⁢ ∫ ℝ N | ∇ ⁡ v | 2 + v 2 ⁢ d ⁢ x - 1 2 ⁢ p ⁢ ∫ ℝ N ( 1 | ⋅ | N - α * | v | p ) ⁢ | v | p ⁢ 𝑑 x .

Then as α → N , the functional J α approaches

J N ⁢ ( v ) = 1 2 ⁢ ∫ ℝ N | ∇ ⁡ v | 2 + v 2 ⁢ d ⁢ x - 1 2 ⁢ p ⁢ ( ∫ ℝ N | v | p ⁢ 𝑑 x ) 2 .

It is easy to see that for p ∈ ( 2 , 2 ⁢ N N - 2 ) the limit functional J N is C 1 on H 1 ⁢ ( ℝ N ) and its Euler–Lagrange equation is

(1.4) { - Δ ⁢ v + v = ( ∫ ℝ N | v | p ⁢ 𝑑 x ) ⁢ | v | p - 2 ⁢ v in ⁢ ℝ N , lim x → ∞ ⁡ v ⁢ ( x ) = 0 .

The existence and properties of a ground state to (1.4) are studied in [15]. More precisely, it is shown in [15] that there exists a positive radial ground state v 0 of equation (1.4). Furthermore, the following properties for ground states to (1.4) are proved:

The ground state energy level of (1.4) satisfies the mountain pass characterization, i.e.,

J N ⁢ ( v 0 ) = min v ∈ H 1 ⁢ ( ℝ N ) ∖ { 0 } ⁡ max t ≥ 0 ⁡ J N ⁢ ( t ⁢ v ) .
Any ground state of (1.4) is sign-definite, radially symmetric up to a translation and strictly decreasing in radial direction.
Any ground state of (1.4) decays exponentially as | x | → ∞ .

Our next result establishes uniqueness and linearized nondegeneracy of the ground state v 0 of (1.4).

Theorem 1.2.

For p ∈ ( 2 , 2 ⁢ N N - 2 ) , let v 0 be a positive radial ground state to (1.4). Then the following assertions hold:

There is no other positive radial ground state to ( 1.4 ).
The linearized equation of ( 1.4 ) at v 0 , given by

(1.5) - Δ ⁢ ϕ + ϕ - p ⁢ ( ∫ ℝ N v 0 p - 1 ⁢ ϕ ⁢ 𝑑 x ) ⁢ v 0 p - 1 - ( p - 1 ) ⁢ ( ∫ ℝ N v 0 p ⁢ 𝑑 x ) ⁢ v 0 p - 2 ⁢ ϕ = 0 in ⁢ ℝ N ,

only admits solutions of the form

ϕ = ∑ i = 1 N c i ⁢ ∂ x i ⁡ v 0 , c i ∈ ℝ ,

in the space L 2 ⁢ ( ℝ N ) .

Using the uniqueness of v 0 , we can obtain an analogous result to Theorem 1.1.

Theorem 1.3.

Fix p ∈ ( 2 , 2 ⁢ N N - 2 ) . Let { u α } be a family of positive radial ground states to (1.1) for α close to N and let v 0 ∈ H 1 ⁢ ( R N ) be a unique positive radial ground state of (1.4). Then one has

lim α → N ⁡ ∥ v α - v 0 ∥ H 1 ⁢ ( ℝ N ) = 0 ,

where v α is a family of rescaled functions given by v α := s ⁢ ( N , α , p ) ⁢ u α .

Remark 1.4.

By applying the standard comparison principle, it is also possible to see that there exist constants C , c > 0 which are independent of α close to N such that

u α ⁢ ( x ) ≤ C ⁢ ( N - α ) 1 2 ⁢ p - 2 ⁢ e - c ⁢ | x | ,

which shows the vanishing profiles of u α .

The limit profiles of ground states to (1.1) lead to the uniqueness and nondegeneracy of them for α either close to 0 or close to N. When N = 3 , α = 2 and p = 2 , these were proved by Lenzmann [5] and Wei and Winter [19]. Xiang [20] extends this result to the case that N = 3 , α = 2 and p > 2 close to 2 by using perturbation arguments.

We say a positive radial ground state u α of (1.1) is nondegenerate if the linearized equation of (1.1) at u α , given by

- Δ ⁢ ϕ + ϕ - p ⁢ ( I α * ( u α p - 1 ⁢ ϕ ) ) ⁢ u α p - 1 - ( p - 1 ) ⁢ ( I α * u α p ) ⁢ u α p - 2 ⁢ ϕ = 0 in ⁢ ℝ N ,

only admits solutions of the form

ϕ = ∑ i = 1 N c i ⁢ ∂ x i ⁡ u α , c i ∈ ℝ ,

in the space L 2 ⁢ ( ℝ N ) . We should assume p ≥ 2 for the well-definedness of the linearized equation.

Theorem 1.5 (Uniqueness and nondegeneracy).

Fix p ∈ [ 2 , N N - 2 ) . Then a positive radial ground state of (1.1) is unique and nondegenerate for α sufficiently close to 0. Fix p ∈ ( 2 , 2 ⁢ N N - 2 ) . Then the same conclusion holds true for α sufficiently close to N.

Remark 1.6.

Here we note that in the case that α is close to 0, the uniqueness and nondegeneracy are proved only when N = 3 , but in the case that α is close to N these are proved for every dimension N ≥ 3 .

It is worth mentioning that unlike the family of ground states u α to (1.1), the family of least energy nodal solutions u ~ α to (1.1) (the minimal energy solution among all nodal solutions) does not converge to any nontrivial solution of the limit equations (1.3) or (1.4), even up to a translation and up to a subsequence. Actually, the asymptotic profile of u ~ α is shown to be

u 0 ( ⋅ - ξ α + ) - u 0 ( ⋅ - ξ α - ) as α → 0

and

( N - α ) 1 2 ⁢ p - 2 ( v 0 ( ⋅ - ξ α + ) - v 0 ( ⋅ - ξ α - ) ) as α → N

for some ξ α + , ξ α - ∈ ℝ N such that lim α → 0 ⁡ | ξ α + - ξ α - | = 0 ; see [15] for the proof. By relying on this fact and the nondegeneracy of the ground state u 0 to (1.3), it is also proved in [15] that u ~ α is odd-symmetric with respect to the hyperplane normal to the vector ξ α + - ξ α - and through the point ( ξ α + - ξ α - ) / 2 when α ∼ 0 or α ∼ N .

The rest of this paper is organized as follows: In Section 2, we collect some useful auxiliary tools and technical results which are frequently invoked when proving the main theorems. Theorem 1.1 is proved in Section 3. Theorem 1.2 and 1.3 are proved in Section 4. In Sections 5 and 6, we prove our uniqueness and nondegeneracy results, respectively.

2 Auxiliary results

In this section, we provide some useful known results and auxiliary tools. We begin with giving sharp information on the best constant of the Hardy–Littlewood–Sobolev inequality. This plays an important role in our analysis.

Proposition 2.1 (Hardy–Littlewood–Sobolev inequality [3, 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

| ∫ ℝ N ∫ ℝ N f ⁢ ( x ) ⁢ g ⁢ ( y ) | x - y | N - α ⁢ 𝑑 x ⁢ 𝑑 y | ≤ C ⁢ ( N , α , p ) ⁢ ∥ f ∥ L p ⁢ ( ℝ N ) ⁢ ∥ g ∥ L r ⁢ ( ℝ N ) .

The sharp constant satisfies

C ⁢ ( N , α , p ) ≤ 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 ) ,

where | S N - 1 | denotes the surface area of the ( N - 1 ) -dimensional unit sphere.

In addition, if p = r = 2 ⁢ N N + α , then

C ⁢ ( N , α , 2 ⁢ N N + α ) = π N - α 2 ⁢ Γ ⁢ ( α / 2 ) Γ ⁢ ( ( N + α ) / 2 ) ⁢ ( Γ ⁢ ( N ) Γ ⁢ ( N / 2 ) ) α N .

The following Riesz potential estimate is equivalent to the Hardy–Littlewood–Sobolev inequality.

Proposition 2.2 ([3, 8]).

Let 1 ≤ r < s < ∞ and 0 < α < N be such that

1 r - 1 s = α N .

Then for any f ∈ L r ⁢ ( R N ) one has

∥ 1 | ⋅ | N - α * f ∥ L s ⁢ ( ℝ N ) ≤ K ⁢ ( N , α , r ) ⁢ ∥ f ∥ L r ⁢ ( ℝ N ) .

Here, the sharp constant K ⁢ ( N , α , r ) satisfies

lim sup α → 0 ⁡ α ⁢ K ⁢ ( N , α , r ) ≤ 2 r ⁢ ( r - 1 ) ⁢ | 𝕊 N - 1 | .

Corollary 2.3.

Let r , s satisfy the assumption in Proposition 2.2 Then for small α > 0 there exists C = C ⁢ ( N , r ) > 0 such that for any f ∈ L r ⁢ ( R N ) ,

∥ I α * f ∥ L s ⁢ ( ℝ N ) ≤ C ⁢ ∥ f ∥ L r ⁢ ( ℝ N ) .

Proof.

This immediately follows from Proposition 2.2 and the fact that Γ ⁢ ( α 2 ) ∼ 1 α as α → 0 . ∎

We denote by H r 1 ⁢ ( ℝ N ) the space of radial functions in H 1 ⁢ ( ℝ N ) . The following compact embedding result is proved in [17].

Proposition 2.4.

The Sobolev embedding H r 1 ⁢ ( R N ) ↪ L p ⁢ ( R N ) is compact if 2 < p < 2 ⁢ N N - 2 .

By combining the Hardy–Littlewood–Sobolev inequality (Proposition 2.1) and the compact Sobolev embedding, it is easy to see that the following convergence holds.

Proposition 2.5.

Let α ∈ ( 0 , N ) and p ∈ ( 1 + α N , N + α N - 2 ) be given. Let { u j } ⊂ H r 1 ⁢ ( R N ) be a sequence converging weakly to some u 0 ∈ H r 1 ⁢ ( R N ) in H 1 ⁢ ( R N ) as j → ∞ . Then

∫ ℝ N ( 1 | ⋅ | N - α * | u j | p ) ⁢ | u j | p ⁢ 𝑑 x → ∫ ℝ N ( 1 | ⋅ | N - α * | u 0 | p ) ⁢ | u 0 | p ⁢ 𝑑 x .

In addition, for any ϕ ∈ H 1 ⁢ ( R N ) ,

∫ ℝ N ( 1 | ⋅ | N - α * | u j | p ) ⁢ | u j | p - 2 ⁢ u j ⁢ ϕ ⁢ 𝑑 x → ∫ ℝ N ( 1 | ⋅ | N - α * | u 0 | p ) ⁢ | u 0 | p - 2 ⁢ u 0 ⁢ ϕ ⁢ 𝑑 x .

It is useful to obtain estimates for I α * ( | u | p - 1 ⁢ u ⁢ ϕ ) as α → 0 and ( 1 / | ⋅ | N - α ) * ( | u | p - 1 ⁢ u ⁢ ϕ ) as α → N when u , ϕ ∈ H 1 ⁢ ( ℝ N ) .

Proposition 2.6.

Let u , ϕ ∈ H 1 ⁢ ( R N ) . Then the following assertions hold:

For every 1 < p < N N - 2 and 0 < α < p - 1 2 ⁢ N , there exists C = C ⁢ ( N ) > 0 independent of α near 0 such that

∥ I α * ( | u | p - 2 ⁢ u ⁢ ϕ ) ∥ L 2 ⁢ ( ℝ N ) < C ⁢ ∥ u ∥ H 1 ⁢ ( ℝ N ) p - 1 ⁢ ∥ ϕ ∥ H 1 ⁢ ( ℝ N )

and

lim α → 0 ⁡ ∥ I α * ( | u | p - 2 ⁢ u ⁢ ϕ ) - | u | p - 2 ⁢ u ⁢ ϕ ∥ L 2 ⁢ ( ℝ N ) = 0 .
For every 2 < p < 2 ⁢ N N - 2 and ( N - 2 ) ⁢ p 2 < α < N , there exists C = C ⁢ ( N , p ) > 0 independent of α near N such that

∥ 1 | ⋅ | N - α * ( | u | p - 2 ⁢ u ⁢ ϕ ) ∥ L ∞ ⁢ ( ℝ N ) < C ⁢ ∥ u ∥ H 1 ⁢ ( ℝ N ) p - 1 ⁢ ∥ ϕ ∥ H 1 ⁢ ( ℝ N )

and

lim α → N ⁡ ∥ ( 1 | ⋅ | N - α * ( | u | p - 2 ⁢ u ⁢ ϕ ) ) - ∫ ℝ N | u | p - 2 ⁢ u ⁢ ϕ ⁢ 𝑑 x ∥ L ∞ ⁢ ( K ) = 0

for any compact set K ⊂ ℝ N .

Proof.

A proof for (i) can be found in [16]. We prove (ii). Observe from the Hölder inequality that

( 1 | ⋅ | N - α * | u | p - 2 ⁢ u ⁢ ϕ ) ⁢ ( x ) ≤ ∫ B 1 ⁢ ( x ) 1 | x - y | N - α ⁢ ( | u | p - 2 ⁢ u ⁢ ϕ ) ⁢ ( y ) ⁢ 𝑑 y + ∫ B 1 c ⁢ ( x ) 1 | x - y | N - α ⁢ ( | u | p - 2 ⁢ u ⁢ ϕ ) ⁢ ( y ) ⁢ 𝑑 y

≤ ( ∫ B 1 ⁢ ( 0 ) 1 | y | ( N - α ) ⁢ 2 * 2 * - p ⁢ 𝑑 y ) 2 * - p 2 * ⁢ ∥ u ∥ L 2 * ⁢ ( ℝ N ) p - 1 ⁢ ∥ ϕ ∥ L 2 * ⁢ ( ℝ N ) + ∥ u ∥ L p ⁢ ( ℝ N ) p - 1 ⁢ ∥ ϕ ∥ L p ⁢ ( ℝ N ) ,

where 2 * denotes the critical Sobolev exponent 2 ⁢ N N - 2 . Note from the condition on α that ( N - α ) ⁢ 2 * 2 * - p < N , so that the integral

∫ B 1 ⁢ ( 0 ) 1 / | y | ( N - α ) ⁢ 2 * 2 * - p ⁢ 𝑑 y

is uniformly bounded for α sufficiently close to N. This proves the former assertion of (ii). To prove the latter, we suppose the contrary. Then there exist a compact set K and sequences α j → N , x j ∈ K such that

(2.1) ∫ ℝ N 1 | x j - y | N - α j ⁢ ( | u | p - 2 ⁢ u ⁢ ϕ ) ⁢ ( y ) ⁢ 𝑑 y ↛ ∫ ℝ N | u | p - 2 ⁢ u ⁢ ϕ ⁢ 𝑑 y as ⁢ j → ∞ .

Define f j ⁢ ( y ) = ( | u | p - 2 ⁢ u ⁢ ϕ ) ⁢ ( y ) / | x j - y | N - α j , so that f j ⁢ ( y ) → ( | u | p - 2 ⁢ ϕ ) ⁢ ( y ) almost everywhere as j → ∞ . We may assume x j → x 0 as j → ∞ for some x 0 ∈ K . We claim that f j is uniformly integrable and tight in ℝ N , i.e., for given ε > 0 there exists δ > 0 such that

∫ E | f j ⁢ ( y ) | ⁢ 𝑑 y < ε

for every E ⊂ ℝ N satisfying | E | < δ and there exists R > 0 such that

∫ B R c ⁢ ( 0 ) | f j ⁢ ( y ) | ⁢ 𝑑 y < ε .

Indeed, we have

∫ E | f j ⁢ ( y ) | ⁢ 𝑑 y ≤ ( ∫ E 1 | x j - y | ( N - α j ) ⁢ 2 * 2 * - p ⁢ 𝑑 y ) 2 * - p 2 * ⁢ ∥ u ∥ L 2 * ⁢ ( E ) p - 1 ⁢ ∥ ϕ ∥ L 2 * ⁢ ( E )

= ( ( ∫ E ∩ B 1 ⁢ ( x ) + ∫ E ∩ B 1 c ⁢ ( x ) ) ⁢ 1 | x j - y | ( N - α j ) ⁢ 2 * 2 * - p ⁢ d ⁢ y ) 2 * - p 2 * ⁢ ∥ u ∥ L 2 * ⁢ ( E ) p - 1 ⁢ ∥ ϕ ∥ L 2 * ⁢ ( E )

≤ ( ∫ B 1 ⁢ ( 0 ) 1 | y | ( N - α j ) ⁢ 2 * 2 * - p ⁢ 𝑑 y + | E | ) 2 * - p 2 * ⁢ ∥ u ∥ L 2 * ⁢ ( E ) p - 1 ⁢ ∥ ϕ ∥ L 2 * ⁢ ( E ) ,

which shows that f j is uniformly integrable. Take also a large R > 0 such that B 2 ⁢ ( x 0 ) ⊂ B R ⁢ ( 0 ) . Then since

∫ B R c ⁢ ( 0 ) | f j ⁢ ( y ) | ⁢ 𝑑 y ≤ ∫ B R c ⁢ ( 0 ) | u ⁢ ( y ) | p - 1 ⁢ | ϕ ⁢ ( y ) | ⁢ 𝑑 y ,

the tightness of f j is proved. Now the Vitali convergence theorem says that ∫ ℝ N f j ⁢ ( y ) ⁢ 𝑑 y → ∫ ℝ N | u | p ⁢ ( y ) ⁢ 𝑑 y , which contradicts (2.1). This completes the proof. ∎

Proposition 2.7.

Fix 1 < p < N N - 2 . Let { α j } > 0 be a sequence converging to 0 and let { u j } ⊂ H r 1 ⁢ ( R N ) be a sequence converging weakly in H 1 ⁢ ( R N ) to some u 0 ∈ H r 1 ⁢ ( R N ) . Then, as j → ∞ , the following holds:

∫ ℝ N ( I α j * | u j | p ) ⁢ | u j | p ⁢ 𝑑 x → ∫ ℝ N | u 0 | 2 ⁢ p ⁢ 𝑑 x ,

∫ ℝ N ( I α j * | u j | p ) ⁢ | u j | p - 2 ⁢ u j ⁢ ϕ ⁢ 𝑑 x → ∫ ℝ N | u 0 | 2 ⁢ p - 2 ⁢ u 0 ⁢ ϕ ⁢ 𝑑 x for any ⁢ ϕ ∈ H 1 ⁢ ( ℝ N ) .

Proof.

For a proof of this proposition, we refer to [16]. ∎

Proposition 2.8.

Fix 2 < p < 2 ⁢ N N - 2 . Let { α j } > 0 be a sequence converging to N and let { u j } ⊂ H r 1 ⁢ ( R N ) be a sequence converging weakly in H 1 ⁢ ( R N ) to some u 0 ∈ H r 1 ⁢ ( R N ) . Then, as j → ∞ , the following holds:

(2.2) ∫ ℝ N ( 1 | ⋅ | N - α j * | v j | p ) ⁢ | v j | p ⁢ 𝑑 x → ( ∫ ℝ N | v 0 | p ⁢ 𝑑 x ) 2 ,

(2.3) ∫ ℝ N ( 1 | ⋅ | N - α j * | v j | p ) ⁢ | v j | p - 2 ⁢ v j ⁢ ϕ ⁢ 𝑑 x → ∫ ℝ N | v 0 | p ⁢ 𝑑 x ⁢ ∫ ℝ N | v 0 | p - 2 ⁢ v 0 ⁢ ϕ ⁢ 𝑑 x for any ⁢ ϕ ∈ H 1 ⁢ ( ℝ N ) .

Proof.

For (2.2), we decompose as

∫ ℝ N ( 1 | ⋅ | N - α j * | v j | p ) ⁢ | v j | p ⁢ 𝑑 x = ∫ ℝ N ( 1 | ⋅ | N - α j * | v j | p ) ⁢ ( | v j | p - | v 0 | p ) ⁢ 𝑑 x + ∫ ℝ N ( 1 | ⋅ | N - α j * ( | v j | p - | v 0 | p ) ) ⁢ | v 0 | p ⁢ 𝑑 x

+ ∫ ℝ N ( 1 | ⋅ | N - α j * | v 0 | p ) ⁢ | v 0 | p ⁢ 𝑑 x

= : A j + B j + C j .

Observe from Proposition 2.6 that

| A j | ≤ C ⁢ ∥ v j ∥ H 1 ⁢ ( ℝ N ) p ⁢ ∥ | v j | p - | v 0 | p ∥ L 1 ⁢ ( ℝ N ) ,

which goes to 0 as j → ∞ by the compact Sobolev embedding H r 1 ⁢ ( ℝ N ) ↪ L p ⁢ ( ℝ N ) . The same argument in the proof of Proposition 2.6 (ii) also applies to show that there exists a constant C > 0 independent of α j such that

| B j | ≤ C ⁢ ( ∥ | v j | p - | v 0 | p ∥ L 2 * / p ⁢ ( ℝ N ) + ∥ | v j | p - | v 0 | p ∥ L 1 ⁢ ( ℝ N ) ) ⁢ ∥ v 0 ∥ L p ⁢ ( ℝ N ) p ,

which also goes to 0 as j → ∞ . Finally, C j goes to ( ∫ ℝ N | v 0 | p ⁢ 𝑑 x ) 2 as j → ∞ by (2.2).

The idea of proof of (2.2) is equally applicable to prove (2.3). We omit it. ∎

3 Limit profile of ground states as α → 0

In this section, we prove Theorem 1.1. We choose an arbitrary p ∈ ( 1 , N N - 2 ) and fix it throughout this section. We denote the ground state energy level of J α by E α . In other word, E α = J α ⁢ ( u α ) , where u α is a ground state solution to (1.1). The ground state energy level E α of (1.1) satisfies the mountain pass characterization, i.e.,

E α := min u ∈ H 1 ⁢ ( ℝ N ) ∖ { 0 } ⁡ max t ≥ 0 ⁡ J α ⁢ ( t ⁢ u ) .

Recall that

J 0 ⁢ ( u ) = 1 2 ⁢ ∫ ℝ N | ∇ ⁡ u | 2 + u 2 ⁢ d ⁢ x - 1 2 ⁢ p ⁢ ∫ ℝ N | u | 2 ⁢ p ⁢ 𝑑 x on ⁢ H 1 ⁢ ( ℝ N ) ,

whose Euler–Lagrange equation is (1.3). We define the mountain pass level of J 0 by

E 0 := min u ∈ H 1 ⁢ ( ℝ N ) ∖ { 0 } ⁡ max t ≥ 0 ⁡ J 0 ⁢ ( t ⁢ u ) .

It is a well-known fact that E 0 is the ground state energy level of J 0 . Namely,

E 0 = min ⁡ { J 0 ⁢ ( u ) ∣ u ∈ H 1 ⁢ ( ℝ N ) , J 0 ′ ⁢ ( u ) = 0 , u ≢ 0 } .

The following lemma is proved in [15, Claim 1 of Proposition 4.1].

Lemma 3.1.

There holds

lim α → 0 ⁡ E α = E 0 .

Choose any sequences { α j } > 0 converging to 0 and { u α j } of positive radial ground states to (1.1).

Lemma 3.2.

There exists a positive radial solution u 0 ∈ H 1 ⁢ ( R N ) to (1.3) such that { u α j } converges to u 0 in H 1 ⁢ ( R N ) up to a subsequence.

Proof.

Multiplying equation (1.1) by u α j and integrating by parts, we get

( 1 2 - 1 2 ⁢ p ) ⁢ ∥ u α j ∥ H 1 ⁢ ( ℝ N ) 2 = J α j ⁢ ( u α j ) ,

so ∥ u α j ∥ H 1 ⁢ ( ℝ N ) is uniformly bounded for j by Lemma 3.1. Then, up to a subsequence, { u α j } weakly converges in H 1 ⁢ ( ℝ N ) to some nonnegative radial function u 0 ∈ H 1 ⁢ ( ℝ N ) . From Proposition 2.7 and the weak convergence of { u α j } , one is able to deduce that u 0 is a weak solution of (1.3). In addition, we again multiply equation (1.1) by u α j , multiply equation (1.3) by u 0 and use Proposition 2.7 to get

∥ u α j ∥ H 1 ⁢ ( ℝ N ) 2 = ∫ ℝ N ( I α j * | u α j | p ) ⁢ | u α j | p ⁢ 𝑑 x → ∫ ℝ N | u 0 | 2 ⁢ p ⁢ 𝑑 x = ∥ u 0 ∥ H 1 ⁢ ( ℝ N ) 2 as ⁢ j → ∞ .

Combining this with the weak convergence of { u α j } , we obtain the strong convergence of { u α j } to u 0 in H 1 ⁢ ( ℝ N ) .

Now, it remains to prove that u 0 is positive. Observe from Corollary 2.3 and the Sobolev inequality that

(3.1) ∥ u α j ∥ H 1 ⁢ ( ℝ N ) 2 = ∫ ℝ N ( I α j * | u α j | p ) ⁢ | u α j | p ⁢ 𝑑 x ≤ C ⁢ ∥ u α j ∥ L 2 ⁢ N ⁢ p / ( N + α j ) ⁢ ( ℝ N ) 2 ⁢ p ≤ C ⁢ ∥ u α j ∥ H 1 ⁢ ( ℝ N ) 2 ⁢ p .

Here, C is a universal constant independent of j. Then, dividing both sides of (3.1) by ∥ u α j ∥ H 1 ⁢ ( ℝ N ) 2 and passing to a limit, we obtain a uniform lower bound for ∥ u α j ∥ H 1 ⁢ ( ℝ N ) which implies that u 0 is nontrivial due to the strong convergence of { u α j } . Since u 0 is nonnegative, it is positive from the maximum principle. This completes the proof. ∎

Then the next lemma follows.

Lemma 3.3.

There holds

J 0 ⁢ ( u 0 ) = E 0 .

In other words, u 0 is a unique positive radial ground state to (1.3).

Proof.

We see from Proposition 2.7, Lemma 3.1 and Lemma 3.2 that

E 0 = lim j → ∞ ⁡ E α j = lim j → ∞ ⁡ J α j ⁢ ( u α j )

= lim j → ∞ ⁡ ( 1 2 ⁢ ∥ u α j ∥ H 1 ⁢ ( ℝ N ) 2 - 1 2 ⁢ p ⁢ ∫ ℝ N ( I α j * | u α j | p ) ⁢ | u α j | p ⁢ 𝑑 x )

= 1 2 ⁢ ∥ u 0 ∥ H 1 ⁢ ( ℝ N ) 2 - 1 2 ⁢ p ⁢ ∫ ℝ N | u 0 | 2 ⁢ p ⁢ 𝑑 x = J 0 ⁢ ( u 0 ) ,

which proves the lemma. ∎

Now, we are ready to complete the proof of Theorem 1.1. Let { u α } ⊂ H r 1 ⁢ ( ℝ N ) be a family of positive radial ground states to (1.1) for α near 0. Suppose { u α } does not converge in H 1 ⁢ ( ℝ N ) to the unique positive radial ground state u 0 of (1.3). Then there exists a positive number ε 0 and a sequence { α j } → 0 such that ∥ u α j - u 0 ∥ H 1 ⁢ ( ℝ N ) ≥ ε 0 , which contradicts Lemma 3.2 and Lemma 3.3.

4 Limit profile of ground states as α → N

In this section, we prove Theorems 1.2 and 1.3. Choose and fix an arbitrary p ∈ ( 2 , 2 ⁢ N N - 2 ) . By deleting the coefficient of the Riesz potential term from (1.1), we obtain the equation

(4.1) { - Δ ⁢ v + v = ( 1 | ⋅ | N - α * | v | p ) ⁢ | v | p - 2 ⁢ v in ⁢ ℝ N , lim x → ∞ ⁡ v ⁢ ( x ) = 0 .

For simplicity, we still denote by J α the energy functional of (4.1). It is clear that the ground state energy level E α of (4.1) also satisfies the mountain pass characterization:

E α := min v ∈ H 1 ⁢ ( ℝ N ) ∖ { 0 } ⁡ max t ≥ 0 ⁡ J α ⁢ ( t ⁢ v ) .

Recall that, as α → N , the functional J α approaches a limit functional

J N ⁢ ( v ) = 1 2 ⁢ ∫ ℝ N | ∇ ⁡ v | 2 + v 2 ⁢ d ⁢ x - 1 2 ⁢ p ⁢ ( ∫ ℝ N | v | p ⁢ 𝑑 x ) 2 on ⁢ H 1 ⁢ ( ℝ N ) ,

whose Euler–Lagrange equation is (1.4).

4.1 Proof of Theorem 1.2

We first prove Theorem 1.2. To prove (i), we let v 1 and v 2 be two positive radial ground states to (1.4). By defining

a 1 = ∫ ℝ N | v 1 | p ⁢ 𝑑 x and a 2 = ∫ ℝ N | v 2 | p ⁢ 𝑑 x ,

they are positive radial solutions of the equations - Δ ⁢ w + w = a 1 ⁢ | w | p - 2 ⁢ w and - Δ ⁢ w + w = a 2 ⁢ | w | p - 2 ⁢ w , respectively. We note that ( a 1 / a 2 ) 1 / ( p - 2 ) ⁢ v 1 satisfies the latter equation. The classical result due to Kwong [4] says that a positive radial solution of the latter (and also the former) equation is unique, so one must have ( a 1 / a 2 ) 1 / ( p - 2 ) ⁢ v 1 ≡ v 2 . Since both of v 1 and v 2 satisfy equation (1.4), we can conclude ( a 1 / a 2 ) 1 / ( p - 2 ) = 1 .

We next prove (ii). Let v 0 be the positive and radial ground state of (1.4). Let a 0 = ∫ ℝ N v 0 p ⁢ 𝑑 x . As discussed above, v 0 is a unique positive radial solution of

(4.2) - Δ ⁢ w + w = a 0 ⁢ | w | p - 2 ⁢ w .

It is a well-known fact that the linearized operator of (4.2) at v 0 , given by

L ⁢ ( ϕ ) := - Δ ⁢ ϕ + ϕ - ( p - 1 ) ⁢ a 0 ⁢ v 0 p - 2 ⁢ ϕ ,

admits only solutions of the form

(4.3) ϕ = ∑ i = 1 N c i ⁢ ∂ x i ⁡ v 0 , c i ∈ ℝ ,

in the space L 2 ⁢ ( ℝ N ) . To the contrary, suppose that (1.5) has a nontrivial solution ϕ ∈ L 2 ⁢ ( ℝ N ) , which is not of the form (4.3). Then we may assume that ϕ is L 2 orthogonal to ∂ x i ⁡ v 0 for every i = 1 , … , N . By denoting λ := p ⁢ ∫ ℝ N v 0 p - 1 ⁢ ϕ ⁢ 𝑑 x , we see L ⁢ ( ϕ ) = λ ⁢ v 0 p - 1 , so λ should not be 0. Observe that

L ⁢ ( λ ( 2 - p ) ⁢ a 0 ⁢ v 0 ) = λ ( 2 - p ) ⁢ a 0 ⁢ L ⁢ ( v 0 ) = λ ( 2 - p ) ⁢ a 0 ⁢ ( - Δ ⁢ v 0 + v 0 - ( p - 1 ) ⁢ a 0 ⁢ v 0 p - 1 ) = λ ⁢ v 0 p - 1 .

This shows L ⁢ ( ϕ - λ ( 2 - p ) ⁢ a 0 ⁢ v 0 ) ≡ 0 , which implies that there are some c i ∈ ℝ such that

(4.4) ϕ - λ ( 2 - p ) ⁢ a 0 ⁢ v 0 = ∑ i = 1 N c i ⁢ ∂ x i ⁡ v 0 .

We claim that c i = 0 for all i. Indeed, by multiplying the left-hand side of (4.4) by ∂ x j ⁡ v 0 and integrating, we get

∫ ℝ N ϕ ⁢ ∂ x j ⁡ v 0 ⁢ d ⁢ x - λ ( 2 - p ) ⁢ a 0 ⁢ ∫ ℝ N v 0 ⁢ ∂ x j ⁡ v 0 ⁢ d ⁢ x = - λ ( 2 - p ) ⁢ a 0 ⁢ ∫ ℝ N 1 2 ⁢ ∂ x j ⁡ ( v 0 2 ) ⁡ d ⁢ x = 0 .

On the other hand, by multiplying (4.4) by ∂ x j ⁡ v 0 and integrating, we get

c j ⁢ ∫ ℝ N ( ∂ x j ⁡ v 0 ) 2 ⁢ 𝑑 x + ∑ i ≠ j c i ⁢ ∫ ℝ N ∂ x i ⁡ v 0 ⁢ ∂ x j ⁡ v 0 ⁢ d ⁢ x = c j ⁢ ∫ ℝ N ( ∂ x j ⁡ v 0 ) 2 ⁢ 𝑑 x + ∑ i ≠ j c i ⁢ ∫ ℝ N x i ⁢ x j r 2 ⁢ v 0 ′ ⁢ ( r ) ⁢ 𝑑 x = c j ⁢ ∫ ℝ N ( ∂ x j ⁡ v 0 ) 2 ⁢ 𝑑 x

since x i ⁢ x j r 2 ⁢ v 0 ′ ⁢ ( r ) is odd in variables x i and x j . Combining these two integrals, the claim follows.

Now, observe that

λ = p ⁢ ∫ ℝ N v 0 p - 1 ⁢ ϕ ⁢ 𝑑 x = p ⁢ λ ( 2 - p ) ⁢ a 0 ⁢ ∫ ℝ N v 0 p ⁢ 𝑑 x = p ⁢ λ 2 - p .

This implies p = 1 , which contradicts the hypothesis for p. This completes the proof of Theorem 1.2.

4.2 Proof of Theorem 1.3

Now it remains to prove Theorem 1.3. Choose an arbitrary positive sequence { α j } → N and an arbitrary sequence { v α j } of positive radial ground states to (4.1). Arguing similarly to the previous section, one can see that the following proposition also holds true.

Proposition 4.1.

By choosing a subsequence, { v α j } converges in H 1 ⁢ ( R N ) to the unique positive radial ground state of (1.4).

Proof.

We follow the same lines as in Section 3. It is proved in [15, Claim 1 of Proposition 5.1] that

lim α → N ⁡ E α = E N ,

where

E N := min v ∈ H 1 ⁢ ( ℝ N ) ∖ { 0 } ⁡ max t ≥ 0 ⁡ J N ⁢ ( t ⁢ v ) .

This implies that ∥ v α j ∥ H 1 is bounded and, consequently, has a weak subsequential limit v 0 ∈ H r 1 ⁢ ( ℝ N ) which is radial and nonnegative. Proposition 2.8 says that v 0 is a solution of (1.4). Again using Proposition 2.8, we have

∥ v α j ∥ H 1 2 = ∫ ℝ N ( 1 | ⋅ | N - α j * v α j p ) ⁢ v α j p ⁢ 𝑑 x = ( ∫ ℝ N v 0 p ⁢ 𝑑 x ) 2 + o ⁢ ( 1 ) = ∥ v 0 ∥ H 1 2 + o ⁢ ( 1 ) ,

which implies the H 1 strong convergence of { v α j } . We now invoke Proposition 2.6 (ii) to see

∥ v α j ∥ H 1 2 = ∫ ℝ N ( 1 | ⋅ | N - α j * v α j p ) ⁢ v α j p ⁢ 𝑑 x ≤ C ⁢ ∥ v α j ∥ H 1 p ⁢ ∥ v α j ∥ L p p ≤ C ⁢ ∥ v α j ∥ H 1 2 ⁢ p ,

where C is independent of j. This shows that v 0 is nontrivial, so that it is positive by the strong maximum principle. Finally, as in Lemma 3.3, we can check J N ⁢ ( v 0 ) = E N , which completes the proof. ∎

Now, we shall complete the proof of Theorem 1.3. Fix p ∈ ( 2 , 2 ⁢ N N - 2 ) . Let { u α } be a family of positive radial ground states to (1.1) for α close to N. Then it is clear that the rescaled functions v α := s ⁢ ( N , α ) ⁢ u α constitute a family of positive radial ground states of (4.1) by a direct computation. Therefore, as in the proof of Theorem 1.1, one may conclude lim α → N ⁡ ∥ v α - v 0 ∥ H 1 ⁢ ( ℝ N ) = 0 , where we denote by v 0 a unique positive radial solution to (1.4).

5 Uniqueness of ground states

We begin this section with a simple elliptic estimate.

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 ) .

Proof.

We multiply the equation - Δ ⁢ u + u = f by u, integrate by parts and apply the Hölder inequality:

∥ u ∥ H 1 ⁢ ( ℝ N ) 2 = ∫ ℝ N f ⁢ u ⁢ 𝑑 x ≤ ∥ f ∥ L q ⁢ ( ℝ N ) ⁢ ∥ u ∥ L q / ( q - 1 ) ⁢ ( ℝ N ) .

Since 2 ≤ q q - 1 ≤ 2 ⁢ N N - 2 , the Sobolev inequality applies to see

∥ u ∥ H 1 ⁢ ( ℝ N ) ≤ C ⁢ ∥ f ∥ L q ⁢ ( ℝ N )

for some C depending only on q and N. Then the density arguments complete the proof. ∎

For p ∈ ( 1 , N N - 2 ) , choose and fix α 0 ∈ ( 0 , N ⁢ ( p - 1 ) 2 ) and define an operator A ⁢ ( α , u ) by

A ⁢ ( α , u ) := { u - ( - Δ + I ) - 1 ⁢ [ ( I α * | u | p ) ⁢ | u | p - 2 ⁢ u ] if ⁢ α ∈ ( 0 , α 0 ) , u - ( - Δ + I ) - 1 ⁢ [ | u | 2 ⁢ p - 2 ⁢ u ] if ⁢ α = 0 .

For p ∈ ( 2 , 2 ⁢ N N - 2 ) , choose and fix α N ∈ ( ( N - 2 ) ⁢ p 2 , N ) and define an operator B ⁢ ( α , v ) by

B ⁢ ( α , v ) := { v - ( - Δ + I ) - 1 ⁢ [ ( 1 | ⋅ | N - α * | v | p ) ⁢ | v | p - 2 ⁢ v ] if ⁢ α ∈ ( α N , N ) , v - ( - Δ + I ) - 1 ⁢ [ ( ∫ ℝ N | v | p ⁢ 𝑑 x ) ⁢ | v | p - 2 ⁢ v ] if ⁢ α = N .

Lemma 5.2.

The operator A is a continuous map from [ 0 , α 0 ) × H r 1 ⁢ ( R N ) into H r 1 ⁢ ( R N ) and is continuously differentiable with respect to u on [ 0 , α 0 ) × H r 1 ⁢ ( R N ) . The same conclusion holds true for B which is a map from ( α N , N ] × H r 1 ⁢ ( R N ) into H 1 ⁢ ( R N ) .

Proof.

We first prove the continuity of A. Let { ( α j , u j ) } be a sequence in ∈ [ 0 , α 0 ) × H r 1 ⁢ ( ℝ N ) converging to some ( α , u ) ∈ [ 0 , α 0 ) × H r 1 ⁢ ( ℝ N ) . We only deal with the case α j ≠ 0 and α = 0 . Then the remaining cases can be dealt with similarly as well as more easily. Lemma 5.1 shows that it is sufficient to prove that ( I α j * | u j | p ) ⁢ | u j | p - 2 ⁢ u j converges to | u | 2 ⁢ p - 2 ⁢ u in L q ⁢ ( ℝ N ) for some q ∈ [ 2 ⁢ N / ( N + 2 ) , 2 ] . We select q = 2 ⁢ p 2 ⁢ p - 1 . Since p ∈ ( 1 , N N - 2 ) , one can easily see that q belongs to the above range. Then

∥ ( I α j * | u j | p ) ⁢ | u j | p - 2 ⁢ u j - | u | 2 ⁢ p - 2 ⁢ u ∥ L 2 ⁢ p / ( 2 ⁢ p - 1 ) ⁢ ( ℝ N )

≤ ∥ ( ( I α j * | u j | p ) - | u j | p ) ⁢ | u j | p - 2 ⁢ u j ∥ L 2 ⁢ p / ( 2 ⁢ p - 1 ) ⁢ ( ℝ N ) + ∥ | u j | 2 ⁢ p - 2 ⁢ u j - | u | 2 ⁢ p - 2 ⁢ u ∥ L 2 ⁢ p / ( 2 ⁢ p - 1 ) ⁢ ( ℝ N )

≤ ∥ ( I α j * | u j | p ) - | u j | p ∥ L 2 ⁢ ( ℝ N ) ⁢ ∥ | u j | p - 1 ∥ L 2 ⁢ p / ( p - 1 ) ⁢ ( ℝ N ) + o ⁢ ( 1 )

≤ C ⁢ ∥ ( I α j * | u | p ) - | u | p + I α j * ( | u j | p - | u | p ) + | u | p - | u j | p ∥ L 2 ⁢ ( ℝ N ) + o ⁢ ( 1 )

(5.1) ≤ C ⁢ ∥ I α j * ( | u j | p - | u | p ) ∥ + o ⁢ ( 1 ) ≤ C ⁢ ∥ | u j | p - | u | p ∥ L 2 ⁢ N / ( N + 2 ⁢ α ) ⁢ ( ℝ N ) + o ⁢ ( 1 ) = o ⁢ ( 1 ) ,

where we used the Hölder inequality, Sobolev inequality and sharp constant estimate in Proposition 2.2.

Differentiating A with respect to u, we get

∂ ⁡ A ∂ ⁡ u ⁢ ( α , u ) ⁢ [ ϕ ] = { ϕ - ( - Δ + I ) - 1 ⁢ [ p ⁢ ( I α * | u | p - 2 ⁢ u ⁢ ϕ ) ⁢ | u | p - 2 ⁢ u + ( p - 1 ) ⁢ ( I α * | u | p ) ⁢ | u | p - 2 ⁢ ϕ ] if ⁢ α ∈ ( 0 , α 0 ) , ϕ - ( - Δ + I ) - 1 ⁢ [ ( 2 ⁢ p - 1 ) ⁢ | u | 2 ⁢ p - 2 ⁢ ϕ ] if ⁢ α = 0 .

Then one can apply essentially the same argument to (5.1) to see that ∂ ⁡ A ∂ ⁡ u is continuous on [ 0 , α 0 ] × H r 1 ⁢ ( ℝ N ) .

We next address the operator B. Let { ( α j , v j ) } be a sequence in ∈ ( α N , N ] × H r 1 ⁢ ( ℝ N ) converging to some ( α , v ) ∈ ( α N , N ] × H r 1 ⁢ ( ℝ N ) . We only deal with the case α = N and α j ≠ N . As above, it is sufficient to show that ( 1 / | ⋅ | N - α j * | v j | p ) ⁢ | v j | p - 2 ⁢ v j converges to ( ∫ ℝ N | v | p ⁢ 𝑑 x ) ⁢ | v | p - 2 ⁢ v in L p / ( p - 1 ) ⁢ ( ℝ N ) for the continuity of B. This follows by arguing similarly to (5.1) with Proposition 2.6. ∎

Lemma 5.3.

Suppose that u 0 is a unique positive radial ground state of (1.3). Then there exists a neighborhood U 0 ⊂ [ 0 , α 0 ) × H r 1 ⁢ ( R N ) of a point ( 0 , u 0 ) ∈ [ 0 , α 0 ) × H r 1 ⁢ ( R N ) such that equation (1.1) admits a unique solution in U 0 . Suppose that v 0 is a unique positive radial ground state of (1.4). Then there exists a neighborhood U N ⊂ ( α N , N ] × H r 1 ⁢ ( R N ) of a point ( N , v 0 ) ∈ ( α N , N ] × H r 1 ⁢ ( R N ) such that equation (4.1) admits a unique solution in U N .

Proof.

We only prove the former assertion. The latter assertion follows similarly. We claim that the linearized operator of A with respect to u at ( 0 , u 0 ) , namely ∂ ⁡ A ∂ ⁡ u ⁢ ( 0 , u 0 ) , is a linear isomorphism from H r 1 ⁢ ( ℝ N ) into H r 1 ⁢ ( ℝ N ) . Observe that

∂ ⁡ A ∂ ⁡ u ⁢ ( 0 , u 0 ) ⁢ [ ϕ ] = ϕ - ( 2 ⁢ p - 1 ) ⁢ ( - Δ + I ) - 1 ⁢ [ u 0 2 ⁢ p - 2 ⁢ ϕ ] .

Since u 0 decays exponentially, the map ϕ ↦ u 0 2 ⁢ p - 2 ⁢ ϕ is compact from H r 1 ⁢ ( ℝ N ) into L 2 ⁢ ( ℝ N ) , so the composite map ϕ ↦ ( - Δ + I ) - 1 ⁢ [ u 0 2 ⁢ p - 2 ⁢ ϕ ] is also compact from H r 1 ⁢ ( ℝ N ) into H r 1 ⁢ ( ℝ N ) . This also shows that ∂ ⁡ A ∂ ⁡ u ⁢ ( 0 , u 0 ) is bounded. One can deduce from the radial linearized nondegeneracy of u 0 that the kernel of ∂ ⁡ A ∂ ⁡ u ⁢ ( 0 , u 0 ) is trivial. Then the Fredholm alternative applies to see that ∂ ⁡ A ∂ ⁡ u ⁢ ( 0 , u 0 ) is an onto map, so the claim is proved. We invoke the implicit function theorem to complete the proof. ∎

Now, we claim that (1.1) admits a unique positive radial ground state for p ∈ ( 1 , N N - 2 ) and α close to 0. Suppose the contrary. Then there exist sequences { α j } > 0 , { u α j 1 } ⊂ H r 1 ⁢ ( ℝ N ) and { u α j 2 } ⊂ H r 1 ⁢ ( ℝ N ) such that α j → 0 as j → ∞ , { u α j 1 } and { u α j 2 } are sequences of positive radial ground states of (1.1), and u α j 1 ≠ u α j 2 for all j. Theorem 1.1 tells us that both of { u α j 1 } and { u α j 2 } converge to a unique positive radial solution u 0 of (1.3) in H 1 ⁢ ( ℝ N ) . This however contradicts Lemma 5.3, and thus shows the uniqueness of a positive radial ground state of (1.1) for p ∈ ( 1 , N N - 2 ) and α close to 0. Note that the analogous conclusion holds for a family of ground states { v α } of (4.1) when p ∈ ( 2 , 2 ⁢ N N - 2 ) and α close to N. By scaling back, this also shows the uniqueness of a positive radial ground state of (1.1) when p ∈ ( 2 , 2 ⁢ N N - 2 ) and α close to N.

6 Nondegeneracy of ground states

6.1 Nondegeneracy for α near 0

Throughout this subsection, we fix N = 3 due to the restriction p ∈ [ 2 , N N - 2 ) . We begin with proving a convergence lemma similar to Proposition 2.7, but slightly different.

Lemma 6.1.

For given p ∈ [ 2 , 3 ) , let u α be a family of the unique positive radial ground states of (1.1) and let u 0 be the positive radial ground state to (1.3). Then, for any { α j } → 0 and { ψ j } , { ϕ j } ⊂ H 1 weakly H 1 converging to ϕ 0 and ψ 0 , there holds

(6.1) ∫ ℝ N ( I α j * ( u α j p - 1 ⁢ ϕ j ) ) ⁢ u α j p - 1 ⁢ ψ j ⁢ 𝑑 x → ∫ ℝ N u 0 2 ⁢ p - 2 ⁢ ϕ 0 ⁢ ψ 0 ⁢ 𝑑 x ,

(6.2) ∫ ℝ N ( I α j * u α j p ) ⁢ u α j p - 2 ⁢ ϕ j ⁢ ψ j ⁢ 𝑑 x → ∫ ℝ N u 0 2 ⁢ p - 2 ⁢ ϕ 0 ⁢ ψ 0 ⁢ 𝑑 x .

Proof.

We first note that u 0 p - 1 ⁢ ϕ j is compact in L 2 due to the uniform decaying property of u 0 . Then one has from the Hölder inequality that

∥ u α j p - 1 ⁢ ϕ j - u 0 p - 1 ⁢ ϕ 0 ∥ L 2 ≤ ∥ ( u α j p - 1 - u 0 p - 1 ) ⁢ ϕ j ∥ L 2 + ∥ u 0 p - 1 ⁢ ϕ j - u 0 p - 1 ⁢ ϕ 0 ∥ L 2

≤ ∥ u α j p - 1 - u 0 p - 1 ∥ L 2 ⁢ p / ( p - 1 ) ⁢ ∥ ϕ j ∥ L 2 ⁢ p + o ⁢ ( 1 )

≤ C ⁢ ∥ | u α j - u 0 | ⁢ ( | u α j | p - 2 + | u 0 | p - 2 ) ∥ L 2 ⁢ p / ( p - 1 ) ⁢ ∥ ϕ j ∥ L 2 ⁢ p + o ⁢ ( 1 )

≤ ∥ u α j - u 0 ∥ L 2 ⁢ p ⁢ ( ∥ u α j ∥ L 2 ⁢ p p - 2 + ∥ u 0 ∥ L 2 ⁢ p p - 2 ) ⁢ ∥ ϕ j ∥ L 2 ⁢ p + o ⁢ ( 1 )

= o ⁢ ( 1 ) ,

from which we deduce that u α j p - 1 ⁢ ϕ j is also compact in L 2 . We decompose the left-hand side of (6.1) as

∫ ℝ 3 ( I α j * ( u α j p - 1 ⁢ ϕ j ) ) ⁢ u α j p - 1 ⁢ ψ j ⁢ 𝑑 x = ∫ ℝ 3 u α j p - 1 ⁢ ϕ j ⁢ u α j p - 1 ⁢ ψ j ⁢ 𝑑 x + ∫ ℝ 3 ( ( I α j * ( u α j p - 1 ⁢ ϕ j ) ) - u α j p - 1 ⁢ ϕ j ) ⁢ u α j p - 1 ⁢ ψ j ⁢ 𝑑 x

= ∫ ℝ 3 u α j p - 1 ⁢ ϕ j ⁢ u α j p - 1 ⁢ ψ j ⁢ 𝑑 x + ∫ ℝ 3 ( I α j * ( u α j p - 1 ⁢ ϕ j - u 0 p - 1 ⁢ ϕ 0 ) ) ⁢ u α j p - 1 ⁢ ψ j ⁢ 𝑑 x

+ ∫ ℝ N ( I α j * ( u 0 p - 1 ⁢ ϕ 0 ) - u α j p - 1 ⁢ ϕ j ) ⁢ u α j p - 1 ⁢ ψ j ⁢ 𝑑 x

= ∫ ℝ 3 u 0 2 ⁢ p - 2 ⁢ ϕ 0 ⁢ ψ 0 ⁢ 𝑑 x + ∫ ℝ 3 ( I α j * ( u α j p - 1 ⁢ ϕ j - u 0 p - 1 ⁢ ϕ 0 ) ) ⁢ u α j p - 1 ⁢ ψ j ⁢ 𝑑 x + o ⁢ ( 1 ) ,

where we used the Hölder inequality, Proposition 2.6 and the L 2 compactness of both { u α j p - 1 ⁢ ϕ j } and { u α j p - 1 ⁢ ψ j } . We now estimate by using Corollary 2.3 that

∫ ℝ N ( I α j * ( u α j p - 1 ⁢ ϕ j - u 0 p - 1 ⁢ ϕ 0 ) ) ⁢ u α j p - 1 ⁢ ψ j ⁢ 𝑑 x ≤ ∥ I α j * ( u α j p - 1 ⁢ ϕ j - u 0 p - 1 ⁢ ϕ 0 ) ∥ L 2 ⁢ N / ( N - 2 ⁢ α j ) ⁢ ∥ u α j p - 1 ⁢ ψ j ∥ L 2 ⁢ N / ( N + 2 ⁢ α j )

≤ C ⁢ ∥ u α j p - 1 ⁢ ϕ j - u 0 p - 1 ⁢ ϕ 0 ∥ L 2 ⁢ ∥ u α j p - 1 ⁢ ψ j ∥ L 1 2 ⁢ α j / N ⁢ ∥ u α j p - 1 ⁢ ψ j ∥ L 2 1 - 2 ⁢ α j / N

= o ⁢ ( 1 ) .

This proves assertion (6.1). The proof of (6.2) follows exactly the same lines. ∎

Now we are ready to prove the nondegeneracy of ground states u α to (1.1) near 0.

Proposition 6.2.

For given p ∈ [ 2 , 3 ) , let u α be a family of unique positive radial ground states of (1.1). Then for α > 0 sufficiently close to 0 the linearized equation of (1.1) at u α , given by

(6.3) - Δ ⁢ ϕ + ϕ - p ⁢ ( I α * ( u α p - 1 ⁢ ϕ ) ) ⁢ u α p - 1 - ( p - 1 ) ⁢ ( I α * u α p ) ⁢ u α p - 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 the space L 2 ⁢ ( R 3 ) .

Proof.

Differentiating (1.1) with respect to x i , we see that ∂ x i ⁡ u α ∈ L 2 ⁢ ( ℝ 3 ) solves (6.3) for all i = 1 , … , N . Define a finite-dimensional vector space

V α := { ∑ i = 1 3 c i ⁢ ∂ x i ⁡ u α | c i ∈ ℝ } .

Arguing indirectly, we suppose there exists a sequence { α j } converging to 0 such that for each j there exists a nontrivial solution ϕ j ∈ L 2 of (6.3) not belonging to V α j . We may assume that ϕ j is L 2 orthogonal to V α j . We claim that any L 2 solution ϕ of (6.3) automatically belongs to H 1 ⁢ ( ℝ 3 ) . Let us define

L ⁢ [ ϕ ] := p ⁢ ( I α * ( u α p - 1 ⁢ ϕ ) ) ⁢ u α p - 1 + ( p - 1 ) ⁢ ( I α * u α p ) ⁢ u α p - 2 ⁢ ϕ .

By elliptic regularity theory, it is enough to show that L ⁢ [ ϕ ] ∈ H - 1 . It is proved in [13] that u α ∈ L ∞ , and so u α ∈ L q for any 2 ≤ q ≤ ∞ by interpolation. Then Proposition 2.2 and Proposition 2.6 imply that for any ψ ∈ H 1 ,

| L ⁢ [ ϕ ] ⁢ ψ | ≤ | ∫ ℝ 3 p ⁢ ( I α * ( u α p - 1 ⁢ ϕ ) ) ⁢ u α p - 1 ⁢ ψ ⁢ 𝑑 x | + | ∫ ℝ 3 ( p - 1 ) ⁢ ( I α * u α p ) ⁢ u α p - 2 ⁢ ϕ ⁢ ψ ⁢ 𝑑 x |

≤ p ⁢ | ∫ ℝ 3 ( I α * ( u α p - 1 ⁢ ψ ) ) ⁢ u α p - 1 ⁢ ϕ ⁢ 𝑑 x | + ( p - 1 ) ⁢ | ∫ ℝ 3 ( I α * u α p ) ⁢ u α p - 2 ⁢ ϕ ⁢ ψ ⁢ 𝑑 x |

≤ p ⁢ ∥ I α * ( u α p - 1 ⁢ ψ ) ∥ L 2 ⁢ ∥ u α ∥ L ∞ p - 1 ⁢ ∥ ϕ ∥ L 2 + ( p - 1 ) ⁢ ∥ I α * u α p ∥ L 3 ⁢ ∥ u α ∥ L ∞ p - 2 ⁢ ∥ ϕ ∥ L 2 ⁢ ∥ ψ ∥ L 6

≤ C ⁢ ( ∥ u α ∥ H 1 p - 1 ⁢ ∥ u α ∥ L ∞ p - 1 ⁢ ∥ ϕ ∥ L 2 + ∥ u α ∥ L 3 ⁢ p / ( 1 + α ) p ⁢ ∥ u α ∥ L ∞ p - 2 ⁢ ∥ ϕ ∥ L 2 ) ⁢ ∥ ψ ∥ H 1 ,

which shows L ⁢ [ ϕ ] ∈ H - 1 . We normalize ϕ j as ∥ ϕ j ∥ H 1 = 1 . As j → ∞ , it is possible to deduce from Lemma 6.1 that ϕ j weakly converges in H 1 to some ϕ 0 ∈ H 1 which satisfies

- Δ ⁢ ϕ 0 + ϕ 0 - ( 2 ⁢ p - 1 ) ⁢ u 0 2 ⁢ p - 2 ⁢ ϕ 0 = 0 ,

where u 0 is a unique positive radial solution of (1.4). Repeatedly applying Lemma 6.1, we also have

1 = ∥ ϕ j ∥ H 1 2

= p ⁢ ∫ ℝ 3 I α j * ( u α j p - 1 ⁢ ϕ j ) ⁢ u α j p - 1 ⁢ ϕ j ⁢ 𝑑 x + ( p - 1 ) ⁢ ∫ ℝ 3 ( I α j * u α j p ) ⁢ u α j p - 2 ⁢ ϕ j 2 ⁢ 𝑑 x

= ( 2 ⁢ p - 1 ) ⁢ ∫ ℝ 3 u 0 2 ⁢ p - 2 ⁢ ϕ 0 2 ⁢ 𝑑 x + o ⁢ ( 1 ) as ⁢ j → ∞ .

This shows that ϕ 0 is nontrivial. Finally, we note that for all i = 1 , 2 , 3 ,

0 = ∫ ℝ 3 ∂ x i ⁡ u α j ⁢ ϕ j ⁢ d ⁢ x → ∫ ℝ 3 ∂ x i ⁡ u 0 ⁢ ϕ 0 ⁢ d ⁢ x as ⁢ j → ∞ .

This means that ϕ 0 is not a linear combination of { ∂ x i ⁡ u 0 ∣ i = 1 , 2 , 3 } . This contradicts the linearized nondegeneracy of u 0 and completes the proof. ∎

6.2 Nondegeneracy for α near N

Arguing as in the proof of Proposition 6.2, we also obtain the nondegeneracy result of α near N. We need lemmas analogous to Lemma 6.1.

Lemma 6.3.

For given p ∈ ( 2 , 2 ⁢ N N - 2 ) , let v α be a family of unique positive radial ground states of (4.1) and let v 0 be the positive radial ground state to (1.4). Then, for any { α j } → 0 and { ψ j } , { ϕ j } ⊂ H 1 weakly H 1 converging to ϕ 0 and ψ 0 , there holds

(6.4) ∫ ℝ N ( 1 | ⋅ | N - α j * ( v α j p - 1 ⁢ ϕ j ) ) ⁢ v α j p - 1 ⁢ ψ j ⁢ 𝑑 x → ( ∫ ℝ N v 0 p - 1 ⁢ ϕ 0 ⁢ 𝑑 x ) ⁢ ( ∫ ℝ N v 0 p - 1 ⁢ ψ 0 ⁢ 𝑑 x ) ,

(6.5) ∫ ℝ N ( 1 | ⋅ | N - α j * u α j p ) ⁢ u α j p - 2 ⁢ ϕ j ⁢ ψ j ⁢ 𝑑 x → ( ∫ ℝ N u 0 p ⁢ 𝑑 x ) ⁢ ( ∫ ℝ N u 0 p - 2 ⁢ ϕ 0 ⁢ ψ 0 ⁢ 𝑑 x ) .

Proof.

From the same argument as in Proposition 6.1 one can see that v α j p - 1 ⁢ ϕ j is compact in L 1 ∩ L 2 * / p , where 2 * = 2 ⁢ N N - 2 . By following the same argument in the proof of Proposition 2.6 (ii), one is able to see

(6.6) ∥ 1 | ⋅ | N - α * f ∥ L ∞ ≤ C ⁢ ∥ f ∥ L 1 ∩ L 2 * / p ,

where f ∈ L 1 ∩ L 2 * / p , α is near N and C is independent of α near N. Using Proposition 2.6 (ii), we then compute

∫ ℝ N ( 1 | ⋅ | N - α j * ( v α j p - 1 ⁢ ϕ j ) ) ⁢ v α j p - 1 ⁢ ψ j ⁢ 𝑑 x

= ∫ ℝ N ( 1 | ⋅ | N - α j * ( v α j p - 1 ⁢ ϕ j - v 0 p - 1 ⁢ ϕ 0 ) ) ⁢ v α j p - 1 ⁢ ψ j ⁢ 𝑑 x ⏟ ( A ) + ∫ ℝ N ( 1 | ⋅ | N - α j * ( v 0 p - 1 ⁢ ϕ 0 ) ) ⁢ v α j p - 1 ⁢ ψ j ⁢ 𝑑 x

= ( A ) + ( ∫ ℝ N v 0 p - 1 ⁢ ϕ 0 ⁢ 𝑑 x ) ⁢ ( ∫ ℝ N v 0 p - 1 ⁢ ψ 0 ⁢ 𝑑 x ) + o ⁢ ( 1 ) ,

and estimate (6.6) implies

| ( A ) | ≤ C ⁢ ∥ v α j p - 1 ⁢ ϕ j - v 0 p - 1 ⁢ ϕ 0 ∥ L 1 ∩ L 2 * / p ⁢ ∥ v α j p - 1 ⁢ ψ j ∥ L 1 = o ⁢ ( 1 ) ,

which proves assertion (6.4).

To prove (6.5), we claim that v 0 p - 2 ⁢ ϕ j ⁢ ψ j is L 1 compact. Observe that

∫ ℝ N ∖ B R | v 0 p - 2 ⁢ ϕ j ⁢ ψ j | ⁢ 𝑑 x ≤ ∥ v 0 ∥ L p ⁢ ( ℝ N ∖ B R ) p - 2 ⁢ ∥ ϕ j ∥ L p ⁢ ( ℝ N ) ⁢ ∥ ψ j ∥ L p ⁢ ( ℝ N ) ,

so that v 0 p - 2 ⁢ ϕ j ⁢ ψ j is tight. Also we note that ∥ v 0 ∥ C 1 ⁢ ( ℝ N ) is finite by the elliptic regularity theory, and for every B R there exists C R such that ∥ v 0 p - 3 ∥ B R ≤ C R because v 0 is continuous and positive everywhere. Then we can see that v 0 p - 2 ⁢ ϕ j ⁢ ψ j ∈ W 1 , 1 ⁢ ( B R ) from the estimate

∥ v 0 p - 2 ⁢ ϕ j ⁢ ψ j ∥ W 1 , 1 ⁢ ( B R )

≤ ∥ v 0 p - 2 ⁢ ϕ j ⁢ ψ j ∥ L 1 ⁢ ( B R ) + ∥ v 0 p - 3 ⁢ ∇ ⁡ v 0 ⁢ ϕ j ⁢ ψ j ∥ L 1 ⁢ ( B R ) + ∥ v 0 p - 2 ⁢ ∇ ⁡ ϕ j ⁢ ψ j ∥ L 1 ⁢ ( B R ) + ∥ v 0 p - 2 ⁢ ϕ j ⁢ ∇ ⁡ ψ j ∥ L 1 ⁢ ( B R )

≤ ∥ v 0 ∥ L p p - 2 ⁢ ∥ ϕ j ∥ L p ⁢ ∥ ψ j ∥ L p + C R ⁢ ∥ ∇ ⁡ v 0 ∥ L ∞ ⁢ ∥ ϕ j ∥ L 2 ⁢ ∥ ψ j ∥ L 2 + ∥ v 0 ∥ L ∞ p - 2 ⁢ ∥ ∇ ⁡ ϕ j ∥ L 2 ⁢ ∥ ψ j ∥ L 2 + ∥ v 0 ∥ L ∞ p - 2 ⁢ ∥ ϕ j ∥ L 2 ⁢ ∥ ∇ ⁡ ψ j ∥ L 2

< ∞ ,

so that v 0 p - 2 ⁢ ϕ j ⁢ ψ j is locally L 1 compact by the compact Sobolev embedding. By combining this with the tightness, we conclude that v 0 p - 2 ⁢ ϕ j ⁢ ψ j is L 1 compact, and consequently v α j p - 2 ⁢ ϕ j ⁢ ψ j is L 1 compact. Now, the remaining part of the proof follows the same lines as the previous one. ∎

Proposition 6.4.

Let p ∈ ( 2 , 2 ⁢ N N - 2 ) and let v α be a family of unique positive radial ground states of (4.1). Then for α < N sufficiently close to N the linearized equation of (4.1) at v α , given by

(6.7) - Δ ⁢ ϕ + ϕ - p ⁢ ( 1 | ⋅ | N - α * ( v α p - 1 ⁢ ϕ ) ) ⁢ v α p - 1 - ( p - 1 ) ⁢ ( 1 | ⋅ | N - α * v α p ) ⁢ v α p - 2 ⁢ ϕ = 0 in ⁢ ℝ N ,

only admits solutions of the form

ϕ = ∑ i = 1 N c i ⁢ ∂ x i ⁡ v α , c i ∈ ℝ ,

in the space L 2 ⁢ ( R N ) .

Proof.

The proof of Proposition 6.4 follows the same lines as the proof of Proposition 6.2. The only thing we need to show is that L ⁢ [ ϕ ] ∈ H - 1 when ϕ is a L 2 solution to (6.7) and

L ⁢ [ ϕ ] := p ⁢ ( 1 | ⋅ | N - α * ( v α p - 1 ⁢ ϕ ) ) ⁢ v α p - 1 + ( p - 1 ) ⁢ ( 1 | ⋅ | N - α * v α p ) ⁢ v α p - 2 ⁢ ϕ .

For ψ ∈ H 1 ⁢ ( ℝ N ) , we compute from Proposition 2.6 that

L ⁢ [ ϕ ] ⁢ ψ = ∫ ℝ N p ⁢ ( 1 | ⋅ | N - α * ( v α p - 1 ⁢ ψ ) ) ⁢ v α p - 1 ⁢ ϕ + ( p - 1 ) ⁢ ( 1 | ⋅ | N - α * v α p ) ⁢ v α p - 2 ⁢ ϕ ⁢ ψ ⁢ d ⁢ x

≤ p ⁢ C ⁢ ∥ v α ∥ H 1 p - 1 ⁢ ∥ ψ ∥ H 1 ⁢ ∥ v α ∥ L 2 ⁢ ( p - 1 ) p - 1 ⁢ ∥ ϕ ∥ L 2 + ( p - 1 ) ⁢ C ⁢ ∥ v α ∥ H 1 p ⁢ ∥ v α ∥ L ∞ p - 2 ⁢ ∥ ϕ ∥ L 2 ⁢ ∥ ψ ∥ L 2 ,

from which we deduce L ⁢ [ ϕ ] ∈ H - 1 . ∎

Now, we shall end the proof of Theorem 1.5. For 2 < p < 2 ⁢ N N - 2 and α < N close to N, let { u α } be a family of unique positive radial ground states of (1.1) and let ϕ α ∈ L 2 ⁢ ( ℝ N ) be a solution of the linearized equation (6.3). Then ϕ α is a solution of (6.7) with v α = s ⁢ ( N , α , p ) ⁢ u α . Then Proposition 6.4 says that ϕ α is a linear combination of the ∂ x i ⁡ v α , which is also a linear combination of the ∂ x i ⁡ u α . This completes the proof of Theorem 1.5.

Funding source: Kyonggi University

Award Identifier / Grant number: Kyonggi University research grant 2016

Funding statement: This work was supported by Kyonggi University Research Grant 2016.

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Received: 2017-08-08

Revised: 2018-01-25

Accepted: 2018-02-03

Published Online: 2018-05-31

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

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https://doi.org/10.1515/anona-2017-0182

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Semilinear elliptic; Choquard; limit profile; uniqueness

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