On the Ground State to Hamiltonian Elliptic System with Choquard’s Nonlinear Term

Wang, Wenbo; Zhou, Jianwen; Li, Yongkun

doi:https://doi.org/10.1155/2020/8358629

Advances in Mathematical Physics

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Research Article | Open Access

Volume 2020 | Article ID 8358629 | https://doi.org/10.1155/2020/8358629

On the Ground State to Hamiltonian Elliptic System with Choquard’s Nonlinear Term

Wenbo Wang,¹Jianwen Zhou,¹and Yongkun Li¹

Academic Editor: Wen-Xiu Ma

Received14 Jan 2020

Accepted23 Mar 2020

Published05 May 2020

Abstract

In the present paper, we consider the following Hamiltonian elliptic system with Choquard’s nonlinear term where is a bounded domain with a smooth boundary, , , and is the primitive of , similarly for . By establishing a strongly indefinite variational setting, we prove that the above problem has a ground state solution.

1. Introduction and Main Results

In this paper, we deal with the existence of ground state solutions for the following Hamiltonian elliptic system with Choquard’s nonlinear term: where is a bounded domain with a smooth boundary, , , , and is the primitive of , similarly for . For a single equation in whole space, is closely related to the Choquard-Pekar equation:

When , , , and , equation (2) has appeared in several contexts of quantum physics. In 1954, Pekar used equation (2) to describe a polaron at rest in quantum theory. In 1976, to model an electron trapped in its own hole, P. Choquard considered (2) as a certain approximation to Hartree-Fock’s theory of one component plasma (see [1, 2]). In some particular cases, (2) is also known as the Schrödinger-Newton equation which was introduced by Penrose in [3] to describe the self-gravitational collapse of a quantum mechanical wave function.

For , , , and , the existence of ground states of (2) was obtained in [4] by variational methods. Later, Moroz and Van Schaftingen [2] investigated the regularity, radial symmetry, and asymptotic behavior at infinity of positive solutions for a generalized Choquard equation. Other results involving existence, multiplicity, and concentration of Choquard’s problems can be found in [5–12] and the references therein.

We also observed that Bhattarai [13] considered the coupled Choquard-type fractional Schrodinger systems: where and appear as the Lagrange multipliers, , , , , , and . Bhattarai looked for minimizers of the -constrained minimization problem via concentration compactness techniques. Recently, Giacomoni et al. [14] were also concerned with the coupled Hardy-Littlewood-Sobolev critical nonlinearity fractional Schrӧdinger system in a smooth bounded domain : where , . By minimizing over a suitable subset of Nehari’s manifold, they proved the existence of at least two nontrivial solutions for a suitable range of and . For problems involving Hardy-Littlewood-Sobolev’s critical exponent problems, please see [15–17] and the references therein. Among them, a strongly indefinite Choquard equation was studied in [17].

Motivated by the papers mentioned above, in particular, the papers [13, 14, 17], the purpose of the present paper is to investigate the ground state solution of problem (1). To the best of our knowledge, there is no work concerning the existence of ground state solutions to the Choquard-type Hamiltonian elliptic system. For the Hamiltonian elliptic system, we refer the readers to the papers [18–22] and the references therein.

Throughout this paper, we will always assume and we suppose that , , and satisfy the following assumptions. (V), lies in a gap of , where is the spectrum of the operator .(H₁).(H₂) as , as .(H₃), , where , satisfy (H₄)There exists , such that : (H₅) for and for .

Before stating our main result, we review the definition of ground state solution about (1). We call as a ground state solution of (1) in work space (see Section 2); if is another weak solution (see Definition 2), then the corresponding energy functional , where will be defined in equation (27) or equation (32).

Theorem 1. Suppose that H₁, H₂, H₃, H₄, and H₅ are satisfied. Then (1) has a ground state solution.

To prove Theorem 1, here we use a minimizing argument based on the ideas developed in [23]. We are concerned with system (1) in whole space involving the Hamiltonian elliptic system with Choquard’s nonlinear term; it is a nonlocal problem, which brings about two obstacles. One is to check the linking structure, and the other one is to prove the boundedness of the corresponding (PS) sequences. To avoid these obstacles, we shall deal with our problem in bounded domain. Indeed, the difficulty is still there for the problem on . It is worth pointing out that the monotonicity condition like [23] is not required on the nonlinear terms and ; it prevents us from using the standard way (see, e.g., [23, 24]) to check that the minimizer is a critical point. Via the basic leitmotiv from Proposition 3.2 of [25], we shall use the deformation lemma to prove it (see Lemma 13.. We end this section by giving our arrangements of this paper. In Section 2, we establish the variational settings about (1). In Section 3, we provide some lemmas and then prove Theorem 1.

2. Variational Settings

For the system if , by Hardy-Littlewood-Sobolev’s inequality (see Theorem 4.3 in [26]), then the term is well defined if for such that . In view of Sobolev’s embedding theorem, it requires that , which leads us to assume that

Since we restrict , and

So it holds that and but or could be supercritical in the sense that

We remark that is used in (38) and similarly for . Since or could be supercritical, we need the fractional Sobolev spaces (see, e.g., [19, 22, 27]). According to H₄, we can choose , with , such that

Denote . Since the effective domain and is symmetric, so is self-adjoined on . Since the self-adjoint operator is closed, according to the polar decomposition theorem (Theorem VIII.32 in [28] and jointly with Theorem 3.2 and 3.3, Ch IV in [29]), it holds that there is a positive self-adjoint operator (in fact ), with and a partial isometry such that and are uniquely determined and

It is well known that and are both self-adjoint operators on . In view of Theorem 3.35 in Chapter V of [30] and Corollary 5.5.6 in [31], there is a unique square root operator such that furthermore, Denote Let

According to [19, 27], we consider a basis of constituted by eigenfunctions of with associated eigenvalues . If for , we write , then the effective domain of and are

They are two Hilbert spaces endowed with the following inner product, respectively, where is an isometric isomorphism. So, has the inverse , and we denote , similarly for . Set , then is a Hilbert space with the inner product and norm for . We recall the embedding theorem (see [32]); for , the embedding is continuous for and is compact for . We also consider the bounded self-adjoint operator defined as follows, for :

A natural question will be asked whether the operator is well defined. Here, we only check if is well defined. In fact, noting that and are self-adjoint operators, we infer that

By the complex interpolation theory (see Chapter 1.15 of [32]), we get

So .

Next, we have that has only two eigenvalues and , whose corresponding eigenspaces are

Clearly, . Indeed, 0 lies in a gap of , and using Theorem 3.3 in Chapter IV of [29], we get where .

Similar to Proposition 1.1 in [19], for , we can easily get

We consider the eigenvalue problem in , which yields that

Here, we have used the fact that and (see [22]).

We consider the corresponding energy functional for (7):

Similarly ([27], p. 61), under our assumptions, using the fact that and are linear, jointly with Lebesgue’s dominated convergence theorem, the Gateaux derivative of in the direction at is defined as

Clearly, is linear bounded about and continuous about . Thus, . And its Fréchet derivative is given by

Definition 2. We say that is a weak solution of (7), if for each , there holds

It is easy to check that the critical point of is a weak solution of (7). Moreover, for , , in view of Lemma 2.1 in [19], it holds that

Therefore,

Remark 3. If , then .

3. Existence of Ground States

Following [23] (in page 3804), we introduce the generalized Nehari manifold

We need to prove that .

For , define given by

It is easy to check the following lemma (see, e.g., Lemma 3.1 in [25]).

Lemma 4. is a critical point of if and only if .
For , set , where .

Lemma 5. For each , there exists , such that

Moreover, .

Proof. By Lemma 4, it suffices to prove that the maximum exists. This follows from the next two lemmas. Indeed, using Lemma 7 and Lemma 8, combining with , it is easy to check that the maximum point exists.

Remark 6. Even though and have satisfied the condition in [23], we cannot have the uniqueness of . So, we cannot get the result similar to Proposition 2.3 in [23].

Lemma 7. There are and such that , where .

Proof. For a given , by H₂, H₃, and H₄, there is a such that By Hardy-Littlewood-Sobolev’s inequality, one has where . Here we use the fact that Similarly, we have Thus, for , it holds that Hence, we can choose some , such that for all .

Lemma 8. For every , there is a such that on .

Proof. If not, there exists such that and .
Without loss of generality, we may assume that , . By doing where . Set Obviously, . Jointly with one has . So, for a subsequence, , , , a.e., in . Hence, . It means that for where . Thus, for , .
It follows from H₂, H₃, and H₅ that there exists , such that and Thus, , it follows from Fatou’s lemma that which would conflict with (38). Here, we use the fact that is bounded.

Lemma 9. contains all nontrivial critical points.

Proof. If , , and , by H₅ and H₆, one has For , it follows that Define

We shall prove that is achieved on and the minimizer is a critical point of .

Lemma 10. (1)There exists such that where .(2) for every .

Proof. First, let us prove assertion (1). For every , with Lemma 7 in hand, we can choose some , small such that Second, for every , which concludes the proof of assertion (2).

Lemma 11. is coercive on .

Proof. If not, there exists a sequence such that and for some . Let . Up to a subsequence, and , a.e., in .
If , one has for , . So it holds that Jointly with Lemma 5, we obtain It is absurd if . So .
Similar to the proof of Lemma 7, it holds that Therefore, which is a clear contradiction.

Lemma 12. is achieved.

Proof. The proof is similar to Lemma 3.6 in [33], but for readers’ convenience, we review the details of the proof.
Let be a minimizing sequence. In the light of Lemma 10, is bounded in . Extracting a subsequence if necessary, we have in . By the continuity of the projections, .
Now let us prove that . If not, by Lemma 5 and the result (2) of Lemma 9, we get It is impossible if is large.
By Lemma 5, there exists such that With Lebesgue’s dominated theorem, it follows that

Lemma 13. If is such that then .

Proof. Assume by contradiction that , then and exist such that where (if necessary, we can choose another and small enough such that ).
Define given by .
Denote , , and . Define given by .
It is clear that For it yields a deformation (see Lemma 2.3 of [34] or A.4 of [35]) such that (a) if ,(b), where .Let us define a and for every , by Lemma 5 and the degree theory now yield that Since , combining with (53), it therefore holds that It follows from (a) that Consequently, we obtain Then, we know that there exists such that . This implies that Noting that and also , in view of (b), one has which is a clear contradiction.

Proof of Theorem 14. The existence of ground state solution of (1) follows directly from Lemma 12 to Lemma 13.

Data Availability

Data are available upon request. All data will be such that no personal identifying information will be included.

Conflicts of Interest

The authors declare that there is no conflict of interest regarding the publication of this paper.

Authors’ Contributions

The authors contributed to each part of this paper equally. The authors read and approved the final manuscript.

Acknowledgments

This work is supported by the National Natural Sciences Foundation of People’s Republic of China under Grants 11901514, 11861072, 11961078, and 11561072.

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