In this paper, we focus on the following two-component Gross-Pitaevskii system coupled with a microwave field and inhomogeneous interactions in ℝ 2 :

where λ ∈ ℝ is the chemical potential, a ∈ ℝ denotes the contact interaction strength between two components, k ( x ) stands for the inhomogeneous interactions, and γ > 0 (resp. < 0) represents that the magnetic field is attractive (resp. repulsive). The system (1.1) originates from Bose-Einstein Condensates (BECs) interacting with a microwave field at low temperature. The microwave field can influence BECs by the local field effect which plays an important role in the process of forming electromagnetic-matter waves. Due to the physical significance of this system, our goal is to research the existence, nonexistence and limiting behavior of the complex-valued states in system (1.1).

Another purpose of studying (1.1) derives from recent research [1]-[5]. In these references [1]-[4], the authors investigate various models, including single-component attractive BECs, two-component attractive BECs and rotating attractive BECs. They research a range of properties such as existence, nonexistence, uniqueness, quality concentration, symmetry breaking, and refined spike profiles of ground states. However, there is relatively little research on planar multi-component BECs interacting with electromagnetic fields. Recently, Wang, Cai and Wang [5] explored (1.1) in the complex-valued case. They obtained results about the existence and nonexistence of central vortex steady states. Now, we add inhomogeneous interactions to (1.1) for further research. For 0 < k ( x ) ≤ 1 , we give the existence of minimizers of e ( γ , a ) in the case of a ∈ ( 0 , 2 a * ) , where a * is the square of L 2 -norm of the unique positive solution of

In this paper, we are interested in the nontrivial solutions of (1.1) under the normalization constraint

Towards this purpose, with the help of the fundamental solution of − Δ , we define the energy functional

Due to the appearance of the harmonic potential term and the logarithmic convolution term, E γ , a is not well-defined on H 1 ( ℝ 2 ) × H 1 ( ℝ 2 ) . Inspired by [3][6][7], we work in the subspace X : = H × H , where

Define the norm on X by

According to ([2] lemma 2.1), we have

We now define the constraint set

and focus on studying the following constrained variational problem:

Throughout the paper, we assume that the potential − 1 ≤ k ( x ) ≤ 1 satisfies the following assumptions:

where C > 0 and b ≥ 2 are some constants, and k ( x ) ∈ C 0 , α ( ℝ 2 ) for some α ∈ ( 0 , 1 ) .

Stimulated by [6][8], we perform that

For simplicity, we use |   ⋅   | q to denote the standard Lebesgue norm on L q ( ℝ 2 ) for q ∈ [ 1 , ∞ ] . Since

We deduce by Hölder inequality that

Applying Hardy-Littlewood-Sobolev inequality (cf. ([9], Theorem 4.3)), we then derive that there exists a constant C > 0 such that

It follows from (1.9)-(1.11) that ∫ ℝ 2 ∫ ℝ 2 ln | x − y | u 1 ( x ) u 2 ( x ) u 1 ( y ) u 2 ( y ) d x d y is well-defined on X . We can get that E γ , a is of class C 1 .

The existence and nonexistence of minimizers of the constrained variational problem (1.6) depend strongly on the following Gagliardo-Nirenberg type inequality (cf. ([2], lemma A. 2])):

where ( u 1 , u 2 ) ∈ H 1 ( ℝ 2 ) × H 1 ( ℝ 2 ) and Q is the unique positive solution of (1.2). For any θ ∈ ( 0 , 2 π ) , the equality in (1.12) can be attained at ( Q sin θ , Q cos θ ) . Note from ([10], Proposition 4.1) that Q decays exponentially as | x | → ∞ in the sense that

Moreover, recall from [11] that Q is the achievement function of the equality in the following classical Gagliardo-Nirenberg inequality:

We then conclude from (1.2) and (1.14) that

Based on the above results, we can establish the existence and nonexistence of minimizers for the constrained variational problem (1.6).

Theorem 1.1Let Q ( x ) = Q ( | x | ) > 0 be the unique positive solution of(1.2)and a * : = | Q | 2 2 , assume that k ( x ) always satisfies(1.7)and(1.8).

1) If − 1 ≤ k ( x ) ≤ 0 , then there exists at least one minimizer of e ( γ , a ) for γ ≠ 0 and a > 0 .

2) If 0 < k ( x ) ≤ 1 , then there exists at least one minimizer of e ( γ , a ) for γ ≠ 0 and 0 < a < 2 a * , and then there is no minimizer of e ( γ , a ) for γ ≠ 0 , a > 2 a * , or γ > 0 , a = 2 a * .

Theorem 1.1 establishes the existence and nonexistence of minimizers of e ( γ , a ) . When − 1 ≤ k ( x ) ≤ 0 , we can get that E γ , a is bounded from below based on the non-positivity of k ( x ) , then we can get the existence of minimizers of e ( γ , a ) . As for 0 < k ( x ) ≤ 1 , then we can also get the existence of minimizers of e ( γ , a ) by rewriting E γ , a into (2.8). Moreover, when a ≥ 2 a * , we use appropriate test function to get that there is no minimizer of e ( γ , a ) . If ( u 1 a , u 2 a ) ∈ M is a minimizer of e ( γ , a ) for γ ≠ 0 and 0 < a < 2 a * , then the variational theory shows that ( u 1 a , u 2 a ) solves

where λ a ∈ ℝ is the Lagrange multiplier associated with (1.3) and satisfies that

The second result in this paper deals with the limiting behavior of minimizers of e ( γ , a ) as a ↗ 2 a * , where γ > 0 is fixed. We always denote by → and ⇀ the strong convergence and the weak convergence, respectively.

Theorem 1.2Let ( u 1 a , u 2 a ) ∈ M be a minimizer of e ( γ , a ) , where γ > 0 is fixed and a ∈ ( 0 , 2 a * ) . Assume that 0 < k ( x ) ≤ 1 and k ( x ) ∈ C 0 , α ( ℝ 2 ) for some α ∈ ( 0 , 1 ) , then there exist a sequence { y ε a } ⊂ ℝ 2 and a point x 0 ∈ ℝ 2 such that

Moreover, lim a ↗ 2 a * λ a ε a 2 = − 1 . Particularly, if ( u 1 a , u 2 a ) ∈ M is non-negative, then 8 π ( 2 a * − a ) γ a *   u i a ( 8 π ( 2 a * − a ) γ a *   x + x i a ) → Q ( x ) 2 a * in H ∩ L ∞ ( ℝ 2 ) as a ↗ 2 a * , where x i a is the unique global maximum point of u i a and x i a → 0 as a ↗ 2 a * .

For any minimizer ( u 1 a , u 2 a ) ∈ M , Theorem 1.2 provides the preliminary limiting behavior as a ↗ 2 a * . Moreover, Theorem 1.2 also gives the more refined convergence information when the minimizer is non-negative. In order to prove Theorem 1.2, we shall show that

The above Theorems show that the system (1.1) has a stable normalized state when 0 < a < 2 a * , and the stable state disappears when a ≥ 2 a * . The inhomogeneous interaction k ( x ) only affects the range of stable states and does not alter the cole role of critical threshold. In short, this result reveals the stability conditions and critical behavior of two-component BECs under microwave field, providing a physical basis for the artificial control of BECs configuration by regulating the interaction strength.

The rest of this paper is organized as follows. In section 2, we prove Theorem 1.1 on the existence and nonexistence of minimizers. In section 3, we analyze the limiting behavior of minimizers and prove Theorem 1.2.

In this section, we prove the existence and nonexistence of minimizers for e ( γ , a ) , where γ ≠ 0 and a > 0 . For the case that 0 < k ( x ) ≤ 1 , the crucial technique is that we rewrite the energy functional E γ , a ( u 1 , u 2 ) into a new form (2.8) and we use an important estimate (2.6).

Proof of Theorem 1.1. 1) For the case that − 1 ≤ k ( x ) ≤ 0 , we prove the existence of minimizers of e ( γ , a ) , where γ ≠ 0 and a > 0 . By Young inequality, we derive from (1.10) that

Recall from [11] the general Gagliardo-Nirenberg inequality: for any v ∈ H 1 ( ℝ 2 ) and q ≥ 2 ,

where Q q is the unique positive solution of

Combining (1.11) with (2.2) yields that there exists a constant C > 0 such that

Following (1.9), (1.12), (2.1) and (2.3), we infer that

which implies that E γ , a is bounded from below on M . Let { ( u 1 , n , u 2 , n ) } ⊂ M be a minimizing sequence of e ( γ , a ) . We get from (2.4) that ∫ ℝ 2 ( | ∇ u 1 , n | 2 + | ∇ u 2 , n | 2 ) d x and ∫ ℝ 2 | x | 2 ( u 1 , n 2 + u 2 , n 2 ) d x are bounded uniformly with respect to n . Combining with the fact | u 1 , n | 2 2 + | u 2 , n | 2 2 = 1 , we then obtain that { ( u 1 , n , u 2 , n ) } is bounded uniformly in X . By (1.5), there is ( u 1 , u 2 ) ∈ X such that

as n → ∞ , which indicates that ( u 1 , u 2 ) ∈ M . By the weak lower semicontinuity, we have

We then claim that

Actually, note that

We denote the norm of any function u in H 1 ( ℝ 2 ) by ‖ u ‖ 1 : = [ ∫ ℝ 2 ( | ∇ u | 2 + u 2 ) d x ] 1 2 for convenience. In view of

we then deduce that

where we use the fact that there exists a constant C > 0 , independent of n ∈ ℤ + and x ∈ ℝ 2 , such that

Similarly, we also have

The claim (2.5) is thus proved.

It follows from Hölder inequality that

then we can get that

We conclude from the above that

which means that e ( γ , a ) = E γ , a ( u 1 , u 2 ) , and hence ( u 1 , u 2 ) is a minimizer of e ( γ , a ) .

2). For the case that 0 < k ( x ) ≤ 1 , we prove the existence and nonexistence of minimizers of e ( γ , a ) .

Case 1: γ ≠ 0 and a ∈ ( 0 , 2 a * ) . For any a ∈ ( 0 , 2 a * ) , there exists a constant β > 0 such that

We rewrite the energy functional E γ , a ( u 1 , u 2 ) as

Following (1.9), (1.12) (2.1), (2.3), (2.7) and (2.8), we infer that

which indicates that E γ , a is bounded from below on M . Using the similar argument as the case that − 1 ≤ k ( x ) ≤ 0 , one can obtain that e ( γ , a ) admits at least one minimizer.

Case 2: γ ≠ 0 and a > 2 a * . For any a > 2 a * , we next prove the nonexistence of minimizers of e ( γ , a ) . Letting 0 ≤ θ ≤ 1 , we consider the test function