Blow-up solutions to the Hartree-Fock type Schrödinger equation with the critical rotational speed

1 Introduction

In this article, we consider the focusing Hartree-Fock type Schrödinger equation with rotation

under the constraint

where t is the time variable, u = u ( x , t ) : R N × R → C is the wave function, N ≥ 3 , b > 0 , λ ∈ R is a Lagrange multiplier, V ( x ) is a harmonic potential of the form V ( x ) = ∣ x ∣ 2 2 , and Ω = 1 is a critical rotational speed. L z ≔ i ( x 2 ∂ x 1 − x 1 ∂ x 2 ) is the angular momentum operator.

The consideration of (1.1) is prompted by recent studies on the nonlinear Schrödinger equation with rotation

where a > 0 , c > 0 , V ( x ) = 1 2 ∑ j = 1 N γ j 2 x j 2 , γ j represent the respective trapping frequencies in each spatial direction and γ = min j = , 1 , 2 , … , N { γ j } > 0 . When the rotational speed Ω = 0 , equation (1.3) is recognized as a model describing the Bose-Einstein condensate under a magnetic trap (see [4,18,45]). When Ω ≠ 0 , equation (1.3) describes the Bose-Einstein condensate with rotation, which appears in a variety of physical settings, such as the description of nonlinear waves and propagation of a laser beam in the optical fiber [16,38]. Many articles studied the significance in theory and applications of (1.3), for example, see [2,3,23,29]. When Ω > γ , the rotational speed is larger than the trapping frequency, the centrifugal force caused by the rotation becomes stronger than the centripetal force created by the harmonic trap and the gas flies apart. In this regime, Bao et al. [7] proved the non-existence of ground state solutions for high rotational speed. When 0 < Ω < γ , the authors showed the existence and stability of prescribed mass standing waves for (1.3) with mass-subcritical 2 < p < 2 + 4 N nonlinearity in [3]. In the mass-critical case p = 2 + 4 N , papers [31,32] established the existence and stability of prescribed mass standing waves for (1.3). Additionally, Guo et al. [21,22] studied the limiting behavior of minimizers for (1.3) when c ↗ M ( Q ) , where

and Q is the unique positive radial solution to

In the mass-supercritical case 2 + 4 N < p < 2 N N − 2 , the energy functional of (1.3) is no longer bounded from below. Luo and Yang [32] proved the existence of local minimizers and stability of the set of local minimizers. Ardila and Hajaiej [5] established the global well-posedness, blow-up and stability of standing waves for (1.3). When Ω = γ 0 ≔ min { γ 1 , γ 2 } , Dinh [13] proved the existence and stability of standing waves of (1.3) with an axially symmetric harmonic potential. In particular, in the mass-critical case, the authors showed that the normalized solutions of (1.3) exist if 0 < c < M ( Q ) , while no normalized solutions exist if c ≥ M ( Q ) . Recently, Dinh et al. [14] obtained the blow-up behavior of the ground state solutions of (1.3) for N = 2 and the critical rotational speed Ω = γ = 1 .

The Hartree-Fock type Schrödinger equation with a potential term has also been the focus of many recent theoretical studies

where α ∈ ( 0 , N ) and J α ( x ) : R N → R is the Riesz potential defined by

When V = 0 , equation (1.5) is commonly referred to as the nonlinear Choquard equation (see [26,30,34]). This equation, with V = 0 , also arises from the model of wave propagation in a media with a large response length [1,25]. Fröhlich [17] investigated the existence of solutions to (1.5) with p = 2 . Ma and Zhao [34] and Xiang [44] established the unique and non-degeneracy positive solutions of (1.5) by using the moving plane method. The existence of positive and nodal solutions to (1.5) was demonstrated in [19,35]. Moroz and Van Schaftingen [36] proved the existence of nontrivial solutions of (1.5) with the lower critical case p = α N + 1 . When V ≠ 0 , Tu and Wang [39] established the existence of the normalized solutions with negative potential. Wang et al. [41] proved the existence of normalized solutions to (1.5) when the potential V is partial confinement. For the case p = 2 and N − α = 2 , Deng et al. [12] considered the following nonlinear Hartree equation:

and proved that (1.6) admits minimizers if and only if

where U = U ( ∣ x ∣ ) > 0 denotes the positive radially symmetric ground state solution of the following equation (see [35]):

Moreover, U is unique when N = 4 , and for other values of N , the uniqueness of U remains unclear (see [43]). There are also many works about the existence and multiplicity of solutions of (1.5), for example, see [6,11,19,35–37] and references therein. Recently, Tu and Wang [40] considered the nonlocal elliptic equation with rotation

and proved the existence and orbital stability of the normalized solutions at the low rotational speed 0 < Ω < γ and the critical rotational speed Ω = min { γ 1 , γ 2 } with γ 0 ≔ γ 1 = γ 2 .

Motivated by the recent works [13,14,40], this article is focused on establishing the existence, nonexistence, and the blow-up behavior of normalized solutions for (1.1) with critical rotational speed and mass critical condition under different value ranges of b . We shall concern first the stationary equation of (1.1):

with Ω = 1 and ‖ u ‖ 2 2 = 1 . Solutions to (1.10) can be obtained by searching critical points of the energy functional

on the constraint

where u ¯ is the conjugate of u and Σ is the functional space:

equipped with the norm

Taking p = 2 , μ 1 = b 2 , α = N − 2 , V ( x ) = ∣ x ∣ 2 2 , and a = 1 in (1.9), system (1.9) becomes (1.10) and γ 0 = 1 . It is well-known that if 0 < Ω < 1 , the following equivalent norm holds (see [13, Lemma 2.1]):

Theorem 1.1 in [40] proved the existence of a normalized ground state of (1.9) for the mass-subcritical N + α N < p < N + α + 2 N case, and the existence of a normalized local minimizer of (1.9) for the mass-critical p = N + α + 2 N and mass-supercritical N + α + 2 N < p < N + α N − 2 case by applying the compact embedding Σ ↪ L q for all 2 ≤ q < 2 * with (1.12). If Ω = 1 (critical rotational speed), the main difficulty to prove the existence of prescribed mass standing waves for (1.1) comes from the fact that the equivalent norm (1.12) is no longer available. To overcome the difficulty, in [40], another norm ‖ ( − i ∇ + P ) u ‖ 2 2 is introduced with P defined in this way

and

Motivated by [32], the authors obtained a local minimizer of (1.9) at the critical rotational speed for the mass-critical case and mass-supercritical case. Similar to [40], it is then equivalent to rewriting the energy functional (1.11) as

where V Ω ( x ) ≔ ∑ i = 3 N x i 2 . The working space is defined as

where H P 1 ( R N ) is the magnetic Sobolev space defined by

The corresponding norm of Σ Ω is

We consider the minimization problem

In order to state our results, we give the following definitions. A set M ⊂ Σ is orbitally stable under the flow associated with (1.1) in the sense that for any ε > 0 , there exists δ > 0 such that for any initial data u 0 ∈ Σ satisfying

the corresponding solution u ( t , ⋅ ) of (1.1) exists globally in time and satisfies

Let the solution u ∈ C ( ( − T 1 , T 2 ) , Σ ) to (1.1). We say that the maximal time of existence satisfies the blow-up alternative: if T 1 < ∞ (resp. T 2 < ∞ ), then lim t ↘ − T 1 ‖ ∇ u ( t ) ‖ 2 = ∞ (resp. lim t ↗ T 2 ‖ ∇ u ( t ) ‖ 2 = ∞ ).

The mass and energy are two important physical quantities, which are formally conserved by the time-evolution associated with (1.1) (see [40, Lemma 3.3]):

Our first result concerns the existence, non-existence and stability of standing waves for (1.1) and (1.2) with prescribed mass.

The following result shows the blow-up behavior of ground states for e Ω , b with the critical rotation speed Ω = 1 when b approaches b * (see (1.7)).

The rest of this article is organized as follows: In Section 2, we introduce the main notations and some preliminary results. In Section 3, we provide the proof of Theorem 1.1. Section 4 is devoted to the proof of Theorem 1.3.

2 Preliminaries

We shall use the following notations and conclusions:

• ‖ ⋅ ‖ H 1 is the norm of H 1 ( R N ) defined by ‖ u ‖ H 1 2 ≔ ∫ R N ( ∣ ∇ u ∣ 2 + ∣ u ∣ 2 ) d x .

• ‖ ⋅ ‖ p is the norm of L p ( R N ) defined by ‖ u ‖ p ≔ ∫ R N ∣ u ∣ p d x 1 ⁄ p .

• ⟨ u , v ⟩ ≔ R e ∫ R N u v ¯ d x , where u , v ∈ L 2 ( R N , C ) and v ¯ denotes the complex conjugate v .

• The embedding H 1 ( R N ) ↪ L q ( R N ) is continuous for 2 < q < 2 * , where 2 * = 2 N N − 2 for N ≥ 3 and 2 * = ∞ for N = 1 , 2 .

In this section, we present some preliminary results. First, we give the following Gagliardo-Nirenberg inequality (see [42, the ineqaulity (2.13)]):

where α ∈ ( 0 , N ) , N + α N < p < N + α N − 2 and S ˜ > 0 is a constant. We recall the following classical Hardy-Littlewood-Sobolev inequality (see [28, Theorem 2.3]):

where 1 < p 1 < q 1 < ∞ , 0 < t < N , and 1 p 1 + 1 q 1 + t N = 2 . By the classical Hardy-Littlewood-Sobolev inequality, there exists a positive constant C ˜ > 0 such that

In addition, the following inequality holds:

where 1 q 1 = 1 p 1 − α N . From [20, Lemma 2.3], we know that if u n ⇀ u in H 1 ( R N ) , then

From [28, Theorem 7.21], we have the diamagnetic inequality

Moreover, as a direct conclusion from Young’s inequality, the rotational term can be controlled as follows: when N ≥ 2 , for any constant δ > 0 ,

As in [35, Proposition 3.1], we know that all solutions of (1.8) satisfy the following Pohozaev identity:

Hence, one deduces from (1.8) and (2.8) that any solution of (1.8) satisfies

When p = 2 and α = N − 2 in (2.1), from [10, Lemma 1.3], we know that the following Gagliardo-Nirenberg inequality

is achieved by any positive radially symmetric ground state solution U = U ( ∣ x ∣ ) of (1.8), where b * is defined in (1.7).

We recall the following lemma, which is crucial for dealing with convergence in Σ Ω (see [40, Lemma 2.3]).

Similarly, if the workspace is Σ , we have the following lemma.

To prove the uniform boundedness of a solution, we need the following local boundedness result [24, Theorem 4.14].

3 Proof of Theorem 1.1