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

−Δu +V x ð Þu = Iα ∗ u j j u j jp−2u, u ∈H1 R  , ð3Þ appears in various physical contexts (see [3–6]). Mathematically, equations of this type have received considerable attention due to the appearance of the nonlocal term ðIα ∗ jujpÞjujp−2u, which makes the problem challenging and interesting. The readers can refer to [4, 7–18] 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:

As we know, when α ⟶ 0, 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 [22][23][24][25][26][27][28] 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 where Then, a solution of system (1) can be found as a critical point of the energy functional E : H ↦ ℝ defined by Set We first show that c λ is attained.
Þ is a sequence satisfying λ n ⟶ 0 + as n ⟶ +∞, then up to a subsequence, andv is a ground state of 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 c λ 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 2 Advances in Mathematical Physics which can be considered an extension of the classical problem By the results that S 1 is attained if and only if where A > 0 is a fixed constant, a ∈ ℝ N , and b ∈ ð0,∞Þ (see ( [1], Theorem 3.1) or ( [2], Theorem 4.3)), and studying the minimum point of a function hðτÞ defined on ½0, +∞Þ by we show that S 0 is attained at 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 c λ . Our second goal is to show the existence of a higher energy vector solution of (1).
Remark 5. For λ > 0 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 λ > 0 is small enough.
Finally, we prove the nonexistence of the nontrivial solution of system (1) by establishing the Pohozaev type identity.
then, system (1) has no nontrivial solutions in H: 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.

Advances in Mathematical Physics
Then, (21) follows from (22) and (25). For the case λ < 0, the conclusions follow by replacing ðU b , τ min U b Þ with ðU b , −τ min U b Þ and repeating the proof previously.
For equations (11) and (12), we set and Then, according to ( [17], Theorem 3 and Theorem 6), we have and B i is achieved, where S 1 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 c λ . By (H3), we have for some C > 0, which suggests that there exists M 1 > 0 such that ∥ðu, vÞ∥>M 1 : Thus, we obtain Second, we show From the assumptions (H1)-(H3), we see that 0 < |λ | <1, and so Theorem 7 holds. For the case λ > 0, by Lemma 8, The last inequality in (34) follows from Theorem 7 and direct calculation. Denote To prove (33), it is enough to show Advances in Mathematical Physics we have Then, by a transformation x = a + bz, 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 U and V, respectively. Since ðU, 0Þ ∈ N and ð0, VÞ ∈ N , we have c λ ≤ min fB 1 , B 2 g: If c λ = min fB 1 , B 2 g, then we see that at least one of ðU, 0Þ and ð0, VÞ is a solution of system (1), which is impossible since λ ≠ 0, so (40) holds.

Proof of Theorem 18
In this section, we study the existence of a higher energy vector solution of system (1) for λ > 0 sufficiently small. We suppose that B 1 ≤ B 2 without loss of generality. Let U, V be ground states of (11) and (12), respectively. Then, we may assume that U and V are positive since |U | and |V | are also ground states of (11) and (12), respectively. Now, we set Advances in Mathematical Physics Then ðU, VÞ ∈ A. Moreover, by a similar argument as that in ( [28], Lemma 12), we obtain the following.

Lemma 12.
Assume that (H1) and (H2) hold. Then, A ⊂ H is compact, and there exist 0 < a 1 < a 2 such that Proof. The proof can be found in ( [28], Lemma 12) and will be omitted here.
Next, we will construct a PS sequence using a perturbation approach.
Proof. We prove indirectly. Suppose that there exists fλ n g satisfying lim n⟶∞ λ n = 0 and fðu n , v n Þg ⊂ E m λ n λ n ∩ ðA d \ A d/2 Þ with ∥E λ n ′ ðu n , v n Þ∥⟶0 as n ⟶ ∞. Then, we see immediately that E λ n ðu n , v n Þ ≤ĉ 0 from Lemma 13, and ðu n , v n Þ ⟶ ðu, vÞ in H for some ðu, vÞ ∈ A by Lemma 14. Thus, ðu n , v n Þ ∈ A d/2 for n sufficiently large, which is in contradiction with fðu n , v n Þg ⊂ E m λ n λ n ∩ ðA d \ A d/2 Þ, so the conclusion holds.
In the sequel, we assume that d, a, λ be fixed such that Lemma 15 holds.