Multiplicity of solutions for a class of critical Schrödinger-Poisson systems on the Heisenberg group

: We deal with multiplicity of solutions to the following Schrödinger-Poisson-type system in this article:

where Δ H is the Kohn-Laplacian and Ω is a smooth bounded region on the first Heisenberg group 1 , μ 1 , and μ 2 are some real parameters, and u satisfying natural growth conditions.By the limit index theory and the concentration compactness principles, we prove that the aforementioned system has multiplicity of solutions for

Introduction and main results
In this article, we are interested in the existence of multiplicity of solutions for the following Schrödinger-Poisson-type system on the Heisenberg group of the form: The condition ( ) Thus, for any > ε 0 and fixed u, there exist ( ) ( ) > a ε b ε , 0 such that Hence, by (1.2) and (1.3), we have for any constants β, fixed u, and ( ) > c ε 0. Furthermore, we assume that ( ) F ξ u v , , also fulfills the following assumption: (F 4 ) There exist < < > p σ 2 4, 0 , and The novelty of our work is that we link several different phenomena to one problem.The characteristics of this article are as follows: (1) We emphasize that no results are available in the current literature for the critical Schrödinger-Poisson system on the Heisenberg group.In this regard, the results demonstrated in this article are completely new.(2) The difficulty in solving Problem (1.1) is the lack of compactness, which can be illustrated by the fact that the embedding of ( ) Before describing the main results of this article, we give some concepts about the Heisenberg group.
be the first Heisenberg group.If x y t , , 1 , then the group law is defined by: A natural group of dilations on 1 is given by ( ) ( ) = δ ξ sx sy s t , , for any positive number s.Hence, ⋅ H is homogeneous of degree one with respect to the dilation δ λ .The left-invariant distance d H on 1 is accordingly defined by: 1 Also, the Heisenberg ball of radius r centered at ξ 0 is the set , .
The natural volume in 1 is the Haar measure, which coincides with the Lebesgue measure L 3 in ; 3 then, . A basis for the Lie algebra of left-invariant vector fields on 1 is given by: and the horizontal gradient Then, the Kohn-Laplacian Δ H is denoted by

H H H
It is well known that Δ H is a very degenerate elliptic operator and Bony's maximum principle is satisfied (see [1]).For more detailed settings about the Heisenberg group, we can refer to [2].
In recent years, the geometric analysis of the Heisenberg group has attracted much attention of many scholars due to its important applications in quantum mechanics, partial differential equations, and other fields.The analysis of the Heisenberg group is very interesting because this space is topologically Euclidean, but analytically non-Euclidean, so there are some basic analytical ideas.The Schrödinger-Poisson system is a standard model in quantum mechanics to describe electron motion on a positive charge background (see [3,4]).The investigation of Problem (1.1) is motivated by the existence of several recent mathematical research.To be more precise, in the Euclidean case, many scholars have studied the following Schrödinger-Poisson system: where . If ( ) = f s s, many researchers have come up with interesting results (see [5][6][7][8]).In particular, Azzollini et al. in [9] proved the existence and nonexistence results of Problem (1.5) when f is subcritical and critical.And just recently, Lei and Suo [10] obtained two positive solutions for the following system: where ( ) ∈ q 1, 2 and ∈ + λ is small enough.In addition, the Schrödinger-Poisson systems with critical growth on 3 were also investigated extensively and we refer the readers to [11][12][13].In [14], the authors studied the following Schrödinger-Poisson-type system where < < q 1 2, by the Green's representation formula and the critical point theory, they obtained at least two positive solutions and a positive ground-state solution.
As we know, the limit index theory proposed by Li [15] is one of the most effective ways to study the existence of infinite solutions of equations in a Euclidean setting.For example, Song and Shi [16] considered the noncooperative critical nonlocal system with the limit index theory.Baldelli et al. [17] proved existence results in N for an elliptic system of ( ) p q , -Laplacian type involving a critical term, nonnegative weights, and a positive parameter λ.Not long after, they studied elliptic systems of (p q , )-Laplacian involving a critical term and a subcritical term in [18].In particular, nonnegative nontrivial weights satisfying some symmetry conditions with respect to a certain group are included in the nonlinearity.
Inspired by the aforementioned literatures, this article mainly studies the existence of multiplicity solutions for Problem (1.1).To the best of our knowledge, this article first deals with this kind of Schrödinger-Poisson system with the Kohn-Laplacian.Furthermore, although some properties are similar between Kohn-Laplacian Δ H and the classical Laplacian Δ, the similarities may be deceitful (see, e.g., [19]).In addition, the critical exponent Q* is equal to 4 on 1 , while 2* is equal to 6 on 3 , which has created us some obstacles in proving the existence of solutions to Problem (1.1).In order to overcome these difficulties, we will use concentration compactness principles on the Heisenberg group.
Critical Schrodinger-Poisson system on the Heisenberg group  3 Definition 1.1.We say that ( . The main result of Problem (1.1) is as follows.
vergence is denoted by ⇀ (resp., → ), and c i is a positive constant and can be determined in concrete conditions.
This article is organized as follows.In Section 2, we present some necessary preliminary knowledge on the Heisenberg group functional setting and collect some properties about the Folland-Stein space ( ) S Ω 0 1 . In Section 3, we prove some preliminary lemmas and Theorem 1.1.Section 4 is devoted to the proofs of Theorem 1.1.

Sobolev spaces and limit index theory
This section will be divided into two parts.First, we briefly review the definitions and list some basic properties of Sobolev spaces.Second, we recall the limit index theory of Li [15].

Sobolev spaces
In this section, we first give some basic results for our space ( ) S Ω 0 1 which will be used later.The Folland-Stein space ( ) S Ω 0 1 is defined as the closure of ( ) with respect to the norm: is a Hilbert space and the embedding ( ) In particular, Jerison and Lee [21] proved that the best Sobolev constant is achieved by the ∞ C function: where c 0 is a suitable positive constant.In other words, the function U is a positive solution to the following equation: and Furthermore, for any > ε 0, the scaling function is alone a solution of (2.2).This deep result of Jerison and Lee [21] is the Kohn-Laplacian counterpart of a celebrated theorem of Talenti [22] for the classical Laplace operator.Since ( ) is separable and reflexive space, there exists the basis { } ( ) , which is characterized by the relationship: (see also [23] for more details).Furthermore, it is easy to see that the following attributes held:

Limit index theory
Next, we review the limit index theory of Li [15].To this end, we introduce the following definitions.
Definition 2.1.[15] The action of a topological group G on a normed space Z is a continuous map The action is isometric if And in this case, Z is called the G-space.
The set of invariant points is defined by: is closed and , be a family of all G-invariant closed subsets of Z, and let Critical Schrodinger-Poisson system on the Heisenberg group  5 be the class of all G-equivariant mappings of Z. Finally, we call the set (where + is the set of all nonnegative integers) such that for all ∈ A B , Σ, and ∈ h Γ, the following conditions are satisfied: Definition 2.3.
[24] An index theory is said to satisfy the d-dimensional property if there is a positive integer d such that , where S 1 is the unit sphere in Z.
Suppose that U and V are G-invariant closed subspaces of Z such that where V is infinite dimensional and where V j is a dn j -dimensional G-invariant subspace of V , = ⋯ j 1, 2, , and and ∀ ∈ A Σ, we and let Definition 2.4.[15] Let i be an index theory satisfying the d-dimensional property.A limit index with respect to ( ) Z j induced by i is a mapping Theorem 2.1.[15] Assume that (B 1 ) There are G-invariant closed subspaces U and V such that V is infinite dimensional and = ⊕ Z U V; (B 3 ) There is a sequence of G-invariant finite dimensional subspaces There is an index theory i on Z satisfying the d-dimensional property; , and by S ρ its relative boundary, that is, , Ω : .
Similar to the proof of Lemma 3.1 in [14], we have the following lemma.
. Then, there exists a unique nonnegative function And the following properties hold: (a) For any nonzero constant s, then and Critical Schrodinger-Poisson system on the Heisenberg group  7 . Moreover, is a solution of Problem (1.1) if and only if is a solution of the following nonlocal problem: Now, we define the functional I : Under the assumptions (F 1 )-(F 3 ), it is easy to prove (see [25]).Therefore, in accordance with the aforementioned argument, we will devote ourselves to the existence critical point of the functional I by using the critical point theory.
Similar to the proof in [25,26], it is easy to obtain the following results.
Then, for every ( , the operator , ˜d  , ,  d.
Define an index γ on Σ by: , if such does not exist.
N Then, we have the following proposition from [26]: γ is an index satisfying the properties given in Definition 2.2.Moreover, γ satisfies the one-dimensional property.According to Definition 2.4, we can obtain a limit index ∞ γ with respect to ( ) X n from γ. Now, we turn to prove the Palais-Smale condition using the following con- centration-compactness principle on the Heisenberg group.
where ( ) Ω is the space of all finite Radon measures on ⊂ Ω 1 , J is an at most countable index set, which can be empty, and family where ∈ ξ Ω j and δ ξj is the Dirac mass at ξ j .Moreover, we have μ Sν for any j J , , 0, .
Lemma 3.5.Let ( ) F 1 -( ) F 3 hold.Then, the functional J satisfies the local ( ) in the following sense: if contains a subsequence converging strongly in X .
Proof.We first show that {( is bounded in X .Note that by ( ) F 1 and ( ) F 3 , we have .
This fact implies that ∥ ∥ u nk is bounded in ( ) S Ω 0 1 . On the one hand, we have Critical Schrodinger-Poisson system on the Heisenberg group  9 .
in (3.8) and together with (3.8), we obtain and C 1 is a some positive number.Thus, (3.9) implies that . Hence, up to a subsequence, . In fact, note that and condition ( ) In the following, we will prove that there exists . Therefore, by Lemma 3.4, we assume that there exist two positive measures μ and ν on Ω such that where j is an at most countable index set, ∈ ξ Ω j , and δ ξj is the Dirac mass at ξ j .Now, we claim that = ∅ J .In fact, we assume that there exists ∈ j J such that ≠ μ 0 j .Then, for > ε 0 small enough, it follows from Lemma 3.2 of [28] that there exists a cut-off function .
Then, by the boundedness of { ( ) } ψ ξ u In addition, it follows from (3.10) and the Hölder inequality that: On the other hand, by using the definition of ψ ε j , and Vitali's convergence theorem, we obtain Critical Schrodinger-Poisson system on the Heisenberg group  11 as → ∞ k .We also observe that the integral goes to 0 as → ε 0. So, by (3.16), we conclude that ≥ ν μ j j .Then, by (3.9), we obtain ≥ μ Sμ j j 1 2 , which implies that If the second case ≥ v S j 2 holds, then by (3.8) and (3.13), we have which contradicts (3.5).Consequently, = ∅ J .Furthermore, we have as → ∞ k .From Lemma 3.6 and (3.18), we obtain H By (3.9), (3.18), and Lemmas 3.1 and 3.6, we also have From (3.20) and (3.21), one has So, from the uniform convexity of ( ) . Thus, we have proved that

. 3
Verification of ( ( ) ) PS c condition Before we begin, let us set a few facts straight.We denote by B ρ the closed ball of radius ρ centered at zero in the Folland-Stein space ( ) S Ω 0 1 from (a) of Lemma 3.1 and (3.10) that

( 3 )
where ρ is to be determined), then by (1.2), we obtain Since all norms are equivalent on the finite-dimensional space Y 1 , we obtain

2 1 2 and I has at least − k 1 0
Critical Schrodinger-Poisson system on the Heisenberg group  13 Let | | = β τ Ω , so we obtain ( ) c in ( ) B 7 .By Lemma 3.7, for any [ condition of ( ) PS * c , then ( ) B 6 in Theorem 2.1 holds.Finally, according to Theorem 2.1, we have that: pair critical points.The proof is thus complete.
Poisson system with the Kohn-Laplacian Δ H .As for the function F , we assume Proof of Theorem 1.1