On a critical Choquard-Kirchhoff p-sub-Laplacian equation in ℍⁿ

1 Introduction and main results

Recently, critical equations on the Heisenberg group H n have been widely studied because of their important applications in quantum mechanics, number theory, partial differential equations, and other fields. It is worth noting that the Heisenberg group is topologically Euclidean, but analytically non-Euclidean. This is why, some basic properties, such as dilatations, need to be developed again [33]. Folland and Stein [8] were the first mathematicians who worked on the subelliptic analysis of the Heisenberg group. As in [7], they consistently were devoted to creating a generalization of the analysis for more general stratified groups. In the celebrated book [9], Rothschild and Stein extended these results for general vector fields satisfying the Hormander’s conditions. The book also laid the foundation of anisotropic analysis.

In this article, we consider the critical Choquard-Kirchhoff problem involving p -sub-Laplacian

where Δ H , p ≔ div H ( ∣ D H u ∣ H p − 2 D H u ) is the p -sub-Laplacian on the Heisenberg group H n , and the number p λ ∗ is the critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality. The Kirchhoff function K and the potential function V satisfy the suitable natural assumptions, which are listed in the following.

The reason of studying equation ( ℰ ) is based on two main reasons. On the one hand, equation ( ℰ ) has several important applications. Indeed, it is well-known that the Heisenberg group appears in various areas of physics and science, such as quantum theory (uncertainty principle, commutation relations). On the other hand, in the development of the theory of partial differential equations in the Heisenberg group and, more generally, in sub-Riemannian manifolds, it is important to explore new challenging problems as well as to figure out, in this emerging variety of results, whether or not the standard methods developed in the Euclidean spaces can be adapted to this new context.

In [34], existence of solutions for critical p -sub-Laplacian equations in the entire Heisenberg group is proved via the concentration-compactness principle. In [35], the results of [34] are completed and existence of entire solutions of critical ( p , q ) equations with Hardy terms on the Heisenberg group is proved. Sun et al. [44] considered existence of infinitely nontrivial solutions for critical Kirchhoff p -sub-Laplacian equations on the Heisenberg group via the concentration-compactness principle and the mountain pass theorem. Liu et al. [24] dealt with existence and multiplicity of nontrivial solutions for a new nonlocal Schrödinger-Poisson system with the superlinear and suberlinear terms on the first Heisenberg group H 1 . For other results, we refer to [14,17,25,31,32,42].

Turning back to the critical Choquard-Kirchhoff problem on the Heisenberg group, we can see that the results are relatively scarce. Via the mountain pass theorem, the linking theorem and other methods, Goel and Sreenadh [12] dealt with the Brézis-Nirenberg-type result of the Dirichlet problem with Choquard-type nonlinearity. With the application of the limit index theory and the concentration-compactness principle for the Choquard equation, Sun et al. [45] considered the following noncooperative Choquard-Kirchhoff system:

where Ω is a smooth bounded domain in H n , and 2 λ ∗ = ( 2 Q − λ ) ⁄ ( Q − 2 ) is the Hardy-Littlewood-Sobolev critical exponent on the Heisenberg group, when p = 2 .

The inspiration of this article comes from some recent results. In the Euclidean setting, what is well known is that the Choquard equation

was first studied in the pioneering work of Pekar [28] for the modeling of quantum mechanics. It was subsequently considered to be an approximation of the Hartree-Fock theory [4]. In 1998, Penrose [29] identified it as a self-gravitational collapse model of a quantum mechanical wave function. The mathematicians who first investigated the existence and symmetry of the solution to equation (1.1) were Lieb [20] and Lions [21]. Since then, equations of type (1.1) have been widely treated (see, e.g., [1,27,50]). In particular, the study of critical Choquard-Kirchhoff-type problems has been widely studied. In [39], the following Schrödinger-Choquard-Kirchhoff-type fractional p -Laplacian equation

where ( − Δ ) p s is the fraction p -Laplacian, and p μ , s ∗ = ( p n − p μ ⁄ 2 ) ⁄ ( n − p s ) is the critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality. With the help of the mountain pass theorem and the Ekeland variational principle, existence of nonnegative solutions to the aforementioned equation is obtained in both the sublinear and superlinear cases. Later, with the help of the concentration-compactness principle in the fractional setting and the Kajikiya new version of the symmetric mountain pass lemma, Wang et al. [47] proved existence of infinitely many solutions for the fractional p -Laplacian equations. Furthermore, there are many contributions in the interesting degenerate Kirchhoff-type problems. Using the Kajikiya new version of the symmetric mountain pass lemma, Liang et al. [19] dealt with existence and multiplicity of solutions of a class of degenerate Kirchhoff-type Schrödinger-Choquard equations involving the fractional p -Laplacian. Then, Song and Shi [41] showed existence of infinitely many solutions for a class of degenerate p -fractional Kirchhoff equations, involving critical Hardy-Sobolev nonlinearities. For recent results on degenerate Kirchhoff stationary problems, we refer to [6,38,49] and references therein.

For critical Choquard equations in unbounded domains, there is no doubt that the concentration-compactness principle for the Choquard equation is the most powerful tool to overcome the difficulties caused by the lack of compactness. Consequently, the study of the critical Choquard equation has a profound connection with the concentration phenomenon, which takes place when one looks for sequences of approximate solutions. Ivanov and Vassilev [15] extended the concentration-compactness principle á la Lions from the Euclidean setting to the general Carnot groups, and so to the Heisenberg group, which is the special interesting subcase of Carnot groups. In [33], the concentration-compactness principle has been given for systems with critical nonlinearities and Hardy terms in the Heisenberg group. Moreover, the article [36] dealt with the concentration-compactness principle involving Hardy terms on the classical and fractional spaces in the Heisenberg group. Sun et al. [43] considered the concentration-compactness principle for the Choquard equation on the Heisenberg group. This article presents a generalization of the results of [43]. To our best knowledge, there do not seem to be concentration-compactness principle theorems for the critical Choquard equation involving p -Laplacian on the Heisenberg group. Moreover, it also seems that there are no applications of these results on the Heisenberg group setting.

Inspired by the aforementioned studies, the main purpose in this article is to prove existence results for equation ( ℰ ). We first establish a version of the concentration-compactness principle for the Choquard equation involving p -Laplacian on the Heisenberg group. Then, we use this result to show that the ( P S ) c condition at special levels c is true. Finally, using the mountain pass theorem, we obtain existence and multiplicity of solutions for equation ( ℰ ) in both the nondegenerate and degenerate cases on the entire Heisenberg group.

Let us now list the key natural assumptions we require in the main results. The potential function V and the Kirchhoff function K : R 0 + → R 0 + are supposed to satisfy

1. V : H n → R + is a continuous non-decreasing function, and there exists V 0 > 0 such that V ≥ V 0 > 0 in H n .

2. There exists k 0 > 0 such that inf t ≥ 0 K ( t ) = k 0 .

3. The Kirchhoff function K is a continuous in R 0 + , and there exists θ ∈ [ 1 , p ∗ ⁄ p ) such that

   K ( t ) t ≤ θ K ( t ) , for all t ∈ R 0 + ,

   where K ( t ) = ∫ 0 t K ( s ) d s and p * = p Q ⁄ ( Q − p ) .

4. The Kirchhoff function K is also non-decreasing in R 0 + , and there exists k 1 > 0 such that

   K ( t ) ≥ k 1 t θ − 1 ,

   for all t ∈ R + and K ( 0 ) = 0 , where θ ∈ [ 1 , p ∗ ⁄ p ) is the number given in ( K 2 ) .

The function f satisfies the following hypotheses:

1. f : H n × R → R is a Carathéodory function such that f is odd with respect to the second variable.

2. There exists an exponent q , with p θ < q < r < p ∗ , such that

   ∣ f ( ξ , t ) ∣ ≤ a ( ξ ) ∣ t ∣ q − 1 , a.e. ξ ∈ H n and all t ∈ R ,

   where 0 ≤ a ∈ L ϑ ( H n ) ∩ L ∞ ( H n ) and ϑ = r ⁄ ( r − q ) .

3. There exists σ ∈ ( p θ , p ∗ ) such that

   0 < σ F ( ξ , t ) ≤ f ( ξ , t ) t , for all t ∈ R + ,

   where F ( ξ , t ) = ∫ 0 t f ( ξ , s ) d s .

The natural solution space for equation ( ℰ ) is denoted by S V 1 , p ( H n ) and is properly defined in Section 2 together with the necessary preliminaries. Let us first state the existence results for equation ( ℰ ) in the non-degenerate case.

Finally, the next existence results for equation ( ℰ ) are also obtained in the degenerate case.

The table of contents of this article is as follows. In Section 2, we present some preliminary results related to the Heisenberg group. Section 3 is devoted to the proof of the concentration-compactness principle for the p -Laplacian Choquard equation on the Heisenberg group, which is useful to prove the Palais-Smale condition at some special energy levels. Section 4 is dedicated to the proofs of the existence and multiplicity Theorems 1.1 and 1.2 for equation ( ℰ ) in the non-degenerate case. Finally, Section 5 deals with the proof of Theorems 1.3 and 1.4, i.e., to the proofs of existence and multiplicity of solutions for equation ( ℰ ) in the degenerate case.

2 Preliminaries

Section 2 starts with a brief review of the basic properties of the Heisenberg group. For a complete treatment on the Heisenberg group, we refer to [11,15,18,26] and references therein.

The Heisenberg group H n is a Lie group of topological dimension 2 n + 1 , i.e., the Lie group that has R 2 n + 1 as a background manifold, endowed with the following group law:

with

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

The Lie algebra of left invariant vector fields is generated by the following vector fields:

It is straightforward to obtain that for all j , k = 1 , 2 , … , n ,

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

Hence, D H u ∈ span { X j , Y j } j = 1 n .

The natural inner product in the span of { X j , Y j } j = 1 n is 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 . The inner product ( ⋅ , ⋅ ) H produces the Hilbertian norm

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 given by

In a similar way, if u ∈ C 2 ( H n ) , then the Kohn-Spencer Laplacian, in other words, equivalently the horizontal Laplacian in H n , or the sub-Laplacian, of u is

Applying the famous Theorem 1.1 due to Hörmander in [13], we see that the operator Δ H is hypoelliptic. In particular, Δ H u = div H D H u for each u ∈ C 2 ( H n ) .

A well-known generalization of the Kohn-Spencer Laplacian is the horizontal p-Laplacian, 1 < p < ∞ , on the Heisenberg group, defined as

for all φ ∈ C c ∞ ( H n ) .

For R > 0 , a natural group of dilation on H n is defined by δ R ( ξ ) = ( R x , R y , R 2 t ) for all ξ = ( x , y , t ) ∈ H n . Hence,

It is easily seen that the Jacobian determinant of dilatations δ s : H n → H n is constant in H n and equal to R 2 n + 2 . This is why the natural number Q = 2 n + 2 is called the homogeneous dimension of H n .

The anisotropic dilation structure on Heisenberg group leads to the Korányi norm, which is defined by

Hence, the homogeneous degree of the Korányi norm with respect to the dilations δ R , R > 0 , is equal to 1. Indeed, for all ξ = ( z , t ) ∈ H n ,

The Korányi distance is defined as follows:

The Korányi open ball with radius R centered at ξ 0 is the set B R ( ξ 0 ) = { ξ ∈ H n : d K o ( ξ 0 , ξ ) < R } . For the sake of simplicity, B R ( O ) is denoted by B R , when ξ 0 = O = ( 0 , 0 ) is the natural origin of H n . Moreover, the complement of B R ( ξ 0 ) , i.e., H n \ B R ( ξ 0 ) , is denoted by B R ( ξ 0 ) c and the complement of B R ( O ) by B R c .

The Lebesgue measure on R 2 n + 1 is invariant under the left translations of the Heisenberg group. The Haar measures d ξ on Lie groups are unique up to constant multipliers, which is the reason why the Haar measure on H n corresponds to the ( 2 n + 1 ) -Lebesgue measure. In what follows, ∣ E ∣ denotes the measure of a measurable subset E of H n . Moreover, the Haar measure on H n is Q -homogeneous with respect to dilations δ R . Therefore, as shown in [18], the topological dimension 2 n + 1 of H n is strictly less than its Hausdorff dimension Q = 2 n + 2 . Indeed, for any measurable subset E of H n and any R > 0 ,

In particular, if E = B R , then ∣ B R ∣ = R Q ∣ B 1 ∣ .

Now, we present some properties on the classical Sobolev spaces on the Heisenberg group H n , recalling that the underlying measure in use here is the Haar measure on H n , which agrees with the Lebesgue measure on R 2 n + 1 . For 1 ≤ p < ∞ , the linear space

endowed with the L p -norm ∣ ⋅ ∣ p , i.e.,

is a separable Banach space and also reflexive, if 1 < p < ∞ .

For 1 ≤ p < ∞ the classical horizontal Sobolev space H W 1 , p ( H n ) on the Heisenberg group consists of all functions u ∈ L p ( H n ) such that the horizontal gradient D H u exists in the sense of distributions and ∣ D H u ∣ H ∈ L p ( H n ) , endowed with the natural norm

The horizontal Sobolev space H W 1 , p ( H n ) is a separable Banach space, while is reflexive, provided that 1 < p < ∞ . Moreover, C c ∞ ( H n ) is dense in H W 1 , p ( H n ) for every p , with 1 ≤ p < ∞ .

Let us assume from now on that 1 < p < ∞ and that condition ( V ) holds. The natural solution space for equation ( ℰ ) is the space

with the norm

Extending the proof of Lemma 10 in [37] from the Euclidean context to the Heisenberg group, we see that S V 1 , p ( H n ) is a separable reflexive Banach space. Moreover, when 1 < p < Q , the embeddings

are continuous for all ℘ , with p ≤ ℘ ≤ p ∗ , where

is the critical Sobolev exponent associated with p in the Heisenberg context.

Furthermore, if 1 ≤ p < Q , then the embedding of S V 1 , p ( H n ) ↪ ↪ L ℘ ( B R ) is compact for all ℘ , with 1 ≤ ℘ < p ∗ and all R > 0 thanks to assumption ( V ) [5,30].

The celebrated Folland-Stein space S 1 , p ( H n ) , 1 ≤ p < Q , is defined as the completion of C c ∞ ( H n ) with respect to the norm

The Folland-Stein inequality says that if 1 < p < Q , then there exists a constant C = C ( p , Q ) > 0 such that

In [46], when 1 < p < Q , Vassilev proved the existence of extremal functions of (2.2) by the concentration-compactness method of Lions. However, the best constant

of (2.2) is not explicitly determined yet.

Proposition 2.1 guarantees that the integral

is well defined if ∣ u ∣ β ∈ L ℘ ( H n ) for some ℘ > 1 such that

The Folland-Stein inequality (2.2) shows that (2.5) is well defined for u ∈ S 1 , p ( H n ) only if

i.e., β has to satisfy

Hence, it is quite natural to call p λ ♯ the lower critical exponent and p λ ∗ the upper critical exponent on the Heisenberg group. Therefore, Proposition 2.1 shows that

where

In our context, we define

where p λ ∗ is given in (2.6). Proposition 2.1 implies also that

where S is the best constant introduced in (2.3).