On the critical Choquard - Kirchho ﬀ problem on the Heisenberg group

: In this paper, we deal with the following critical Choquard - Kirchho ﬀ problem on the Heisenberg group of the form:

Abstract: In this paper, we deal with the following critical Choquard-Kirchhoff problem on the Heisenberg group of the form: where M is the Kirchhoff function, Δ is the Kohn Laplacian on the Heisenberg group N , f is a Carathéodory function, > μ 0 is a parameter and =

Introduction and main results
In this paper, we are interested in the existence of solutions to the following Choquard-Kirchhoff type problem on the Heisenberg group of the form: where M is the Kirchhoff function, u Δ is the Kohn Laplacian on N , f is a Carathéodory function, is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality (see (2.5) in Section 2).
In the Euclidean, we note that there has been an increasing amount of research on the Choquard equation.It is well known that the following Choquard equation was first established in the pioneering work of Fröhlich [14] and Pekar [34] for the modeling of quantum polaron.At the same time, the Choquard equation is also the celebrated Schrödinger-Newton equation in models coupling the Schrödinger equation of quantum physics combining with nonrelativistic Newtonian gravity.Moreover, as described by Fröhlich [14] and Pekar, this model is consistent with the study of the interaction of free electrons in ionic lattices with phonons related to lattice deformations of lattices or with the polarization produced on the media (interaction of an electron with its own hole).For more details, we refer to [10,23,46].
Recently, there has been a lot of interest in studying critical Choquard type equations.Indeed, the critical problem was initially studied in the seminal paper of Brézis and Lieb [7], which was used to deal with the Laplacian equations.After that, there are many generalizations of reference [7] in many directions.In particular, the study of critical Kirchhoff-type problems is more meaningful.For example, Liang et al. [25] considered multiplicity results for Choquard-Kirchhoff type equations with Hardy-Littlewood-Sobolev critical exponents, where Δ is the Kohn Laplacian on the Heisenberg group N and Ω is a smooth bounded subset of the Heisenberg group N with C 2 boundary.Here, and a is a positive real parameter.They proved the Brézis-Nirenberg type result for the aforementioned problem and the regularity of solutions and nonexistence of solutions depending on the range of a by the mountain pass theorem, the linking theorem and iteration techniques and boot-strap method.
When people consider the critical Choquard equation over unbounded domain, it is crucial for people to think of using the concentration-compactness principle for the Choquard equation to solve difficulties caused by the lack of compactness.Therefore, the study of critical Choquard equations is deeply associated with the concentration phenomena, which can take place when one looks for the sequences of approximated solutions.Lions established the celebrated concentration-compactness principles in classical Sobolev space [28][29][30] to describe the lack of compactness for the injection from ( ) More precisely, Lions [28] stated the concentration-compactness principles at finite points in the Euclidean.Later, the concentration-compactness principle at infinity was extended by Chabrowski [9], Bianchi et al. [5], Ben-Naoum et al. [4], which provided some quantitative information about the loss of mass of a sequence at infinity in the Euclidean.To prove the existence of solutions for the critical Choquard equation with upper critical exponent * 2 μ , Gao et al. [15] established the concentration-compactness principle invol- ving the convolution type nonlinearities.However, the aforementioned concentration-compactness principles are established on the Euclidean Spaces.To the best of our knowledge, there seems to be no relevant results that describe the possible concentration properties of a weakly convergent sequence both at finite points and infinity on the Heisenberg group.Moreover, there also seems no application of such a concentration-compactness principle for studying the critical Choquard equation on the Heisenberg group.
Inspired by the aforementioned papers, we are devoted to proving the existence results for problem (1.1).To achieve the purposes, we first establish a version of concentration-compactness principle for the Choquard equation on the Heisenberg group that be able to verify the ( ) PS c condition at special levels c.Then, together with the mountain pass theorem, we obtain the existence and multiplicity of solutions for problem (1.1) in the nondegenerate case and the degenerate case on the Heisenberg group.
To this end, we suppose that the Kirchhoff function → + + M : 0 0 is a continuous and nondecreasing function, and f , V satisfy the following assumptions: Here, we call problem is a continuous function, and there exists Now, we first state the following main results for problem (1.1) in the nondegenerate case. .Moreover, we also obtain the following existence results for problem (1.1) in the degenerate case.
The paper is organized as follows.In Section 2, we give some notations and preliminary results connected to the Heisenberg setting.In Section 3, we shall give the proof of the concentration-compactness principle for the Choquard equation on the Heisenberg group.In Section 4, we are devoted to proving the Palais-Smale condition at some special energy levels.In Section 5, we deal with the existence and multiplicity results for problem (1.1) in the nondegenerate case.In Section 6, we are interested in the proof of Theorems 1.3 and 1.4, that is, to the proof of existence and multiplicity of solutions for problem (1.1) in the degenerate case.

The Heisenberg group
First, we give some basic properties of the Heisenberg group.We also can refer to [16,20,24,33] for a complete treatment on the Heisenberg group functional setting.Let be the Heisenberg Lie group of topological dimension 1, that is, the Lie group that has + N 2 1 as a background manifold, endowed with the non-Abelian group law: n The inverse is given by , , j j j j j j establishes a basis for the Heisenberg regular commutative relation satisfying the real Lie algebra of left invariant vector fields on the Heisenberg group N .That means, 1 is called horizontal.
On the critical Choquard-Kirchhoff problem on the Heisenberg group  213 The horizontal gradient of a C 1 function Obviously, D u is an element of the span of { } = X Y , j j j N 1 .We consider the natural inner product in the span of 1 Therefore, the corresponding Hilbertian norm is defined as follows: for the horizontal vector field D u.
For any horizontal vector field function the horizontal divergence of X is defined by , then the second-order self-adjoint operator u Δ in N is given by is usually called subelliptic Laplacian or Kohn Laplacian on N .According to the celebrated Theorem 1.1, thanks to Hörmander [19], we know that the operator Δ is hypoelliptic.Notably, . At the same time, inspired by the seminal paper of Hörmander [19] on the sum of squares of vector fields and type operators, Folland [12] obtained the fundamental solution of the operator Δ .
As we all know, the generalization of the Kohn-Spencer Laplacian is the horizontal p-Laplacian on the Heisenberg group, , given by For > s 0, there is a dilation naturally associated with the Heisenberg group structure, which is usually defined by ( ) ( ) The Korányi norm is derived from an anisotropic dilation on the Heisenberg group and defined as follows: Thus, the homogeneous degree of the Korányi norm is equal to 1, with respect to dilations We shall define Korányi distance by , .
The Korányi open ball of radius R centered at . For the sake of simplicity, we denote , where ( ) represents the Korányi open ball centered at the natural origin with radius R. For any measurable set | | E , the Haar measure ξ d of Heisenber group N that consists with the ( ) 1 -Lebesgue measure and is invariant under left translations.Therefore, as shown in [24], the topological dimension Moreover, it is Q-homogeneous with respect to dilations: 2 Classical Sobolev spaces in the Heisenberg group be the space of all continuous functions u on Ω such that X u Yu X u , , 2 and Y u j 2 are all continuous in Ω, which can be continuously extended up to ∂Ω.Folland and Stein [11] introduced the space ( ) S N 1 2 , which is connected to Vector fields X j and Y j and the space ( ) S N 1 2 is similar to space ( ) H N

1
. More precisely, the space , for all 1,2, , is a Hilbert space with inner product Therefore, the norm in ( ) S N 1 2 is defined as follows: The space ( ) . Therefore, it follows from a Poincare type inequality that . At the same time, we aslo define a function space N equipped with the norm Folland and Stein [11] obtained the following Sobolev type inequality: there exists a positive constant . In addition, these embeddings are compact for < * ν Q and the embedding is not compact for = * ν Q .The best constant for the embedding ( ) ( ) is defined as follows: By (2.1), we know that > S 0. Also, for any nonempty open set ⊆ Ω N , we obtain That means, infimum is achieved in the case = Ω N .
In [21,22], Jersion and Lee showed that the function: , where B 0 is a positive constant.More precisely, any minimizer of S takes the following form: for suitably > β 0 and ∈ ω N .In [18], the best constant S HG is defined by Lemma 2.1.
[18] The best constant S HG is achieved if and only if where where S is the best constant defined in ( then W is unique minimizer of S HG and satisfies the following: where the function k is a real valued, even and homogeneous of degree , where Then ‖⋅‖ FL defines a norm on where Γ is the usual Gamma function.Equality holds in (2.5) if and only if and U is defined as follows: (2.6) By using Proposition 2.1, the following integral , then with the help of Sobolev-type inequality, (2.7) is defined only if That is, the upper critical exponent on the Heisenberg group.
Together with the assumption is a reflexive Banach space, see [36,Lemma 10] for the proof in the Euclidean context, with minor changes in the Heisenberg setting, and the continuous embedding of ( ) In particular, we have the following theorem thanks to the assumption (V ) (see [6,41]).
, there is a compact embedding

The concentration-compactness principle
In this section, we first give the results of the concentration compactness principle involving convolution type nonlinearities on the Heisenberg group.
converging weakly and a.e. to some u and ω ζ , , be the bounded nonnegative measures.Assume that weakly in the sense of measure, where ν is a bounded positive measure, ω and ζ are bounded nonnegative measures and ∞ ∞ ω ζ , are real numbers on N and define Then, there exists a countable sequence of points { } ⊂ ∈ z j j J N , and families of positive numbers { } ∈ ν j J : , where δ ξ is the Dirac-mass of mass 1 concentrated at ∈ ξ N .
For the energy at infinity, we have , then ν is concentrated at a single point.
Remark 3.1.In [20], it is worth noting that the concentration-compactness principle is extended for the first time from the Euclidean setting to the general context of Carnot groups, which gives us much inspiration into the concentration-compactness results on Heisenberg group.Notably, Heisenberg group is the special class of Carnot groups.In Theorem 3.1, we extends the concentration-compactness principle involving the convolution type nonlinearities from the Euclidean setting to the Heisenberg group.The strategy is similar to the one in the papers of [15]; however, there are some complications due to the non-Euclidean context.At the same time, the whole Heisenberg group N is endowed with noncompact families of dilations and translations, which makes a loss of compactness occur due to the concentration at infinity.To deal with this type of phenomena, we also prove a variant of the concentration-compactness principle of Lions [28], that is the concentration-compactness principle involving the convolution type nonlinearities at infinity on the Heisenberg group.
converging weakly to u, denote by . Applying Lemma 2.4, in the sense of measure, we obtain .
To prove the possible concentration at finite points, we show that where ( ) In fact, we denote N , we have for every > δ 0 there exists > M 0 such that Combining the Riesz potential defines a linear operator and the fact that ( ) → v ξ 0 n a.e. in N , we obtain that For almost all ξ , there exists > R 0 large enough such that On the critical Choquard-Kirchhoff problem on the Heisenberg group  219 where M is given in (3.10).Obviously, for > R 0 large enough and so, we have Therefore, we can obtain for > θ 0 small enough By this and ( ) → ξ Ψ 0 n a.e. in N , we have Combining this and (3.10), we have n N By the Hardy-Littlewood-Sobolev inequality, we have According to (3.9), we obtain Passing to the limit as → ∞ n , we obtain Applying Lemma 1.2 in [28], we know (3.3) holds.
Taking { } = ∈ φ χ z j J j , in (3.12), we have Combining with definition of S HG , we also have By (3.9) and → v 0 Passing to the limit as → ∞ n , we have Using Lemma 1.2 in [28] again, we obtain that (3.2) holds.Now, by taking , in (3.13), we obtain Thus, we proved (3.1) and (3.4) hold.
In the following, we shall prove the possible loss of mass at infinity.Using the same technique as in [4], one can prove (3.6) and (3.7).Let ( ) When → ∞ R , by Lebesgue's theorem, we can obtain that Moreover, by the Hardy-Littlewood-Sobolev inequality, we have On the critical Choquard-Kirchhoff problem on the Heisenberg group  221 which implies Similarly, by the definition of S HG and ∞ ν , we have which means that Then the Hölder inequality and (3.13) show that, for ( ) Therefore, we can deduce that It follows from (3.13) that, for ( ) Hence, for each open set Ω, we have It follows that ν is concentrated at a single point.□

Verification of PS c ( ) condition
The functional associated with problem (1.1) is defined as follows: In what follows, we give the definition of (weak) solutions for problem (1.1) for any To obtain the existence of solutions of problem (1.1), we are able to recover the lack of compactness by applying Theorem 3.1.Some techniques for finding the solutions are borrowed from [17].
where S HG comes from Theorem 3.1, then there exists a subsequence of { } u n n strongly convergent in ( ) On the critical Choquard-Kirchhoff problem on the Heisenberg group  223 and this fact together with such that, up to a subsequence, it follows that as → ∞ n .Therefore, we obtain that In fact, it follows from Theorem 3.1 that there exists an at most countable set of distinct points { } ∈ ξ j j J , nonnegative numbers { } ∈ ν j j J , { } ∈ ω j j J , such that Next, to prove (4.4), we proceed by three steps.
Step 1. Fix ∈ j J. Then we prove that either = ν 0 j or Observe that It is easy to see that and Combining (4.9), ( ) , we obtain that By ( ) M 1 , we obtain ≥ ν m ω j j 0 .For all ∈ j J, it follows from Step 2. We claim that (4.6) cannot occur, so = ν 0 j for all ∈ j J.By contradiction, we assume that there exists j such that (4.6) holds true.Since ( ) Moreover, by ( ) M 2 and ( ) f 3 , we have is given earlier.Combining (4.11) with (4.12), we obtain that from which, by letting → + ε 0 and using (4.5), it implies that which contradicts the assumption.Hence, = ν 0 j for any ∈ j J.
We show that → As in the proof of Theorem 3.1, we have and Thus, a similar discussion as in the proof of Theorem 3.1 gives that .
Let's estimate each of (4.16).We deduce from (4.13) and (4.14) that Clearly, and λ are both convergent.Therefore, by (4.13) and (4.14), we have that Furthermore, by Lemma 2.4, we obtain that Now we define an operator as follows: . Clearly, L is a bounded linear operator, being by the Hölder inequality.Hence, the weak convergence of Clearly, ( ) Therefore, by (4.3), we obtain that . Therefore, (4.4) holds true.The proof is thus completed.□ On the critical Choquard-Kirchhoff problem on the Heisenberg group  227 5 Non-degenerate case for problem (1.1)

Proof of Theorem 1.1
Now we state the general version of the Mountain Pass theorem in [1,2], which will be used later.
Theorem 5.1.Let E be a Banach space and ( ) and there exists a ( ) .
Next, we show that I μ satisfies geometric properties ( ) i and ( ) ii of Theorem 5.1.
Proof.For each > μ 0, by the Sobolev embedding , where C is a positive constant.Then, we can take λ and > μ 0. Therefore, ( ) i in Theorem 5.1 holds true.
In the following, we verify condition ( ) Then by ( ) Therefore, there exists t 0 large enough such that ( ) < I tv 0. (5.2) for large enough μ.Now we assume (5.2) holds true, then Lemmas 4.1, 5.1 and Theorem 5.1 give the existence of nontrivial critical points of , such that ‖ ‖ = v 1 0 and for given by (ii) in Lemma 5.1, we have ( ) → −∞ →∞ I tv lim t μ 0 . Then there exists t μ such that for some 0.
Let us first claim that { } t μ μ is bounded.From (5.3), we immediately obtain due to > μ 0 and assumption ( ) by the continuity of ( ) ⋅ f ξ , .Therefore, Lebesgue's dominated conver- gence theorem yields Hence, (5.5) gives Furthermore, it follows from (5.3) that Then there exists μ 1 such that for any ≥ μ μ 1 , If we take = e Tv 0 , with T large enough to verify ( ) < I e 0 μ , then we obtain for μ large enough.The proof of Theorem 1.1 is hence completed.□

Proof of Theorem 1.2
We shall use the Krasnoselskii's genus theory introduced by Krasnoselskii in [44] to prove Theorem 1.2.Let X be a Banach space and denote by Λ the class of all closed subsets { } ⊂ A X\ 0 that are symmetric with respect to the origin, that is, ∈ u A implies − ∈ u A.
Assume furthermore that Y is k dimensional and id and ψ is odd Ω , : Proof of Theorem 1.2.We will apply Theorem 5.2 to I μ .We know that ( ) . From (4.1), the functional I μ satisfies ( ) = I 0 0 μ .We shall divide the proof into the next four steps.
Step 1.The proof is similar to the proof of ( ) i and ( ) ii in Theorem 5.1.We can obtain that I μ satisfies ( ) a and ( ) c of Theorem 5.2.
Step 2. We claim that there exists a sequence ( ) To this aim, using an argument given in [48], according to the definition of c n μ , we obtain , by the definition of Γ n .
Step 3. We claim that problem (1.1) has at least k pairs of weak solutions.To achieve this goal, we distinguish two cases: Case I. Fix > μ 0. Choosing m 0 so large that Case II.Using a similar discussion as in (5.2), there exists for all > μ μ 2 .Therefore, in any case, we have An application of [44] guarantees that the levels 1, so by Theorem 4.2 and Remark 2.12 in [1], the set K c j μ contains infinitely many distinct points and hence problem (1.1) has infinitely many weak solutions.Therefore, problem (1.1) has at least k pairs of solutions.This ends the proof.□ 6 Degenerate case for problem (1.1) In this section, we are devoted to investigate the degradation of problem (1.1).To this end, we always suppose that M satisfies ( ) M 2 and ( ) M 3 , and f verifies ( ) f 1 -( ) f 3 .We shall give the following lemma, which is crucial to prove the existence results for problem (1.1).Lemma 6.1.Suppose that the functions M, V and f satisfy

If
On the critical Choquard-Kirchhoff problem on the Heisenberg group  231 then there exists a subsequence of { } u n n strongly convergent in ( ) Proof.Since the degenerate nature of problem (1.1), two situations must be considered: Case 1: . Here, either 0 is an accumulation point for the real sequence { } u n n and so there is a subsequence of { } u n n strongly converging to = u 0, or 0 is an isolated point of { } u n n and so there exists a subsequence, still denoted by { } u n n , such that . The first case cannot occur since it implies that the trivial solution is a critical point at level c μ .This is impossible, being 0 . Therefore, only the latter case can occur, so that there is a subsequence, still denoted by { } Therefore, by (6.3) and (6.4), we have ≥ ν m ω .Assume by contradiction that (6.8) holds true.The similar to the proof of Lemma 4.1, conditions ( ) M 2 and ( ) Moreover, .
This contradicts the assumption (6.1).So = ν 0 j for any ∈ j J.In the following, we will proof that (6.12) By contradiction, we suppose that (6.12) holds true.Similar to the proof of (4.12), we have .
The rest of the proof is similar to that of Theorem 1.1.□ Proof of Theorem 1.4.The proof of Theorem 1.4 is similar to that of Theorem 1.2.□

μ
Then we take = e t v 0 and ( ) < I e 0 μ .Hence, ( ) ii of Theorem 5.1 holds true.This completes the proof.□Proof of Theorem 1.1.We claim that

Theorem 5 . 2 . [ 44 ]
Let X be an infinite dimensional Banach space and ( Assuming that = ⊕ X Y Z, where Y is finite dimensional, and that I satisfies (a) There exists constant > There exists > Θ 0 such that I satisfies the ( ) PS c condition for all c, with ( ) ∈ c 0, Θ ; (c) For any finite dimensional subspace ͠ ⊂ X X, there is

2 , 3 ) 5 )Inserting ( 6 . 6 )
this fact shows that sequence { } u n n is bounded in ( )Taking a smooth cut-off function φ ε as Lemma 4.1.Obviously, { } u n n is bounded in ( )Using a similar discussion as (4.9) in Lemma 4.1 and (6.2), we can obtain that Next, we shall analyze each term at the right-hand side of (6.5).In fact, by Theorem 3.1, we have and (6.7) into (6.5),we have

.
Choosing a smooth cut-off function ψ R as in Lemma 4.1.Since ( )

2 2 2 ( 6 . 13 ) 0 . 2 .
This is impossible.Thus, = ∞ ν On the critical Choquard-Kirchhoff problem on the Heisenberg group  233In view of = ∅ J , we can obtain that Then ( ) i in Theorem 5.1 holds true.Similar to Lemma 5.1, we can obtain that ( ) ii of Theorem 5.1 still holds.□Proof of Theorem 1.3.By using the same discussion as the proof for Theorem 5.1, we have that

.
Theorem 1.2.Let (V) hold.Under the assumptions of Theorem 1.1, and assume one of the following conditions Let (V) hold.Under the assumptions of Theorem 1.3, and suppose one of the following condi- 1.Then, problem (1.1) has at least n pairs of nontrivial weak solutions in ( ) On the critical Choquard-Kirchhoff problem on the Heisenberg group  229 N