On the critical Choquard-Kirchhoff problem on the Heisenberg group

2.1 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 H N = R N × R N × R , N ∈ N be the Heisenberg Lie group of topological dimension 2 N + 1 , that is, the Lie group that has R 2 N + 1 as a background manifold, endowed with the non-Abelian group law:

with

The inverse is given by ξ − 1 = − ξ , so that ( ξ ∘ ξ ′ ) − 1 = ( ξ ′ ) − 1 ∘ ξ − 1 .

The vector fields for j = 1 , … , N ,

establishes a basis for the Heisenberg regular commutative relation satisfying the real Lie algebra of left invariant vector fields on the Heisenberg group H N . That means,

A vector field in the span of [ X j , Y j ] j = 1 N is called horizontal.

The horizontal gradient of a C 1 function u : H N → R is defined by

Obviously, D H u is an element of the span of { X j , Y j } j = 1 N . We consider the natural inner product in the span of { X j , Y j } j = 1 N given by

where X = { x j X j + x ˜ j Y j } j = 1 N and Y = { y j X j + y ˜ j Y j } j = 1 N . Therefore, the corresponding Hilbertian norm is defined as follows:

for the horizontal vector field D H u .

For any horizontal vector field function X = X ( ξ ) , X = { x j X j + x ˜ j Y j } j = 1 N of class C 1 ( H N , R 2 N ) , the horizontal divergence of X is defined by

Analogously, if u ∈ C 2 ( H N ) , then the second-order self-adjoint operator Δ H u in H N is given by

is usually called subelliptic Laplacian or Kohn Laplacian on H N . According to the celebrated Theorem 1.1, thanks to Hörmander [19], we know that the operator Δ H is hypoelliptic. Notably, Δ H u = div H D H u for each u ∈ C 2 ( H N ) . 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 H .

As we all know, the generalization of the Kohn-Spencer Laplacian is the horizontal p -Laplacian on the Heisenberg group, p ∈ ( 1 , ∞ ) , given by

for all φ ∈ C c ∞ ( H N ) .

For s > 0 , there is a dilation naturally associated with the Heisenberg group structure, which is usually defined by δ s ( ξ ) = ( s x , s y , s 2 t ) . Therefore, δ s ( ξ 0 ∘ ξ ) = δ s ( ξ 0 ) ∘ δ s ( ξ ) . Moreover, for all ξ = ( x , y , t ) ∈ H N , it is easy to verify that the Jacobian determinant of dilatations δ s : H N → H N is constant and equal to R 2 N + 2 . That is why the natural number Q = 2 N + 2 is called homogeneous dimension of H N and the homogeneous dimension Q acts as the equivalent topological dimension in Euclidean spaces.

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 ξ 0 is { B R ( ξ 0 ) = ξ ∈ H N : d H ( ξ , ξ 0 ) < R } . For the sake of simplicity, we denote B R = B R ( O ) , where B R ( O ) represents the Korányi open ball centered at the natural origin with radius R .

For any measurable set ∣ E ∣ , the Haar measure d ξ of Heisenber group H N that consists with the ( 2 N + 1 ) -Lebesgue measure and is invariant under left translations. Therefore, as shown in [24], the topological dimension 2 N + 1 of H N is strictly less than its Hausdorff dimension Q = 2 N + 2 . Moreover, it is Q -homogeneous with respect to dilations:

2.2 Classical Sobolev spaces in the Heisenberg group

Let Γ 2 ( Ω ) be the space of all continuous functions u on Ω such that X j u , Y j u , X j 2 u and Y j 2 u are all continuous in Ω , which can be continuously extended up to ∂ Ω . Folland and Stein [11] introduced the space S 1 2 ( H N ) , which is connected to Vector fields X j and Y j and the space S 1 2 ( H N ) is similar to space H 1 ( R N ) . More precisely, the space

is a Hilbert space with inner product

Therefore, the norm in S 1 2 ( H N ) is defined as follows:

The space S ˚ 1 2 ( Ω ) is the closure of C c ∞ ( Ω ) in S 1 2 ( H N ) . Therefore, it follows from a Poincare type inequality that S ˚ 1 2 ( Ω ) is a Hilbert space equipped with the norm ‖ u ‖ S ˚ 1 2 ( Ω ) 2 = ∫ Ω ∣ ∇ H u ∣ 2 d ξ . At the same time, we aslo define a function space

equipped with the norm

Folland and Stein [11] obtained the following Sobolev type inequality: there exists a positive constant C Q such that

where ∣ ⋅ ∣ Q ∗ for the L Q ∗ ( H N ) -norm and Q ∗ = 2 Q Q − 2 . Moreover, if Ω is bounded then the embedding S ˚ 1 2 ( Ω ) ↪ L ν ( Ω ) is continuous for all ν ∈ [ 1 , Q ∗ ] . In addition, these embeddings are compact for ν < Q ∗ and the embedding is not compact for ν = Q ∗ .

The best constant for the embedding S 1 2 ( H N ) ↪ L Q ∗ ( H N ) is defined as follows:

By (2.1), we know that S > 0 . Also, for any nonempty open set Ω ⊆ H N , we obtain

That means, infimum is achieved in the case Ω = H N .

In [21,22], Jersion and Lee showed that the function:

is such that S Q 2 = ‖ Z ‖ 2 = ∣ Z ∣ Q ∗ Q ∗ , where B 0 is a positive constant. More precisely, any minimizer of S takes the following form:

for suitably β > 0 and ω ∈ H N . In [18], the best constant S H G is defined by

where Q λ ∗ = 2 Q − λ Q − 2 .

If

then W is unique minimizer of S H G and satisfies the following:

and

By using Proposition 2.1, the following integral

is well defined if ∣ u ∣ α ∈ L^r for some r > 1 such that 2 r + λ Q = 2 . It implies that if u ∈ S 1 2 ( H N ) , then with the help of Sobolev-type inequality, (2.7) is defined only if

That is,

Due to this fact, it is quite natural to call 2 Q − λ Q the lower critical exponent and Q λ ∗ ≔ 2 Q − λ Q − 2 the upper critical exponent on the Heisenberg group.

Together with the assumption V ( ξ ) ≥ V 0 > 0 implies that S V , 1 2 ( H N ) 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 S V , 1 2 ( H N ) ↪ S 1 2 ( H N ) ↪ L ν ( H N ) for all 2 ≤ ν ≤ Q ∗ . In particular, we have the following theorem thanks to the assumption ( V ) (see [6,41]).