Qualitative analysis for the nonlinear fractional Hartree type system with nonlocal interaction

1 Introduction and main results

In the present paper we study the coupled nonlinear fractional Hartree system in the following form

where N≥3,γ∈(0,N),1≤p≤N+γN−α and

Recall that the operator (−Δ)α2 is the fractional Laplacian in R^N which is defined as a nonlocal pseudo differential operator

for each u∈Cloc1,1(ℝN)∩Fα(ℝN), where the constant CN,α=(∫ℝN1−cos(2πω1)|ω|N+αdω)−1 and ω = (ω₁, · · ·, ω_N)(see [3, 6, 7, 9, 49]). Note that both the fractional Laplacians (−Δ)α2 and the Hartree type nonlinearities are nonlocal in the system (1.1). Moreover,we can check that the system is closely related to the following integral equation

where ϕw(y)=∫ℝN|w(z)|p|y−z|N−γdz and RN,α=Γ(N−α2)πN/22αΓ(α2).

One of the motivation to study the system (1.1) is motivated by recent studies on the nonlinear fractional Schrödinger equation with nonlocal interaction

If α = 2, N = 3 and p = 2, the system 1.5 is related to the condensate in the mean field regime. Indeed, concerning the movement of the identical and non-relativistic basic particles (such as bosons or electrons), under the influence of an external potential, the interaction between two particles is governed by the nonlinear nonlocal Hatree equation(see [17, 18, 23, 24]). Here the function ψ is a radially symmetric two-body potential function defined and * denotes the convolution in R³. The most typical external potential is the Coulomb function C(x) = |x|⁻¹. On the other hand, the system 1.5 is also used in the description of the Bose-Einstein condensates, in which V is the trapping potential and the nonlocal interaction also describes the interaction between the bosons in the condensate [14, 44, 47]. When V = 0, 1.5 is also known as nonlinear Choquard equation [33, 37, 40], and the equation 1.5 with V = 0 also arises from the model of wave propagation in a media with a large response length [1, 26]. Many papers considered the general interaction case. That is,

Then the stationary system of 1.5 reduces to

If α = 2, the paper [41] proved the existence and some properties of solution of 1.7. Recently, the paper [20] prove the existence of nodal solution of 1.7. For more general information one can refer to the papers [19, 20, 30, 31, 40, 42, 46] and references therein. For the fractional case 0 < α < 2, the paper [12] proved the regularity and classification of the solution of 1.7. Recently, by using the direct moving plane methods, the paper [11] make the classification of the positive solution of 1.7 when p = 2 and N − γ = 2α. The paper [28] make the classification of the positive solution of 1.7 for the general case.

The system (1.1) was studied in the sevral recent papers [51, 52, 53, 54, 56] for the case α = 2. This kind of systems are considered in the basic quantum chemistry model of small number of electrons interacting with static nucleii which can be approximated by Hartree or Hartree-Fock minimization problems (see [29, 35, 39]). In fact, the Euler-Lagrange equations corresponding to such Hartree problem are

where k ∈ N, V(x) describes the attractive interaction between the electrons and the nucleii, the integral term shows the repulsive Coulomb interaction between the electrons, and −ε_i are the Lagrange multipliers. Following the discussion in [39], we usually consider the case that some components are set to be equal in 1.8. For example, when k = 2 and u₁ = u₂, then 1.8 is reduced to a scalar equation

The solutions of 1.9 were considered in, for example, [19, 38, 39]. We notice that in 1.8, the interaction between electrons is repulsive while the one in 1.5 is attractive. On the other hand, if k = 4, u₁ = u₂ and u₃ = u₄, then we can also obtain (1.1)when V = 0, p = q = 2, N = 3 and γ = α = 2. Recent years, the following nonlocal system

has been studied by many literatures, where ϕ_u(x) = _RN C(x − y)|u(y)|^p dy and C(x) is given in 1.6. Note that the system 1.10(or (1.1)) has two semitrivial solutions (u, 0) and (0, v), where u, v are solutions of 1.7. In order to make clear statement one gives the following definitions. We say (u, v) is a nontrivial solution of 1.10(or (1.1)) if u ≠ 0 and v ≠ 0. If u, v > 0, we say (u, v) is a positive solution of 1.10(or (1.1)). A solution is called a nontrivial ground state solution(or positive ground state solution) if its energy is minimal among all the nontrivial solutions(or all the positive solutions) of 1.10(or (1.1)). The paper [56] considered the semi-classical case. Under some conditions for the potential function λ_i(x), i = 1, 2, the existence of a ground state solution of 1.10 for ε > 0 small and β > 0 sufficiently large was proved. Later, the paper [53] studied the case λ_i(x) = λ_i = constant, ε = 1, p = 2, N = 3 and γ = 2. The authors proved the existence and nonexistence of positive ground state solutions of 1.10. Moreover, various qualitative properties of ground state solutions are also obtained. Recently, the papers [51, 54] studied the existence and properties of normolized solution of 1.10 with general interaction. Very recently, the paper [52] proved the existence of nontrivial solutions of the more general nonlocal interaction case of 1.10.

Motivated by the previous works, in the present paper we shall study the general case of the system 1.10 in fractional situation. Precisely, the main purpose of this paper is the following three parts. First, we use the direct moving plane methods to establish the maximum principle(Decay at infinity and Narrow region principle) and prove the symmetry and nonexistence of positive solution of this nonlocal system. Second, we make complete classification of positive solutions of the system for the critical case when μ₁ = μ₂. Finally, we prove the existence of multiple nontrivial solutions of the critical case for μ₁ ≠ μ₂ by using variational methods. To accomplish the first two conclusions, we shall use a variant (for nonlocal nonlinearity) of the direct method of moving planes for fractional Laplacians due to the paper [6, 7] to obtain symmetry, monotonicity, nonexistence and classification of the positive solutions to (1.1). The methods of moving planes was initially invented by Alexanderoff in the early 1950s. Later, it was further developed by Serrin [48], Gidas, Ni and Nirenberg [21, 22], Caffarelli, Gidas and Spruck [2], Chen and Li [4], Li and Zhu [32] and many others. For more literatures on this direction, one can see the papers [4, 7, 8, 13, 16, 32, 36, 55] and the references therein. In this paper we establish the maximum principle(Decay at infinity and Narrow region principle, see Theorems 2.6-2.7 below) for the fractional nonlocal system. This is new and we believe that it will be useful to study the other nonlocal problems. To accomplish the third result, we should make careful study the uniqueness of the synchronous solutions of (1.1) and use the perturbation methods to obtain the nontrivial ground state solution of (1.1).

Then we first have the following main results for the subcritical and critical cases.

2 Main conclusions from the direct methods of moving planes

Throughout the paper, we use the following notations:

1. | · |_p is the norm of L_p(R^N) defined by |u|p=(∫ℝN|u|p)1/p for 0<p<∞.

2. Let c > 0 be an arbitrary constants.

In this section we shall establish the main maximum principle theorems(Decay at infinity and Narrow region principle) for the fractional nonlocal system by using the direct methods of moving planes. We first recall the following classical Hardy-Littlewood-Sobolev inequality(see [34, Theorem 4.3]) and Cauchy-Schwarz inequality(see [34, Theorem 5.9] or [20, Inequality 3.3]) for nonlocal problem.