Segregated solutions for nonlinear Schrödinger systems with a large number of components

Haixia Chen; Angela Pistoia

doi:10.1515/ans-2022-0076

Article Open Access

Segregated solutions for nonlinear Schrödinger systems with a large number of components

Haixia Chen and Angela Pistoia

Published/Copyright: March 12, 2024

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From the journal Advanced Nonlinear Studies Volume 24 Issue 2

Abstract

In this paper we are concerned with the existence of segregated non-radial solutions for nonlinear Schrödinger systems with a large number of components in a weak fully attractive or repulsive regime in presence of a suitable external radial potential.

Keywords: Schrödinger systems; segregated solutions; large number of components

2010 Mathematics Subject Classification: 35B44; 35J47 (primary); 35B33 (secondary)

1 Introduction

The well-known Gross–Pitaevskii system

(1.1) − ι ∂ t ϕ i = Δ ϕ i − V i ( x ) ϕ i + μ i | ϕ i | 2 ϕ i + ∑ j = 1 j ≠ i m β i j | ϕ j | 2 ϕ i , i = 1 , … , m

has been proposed as a mathematical model for multispecies Bose–Einstein condensation in m different states. We refer to Refs. [1], [2], [3], [4], [5] for a detailed physical motivation. Here the complex valued functions ϕ _i’s are the wave functions of the ith condensate, |ϕ _i| is the amplitude of the ith density, μ _i describes the interaction between particles of the same component and β _ij, i ≠ j, describes the interaction between particles of different components, which can be attractive if β _ij > 0 or repulsive if β _ij < 0. To obtain solitary wave solutions of the Gross–Pitaevskii system (1.1) one sets ϕ i ( t , x ) = e ι λ i t u i ( x ) and the real functions u _i’s solve the system

(1.2) − Δ u i + λ i u i + V i ( x ) u i = μ i u i 3 + u i ∑ j = 1 j ≠ i m β i j u j 2 in R n , i = 1 , … , m

where μ _i > 0, λ _i > 0, β i j = β j i ∈ R , V i ∈ C 0 ( R n ) and n ≥ 2.

We are interested in finding solutions whose components u _i do not identically vanish for any index i = 1, …, m.

In the last decades, the nonlinear Schrödinger system (1.2) has been widely studied. Most of the work has been done in the autonomous case (i.e. V _i is constant) and in the sub-critical regime (i.e. n = 2 or n = 3). We refer the reader to a couple of recent papers [6], [7] where the authors provide an exhaustive list of references. There are a few results concerning the non-autonomous case, which have been recently obtained by Peng and Wang [8], Pistoia and Vaira [9] and Li, Wei and Wu [10].

In the critical regime (i.e. n = 4) the existence of solutions to (1.2) is a delicate and much more difficult issue. While there are some results concerning the autonomous case obtained by Clapp and Pistoia [11], Clapp and Szulkin [12] and Chen, Medina and Pistoia [13], a very few is known about the non-autonomous one. As far as we know the first result is due to Chen, Pistoia and Vaira [14], where system (1.2) is studied in a fully symmetric regime, namely all the coupling parameters β _ij’s are equal to a real number β and all the potentials V _i’s coincide with a positive and radially symmetric function V, namely the system (1.2) reduces to the system

(1.3) − Δ u i + V ( x ) u i = u i 3 + β ∑ j ≠ i u i u j 2 in R 4 , i = 1 , … , m .

In particular, using this symmetric setting, if

(1.4) R i ≔ cos 2 ( i − 1 ) π m sin 2 ( i − 1 ) π m 0 0 − sin 2 ( i − 1 ) π m cos 2 ( i − 1 ) π m 0 0 0 0 1 0 0 0 0 1 for any i = 1 , … , m ,

the authors build a solution u i ( x ) = u ( R i x ) to the system (1.3) via a solution u to the non-local equation

(1.5) − Δ u + V ( x ) u = u 3 + β u ∑ i = 2 m u 2 ( R i x ) in R 4 .

Then, using a Ljapunov–Schmidt procedure and adapting the argument developed by Peng, Wang and Yan [15], they build a solution to (1.5) using the bubbles

(1.6) U δ , ξ ( x ) = 1 δ U x − ξ δ with U ( x ) = c 1 + | x | 2 , c = 2 2 ,

which are all the positive solutions to the critical equation

− Δ u = u 3 in R 4 .

The solution to (1.5) they build looks like the sum of a large number k of bubbles, i.e. ∑ ℓ = 1 k U δ , ξ ℓ , whose peaks ξ _ℓ are located at the vertices of a regular polygon placed in the sphere S ( r k ) ≔ ( x 1 , x 2 , 0,0 ) | x 1 2 + x 2 2 = r k 2 . Moreover, the radius r _k as k → ∞ approaches a non-degenerate critical point r ₀ of the function r → r ² V(r). As a result, the solution to the system (1.3) are of segregated type in the sense that different components blow-up at different points as k is large enough. Indeed, the shape of the function u _i resembles k copies of bubbles whose k peaks are the points ξ ℓ i = ( R i ) − 1 ξ ℓ , ℓ = 1 , … , k (see (1.4)).

The segregation phenomena has been widely studied by Terracini and her collaborators in a series of papers (see for example [16], [17], [18], [19], [20] and references therein) and naturally appears in the strongly repulsive case (i.e. β _ij → −∞). In [14] the existence of this kind of solutions is not affected at all by the presence of the coupling parameter β: they do exist in both fully repulsive (i.e. β < 0) and attractive regime (i.e. β > 0). This is due to the fact that the regime of the system which is entirely encrypted in the non-local term of (1.5) does not appear in the main terms of the reduced problem. Therefore, a natural question arises.

(Q): do there exist any functions V (possibly changing-sign) such that existence of solutions to the system (1.3) is affected by the sign of the parameter β?

In this paper, we give a positive answer.

We assume that V is radially symmetric, V ∈ C 2 ( R 4 ) ∩ L 2 ( R 4 ) , either V ≥ 0 or ‖ V ‖ L 2 ( R 4 ) ≤ 4 3 π 2 (see Remark 1.4), and

(1.7) r 0 > 0 is a non − degenerate critical point of r → r 2 V ( r ) and V ( r 0 ) ≠ 0 .

Moreover we will assume that the coupling parameter β = β(m) → 0 as m → +∞ so that

(1.8) B ≔ lim m → + ∞ m α β ≠ 0 for some α ∈ ( 1,2 ) .

Our main result reads as follows.

Theorem 1.1.

Assume (1.7) and (1.8) with B ⋅ V ( r 0 ) > 0 . There exists m ₀ > 0 such that for any m ≥ m ₀ the problem (1.5) has a solution u ∈ D 1,2 ( R 4 ) such that as m → +∞,

u ( x ) ∼ U δ , ζ , with ζ = ( ρ , 0,0,0 ) , ρ ∼ r 0 and δ ∼ r 0 2 c V ( r 0 ) B m α − 4 2

for some positive constant c (see (2.31)).

According to the previous discussion, as an immediate consequence of Theorem 1.1 we get the following existence result of solutions for the system (1.3).

Theorem 1.2.

Assume (1.7) and (1.8) with B ⋅ V ( r 0 ) > 0 . There exists m ₀ > 0 such that for any m ≥ m ₀ the system (1.3) has a solution u i ( x ) = u ( R i x ) , i = 1, …, m where u is given in Theorem 1.1 and R i is defined in (1.4).

Let us make some comments.

Remark 1.3.

If we want to find solutions depending on the sign of the parameter β, the coupling term β u ∑ i = 2 m u 2 ( R i x ) must be present in the reduced problem (2.15). To achieve this goal, we introduce the number m of components as a large parameter. Unfortunately, the choice of a large number of interaction has to be balanced by a suitable small parameter β which depends on m. It would be extremely interesting to understand if this is a purely technical issue or not.

Remark 1.4.

In order to build the solutions in the competitive regime, i.e. β < 0, the function V has to be negative somewhere. In this case, we require that the linear operator − Δ + V I d is coercive and this is true if

‖ V ‖ L 2 ( R 4 ) < min u ∈ D 1,2 ( R 4 ) \ { 0 } ∫ R 4 | ∇ u | 2 ∫ R 4 | u | 4 1 2 = 4 3 π 2

as a direct computation shows (the minimum is achieved at the bubbles (1.6)).

Remark 1.5.

The system (1.2) in low dimensions, i.e. n = 2 or n = 3, has been recently studied by Pistoia and Vaira in [9] in the same fully symmetric regime (see also [10]). They are lead to find solutions to the non-local subcritical Equation (1.3) defined in R 2 or R 3 (instead of R 4 ). The existence of solutions in the competitive (i.e. β < 0) or cooperative regime (i.e. β > 0) strongly depends on the fact that the radial potential V has a maximum or a minimum at infinity, respectively. We believe that the idea of using the number of components as a large parameter could also be applied in this setting to produce a new kind of solutions.

Notations. In this paper we agree that f ≲ g or f = O ( g ) means |f| ≤ C|g| for some positive constant C independent of m and f ∼ g means f = g + o(g).

2 Proof of Theorem 1.1

2.1 The ansatz

We will find solutions of (1.5) in the space

X ≔ { u ∈ D 1,2 ( R 4 ) : u satisfies ( 2.1 ) } with ‖ u ‖ ≔ ∫ R 4 | ∇ u | 2 d x 1 2

where

(2.1) u ( x 1 , x 2 , x 3 , x 4 ) = u ( x 1 , − x 2 , x 3 , x 4 ) = u ( x 1 , x 2 , − x 3 , x 4 ) = u ( x 1 , x 2 , x 3 , − x 4 ) .

In particular, we are going to build a solution to (1.5) as

u = χ U δ , ρ ξ 1 + φ ,

where the bubbles U _δ,ξ is defined in (1.6), χ(x) = χ(|x|) is a radial cut-off function whose support is close to the sphere |x|≔r = r ₀ (r ₀ is given in (1.7)), namely

χ = 1 in | r − r 0 | ≤ σ and χ = 0 in | r − r 0 | > 2 σ for some σ > 0 small

and φ is an higher order term. Moreover, the blow-up point is

ρ ξ 1 ≔ ρ ( 1,0,0,0 ) , | ρ − r 0 | ≤ ϑ

for some small ϑ > 0 and the blow-up rate satisfies (see (1.8))

(2.2) δ = d | B | 1 2 m α − 4 2 with α ∈ ( 1,2 ) and d > 0 .

As it is usual, the precise choice of the concentration parameter will be done in the last step of the proof (see Section 2.6). In particular, by (2.28) we will deduce that δ 2 β m 4 = O ( 1 ) and this will force the choice of δ in terms of β and m or equivalently the choice made in (2.2) because of the assumption (1.8) on the decay of β. Moreover, we also need to point out that the reduction argument works if both β and δ are of order o(m ⁻¹) as m → +∞ and that is why we have to choose α ∈ (1, 2).

Finally, let us point out that the function u _i defined by u i ( x ) = u ( R i x ) (see (1.4)) blows up at a single point

ρ ξ i ≔ ρ R i − 1 ξ 1 = ρ cos 2 ( i − 1 ) π m , sin 2 ( i − 1 ) π m , 0,0 .

2.2 Rewriting the non-local equation via the Lyapunov Schmidt reduction method

Subtituting u = χ U δ , ρ ξ 1 + φ into the non-local problem (1.5), it can be rewritten as

(2.3) L ( φ ) = E + N ( φ ) in R 4

where L ( φ ) , E , N ( φ ) are defined as

(2.4) L ( φ ) ≔ − Δ φ + V ( x ) φ − 3 ( χ U δ , ρ ξ 1 ) 2 φ − β φ ∑ i = 2 m ( χ U δ , ρ ξ i ) 2 ,

(2.5) E ≔ ( χ U δ , ρ ξ 1 ) 3 + Δ ( χ U δ , ρ ξ 1 ) − V ( x ) χ U δ , ρ ξ 1 + β χ 3 U δ , ρ ξ 1 ∑ i = 2 m U δ , ρ ξ i 2 ,

(2.6) N ( φ ) ≔ φ 3 + 3 χ U δ , ρ ξ 1 φ 2 + β φ ∑ i = 2 m φ 2 ( R i x ) + 2 β χ φ ∑ i = 2 m U δ , ρ ξ i φ ( R i x ) + β χ U δ , ρ ξ 1 ∑ i = 2 m φ 2 ( R i x ) + 2 β χ 2 U δ , ρ ξ 1 ∑ i = 2 m U δ , ρ ξ i φ ( R i x ) .

As it is usual, to solve (2.3) we will follow the classical steps of the Ljapunov–Schmidt procedure:

we show there exists φ ∈ X solution to the problem
(2.7) L ( φ ) = E + N ( φ ) + ∑ l = 0 1 c l ( δ , ρ ) ( χ U δ , ρ ξ 1 ) 2 Z δ , ρ ξ 1 l in R 4 , ∫ R 4 ( χ U δ , ρ ξ 1 ) 2 Z δ , ρ ξ 1 l φ d x = 0 , l = 0,1
where
Z δ , ρ ξ 1 0 = ∂ ( χ U δ , ρ ξ 1 ) ∂ δ and Z δ , ρ ξ 1 1 = ∂ ( χ U δ , ρ ξ 1 ) ∂ ρ .
we find δ > 0 and ρ close to r ₀ such that c 0 ( δ , ρ ) = c 1 ( δ , ρ ) = 0 .

At the beginning, we put forward two lemmas which are used frequently throughout the paper. The proofs can be founded by the similar arguments as Appendix A in [21] and Appendix A in [22] respectively.

Lemma 2.1.

For any 0 < τ ≤ min{τ ₁, τ ₂}, i ≠ j, it holds true that

1 ( 1 + | y − x i | ) τ 1 1 ( 1 + | y − x j | ) τ 2 ≲ 1 | x i − x j | τ 1 ( 1 + | y − x i | ) τ 1 + τ 2 − τ + 1 ( 1 + | y − x j | ) τ 1 + τ 2 − τ for any y ∈ R 4 .

Lemma 2.2.

There exists a positive constant C _τ such that

(2.8) ∑ i = 2 m 1 | ξ 1 − ξ i | τ ∼ C τ m τ , for any τ > 1

for some positive constant C _τ.

In the following proof, we will use the Jensen’s inequality

1 m ∑ i = 1 m t i p 1 p ≤ 1 m ∑ i = 1 m t i q 1 q for any t i ≥ 0 , 0 < p < q .

When p = 1 and q > 1, it becomes

(2.9) ∑ i = 1 m t i q ≤ m q − 1 ∑ i = 1 m t i q for any t i ≥ 0 .

Let us denote η _i = ξ _i/δ in the following.

2.3 The size of the error term E

Proposition 2.3.

Let E be defined as in (2.5), then

‖ E ‖ L 4 3 ( R 4 ) ≲ δ + | β | δ 2 m 9 4 ≲ δ ( because of ( 2.2 ) ) .

Proof.

Notice that

(2.10) E = ( χ U δ , ρ ξ 1 ) 3 + Δ ( χ U δ , ρ ξ 1 ) ︸ ≔ E 1 − V ( x ) χ U δ , ρ ξ 1 ︸ ≔ E 2 + β χ 3 U δ , ρ ξ 1 ∑ i = 2 m U δ , ρ ξ i 2 ︸ ≔ E 3 .

On one side, it gives from direct computations

∫ R 4 E 1 4 3 d x ≤ ∫ R 4 χ U δ , ρ ξ i 3 − χ U δ , ρ ξ i 3 + 2 ∇ χ ∇ U δ , ρ ξ 1 + Δ χ U δ , ρ ξ 1 4 3 ≲ δ 4 3 ,

∫ R 4 E 2 4 3 d x ≲ ∫ | r − r 0 | ≤ σ U δ , ρ ξ i 4 3 d x ≲ δ 4 3 .

And then using (2.9) and Lemma 2.1, one can get

(2.11) ∫ R 4 E 3 4 3 d x ≲ | β | 4 3 ∫ | r − r 0 | ≤ σ ∑ i = 2 m δ δ 2 + | x − ρ ξ 1 | 2 δ 2 δ 2 + | x − ρ ξ i | 2 2 4 3 d x ≲ | β | 4 3 ∫ | r − r 0 | ≤ σ δ m 1 3 ∑ i = 2 m 1 ρ 8 3 | η 1 − η i | 8 3 1 ( 1 + | y − ρ η 1 | ) 16 3 + 1 ( 1 + | y − ρ η i | ) 16 3 d y ≲ | β | 4 3 δ 8 3 m 3 .

Therefore, it follows ‖ E ‖ L 4 3 ( R 4 ) ≲ δ + | β | δ 2 m 9 4 . □

2.4 The linear theory

Proposition 2.4.

Let L ( φ ) be defined as in (2.4). For any 0 < d ₁ < d ₂ < +∞ and ϑ > 0 small enough, there exists m ₀ > 0 such that for any m ≥ m ₀, d ∈ [d ₁, d ₂], ρ ∈ (r ₀ − ϑ, r ₀ + ϑ) and h ∈ L 4 3 ( R 4 ) satisfying (2.1), the linear problem

(2.12) L ( φ ) = h + ∑ l = 0 1 c l ( χ U δ , ρ ξ 1 ) 2 Z δ , ρ ξ 1 l , in R 4 , ∫ R 4 ( χ U δ , ρ ξ 1 ) 2 Z δ , ρ ξ 1 l φ d x = 0 , l = 0,1

admits a unique solution φ ∈ X and c 0 , c 1 satisfying

(2.13) ‖ φ ‖ ≲ ‖ h ‖ L 4 3 ( R 4 ) and c 0 , c 1 = O δ ‖ h ‖ L 4 3 ( R 4 ) .

Proof.

Step 1: Assume first that (2.13) holds true. Define

X ̃ ≔ φ ∈ X , ∫ R 4 ( χ U δ , ρ ξ 1 ) 2 Z δ , ρ ξ 1 l φ d x = 0 , l = 0,1 .

The first equation in (2.12) can be rewritten as

φ + ( − Δ ) − 1 V ( x ) φ − 3 ( χ U δ , ρ ξ 1 ) 2 φ − β φ ∑ i = 2 m ( χ U δ , ρ ξ i ) 2 ︸ ≔ K φ = ( − Δ ) − 1 h + ∑ l = 0 1 c l ( χ U δ , ρ ξ 1 ) 2 Z δ , ρ ξ 1 l ︸ ≔ f , φ ∈ X ̃

where there exist c 0 , c 1 such that K : X ̃ → X ̃ , f ∈ X ̃ and K is a compact operator for each fixed m since U δ , ρ ξ 1 2 , ∑ i = 2 m U δ , ρ ξ i 2 ∈ L 2 ( R 4 ) , U δ , ρ ξ i = O ( 1 | x | 2 ) if |x| → +∞, V ∈ C 2 ( R 4 ) and ‖ V ‖ L 2 ( R 4 ) is bounded if V is non-negative otherwise ‖ V ‖ L 2 ( R 4 ) ≤ 4 3 π 2 (see Lemma 2.3 in [23]). Thus there is a unique φ ∈ X ̃ such that φ + K φ = f with f ∈ X ̃ by Fredholm-alternative theorem.

Now, what is left is to show that (2.13) holds ture. Suppose that there exist m _n → +∞, φ _n satisfying (2.12) with h = h _n such that ‖ φ n ‖ = 1 , ‖ h n ‖ L 4 3 ( R 4 ) → 0 as n → +∞.

Letting φ ̃ n ( x ) = δ n φ n ( δ n x + ρ n ξ 1 ) , up to a subsequence, we abtain

φ ̃ n → φ * weakly in D 1,2 ( R 4 ) , strongly in L loc p ( R 4 ) for p ∈ [ 2,4 ) .

Step 2: Let us prove c l = O ( δ n ‖ φ n ‖ ) + O ( δ n ‖ h n ‖ L 4 3 ( R 4 ) ) , l = 0,1 . For simplicity, we drop the subscript n in this step. Multiplying the first equation in (2.12) by Z δ , ρ ξ 1 j , j = 0,1 , one can get

∑ l = 0 1 c l ∫ R 4 ( χ U δ , ρ ξ i ) 2 Z δ , ρ ξ 1 l Z δ , ρ ξ 1 j d x = ∫ R 4 − Δ φ + V ( x ) φ − 3 ( χ U δ , ρ ξ 1 ) 2 φ − β φ ∑ i = 2 m ( χ U δ , ρ ξ i ) 2 − h Z δ , ρ ξ 1 j d x .

Using straightforward computations and notice that

∂ U δ , ρ ξ i ∂ δ L 4 3 ( R 4 ) ≤ C and ∂ U δ , ρ ξ i ∂ δ L 4 ( R 4 ) ≲ 1 δ ,

we have for j = 0 (j = 1 is similar)

∫ R 4 ( − Δ φ − 3 ( χ U δ , ρ ξ 1 ) 2 φ ) Z δ , ρ ξ 1 0 d x ≲ ∫ R 4 3 χ U δ , ρ ξ 1 2 ∂ U δ , ρ ξ 1 ∂ δ − 3 χ 3 U δ , ρ ξ 1 2 ∂ U δ , ρ ξ 1 ∂ δ φ d x + ∫ R 4 ( − Δ χ ) ∂ U δ , ρ ξ 1 ∂ δ φ d x − ∫ R 4 2 ∇ χ ∇ ∂ U δ , ρ ξ 1 ∂ δ φ d x ≲ ‖ φ ‖ ,

∫ R 4 ( V ( x ) φ − h ) Z δ , ρ ξ 1 j d x ≲ ‖ φ ‖ ‖ Z δ , ρ ξ 1 j ‖ L 4 3 ( R 4 ) + ‖ h ‖ L 4 3 ( R 4 ) ‖ Z δ , ρ ξ 1 j ‖ L 4 ( R 4 ) ≲ ‖ φ ‖ + ‖ h ‖ L 4 3 ( R 4 ) δ .

Similar to (2.11), one can deduce

∫ R 4 β φ ∑ i = 2 m ( χ U δ , ρ ξ i ) 2 Z δ , ρ ξ 1 j d x ≲ | β | ‖ φ ‖ ∑ i = 2 m ∫ R 4 χ 2 U δ , ρ ξ i 2 Z δ , ρ ξ 1 j 4 3 d x 3 4 ≲ | β | ‖ φ ‖ ∑ i = 2 m 1 δ ∫ | r − r 0 | ≤ σ U δ , ρ ξ i 2 U δ , ρ ξ 1 4 3 d x 3 4 ≲ | β | ‖ φ ‖ δ m 2

due to | Z δ , ρ ξ 1 j | ≲ U δ , ρ ξ 1 δ . In addition, it is easy to check that there exist constants B _l such that

∫ R 4 ( χ U δ , ρ ξ 1 ) 2 Z δ , ρ ξ 1 l Z δ , ρ ξ 1 j d x = B l δ − 2 ( 1 + o ( 1 ) ) , l = j , 0 , l ≠ j .

So we have c l = O ( ( δ 2 + | β | δ 3 m 2 ) ‖ φ ‖ ) + O ( δ ‖ h ‖ L 4 3 ( R 4 ) ) = O ( δ ( ‖ φ ‖ + ‖ h ‖ L 4 3 ( R 4 ) ) ) , l = 0,1 , which ends this step.

Thus φ* satisfies

− Δ φ * = 3 U 2 φ * , in R 4 .

Step 3: Let us prove φ* = 0. Denote Z i = ∂ U ∂ x i , i = 1 , … , 4 . Recalling the second equation in (2.12), it suffices to show that

I i n ≔ ∫ R 4 φ ̃ n U 2 Z i d x = 0 , i = 1 , … , 4 ,

Since φ _n satisfies (2.1) and ξ ₁ = (1, 0, 0, 0), it follows I _2n = I _3n = I _4n = 0.

Thanks to ∫ R 4 ( χ U δ n , ρ n ξ 1 ) 2 Z δ n , ρ n ξ 1 1 φ n d x = 0 , it derives I _in = 0, i = 1, …, 4.

Step 4: Let us prove ∫ R 4 | ∇ φ n | 2 d x → 0 as n → +∞.

Firstly, since ‖ V ‖ L 2 ( R 4 ) is bounded if V is non-negative, otherwise ‖ V ‖ L 2 ( R 4 ) ≤ 4 3 π 2 , we have ‖ φ ‖ V ≔ ( ∫ R 4 | ∇ φ | 2 + V ( x ) φ 2 d x ) 1 2 ≲ ‖ φ ‖ ≲ ‖ φ ‖ V .

Testing the first equation in (2.12) by φ _n, one deduces from the second equation in (2.12),

∫ R 4 | ∇ φ n | 2 + V ( x ) φ n 2 d x = ∫ R 4 3 φ n 2 ( χ U δ n , ρ n ξ 1 ) 2 d x + ∫ R 4 β ∑ i = 2 m ( χ U δ n , ρ n ξ i n ) 2 φ n 2 d x + ∫ R 4 h n φ n d x .

Since φ ̃ n → 0 weakly in L 4 ( R 4 ) , it follows

∫ R 4 3 φ n 2 ( χ U δ n , ρ n ξ 1 ) 2 d x ∼ 3 ∫ R 4 φ ̃ n 2 U 2 d x → 0 .

Reminding that β = o(m ⁻¹), one can get

β ∫ R 4 ∑ i = 2 m ( χ U δ n , ρ n ξ i n ) 2 φ n 2 d x ≤ | β | ⋅ m ‖ U δ n , ρ n ξ i n ‖ L 4 ( R 4 ) 2 ‖ φ n ‖ 2 → 0 ,

and according to ‖ h n ‖ L 4 3 ( R 4 ) → 0 , ‖ φ n ‖ = 1 , it yields

∫ R 4 h n φ n d x → 0 ,

i.e. ∫ R 4 | ∇ φ n | 2 + V ( x ) φ n 2 d x → 0 and then ‖φ _n‖ → 0 as n → +∞, which contradicts with ‖φ _n‖ = 1. □

We are going to solve the non-linear problem (2.7).

Proposition 2.5.

For any 0 < d ₁ < d ₂ < +∞ and ϑ > 0 small enough, there exists m ₀ > 0 such that for any m ≥ m ₀, d ∈ [d ₁, d ₂], ρ ∈ (r ₀ − ϑ, r ₀ + ϑ) and h ∈ L 4 3 ( R 4 ) satisfying (2.1), the problem (2.7) has a unique solution φ ∈ X ̃ satisfying

(2.14) ‖ φ ‖ ≲ δ .

Moreover, the c 0 , c 1 = o ( 1 ) .

Proof.

The proof relies on a standard contraction mapping argument combined with Propositions 2.4 and 2.3. □

2.5 The reduced problem

Let L ( φ ) , E , N ( φ ) be defined as in (2.4)–(2.6)

Proposition 2.6.

It holds true that:

(2.15) ∫ R 4 ( L ( φ ) − E − N ( φ ) ) Z δ , ρ ξ 1 0 d x = − 8 V ( ρ ) δ ( ln ⁡ δ ) + b β δ 3 ( ln ⁡ δ m ) m 4 ρ 4 + Θ 0

where b is a positive constant (see (2.20)) and

Θ 0 = O δ + | β | m ‖ φ ‖ 2 δ + ‖ φ ‖ 2 δ + | β | ‖ φ ‖ 2 δ | ln ⁡ δ | 1 2 m 2 + | β | ‖ φ ‖ δ m 2 + ‖ φ ‖

uniformly with respect to d in compact sets of (0, ∞) and ρ in (r ₀ − ϑ, r ₀ + ϑ).

Proof.

Notice that

(2.16) ∫ R 4 ( L ( φ ) − E − N ( φ ) ) Z δ , ρ ξ 1 0 d x = − ∫ R 4 E 1 Z δ , ρ ξ 1 0 d x ︸ ≔ Q 1 + ∫ R 4 E 2 Z δ , ρ ξ 1 0 d x ︸ ≔ Q 2 − ∫ R 4 E 3 Z δ , ρ ξ 1 0 d x ︸ ≔ Q 3 − ∫ R 4 N ( φ ) Z δ , ρ ξ 1 0 d x ︸ ≔ Q 4 + ∫ R 4 L ( φ ) Z δ , ρ ξ 1 0 d x ︸ ≔ Q 5

where E 1 , E 2 , E 3 are defined in (2.10).

Let us estimate Q 1 . It is immediate to get

(2.17) Q 1 = ∫ R 4 χ U δ , ρ ξ i 3 − χ U δ , ρ ξ i 3 + 2 ∇ χ ∇ U δ , ρ ξ 1 + Δ χ U δ , ρ ξ 1 Z δ , ρ ξ 1 0 d x ≲ δ .

Let us estimate Q 2 . We obtain

(2.18) Q 2 = V ( ρ ) ∫ | r − r 0 | ≤ σ U δ , ρ ξ 1 ∂ U δ , ρ ξ 1 ∂ δ d x + O ( δ ) = V ( ρ ) δ ∫ B 0 , σ δ − U ( ⟨ y , ∇ U ⟩ + U ) d y + O ( δ ) = V ( ρ ) δ ∫ B 0 , σ δ U 2 d y + O ( δ ) ∼ − 8 V ( ρ ) δ ⁡ ln ⁡ δ

where we use c 2 = 8 > 0 (see (1.6)).

Let us estimate Q 3 . We claim

(2.19) ∫ R 4 E 3 Z δ , ρ ξ 1 0 d x ∼ β ∑ i = 2 m ∫ | r − r 0 | ≤ σ U δ , ρ ξ 1 ∂ U δ , ρ ξ 1 ∂ δ U δ , ρ ξ i 2 d x ∼ β ∑ i = 2 m ∫ | x − ρ ξ 1 | ≤ ρ | ξ 1 − ξ i | 2 … ︸ I 1 + ∫ | x − ρ ξ i | ≤ ρ | ξ 1 − ξ i | 2 … ︸ I 2 + ∫ | x − ρ ξ 1 | ≥ ρ | ξ 1 − ξ i | 2 ρ | ξ 1 − ξ i | 2 ≤ | x − ρ ξ i | ≤ 2 ρ | ξ 1 − ξ i | … ︸ I 3 + ∫ | x − ρ ξ 1 | ≥ ρ | ξ 1 − ξ i | 2 | x − ρ ξ i | ≥ 2 ρ | ξ 1 − ξ i | … ︸ I 4 ∼ 2 β ∑ i = 2 m c 4 δ 3 ρ 4 | ξ 1 − ξ i | 4 ln ρ | ξ 1 − ξ i | 2 δ + O ∑ i = 2 m δ 3 | β | ρ 4 | ξ 1 − ξ i | 4 ∼ − b β δ 3 ( ln ⁡ δ m ) m 4 ρ 4

where

(2.20) b ≔ 2 c 4 C 4 = 128 C 4 , with ( see ( 2.8 ) ) C 4 ≔ lim m → + ∞ 1 m 4 ∑ i = 2 m 1 | ξ 1 − ξ i | 4 ,

because ∑ i = 2 m 1 | ξ 1 − ξ i | 4 ln ( m | ξ 1 − ξ i | ) = O ( m 4 ) .

Indeed, on one hand, by the Lagrange Mean Value Theorem, for any y ∈ B ( 0 , ρ | ξ i − ξ j | 2 δ ) , ξ i ≠ ξ j , we have

U δ , ρ ξ j ( δ y + ρ ξ i ) = c δ ρ 2 | ξ j − ξ i | 2 1 − 2 δ ρ ⟨ y , ξ i − ξ j ⟩ ρ 2 | ξ i − ξ j | 2 + O δ 2 ( 1 + | y | 2 ) ρ 2 | ξ j − ξ i | 2 uniformly ,

(2.21) I 1 ∼ c 2 δ 2 1 ρ 4 | ξ i − ξ 1 | 4 ∫ B ρ ξ 1 , ρ | ξ 1 − ξ i | 2 U δ , ρ ξ 1 ∂ U δ , ρ ξ 1 ∂ δ d x ∼ c 2 δ 3 1 ρ 4 | ξ i − ξ 1 | 4 ∫ B 0 , ρ | ξ 1 − ξ i | 2 δ U 2 d y ∼ c 4 δ 3 ρ 4 | ξ 1 − ξ i | 4 ln ρ | ξ 1 − ξ i | 2 δ .

And one can check that

(2.22) I 2 = c 4 ∫ | x − ρ ξ i | ≤ ρ | ξ i − ξ 1 | 2 δ 3 | x − ρ ξ 1 | 2 − δ 2 | x − ρ ξ 1 | 2 + δ 2 3 1 | x − ρ ξ i | 2 + δ 2 2 d x set x = δ y + ρ ξ i = c 4 ∫ | y | ≤ ρ | ξ i − ξ 1 | 2 δ δ 3 | δ y + ρ ξ i − ρ ξ 1 | 2 − δ 2 | δ y + ρ ξ i − ρ ξ 1 | 2 + δ 2 3 1 | y | 2 + 1 2 d y = c 4 ∫ | y | ≤ ρ | ξ i − ξ 1 | 2 δ δ 3 1 | y | 2 + 1 2 1 | ρ ξ 1 − ρ ξ i | 4 d y + c 4 ∫ | y | ≤ ρ | ξ i − ξ 1 | 2 δ δ 3 1 | y | 2 + 1 2 | δ y + ρ ξ i − ρ ξ 1 | 2 − δ 2 | δ y + ρ ξ i − ρ ξ 1 | 2 + δ 2 3 − 1 | ρ ξ 1 − ρ ξ i | 4 d y ∼ c 4 δ 3 ρ 4 | ξ 1 − ξ i | 4 ln ρ | ξ 1 − ξ i | 2 δ + O δ 3 ρ 4 | ξ 1 − ξ i | 4

since |θδy + ρξ _i − ρξ ₁|≥ cρ|ξ ₁ − ξ _i| for some θ ∈ (0, 1), c > 0,

| δ y + ρ ξ i − ρ ξ 1 | 2 − δ 2 | δ y + ρ ξ 1 − ρ ξ i | 2 + δ 2 2 − 1 | ρ ξ 1 − ρ ξ i | 4 = δ θ δ + ⟨ θ δ y + ρ ξ 1 − ρ ξ i , y ⟩ | θ δ y + ρ ξ 1 − ρ ξ i | 2 + ( θ δ ) 2 3 ≲ δ 2 ( ρ | ξ 1 − ξ i | ) 6 + δ 2 | y | 2 + δ ρ | ξ 1 − ξ i ‖ y | ( ρ | ξ 1 − ξ i | ) 6 .

On the other hand, it shows from direct calculations that

(2.23) | I 3 | ≲ δ ρ 4 | ξ 1 − ξ i | 4 ∫ ρ | ξ 1 − ξ i | 2 ≤ | x − ρ ξ i | ≤ 2 ρ | ξ 1 − ξ i | δ 2 δ 2 + | x − ρ ξ i | 2 2 d x ≲ δ 3 ρ 4 | ξ 1 − ξ i | 4 ln 2 ρ | ξ 1 − ξ i | δ − ln ρ | ξ 1 − ξ i | 2 δ ≲ δ 3 ρ 4 | ξ 1 − ξ i | 4

and

(2.24) | I 4 | ≲ ∫ | x − ρ ξ 1 | ≥ | ρ ξ 1 − ρ ξ i | 2 | x − ρ ξ i | ≥ 2 | ρ ξ 1 − ρ ξ i | δ 3 1 | x − ρ ξ 1 | 4 1 | x − ρ ξ i | 4 d x set x = ρ | ξ 1 − ξ i | y + ρ ξ i ≲ ∫ | y | ≥ 2 | y + ( ξ 1 − ξ i ) / | ξ 1 − ξ i ‖ ≥ | y | 2 δ 3 1 | y | 4 1 | y + ( ξ 1 − ξ i ) / | ξ 1 − ξ i ‖ 4 d y ≲ δ 3 .

Combining (2.21)–(2.24), then (2.19) as desired.

Let us estimate Q 4 . Indeed,

(2.25) ∫ R 4 N ( φ ) Z δ , ρ ξ 1 0 d x ≲ ‖ φ ‖ 3 δ + | β | m ‖ φ ‖ 3 δ + ‖ φ ‖ 2 δ + | β | m ‖ φ ‖ 2 δ + | β | ‖ φ ‖ 2 ∑ i = 2 m ∫ R 4 χ U δ , ρ ξ i Z δ , ρ ξ 1 0 2 d x 1 2 + | β | ‖ φ ‖ ∑ i = 2 m ∫ R 4 χ 2 U δ , ρ ξ 1 U δ , ρ ξ i Z δ , ρ ξ 1 0 4 3 d x 3 4 ≲ | β | m ‖ φ ‖ 2 δ + ‖ φ ‖ 2 δ + | β | ‖ φ ‖ 2 δ | ln ⁡ δ | 1 2 m 2 + | β | ‖ φ ‖ δ m 2

since from (2.9) and Lemma 2.1

∑ i = 2 m ∫ R 4 χ U δ , ρ ξ i Z δ , ρ ξ 1 0 2 d x 1 2 ≲ ∑ i = 2 m 1 δ 2 ∫ | r − r 0 | ≤ σ δ 1 ρ 4 | η 1 − η i | 4 1 ( 1 + | y − ρ η 1 | ) 4 + 1 ( 1 + | y − ρ η i | ) 4 d y 1 2 ≲ δ m 2 | ln ⁡ δ | 1 2

and

∑ i = 2 m ∫ R 4 χ 2 U δ , ρ ξ 1 U δ , ρ ξ i Z δ , ρ ξ 1 0 4 3 d x 3 4 ≲ ∑ i = 2 m 1 δ 4 3 ∫ | r − r 0 | ≤ σ δ 1 ρ 8 3 | η 1 − η i | 8 3 1 ( 1 + | y − ρ η 1 ) 16 3 + 1 ( 1 + | y − ρ η i ) 16 3 d y 3 4 ≲ δ m 2 .

Let us estimate Q 5 . We derive from step 2 in Proposition 2.4

(2.26) ∫ R 4 − Δ φ + V ( x ) φ − 3 ( χ U δ , ρ ξ 1 ) 2 φ − β φ ∑ i = 2 m ( χ U δ , ρ ξ i ) 2 Z δ , ρ ξ 1 0 d x ≲ ‖ φ ‖ + | β | ‖ φ ‖ δ m 2 .

Therefore one can end with (2.16)–(2.19), (2.25), (2.26) immediately. □

Proposition 2.7.

Denote D _ɛ≔{|r − r ₀| ≤ ɛ, ɛ ∈ (2σ, 5σ)}. It holds true that

(2.27) ∫ D ε ( L ( φ ) − E − N ( φ ) ) ⟨ x , ∇ u ⟩ d x = 1 ρ ∂ ( ρ 2 V ( ρ ) ) ∂ ρ 4 δ 2 ⁡ ln ⁡ δ + Θ 1 ,

where

Θ 1 = O ( δ 2 + ‖ φ ‖ 2 + | β | m ‖ φ ‖ 4 )

uniformly with respect to d in compact sets of (0, ∞) and ρ in (r ₀ − ϑ, r ₀ + ϑ).

Proof.

By the similar arguments as Proposition 2.10 of [14], we immediately get

∫ D ε − Δ u + V ( x ) u − u 3 − β u ∑ i = 2 m u 2 ( R i x ) 〈 x , ∇ u 〉 d x = ∫ D ε ( − Δ u + V ( x ) u − u 3 ) 〈 x , ∇ u 〉 d x + ∫ D ε β u 2 ∑ i = 2 m u 2 ( R i x ) d x + O ( β m ‖ φ ‖ 4 ) = ∫ D ε − | ∇ u | 2 − V ( x ) u 2 − V ( x ) + 1 2 〈 ∇ V , x 〉 u 2 + u 4 + β u 2 ∑ i = 2 m u 2 ( R i x ) d x + O ( ‖ φ ‖ 2 + β m ‖ φ ‖ 4 ) = ∫ D ε − V ( x ) + 1 2 〈 ∇ V ( x ) , x 〉 u 2 d x + O ( ‖ φ ‖ 2 + β m ‖ φ ‖ 4 ) + o ( δ 2 ) = ∫ D ε − 1 2 r ∂ ( r 2 V ( r ) ) ∂ r u 2 d x + O ( ‖ φ ‖ 2 + β m ‖ φ ‖ 4 ) + o ( δ 2 ) = 1 ρ ∂ ( ρ 2 V ( ρ ) ) ∂ ρ 4 δ 2 ⁡ ln ⁡ δ + O ( δ 2 + ‖ φ ‖ 2 + β m ‖ φ ‖ 4 )

where we use 1 2 c 2 = 4 . Thus we are done. □

2.6 Proof of Theorem 1.1: completed

Similar to Proposition 2.8 in [14], by (2.15) and (2.27), the problem reduces to find (δ, ρ) such that

(2.28) − 8 δ ⁡ ln ⁡ δ V ( ρ ) + b β δ 3 ⁡ ln ( δ m ) m 4 ρ 4 = O δ + ‖ φ ‖ 2 | β | m δ + 1 δ + | β | δ | ln ⁡ δ | 1 2 m 2 + ‖ φ ‖ ( | β | δ m 2 + 1 )

and

(2.29) 4 δ 2 ⁡ ln ⁡ δ 1 ρ ∂ ( ρ 2 V ( ρ ) ) ∂ ρ = O ( δ 2 + ‖ φ ‖ 2 + β m ‖ φ ‖ 4 ) .

Now, by (2.14), by the decay of β in (1.8) and by the choice of the parameter δ in (2.2)

O δ + ‖ φ ‖ 2 | β | m δ + 1 δ + | β | δ | ln ⁡ δ | 1 2 m 2 + ‖ φ ‖ ( | β | δ m 2 + 1 ) = o ( δ | ln ⁡ δ | )

and

O ( δ 2 + ‖ φ ‖ 2 + β m ‖ φ ‖ 4 ) = o ( δ 2 | ln ⁡ δ | ) .

Therefore, solving the Equations (2.28) and (2.29) is equivalent to find d = d(m) > 0 (see (2.2)) and ρ = ρ(m) > 0 such that

(2.30) c V ( ρ ) − B d 2 ρ 4 = o ( 1 ) and ∂ ( ρ 2 V ( ρ ) ) ∂ ρ = o ( 1 ) ,

where

(2.31) c ≔ 8 | B | ( α − 4 ) b ( α − 2 ) ( see also ( 2.20 ) )

is a positive constant. Finally, since the function r → r ² V(r) has a non-degenerate critical point r ₀ and V(r ₀) and B have the same sign (i.e. V ( r 0 ) ⋅ B > 0 ), the two equations in (2.30) have a solution ρ(m) ∼ r ₀ and d ( m ) ∼ r 0 2 c V ( r 0 ) B as m → +∞. That completes the proof. □

Corresponding author: Angela Pistoia, Dipartimento di Scienze di Base e Applicate per l’Ingegneria, Sapienza Università di Roma, Via Scarpa 16, 00161 Roma, Italy, E-mail: angela.pistoia@uniroma1.it

Funding source: H. Chen is partially supported by the NSFC grants

Award Identifier / Grant number: (No. 12071169)

Funding source: China Scholarship Council

Award Identifier / Grant number: (No. 202006770017)

Funding source: A. Pistoia is also partially supported by INDAM-GNAMPA funds

Acknowledgments

The authors thank the referees for the comments and suggestions which definitely help to improve the readability and quality of the paper.

Research ethics: Not applicable.
Informed consent: Not applicable.
Author contributions: The authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Competing interests: The authors state no conflict of interest.
Research funding: H. Chen is partially supported by the NSFC grants (No. 12071169) and the China Scholarship Council (No. 202006770017). A. Pistoia is also partially supported by INDAM-GNAMPA funds.
Data availability: Not applicable.

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Received: 2022-12-28

Accepted: 2023-05-02

Published Online: 2024-03-12

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Schrödinger systems; segregated solutions; large number of components

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