In this paper, we consider the three-dimensional Hartree equation with a constant magnetic field

where u = u ( t , x ) is a complex-valued function. The vector potential A = A ( x ) = b 2 ( − x 2 , x 1 , 0 ) describes a constant magnetic field B = ∇ × A = ( 0 , 0 , b ) with b ≠ 0 . Here ∗ denotes the convolution, | x | − α is the Riesz potential, and 0 < α < 3 .

The Schrödinger equation with a constant magnetic field provides an effective model for describing the dynamics of a single non-relativistic quantum particle in

an electromagnetic field (see, e.g., [1]). Replacing | u | 2 u with ( | x | − α ∗ | u | 2 ) u

leads to the magnetic Hartree Equation (1), which arises as a mean-field model for initially factorized bosonic states (see, e.g., [2]-[4]).

Equation (1) admits a class of special solutions known as standing waves. These solutions take the form u ( t , x ) = e i ω t ϕ ( x ) , where ω ∈ ℝ is a frequency and ϕ is a complex-valued function. The standing wave profile ϕ satisfies the elliptic equation

Solutions of (2) are critical points of the action functional defined on the magnetic Sobolev space H A 1 ( ℝ 3 ) . This action functional is given by

Equation (1) is locally well-posed in the energy space H A 1 ( ℝ 3 ) (see [5][6]). More precisely, for 0 < α < 3 and u 0 ∈ H A 1 ( ℝ 3 ) , there exists T max = T max ( u 0 ) ∈ ( 0 , + ∞ ] and a unique solution

where H A − 1 ( ℝ 3 ) is the dual space of H A 1 ( ℝ 3 ) . The solution exhibits the following blow-up alternative: either T max = + ∞ (global existence), or T max < + ∞ and lim t ↗ T max ‖ u ( t ) ‖ H A 1 = + ∞ (finite-time blow-up).Furthermore, the solution u ( t ) satisfies the following conservation laws:

If the nonlocal term ( | x | − α ∗ | u | 2 ) u is replaced by the local nonlinearity | u | 2 u ,

then the nonlinear Schrödinger Equation (1) has been extensively studied, and there exists a large body of literature on the Cauchy problem for this equation (see, e.g., [5][7] and the references therein). In particular, the blow-up and global existence of solutions to the magnetic Schrödinger equation were investigated in [8]-[10]. Moreover, the stability and instability of standing waves for this equation were studied in [9][11]. For the linear magnetic Schrödinger operator, we refer the reader to the works of Avron, Herbst, and Simon [12]-[14]. Furthermore, Dinh revisited the Cauchy problem for the three-dimensional nonlinear Schrödinger equation with a constant magnetic field in [15].

We next turn to the Hartree Equation (1). In the case A ≡ 0 , the Hartree equation has been widely studied. In particular, there exist many results on the Cauchy problem and the asymptotic behavior of solutions (see, e.g., [16]-[20]). For further studies related to the Hartree equation, we also refer the reader to [21][22] and the references therein.

We now consider the magnetic Hartree Equation (1). Cazenave [5][6] established the local well-posedness of this equation. More recently, the existence and orbital stability of normalized standing waves have been studied in [23]. But as we know, there is no result on the strong instability of ground state standing waves in the magnetic Hartree Equation (1).

Compared with the existing literature, the present work contains several new features. First, the presence of the magnetic field together with the nonlocal Hartree interaction creates additional difficulties in the variational analysis and in the derivation of virial-type estimates. In particular, the interaction between the magnetic operator and the nonlocal convolution term requires a more delicate treatment under the L 2 -invariant scaling.

Second, the instability analysis combines the variational characterization of ground states, conservation laws associated with the magnetic flow, and a refined virial argument adapted to the Hartree nonlinearity.

The main purpose of this paper is to investigate the strong instability of standing waves for (1). Motivated by the approach of Berestycki and Cazenave [24], we first establish the existence and variational characterization of ground states. We then prove finite-time blow-up of solutions with initial data arbitrarily close to the ground states, which yields the strong instability of standing waves. We now state our main results.

Theorem 1.1. Assume 0 < α < 3 and ω > − | b | . Then there exists a ground state solution of (2). In particular, u ( t , x ) = e i ω t ϕ ( x ) is a solution to (1). Moreover, the set of ground states G ( ω ) is characterized by

here

with the Nehari functional

Remark 1.We characterize the ground state as a minimizer of the action functional S ω on the Nehari manifold N ω = { f ≠ 0 : K ω ( f ) = 0 } . Under the constraint K ω ( f ) = 0 , by combining the Hardy-Littlewood-Sobolev inequality, the Gagliardo-Nirenberg inequality, and the equivalent norm relation (19), we derive that every minimizing sequence for d ( ω ) has a uniform positive lower bound in the L 12 6 − α -norm, thereby ruling out vanishing of minimizing sequences.

Using the compactness lemma (Lemma 2.7), we then obtain a nontrivial weak limit and verify that it satisfies the Euler-Lagrange equation, yielding the existence of a ground state. This approach avoids the compactness difficulties in the fixed-mass framework, where the classical scaling argument cannot be directly applied in the presence of a magnetic field. The argument is inspired by Dinh [15].

Theorem 1.2. Let 2 < α < 3 , ω > − | b | , and let ϕ ∈ G ( ω ) . If

where

then the ground state standing wave e i ω t ϕ ( x ) is strongly unstable. More precisely, for any ε > 0 , there exists u 0 ∈ Σ A ( ℝ 3 ) such that ‖ u 0 − ϕ ‖ Σ A < ε , and the solution of (1) with initial data u 0 blows up in finite time.

Remark 2.Condition(1.6)describes the behavior of the action functional along the scaling direction ϕ λ ( x ) = λ 3 2 ϕ ( λ x ) . In particular, the condition

implies that the action functional decreases locally along the scaling orbit near λ = 1 . Combined with the variational characterization of ground states and the virial identity, this allows us to construct suitable perturbations of the standing wave, which play a crucial role in proving finite time blow-up. Similar conditionsnaturally appear in the instability theory developed by Ohta for nonlinearSchrödinger and Hartree-type equations; see[25].

This paper is organized as follows. In Section 2, we present some useful lemmas, including the local well-posedness of Equation(1) and several inequalities. Section 3 is devoted to the existence of ground state standing waves. Finally, in Section 4, we study the strong instability of ground state standing waves.

In this section, we collect several preliminary results that will be repeatedly used in the subsequent analysis. For convenience, Equation (1) can be rewritten as

where

denotes the third component of the angular momentum operator.

First, we recall that the local well-posedness of Equation (1) in H A 1 ( ℝ 3 ) was established by Cazenave and Esteban in [6]. Here, H A 1 ( ℝ 3 ) : = { f ∈ L 2 ( ℝ 3 ) : | ( ∇ + i A ) f | ∈ L 2 ( ℝ 3 ) } is a Hilbert space equipped with the norm

Lemma 2.1.[7] Let A ∈ L loc 2 ( ℝ 3 , ℝ 3 ) . Then H A 1 ( ℝ 3 ) equipped with the inner product

is a Hilbert space.

To state blow-up result for (1),let us introduce the following Hilbert space Σ A

is equipped with the norm

As shown in ([26], Lemma 2.2] we have Σ A ( ℝ 3 ) ≡ Σ ( ℝ 3 ) , where Σ ( ℝ 3 ) is the standard weighted Sobolev space

In addition, we have the following useful identity

is called the angular momentum with L b as in (9). Since A ( x ) = b 2 ( − x 2 , x 1 , 0 ) and the Hartree kernel | x | − α are invariant under rotations around the z -axis, the Hamiltonian associated with (1) commutes with the angular momentum operator L b . Hence the angular momentum R ( u ( t ) ) is conserved along the flow of (1). See Avron, Herbst and Simon [12] for a rigorous proof.

We recall that if the initial data u 0 ∈ Σ A ( ℝ 3 ) , then the corresponding solution of (1) remains in Σ A ( ℝ 3 ) on its maximal existence interval; see [15]. Therefore, the following virial identity is well defined.

To proceed with the variational analysis, we recall several basic properties of the magnetic Sobolev space.

Lemma 2.2.[7] Let A ∈ L loc 2 ( ℝ 3 , ℝ 3 ) . Then the following properties hold:

1) C 0 ∞ ( ℝ 3 ) is dense in H A 1 ( ℝ 3 ) .

2) H A 1 ( ℝ 3 ) is continuously embedded in L r ( ℝ 3 ) for all 2 ≤ r ≤ 6 .

3) Assume that A is linear, i.e., A ( x + y ) = A ( x ) + A ( y ) for all x , y ∈ ℝ 3 . Let y ∈ ℝ 3 , f ∈ H A 1 ( ℝ 3 ) , and set

Then, ( ∇ + i A ) f ˜ ( x ) = e i A ( y ) ⋅ x ( ∇ + i A ) f ( x + y ) . In particular,

4) If A ∈ L loc 3 ( ℝ 3 , ℝ 3 ) , then H A 1 ( ℝ 3 ) is continuously embedded in H loc 1 ( ℝ 3 ) . In particular, H A 1 ( ℝ 3 ) is compactly embedded in L loc r ( ℝ 3 ) for all 2 ≤ r < 6 .

Remark 3.This lemma provides the basic embedding and translation properties needed later in Section 3.

Our further considerations need the following inequalities.

Lemma 2.3.[12] Let A ∈ W loc 1 , ∞ ( ℝ 3 , ℝ 3 ) and j , k ∈ { 1 , 2 , 3 } . Then, for any f ∈ C 0 ∞ ( ℝ 3 ) , we have

In particular, if A = b 2 ( − x 2 , x 1 , 0 ) , then

Lemma 2.4.(Diamagnetic inequality [27]) Let A ∈ L loc 2 ( ℝ 3 , ℝ 3 ) and f ∈ H A 1 ( ℝ 3 ) . Then, | f | ∈ H 1 ( ℝ 3 ) . In particular, we have

Combining the Diamagnetic inequality with the classical Gagliardo-Nirenberg inequality, we obtain the following magnetic version, which will be used to control the nonlinear Hartree term.

Lemma 2.5.(Magnetic Gagliardo-Nirenberg inequality[23]). For A ∈ L loc 2 ( ℝ 3 , ℝ 3 ) and 2 < r < 6 ,

where C r > 0 is the optimal constant.

Lemma 2.6.(Hardy-Littlewood-Sobolev inequality [27]). Let 0 < α < 3 and s , r > 1 be constants satisfying 1 r + 1 s + α 3 = 2 . Assume that f ∈ L r ( ℝ 3 ) and g ∈ L s ( ℝ 3 ) . Then, there exists a positive constant C ( s , α ) depending on s and α such that

In order to prove the existence of ground states, we need the following compactness lemma.

Lemma 2.7.[15]. Let A ∈ L loc 3 ( ℝ 3 , ℝ 3 ) be linear. Let ( f n ) n ≥ 1 be a bounded sequence in H A 1 ( ℝ 3 ) , i.e.,

Assume that there exists ε 0 > 0 such that

for some 2 < r < 6 . Then up to a subsequence, there exist f ∈ H A 1 ( ℝ 3 ) \ { 0 } and ( y n ) n ≥ 1 ⊂ ℝ 3 such that

Remark 4.The proof of this compactness lemma is based on an argument of [28].

We also require splitting formulas for the magnetic kinetic energy and the Hartree interaction.

Lemma 2.8.[15] Let A ∈ L loc 2 ( ℝ 3 , ℝ 3 ) and ( f n ) n ≥ 1 be a bounded sequence in H A 1 ( ℝ 3 ) . Assume that f n ⇀ f weakly in H A 1 ( ℝ 3 ) . Then, we have

Lemma 2.9.[28] Let 0 < α < 3 , 2 < p < + ∞ . Assume that f n and f are functions on ℝ 3 such that

and

Then,

Lemma 2.10.[23] Let A ∈ L loc 2 ( ℝ 3 , ℝ 3 ) and ( f n ) n ≥ 1 be a bounded sequence in H A 1 ( ℝ 3 ) . Assume that f n ⇀ f weakly in H A 1 ( ℝ 3 ) . Then, we have

Next, we need the following variance identity.

Lemma 2.11. Let 0 < α < 3 and u 0 ∈ Σ A ( ℝ 3 ) . Let u : [ 0 , T max ) × ℝ 3 → ℂ be the corresponding solution to (1). Set

then

Proof. Clearly there holds

Noticing that i ∂ t u = − ( ∇ + i A ) 2 u − ( V ∗ | u | 2 ) u with V ( x ) = | x | − α , we obtain

Differentiating again yields

We decompose this expression into three parts:

so that I ″ ( u ( t ) ) = 4 Re ( A + B + C ) . Direct computations show that:

For part C, let K ( x ) = ( V ∗ | u | 2 ) ( x ) = ∫ ℝ 3 | u ( y ) | 2 | x − y | α d y . Then

Using symmetry by interchanging x and y , we obtain

Combining all parts:

□

This section is devoted to the proofs of the existence of ground states for (2) in the mass-supercritical case. A nonzero solution ϕ of (2) is referred to as a ground state associated with (2) if it minimizes the action functional S ω ( f ) over all nontrivial solutions of (2). Note that (2) can be expressed as S ′ ω ( ϕ ) = 0 . We accordingly define

as the set of all nontrivial solutions to (2), and introduce

as the corresponding set of ground states.

Before proceeding to the proof of Theorem 1.1, we establish the following preliminary.

Lemma 3.1. Let A ( x ) = b 2 ( − x 2 , x 1 , 0 ) and ω > − | b | . Then

Proof. In fact, by (14), we see that

It follows that

On the other hand, we see that

Lemma 3.2. Let A ( x ) = b 2 ( − x 2 , x 1 , 0 ) , 0 < α < 3 , and ω > − | b | . Then there exists f ∈ H A 1 ( ℝ 3 ) such that K ω ( f ) = 0 .

Indeed, for f ∈ C 0 ∞ ( ℝ 3 ) . We have

It follows that K ω ( λ 0 f ) = 0 with λ 0 = ( H ω ( f ) ∫ ℝ 3 ( | x | − α ∗ | f | 2 ) | f | 2 d x ) 1 2 .

Next we have the following existence of minimizers for d ( ω ) > 0 .

Lemma 3.3. Let A ( x ) = b 2 ( − x 2 , x 1 , 0 ) , 0 < α < 3 , and ω > − | b | . Then there exists a minimizer for d ( ω ) and hence the set of all minimizers for d ( ω ) defined by

is not empty.

Proof. The proof is done by several steps.

Step 1. We first show that d ( ω ) > 0 . Let f ∈ H A 1 ( ℝ 3 ) be such that K ω ( f ) = 0 .

By the Hardy-Littlewood-Sobolev inequality (Lemma 2.7) and magnetic Gagliardo-Nirenberg inequality (Lemma 2.6), we have

which implies H ω ( f ) ≥ C > 0 . It follows that

Taking the infimum over all f ∈ H A 1 ( ℝ 3 ) \ { 0 } satisfying K ω ( f ) = 0 , we obtain d ( ω ) > 0 .

Step 2. We next show that there exists a minimizer for d ( ω ) . Let ( f n ) n ≥ 1 be a minimizing sequence for d ( ω ) . We have

which, by (3.1), implies that ( f n ) n ≥ 1 is a bounded sequence in H A 1 ( ℝ 3 ) . As K ω ( f n ) = 0 , we have

Thus up to a subsequence, we have inf n ≥ 1 ∫ ℝ 3 ( | x | − α ∗ | f n | 2 ) | f n | 2 d x ≥ C > 0 , by the Hardy-Littlewood-Sobolev inequality, this implies inf n ≥ 1 ‖ f n ‖ L 12 6 − α ≥ ϵ 0 > 0 . Applying Lemma 2.7, there exist f ∈ H A 1 ( ℝ 3 ) \ { 0 } and ( y n ) n ≥ 1 ⊂ ℝ 3 such that

Thanks to Lemma 2.8 and Lemma 2.9, we have

We will show that K ω ( f ) = 0 . Indeed, if K ω ( f ) < 0 , then there exists λ 0 ∈ ( 0 , 1 ) such that K ω ( λ 0 f ) = 0 . From the definition of d ( ω ) , we have

which is a contradiction. If K ω ( f ) > 0 , then, by (21) and the fact that K ω ( f ˜ n ) = K ω ( f n ) = 0 , we have K ω ( f ˜ n − f ) < 0 for n sufficiently large. Thus there exists ( λ n ) n ≥ 1 ⊂ ( 0 , 1 ) such that K ω ( λ n ( f ˜ n − f ) ) = 0 . It follows that