Critical equations with Hardy terms in the Heisenberg group

In this paper, we are concerned with the study of a critical (p, q) equation with Hardy terms on the Heisenberg group. Existence of entire solutions is obtained via an application of some concentration–compactness type results and the mountain pass theorem. Our results are presented in the model case of the (p, q) horizontal Laplacian equations, but the method can be extended to deal with a more general class of problems with operators of (p, q) growth.


Introduction
In this paper, we complete the study started in [33] by giving some applications of the results contained therein. More precisely, we consider the critical equation with Hardy terms in ℍ n 1 3 −∆ H,p u − ∆ H,q u + |u| p−2 u + |u| q−2 u − σψ q |u| q−2 u r q = λf (ξ, u)+ |u| q * −2 u, (E) where and > 0 are real parameters and Q = 2n + 2 is the homogeneous dimension of ℍ n . The exponents p and q are such that 1 < p < q < Q , where q * = qQ∕(Q − q) is the critical exponent related to q. Moreover, the operator H,℘ , with ℘ ∈ {p, q} , appearing in equation (E ), is the well known horizontal ℘-Laplacian on the Heisenberg group, which is defined as The function r in ( E ) denotes the Korányi norm of the Heisenberg group ℍ n , given by where = (z, t) and z = (x, y) ∈ ℝ n × ℝ n , t ∈ ℝ , and |z| is the Euclidean norm in ℝ 2n of z. Finally, the weight function in (1.3) is related to the horizontal Hilbertian norm of the horizontal gradient of r, in short = |D H r| H in ℍ n ⧵ {O} . Its presence in equation ( E ) will be explained later on, by means of the Hardy inequality in ℍ n . For further details we refer to Sect. 2.
The study of critical equations in the context of the stratified Lie groups is a fast growing and fascinating topic. The main reason behind this interest is the strong connection between this subject and the Yamabe problem on CR manifolds. We refer to [8,20,24,25] and to the references therein for many details in the special case ℘ = 2 on Carnot groups. We also cite [3,34] for the general case when 1 < ℘ < 2 in the Heisenberg setting.
On the other hand, only few recent works deal with (p, q) horizontal Laplacian problems in the whole ℍ n . The so-called (p, q) operators are introduced by Marcellini in [22,23] in the Euclidean context. Marcellini considers functions with different growth near the origin and at infinity (unbalanced growth). Since then, the topic has been extensively studied and we just mention few recent contributions which are very relevant to the present paper. More precisely, when p = 2 and 2 < q < N , the (p, q) equations were studied in [28][29][30]. In particular, in [28] the authors prove existence and multiplicity of solutions of a parametric nonlinear nonhomogeneous Dirichlet problem in a bounded domain ⊂ ℝ n , with of class C 2 . In [27], the authors give existence and multiplicity of solutions for a parametric (p, q)-equations with sign-changing reaction and Robin boundary conditions. These contributions are also related to the works of Zhikov [40,41], where the so-called double phase operators are studied in connection with phenomena arising in nonlinear elasticity. We mention the paper [1] for further details.
In order to state the main results of the present paper, it is crucial to introduce the best constant in the Folland-Stein inequality. By [12], we know that for all ℘ , with 1 < ℘ < Q , there exists a positive constant C ℘ * = C ℘ * (℘, Q) , related to the associated critical exponent ℘ * = ℘Q∕(Q − ℘) , such that for all ∈ C ∞ c (ℍ n ) where the horizontal gradient of is the vector r( ) = r(z, t) = (|z| 4 + t 2 ) 1∕4 , and {X j , Y j } n j=1 is the basis of horizontal left invariant vector fields on ℍ n , that is By [16], we know that the best constant of the Folland-Stein inequality is achieved in the so called Folland-Stein space S 1,℘ (ℍ n ) , 1 < ℘ < Q , which is defined as the completion of C ∞ c (ℍ n ) with respect to the norm Hence, we can write the best constant C ℘ * of the Folland-Stein inequality as and clearly C ℘ * > 0. For our purposes, it is also crucial to introduce the best Hardy-Sobolev constant The weight function appearing in (1.3) is defined as = |D H r| H . For further details we refer to Sect. 2.
The main difficulty of working with Hardy terms is that the Hardy embedding is continuous, but not compact, that is On f in ( E ) we assume the following condition: (F) f is a Carathéodory function, with f (⋅, u) = 0 for all u ≤ 0 and f (⋅, u) > 0 for all u > 0 , satisfying the two properties (f 1 ) there exist and m, with p < < m < q * , such that for every > 0 there exists C > 0 for which the inequality holds for a.e. ∈ ℍ n ; (f 2 ) there exists , with q < < q * , such that the inequality holds for a.e. ∈ ℍ n , where F( , u) = ∫ u 0 f ( , v)dv for a.e ∈ ℍ n and all u ∈ ℝ.
Clearly, the natural space where finding solutions of (E ) is endowed with the norm  [10,11] in ℝ n and [3,[31][32][33][34] in ℍ n . Existence is obtained via the mountain pass lemma of Ambrosetti and Rabinowitz and follows somehow the ideas of [10,11]. A very delicate step is the tricky compactness Theorem 3.1, which extends to the Heisenberg group analogous results obtained in the Euclidean setting in Lemma 2.3 of [5], see also Lemma 3.3 of [7], Lemma 2.2 of [6] and Theorem 2.8 of [11]. Moreover, the "triple loss of compactness" in (E ), caused by the simultaneous presence of the Hardy and critical terms in the whole Heisenberg group ℍ n , forces to study the exact behavior of the Palais-Smale sequences, in the spirit of Lions. This analysis is deeply connected with the concentration phenomena taking place and strongly relies on the results in [33]. More precisely, existence is based on Theorems 1.1 and 1.2 of [33], where the concentration-compacntess principle of Lions [18,19], or CC principle, and its variant, that is, the CC principle at infinity of Chabrowski [4], both are proved involving the Hardy-Sobolev embedding in the Folland-Stein space.
Actually, applying the same arguments of Theorem 1.1, see Sect. 3, we could obtain existence of solutions for a much more general class of the (p, q) operators, of the type considered in [10,11,31]. More precisely, we could replace the sum of the p and q horizontal Laplacian operators in equation (E ) by a more general divergence type operator of the form where A ∶ ℝ + → ℝ + , ℝ + = (0, ∞) , is a strictly positive and strictly increasing function of class C 1 (ℝ + ) satisfying the assumptions described in [31] and div H denotes the horizontal divergence, which is defined along any horizontal vector field X = {x j X j +x j Y j } n j=1 , of class C 1 (ℍ n , ℝ 2n ) as Furthermore, A is assumed to be such that tA(t) → 0 as t → 0 + . The function A denotes the potential of t ↦ tA(t) , which is 0 at 0. Without entering into further details, we just give some examples covered by A, when 1 < p < q < Q . If A(t) = t p ∕p + t q ∕q , we recover the (p, q) horizontal Laplacian operator. Moreover, if A(t) = t p ∕p + 1 2 (1 + t q ) 2∕q , we get The range of examples is actually very wide and for more details we refer to [11,31]. However, for notational simplicity, here we stick to the model case of the (p, q) horizontal Laplacian operator.
The paper is organized as follows. Section 2 is dedicated to a short recap of the main definitions and properties of the Heisenberg group. In Sect. 3, we introduce the functional setting of equation ( E ) and we present some useful tools which will be crucial in the paper, such as Theorem 3.1. Finally, Sect. 4 is devoted to the proof of the main Theorem 1.1, via a combination of standard variation methods and a more delicate analysis of the exact behavior of the Palais-Smale sequences, in the spirit of Lions.

The Heisenberg group ℍ n
In this section we present the basic properties of the Heisenberg group ℍ n . For a complete and exhaustive treatment we refer, e.g., to [13,14,16,38].
Let ℍ n be the Heisenberg group of topological dimension 2n + 1 , that is the Lie group which has ℝ 2n+1 as a background manifold and whose group structure is given by the non-Abelian law for all , � ∈ ℍ n , with The 2n + 1 left-invariant vector fields on ℍ n for j = 1, … , n , form a basis for the real Lie algebra of ℍ n of left-invariant vector fields. This basis satisfies the Heisenberg canonical commutation relations Moreover, all the commutators of length greater than two vanish, and so ℍ n is a nilpotent graded stratified group of step two. A left invariant vector field X that belongs to the span of {X j , Y j } n j=1 , is called horizontal. For each real number R > 0 , we consider the dilation R ∶ ℍ n → ℍ n naturally associated with the Heisenberg group structure, which is defined by It is easy to verify that the Jacobian determinant of dilatations R is constant and equal to R 2n+2 , where the natural number Q = 2n + 2 is the homogeneous dimension of ℍ n .
The Korányi norm on ℍ n is given by Consequently, the Korányi norm is homogeneous of degree 1, with respect to the dilations R , R > 0 , that is for all = (z, t) ∈ ℍ n . The corresponding distance, the so called Korányi distance, is Throughout the paper, we denote by The Lebesgue measure on ℝ 2n+1 is invariant under the left translations of the Heisenberg group. Thus, since the Haar measures on Lie groups are unique up to constant multipliers, we denote by d the Haar measure on ℍ n that coincides with the (2n + 1)-Lebesgue measure and by |U| the (2n + 1)-dimensional Lebesgue measure of any measurable set U ⊆ ℍ n . Furthermore, the Haar measure on ℍ n is Q-homogeneous with respect to dilations R . Consequently, We define the horizontal gradient of a C 1 function u ∶ ℍ n → ℝ by . In span {X j , Y j } n j=1 ≃ ℝ 2n we consider the natural inner product given by . The inner product ⋅, ⋅ H produces the Hilbertian norm for the horizontal vector field X.
For any horizontal vector field function , of class C 1 (ℍ n , ℝ 2n ) , we define the horizontal divergence of X by Similarly, if u ∈ C 2 (ℍ n ) , then the Kohn-Spencer Laplacian, or equivalently the horizontal Laplacian, or the sub-Laplacian, in ℍ n , of u is According to the celebrated Theorem 1.1 due to Hörmander in [15], the operator H is hypoelliptic. In particular, 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 ℘-Laplacian on the Heisenberg group, ℘ ∈ (1, ∞) , defined by Let us now review some classical facts about the first-order Sobolev spaces on the Heisenberg group ℍ n . We just consider the special case in which 1 ≤ ℘ < Q and is an open set in ℍ n . Denote by HW 1,℘ ( ) the horizontal Sobolev space consisting of the functions u ∈ L ℘ ( ) such that D H u exists in the sense of distributions and |D H u| H ∈ L ℘ ( ) , endowed with the natural norm . Thanks to [12] we know that if 1 ≤ ℘ < Q , then the embedding is continuous.
Let us also briefly recall a version of the Rellich theorem in the Heisenberg group. We refer to [13,14,16,21], where this topic is extensively treated and we just recall that, if 1 ≤ ℘ < Q and B R ( 0 ) is any Korányi ball, then the embedding is compact, provided that 1 ≤ s < ℘ * .
Finally, following [13], we denote by Note that the density function 2 is homogeneous of degree zero with respect to the fam- In the Euclidean space the presence of the density is outshone by the flat geometry of ℝ n , which yields ≡ 1.
The Hardy inequality in ℍ n was obtained in [13] when ℘ = 2 and the extended in [26] for all ℘ > 1 , see also [9]. We report here this second version.

Then, the following inequality holds
Clearly, by a density argument inequality (2.4) is still valid for in S 1,℘ (ℍ n ) . Moreover, (2.4) implies that continuously. However, as already noted, the embedding (2.5) is not compact, even locally in any neighborhood of O.
For computational simplicity, we prefer to use the Korányi norm and distance instead of the Carnot-Carathéodory distance, though the Korányi distance does not reflect the sub-Riemannian structure of the Heisenberg group. However, the two metrics are closely related. Interestingly, in the setting of the Heisenberg group it was shown by Yang in [39] that the L-gauge d(x) -sometimes also called the Korányi-Folland or Kaplan gauge -can be replaced by the Carnot-Carathéodory distance, and the Hardy inequality in the Heisenberg group remains valid with the same best constant ℘∕(Q − ℘) . For further comments and related results we refer to [35].
Let us conclude the section introducing a suitable sequence of mollifiers in the Heisenberg group. To this aim, we first define the group convolution as follows. If is called convolution of u and v. By the analogue of the Young theorem u * v belongs to L ℘ (ℍ n ) and For further details we refer to [12,17,37]. Using the convolution, it is possible to generate a sequence of mollifiers ( k ) k on ℍ n , with the properties

Functional setting and preliminary results
In this section we present some useful results and comments and from now on we assume that the structural assumptions required in Theorem 1.1 hold. Clearly, ( E ) has a variational structure and the Euler-Lagrange functional I ∶ W → ℝ associated to (E ) is given by for all u ∈ W . Indeed, I is well defined and of class C 1 (W) by the assumption (F) , and Hence, the (weak) solutions of (E ) are exactly the critical points of I. From the main properties summarised in Sect. 2 we easily get the next result. Before studying equation ( E ), let us recall the crucial inequality, originally proved by Simon in [36]. For all s ∈ (1, ∞) there exists > 0 , depending only on s, such that . The structural assumptions of Theorem 1.1 lead to geometry of the mountain pass theorem of Ambrosetti and Rabinowitz for the functional I at special levels. The proof of Lemma 3.2 is standard and again very similar to the demonstration which first appears in Lemma 2.4 of [11] and Lemma 4.1 of [32] and so there is no reason to produce it here.
I(u) ≥ for any u ∈ W with ‖u‖ = .
Now, for fixed ∈ (−∞, H q ) and > 0 , thanks to the geometry given in Lemma 3.2, we introduce the special levels of I Obviously, c , > 0 thanks to Lemma 3.2, since ‖e‖ > . The next asymptotic property of the levels c , as → ∞ is crucial to overcome the difficulties due to the presence of the Hardy terms and the critical nonlinearities. This result was observed in the Euclidean space in [10] and in [11] for the scalar and vectorial case, respectively. We also refer to [31] for a similar feature in the Heisenberg context.

Lemma 3.3 For any ∈ (−∞, H q ) it results
Again the proof of Lemma 3.3 follows directly from that of Lemma 2.5 of [11] and it is not reported here.
Let us now prove a tricky compactness theorem of independent interest. The argument we use first appears in the Euclidean context in the proof of Lemma 2.3 in [5], see also Lemma 3.3 of [7] and Lemma 2.2 of [6] and Theorem 2.7 of [11]. Theorem 3.1 Let ℘ ∈ (p, q * ) and let (u k ) k be a bounded sequence in W. Then there exists u ∈ W such that, up to a subsequence, u k → u strongly in L ℘ (ℍ n ) as k → ∞.
Proof Fix ℘ and (u k ) k as in the statement. Then, there exists u ∈ W such that, up to a subsequence, for any ∈ [1, q * ) and radius R.
We first claim that To this aim fix > 0 . There exist positive numbers s 0 , S 0 such that 0 < s 0 < S 0 and |s| ℘ ≤ |s| p for all s ∈ ℝ , with |s| ≤ s 0 , while |s| ℘ ≤ |s| q * for all s ∈ ℝ , with |s| ≥ S 0 , since Let us first show that Otherwise, there exist k 0 ∈ ℕ , > 0 and radii R j ↑ ∞ as j → ∞ such that Hence, (3.8) holds. On the other hand, (3.4) gives that u ∈ HW 1,p (ℍ n ) ∩ HW 1,℘ (ℍ n ) and u k → u a.e. in ℍ n . Hence, there exists R 0 = R 0 ( ) ≥ 1 such that for all R ≥ R 0 Clearly, D is a bounded domain for all ∈ ℕ , and D ∩ D m = � if ≠ m . It is possible to apply the reverse Fatou lemma to the sequence Therefore, ) for all R ≥ R 0 and k ∈ ℕ . Hence, the claim lim R→∞ |A k ∩ B c R | = 0 uniformly in k ∈ ℕ is proved by (3.8) and (3.9). In particular, for any > 0 there exist R 0 ≥ 1 and 0 ∈ (0, CS Hence, (3.6) and (3.7) yield Now, (3.4) implies that u k → u in L ℘ (B R 0 ) . Therefore, for all R ≥ R 0 = R 0 ( ) by (3.10) Letting R → ∞ we have and so as → 0 we obtain This, together with (3.4), gives at once the validity of (3.5). Finally, applying the Clarkson and Radon theorems we obtain

Proof of Theorem 1.1
In this section we prove the existence of nontrivial solutions for (E ), and so all the structural assumptions required in Theorem 1.1 are supposed to hold. Let us now prove the following crucial lemma.
Lemma 4.1 Let (u k ) k ⊂ W be a Palais-Smale sequence of I at the level c , for all ∈ (−∞, H q ) and > 0 . Then, (ii) there exists * = * ( ) such that the weak limit u is a critical point of I for any ≥ * . Assume by contradiction that ‖u k ‖ → ∞ as k → ∞ . Now we have three possibilities as k → ∞ 1. ‖u k ‖ HW 1,p → ∞ and ‖u k ‖ HW 1,q → ∞; 2. ‖u k ‖ HW 1,p → ∞ and ‖u k ‖ HW 1,q is bounded; 3. ‖u k ‖ HW 1,p is bounded and ‖u k ‖ HW 1,q → ∞.
In the first case, for k sufficiently large obviously ‖u k ‖ q HW 1,q ≥ ‖u k ‖ p HW 1,q , being p < q . Therefore, we get which implies, as k → ∞ , that 0 ≥ 2 1−p > 0 . This is obviously impossible.
In the second case, we know that and so dividing both sides by ‖u k ‖ p HW 1,p , we obtain This in turn implies 0 ≥ > 0 as k → ∞ , and gives the required contradiction. Finally, the third case is analogous to the second one. Therefore, since we ruled out all the three possibilities, we conclude that (u k ) k is bounded in W. Thus, since W is a reflexive Banach space, there exists u ∈ W such that u k ⇀ u in W. For simplicity, in what follows we denote by (u k ) k every subsequence extracted from the original sequence.
Step 1. Let us first show that j = 0 for all j ∈ J ∪ {0} . Assume by contradiction that j > 0 for some j ∈ J ∪ {0} , that is j is a singular point of the measure (note that we put 0 = O). Thus, (4.4) and (4.13) imply C q * q∕q * j ≤ j ≤ j and, since by contradiction j > 0 , this yields j ≥ C Q∕q q * . Moreover, by (4.4) and (4.6) we know that as k → ∞ since ,j ≤ 1 and so, sending k → ∞ and → 0 + , we have for all ≥ * This is an obvious contradiction by (4.14). Hence, j = j = 0 for all j ∈ J and for all ≥ * , as desired. Similarly, when the center of the ball is O, then (4.13) gives I q∕q * 0 Assume by contradiction that 0 ≠ 0 . Then, 0 ≥ I Q∕q . As above, by (4.6) we obtain as k → ∞ and → 0 + which is again a contradiction by (4.14). Therefore, 0 = 0 and so 0 = 0 for all ≥ * .
Step 2. Let us now prove that 0 = 0 . Assume again by contradiction that 0 > 0 . By (1.3) and differentiation, we get for all > 0 the existence of a constant C such that where by (4.3), the Hölder inequality and a change of variable, we can estimate the last term as follows where c was defined above. Consequently, from which, using (4.3) and (4.15), we get for all > 0 which yields, sending → 0 + , that H q 0 ≤ 0 . Thus, we obtain from the previous step This is impossible. Hence 0 = 0 for all ≥ * , as desired. In summary, we have shown that there exists * = * ( ) such that for all ≥ * as k → ∞ , that is for all ∈ C ∞ 0 (ℍ n ) it results From now on in the proof we assume that is fixed, with ≥ * . Our goal is now to prove that for all R > 0 we have To this aim, take R > 0 and ∈ C ∞ 0 (ℍ n ) such that 0 ≤ ≤ 1 in ℍ n , ≡ 1 in B R , ≡ 0 in B c 2R and ‖D H ‖ ∞ ≤ 2 . Clearly, Similarly, which yields, again by (4.3), Moreover, by (f 1 ) , the Hölder inequality and (4.3), as k → ∞ since 1 < p < < m < q * , where C = sup k∈ℕ ‖u k ‖ −1 + m C 1 sup k∈ℕ ‖u k ‖ m−1 and C < ∞ by Lemma 3.1. Finally, as k → ∞ and by (2.2), (4.3) and (4.16), since |u k | q * −2 u k ⇀ |u| q * −2 u in L q * (ℍ n ) and, similarly, |u k | q−2 u k ⇀ |u| q−2 u in L q � (ℍ n , q r −q ) in virtue of Proposition A.8 of [2], which can be applied in this contest, since the weight function q r −q is of class L 1 loc (ℍ n ) , being = | | ≤ 1 and 1 < q < Q. Thus, combining (4.18)-(4.23), we have as k → ∞ Now, using the notations of (3.2), by convexity a.e. in ℍ n and for all k. Let us now distinguish two different cases.
Case q ≥ 2 . By (3.2), there exists > 0 , depending only on q, such that 2) with s = q and the Hölder inequality with = 2∕q and � = 2∕(2 − q) , there exists > 0 , depending only on q, such that as k → ∞ by (4.24), since ‖D H u k ‖ q q + ‖D H u‖ q q ≤ M for all k and some positive constant M, as stated. Therefore (4.17) is proved for all q, with 1 < q < Q . Hence, D H u k → D H u in L q (B R ;ℝ 2n ) for all R > 0 . Consequently, up to subsequences, not relabelled, we get that and for all R > 0 there exists a function h R ∈ L q (B R ) such that |D H u k | ≤ h R a.e. in B R for all k.
Fix in C ∞ 0 (ℍ n ) and let R > 0 so large that supp ⊂ B R . Since by assumption ⟨I � (u k ), ⟩ = o(1) as k → ∞ , we have  Finally, from an application of the Brézis and Lieb lemma we conclude, thanks to (4.45), that as k → ∞ . Consequently, u k → u in W as k → ∞ . ◻ Proof of Theorem 1.1 First, Lemma 3.2 guarantees that for any ∈ (−∞, H q ) and any ≥ * the functional I has the geometry of the mountain pass lemma. Thus, I admits a Palais-Smale sequence (u k ) k at level c , which, in virtue of Lemma 4.1 part (i), up to a subsequence, weakly converges to some limit u ∈ W . Moreover, by Lemma 4.1 part (ii), the weak limit u ∈ W is a critical point of I. Now, for all ∈ (−∞, H q ) and all ≥ * the functional I satisfies the (PS) c , condition, as asserted in Lemma 4.2. Therefore, up to a subsequence, u k → u in W as k → ∞ and so u is a nontrivial solution of equation ( E ). ◻ Funding Open access funding provided by Università degli Studi di Perugia within the CRUI-CARE Agreement.
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