Advances in Mathematical Physics

Advances in Mathematical Physics / 2020 / Article

Research Article | Open Access

Volume 2020 |Article ID 6623902 | https://doi.org/10.1155/2020/6623902

Huiling Wu, "Vector Solutions for Linearly Coupled Choquard Type Equations with Lower Critical Exponents", Advances in Mathematical Physics, vol. 2020, Article ID 6623902, 12 pages, 2020. https://doi.org/10.1155/2020/6623902

Vector Solutions for Linearly Coupled Choquard Type Equations with Lower Critical Exponents

Academic Editor: Jacopo Bellazzini
Received17 Oct 2020
Revised21 Nov 2020
Accepted24 Nov 2020
Published21 Dec 2020

Abstract

The existence, nonexistence, and multiplicity of vector solutions of the linearly coupled Choquard type equations are proved, where , , are positive functions, and denotes the Riesz potential.

1. Introduction

We deal with the linearly coupled Choquard type equations: where , , and are positive functions, is the lower critical exponent with respect to a Hardy-Littlewood-Sobolev inequality (see ([1], Theorem 3.1) or ([2], Theorem 4.3)), and denotes the Riesz potential defined on by

The single equation appears in various physical contexts (see [36]). Mathematically, equations of this type have received considerable attention due to the appearance of the nonlocal term , which makes the problem challenging and interesting. The readers can refer to [4, 718] and references therein for research on related problems.

Recently, Chen and Liu [19] established the existence and asymptotic behavior of the vector ground state of the linearly coupled system: where , . Xu, Ma and Xing [20] extended the results in [19] to (4) in the case that and are replaced with general subcritical nonlinearities and , respectively. Yang et al. [21] obtained the existence of the vector ground state of (4) in the following three cases:

They also proved that (4) has no nontrivial solutions if or .

As we know, when , the local system which has application in a large number of physical problems such as in nonlinear optics, can be regarded as a limiting system of (4). Systems of this type have received great attention in recent years (see [2228] for instance). However, linearly coupled systems with nonlocal nonlinearities have been less studied.

In this paper, we are interested in the existence, nonexistence, and multiplicity of solutions of system (1) with positive nonconstant potentials. We assume that

(H1) and

(H2) ,

(H3)

For simplicity, the integral is denoted by . According to (H1), the norm in can be defined by where

Then, a solution of system (1) can be found as a critical point of the energy functional defined by

Set

We first show that is attained.

Theorem 1. Assume that (H1), (H2), and (H3) hold. Then, there exists a vector ground state of system (1). Additionally, if is a sequence satisfying as , then up to a subsequence, either or in as , where is a ground state of and is a ground state of

Remark 2. We call a solution of system (1) a nontrivial solution if and a vector solution if and . A nontrivial solution satisfying for any nontrivial solutions of system (1) is called a ground state.

Remark 3. Under assumptions (H1) and (H2), the existence of ground states of equations (11) and (12) has been proved by Moroz and Van Schaftingen ([17], Theorem 3 and Theorem 6).

To prove Theorem 1, it is crucial to give an estimate of the upper bound of the least energy due to the lack of compactness. In our case, the estimate is quite involved, since we are dealing with a coupled system, which is more complex than a single equation. The method we follow can be sketched as follows. We first study the minimizing problem which can be considered an extension of the classical problem

By the results that is attained if and only if where is a fixed constant, , and (see ([1], Theorem 3.1) or ([2], Theorem 4.3)), and studying the minimum point of a function defined on by we show that is attained at if and at if (see Theorem 7 in Section 2), which combined with the existence of ground states for equations (11) and (12) enables us to obtain the precise upper bound of .

Our second goal is to show the existence of a higher energy vector solution of (1).

Theorem 4. Assume that (H1) and (H2) hold. Then, for some , there exists a vector solution of system (1) if . Additionally, if is a sequence satisfying as , then up to a subsequence, in , where is a ground state of (11) and is a ground state of (12).

Remark 5. For sufficiently small, it is trivial to see that the solutions obtained in Theorem 1 and Theorem 4 are different, which implies that there exists at least two vector solutions of system (1) if is small enough.

Finally, we prove the nonexistence of the nontrivial solution of system (1) by establishing the Pohozaev type identity.

Theorem 6. Assume that (H3) holds. If and then, system (1) has no nontrivial solutions in

This paper is structured as follows. Some preliminary results are provided in Section 2. The proofs of Theorems 1 and 4 are presented in Section 3 and Section 4, respectively. In Section 5, we show the nonexistence of nontrivial solutions.

2. Preliminary Results

In this section, we show the sharp constant defined in (13) is attained and give an estimate of the upper bound of

Theorem 7. If , then is attained. Moreover, (or ) is a solution of (13) for (or ), where is a minimum point of defined on by

Proof. First, we show that there exists such that Calculating directly, we have Set It can be easily seen that as , and as Then, there is such that and
In the next step, we prove where is defined in (14). We employ the idea in ([29], Theorem 5) to prove (21). For the case , taking gives Let be a minimizing sequence for Set , where Then, Collecting (23) and (24) leads to Then, (21) follows from (22) and (25). For the case , the conclusions follow by replacing with and repeating the proof previously.

Lemma 8. Assume that (H1) and (H3) holds, then for any , there exists such that and

Proof. This result is standard and the proof can be found in ([30], Lemma 12). We omit it.

For equations (11) and (12), we set and where

Then, according to ([17], Theorem 3 and Theorem 6), we have and is achieved, where is defined in (14). By Theorem 7 and ([17], Theorem 3 and Theorem 6), we are able to get the following estimate.

Lemma 9. Assume that (H1), (H2), and (H3) hold. Then,

Proof. We first show the positivity of . By (H3), we have for some , which suggests that there exists such that Thus, we obtain Second, we show From the assumptions (H1)–(H3), we see that , and so Theorem 7 holds. For the case , by Lemma 8, there is such that ; then, we have The last inequality in (34) follows from Theorem 7 and direct calculation. Denote To prove (33), it is enough to show for some . Since we have Then, by a transformation , we get Taking the assumption (H2) into consideration, we see that that (36) holds. Then, (33) follows from (34).
Now, it remains to show Denote ground states of (11) and (12) by and , respectively. Since and , we have If , then we see that at least one of and is a solution of system (1), which is impossible since , so (40) holds.

3. Proof of Theorem 11

Lemma 10. Assume that (H1), (H2), and (H3) hold. Then, there exists a vector ground state of system (1).

Proof. According to Ekeland’s variational principle, there exists such that For simplicity, we denote . First, we prove . Indeed, for some and sufficiently large . Particularly, From the proof of Lemma 9, we observe that there exists such that . Then, we have Taking (43) into consideration, we obtain that as . Then, from (42), we get
We may assume that Then, To complete the proof, it is sufficient to prove that Actually, if By Fatou’s lemma, we get Furthermore, since and , we see from (1) that and , that is, is a vector ground state of (1).
Suppose the assertion is false, that is, On the one hand, we know from (H1) that as . Then, it follows as . Using Theorem 7, we obtain On the other hand, Collecting (49) and (50) yields which contradicts Lemma 9. Thus,

Proof of Theorem 11. By Lemma 10, we need only show the asymptotic behavior of the vector ground state when First, we claim that decreases strictly monotonically with respect to Indeed, fix with Denoting a vector ground state of system (1) when by and letting be the constant such that , we obtain Then, by , we deduce that , which gives