Synchronized vector solutions for the nonlinear Hartree system with nonlocal interaction

1 Introduction and main results

The two-component system with Hartree type nonlinearities

describes the dynamics of boson stars in mean-field theory [1], [2] and attracts much attention in recent years. Here, Φ i : R + × R N → C , β > 0 is the scattering length describing the attractive interaction, V _i are the external potentials, and W is the response function which possesses information on the mutual interaction between the particles. It is well known that the nonlocal nonlinearities play important roles in the theory of Bose–Einstein condensation (cf. [1]).

It is interesting to study the standing wave solutions of (1.1) with a response function of Riesz potential, i.e., V(x) = |x|^−μ . Clearly, Φ 1 ( t , x ) = e − i E 1 t u ( x ) and Φ 2 ( t , x ) = e − i E 2 t v ( x ) solve (1.1) if and only if (u(x), v(x)) is a solution of the Hartree system

where P _i (x) = V _i (x) − E _i for i = 1, 2. In [3], Wang and Shi studied (1.2) with positive constant potentials and proved the existence and nonexistence of positive ground state solutions. In [4], the authors proved the existence of ground state solution for a singular perturbed problem related to (1.2) when the coupling constant β is large. Later, Gao et al. [5] studied the existence of positive solutions of high energy for the critical coupled Hartree system (1.2) with N ≥ 5, μ = 4, β > max{α ₁, α ₂} ≥ min{α ₁, α ₂} > 0, and P 1 , P 2 ∈ L N / 2 ( R N ) ∩ L loc ∞ ( R N ) . Recently, Ye et al. [6] considered the couple critical interaction system (1.2) with axisymmetric potentials and constructed infinitely many solutions by employing the Lyapunov-Schmidt reduction argument and Hatree-type local Pohožaev identities.

Our another motivation comes from the study of existence of positive solution to the single nonlinear Hartree equation. In precise, let u = 0 or v = 0 in (1.2), it goes back to the single Hartree type equation

The existence and uniqueness of ground state solutions to (1.3) has been studied in [7]–[12]. Ackermann [13] proved the existence of infinitely many geometrically distinct weak solutions. Moroz and Van Schaftingen [14] proved the existence, regularity and positivity of radial symmetric ground state.

The nondegeneracy of the ground state solution to the nonlinear Hartree equation has been obtained in a series of papers [7], [12], [15]–[17]. It is a key ingredient in the Lyapunov–Schmidt reduction method of constructing semiclassical or blow-up solutions to the partial differential equation. In [17], Wei and Winter studied the equivalent local Schrödinger–Newton system instead and applied the expansion of the equations by spherical harmonics. Then they use the Liapunov-reduction scheme to construct multibump solutions of the nonlocal partial differential equation with strongly interacting bumps. In [7], Lenzmann found a method that relied on spectral analysis of the linearized operators at the ground states. Later, Chen [15] generalized the nondegeneracy result to higher dimensions and constructed multiple semiclassical solutions to Hartree equation with an external potential by using Lyapunov-Schmidt reduction argument. For other related results, we refer the readers to [13], [18], [19] [13], [18]–[20] for some progress on the topic of Hartree equation. In particular, the authors in [20] proved the existence of infinitely many non-radial solutions for nonlinear Hartree equation (1.3) under some proper assumptions on the function P. Therefore, a natural question to ask is in the following: For the following coupled Hartree system, are there infinitely many non-radial positive synchronized solutions? If yes, can the energy of these solutions be arbitrarily large?

where P ₁(|x|) and P ₂(|x|) are positive radial potentials, α ₁ > 0, α ₂ > 0 and β ∈ R is a coupling constant. In this paper we answer these questions. We rigorously prove that Hartree system (1.4) has an unbounded sequence of non-radial synchronized vector solutions by employing the Lyapunov-Schmidt reduction argument.

To state our result, we need to introduce potential function P ₁(|x|) and P ₂(|x|) and satisfy the following condition:

(P ₁) There are constants a > 0, m ≥ 3 and θ > 0 such that as r → +∞,

(P ₂) There are constants b > 0, n ≥ 3 and ϑ > 0 such that as r → +∞,

The main result of this paper reads as follows:

We use W to denote the solution of the following problem

In [14] the authors showed radial symmetry of solution W yields that W is either positive or negative, and there exist x 0 ∈ R 3 and a monotone function v ∈ C ^∞ (0, ∞) such that for every x ∈ R 3 , W(x) = v(|x − x ₀|). Without loss of generality, we can suppose W > 0 and x ₀ = 0, that is, W(x) = W(|x). Chen [15] applied the expansion of the nonlocal term by spherical harmonics to obtain the nondegeneracy of positive bubble solutions W for nonlinear Hartree equation (1.5). More precisely, we summarize this result as follows:

Note that the limit system for (1.4) is

and that

solves (1.6), provided − α 1 α 2 < β < min { α 1 , α 2 } or β > max{α ₁, α ₂}, where h 0 = β − α 2 β 2 − α 1 α 2 , l 0 = β − α 1 β 2 − α 1 α 2 .

Since P ₁(r) → 1 as r → ∞ and P ₂(r) → 1 as r → ∞, we will use (U, V) to build up the approximate solutions for (1.4) with large number of bumps near the infinity. As in [21], let

and

where m is the constant in the expansion for P ₁, n is the constant in the expansion for P ₂, and δ > 0 is a small constant.

The Sobolev space H P i 1 ( R 3 ) is endowed with the standard norm

Define H to be the product space H P 1 1 ( R 3 ) × H P 2 1 ( R 3 ) with the norm

Set x = (x′, x″), x ′ ∈ R 2 and x ″ ∈ R . Define

where i = 1, 2.

We denote

where U z j ( x ) = U ( x − z j ) and V z j ( x ) = V ( x − z j ) . It is easy to check that (U _r , V _r ) is in H P 1 1 ( R 3 ) × H P 2 1 ( R 3 ) .

Before concluding this introduction, we would like to mention some related results to our problem. Wei and Yan in [21] use a Lyapunov–Schmidt reduction argument to consider the nonlinear Schrödinger equation

with V(r) → V ₀ > 0 as r → ∞ and they proved the existence of infinitely many positive non-radial solutions for such equation. This technique has also been applied successfully for the study of different problems, for example, Schrödinger-Poisson type system (see [22]), nonlinear Schrödinger system (see [23]), nonlinear Hartree equation (see [20], [24]–[26]). Results on the existence of infinitely many solutions for other elliptic problems can also be found in [27]–[30] and the references therein. In contrast with the single nonlinear Schrödinger equations (1.3) or (1.8), since we deal with a system, we face additional difficulties. Moreover, more involved technical difficulties is needed to deal with the nonlocal term in (1.4) and a more careful analysis of the interaction between the bumps is required

The paper is organized as follows. In Section 2, we give a nondegeneracy result for the limit system (1.6). In Section 3, we carry out the reduction. Section 4, we study the reduced finite dimensional problem and prove Theorem 1.4.

2 Nondegeneracy

Let us consider the following eigenvalue problem

By the Hardy-Littlewood-Sobolev inequality, Hölder inequality and Sobolev inequality, we can obtain for all v ∈ H 1 ( R 3 ) , it holds that

and

Then we may consider the eigenvalues of problem (2.1) as the following:

Choosing v = W in (2.2), since W is the solution of equation (1.5) then we have

Then from Proposition 1.3 regarding the nondegeneracy of W, we have

For convenience, we denote

Proof. We follow the arguments in [31] or [23]. Now for − α 1 α 2 < β < 0 or 0 < β < min{α ₁, α ₂}, linearization of equations (1.6) at (U, V) gives us

We see at once that α ₁ h ₀ + βl ₀ = α ₂ l ₀ + βh ₀ = 1 then

where

For simplicity, we rewrite system (2.3) as follow

Let δ ± = − e − d 2 c ± 1 2 c ( e − d ) 2 + 4 c 2 . For β < 0, a direct computation shows that

Then we have

Returning to the equation for ψ we get that

Set ψ = ∑ j = 1 ∞ γ j e j and f(β) = e + cδ ₊. Assume that f(β) ≠ λ _k for any k, where λ _k is defined in Definition 2.1. Using orthogonality we see easily that γ _j = 0 for j ≠ τ ₀ + 2, τ ₀ + 3, τ ₀ + 4, and γ j = 2 c c τ 0 + j 1 − 2 f ( β ) for j = τ ₀ + 2, τ ₀ + 3, τ ₀ + 4. Thus, the kernel at (U, V) is a three dimensional space and it can be given by

We may take θ ( β ) = δ + + 1 − 2 f ( β ) 2 c . Since c ≠ 0 and 1 − 2e ≠ 0, we have θ(β) ≠ 0. If f(β) = λ _j for some j, then similarly γ _j = 0 for j ≠ τ ₀ + 2, τ ₀ + 3, τ ₀ + 4. It is easy to check that e _k is in the kernel when λ _k = λ _j . Thus the kernel is generated by

and span{e _k : λ _k = λ _j }. For min{α ₁, α ₂} > β > 0 we use δ ₋ instead of δ ₊ to get the same result. From [31], f ( β ) = 1 2 if and only if β = 0 and f(β) is monotone decreasing. Also, from [31] there exists a decreasing sequence β _k in ( − α 1 α 2 , 0 ) such that f(β) = λ _k if and only if β = β _k . Thus, for β∉{β _k }, f(β) = λ _j for any j. The same arguments can be used to treat the case β > max{α ₁, α ₂}, so we omit it. □

3 The reduction argument

Let

where z _j is given in (1.7). Then, we have the following basic estimates:

Denote

and

By the energy functional J : H → R associated with equation (1.4),

we denote

Expand I(φ, ϕ) as follows:

where

and