Critical Schrödinger-Hardy systems in the Heisenberg group

The paper is focused on existence of nontrivial solutions of a Schrodinger-Hardy system in the Heisenberg group, involving critical nonlinearities. Existence is obtained by an application of the mountain pass theorem and the Ekeland variational principle, but there are several difficulties arising in the framework of Heisenberg groups, also due to the presence of the Hardy terms as well as critical nonlinearities.

1. Introduction. In this paper we prove existence of nontrivial solutions for the elliptic Schrödinger-Hardy system in the Heisenberg group H n        − ∆ p H n u + V (q)|u| p−2 u − γψ p |u| p−2 u r(q) p = λH u (q, u, v) + α p * |v| β |u| α−2 u, where γ and λ are real parameters, Q = 2n + 2 is the homogeneous dimension of the Heisenberg group H n , 1 < p < Q, the exponents α > 1 and β > 1 are such that α + β = p * , p * = pQ/(Q − p), and ∆ p H n is the p-Laplacian operator on H n , which is defined by ∆ p H n ϕ = div H (|D H n ϕ| p−2 H n D H n ϕ) along any ϕ ∈ C ∞ 0 (H n ), that is ∆ p H n is the familiar horizontal p-Laplacian operator. The potential function V verifies (V) V ∈ C(H n ) and inf x∈H n V (x) = V 0 > 0.
Moreover, r denotes the Heisenberg norm r(q) = r(z, t) = (|z| 4 + t 2 ) 1/4 , z = (x, y) ∈ R n × R n , t ∈ R, |z| the Euclidean norm in R 2n , 376 PATRIZIA PUCCI the horizontal gradient as in (2.2), {X j , Y j } n j=1 the basis of horizontal left invariant vector fields on H n , that is for j = 1, . . . , n. The weight function ψ appearing in (1.1) is defined as ψ = |D H n r| H n . We emphasize that ψ is identically 1 in the Euclidean canonical case and we refer to Section 2 for further details.
The nonlinearities H u and H v denote the partial derivatives of H with respect to the second variable and the third variable, respectively, and H satisfies (H) H ∈ C 1 (H n × R 2 , R + ), H z (q, 0, 0) = 0 for all q ∈ H n and there exist µ and s such that p < µ ≤ s < p * and for every ε > 0 there exists C ε > 0 for which the inequality |H w (q, w)| ≤ µε|w| µ−1 + qC ε |w| s−1 , w = (u, v), |w| = u 2 + v 2 , holds for any (q, w) ∈ H n × R 2 , where H w = (H u , H v ), and also 0 ≤ µH(q, w) ≤ H w (q, w) · w for all (q, w) ∈ H n × R 2 is valid. Finally, for all measurable set E of H n , with positive measure, H(q, u, v) > 0 for all q ∈ E and (u, v) ∈ R + × R + .
Throughout the paper, statements involving measure theory are always understood to be with respect to Haar measure on H n , which coincides with the (2n + 1)dimensional Lebesgue measure. We refer to Section 2 for further details.
A similar problem was recently studied in [13] and [27], for the fractional p-Laplacian operator, in the context of the Euclidean space. In [27], the Hardy terms were not considered.
In order to handle system (1.1), it is crucial to introduce the Hardy-Sobolev inequality. Since 1 < p < Q, by [14,15,31,32], we know that for all ϕ ∈ C ∞ 0 (H n ) ϕ p * ≤ C Q,p D H n ϕ p , p * = p Q Q − p , (1.2) where C Q,p is a positive constant depending only on Q and p. Theorem 1 of [28] gives that for all ϕ ∈ C ∞ 0 (H n \ {O}), with O = (0, 0) the natural origin in H n . The above Hardy inequality was obtained in [18] when p = 2 and, in another version, in [10] for all p > 1. The best Hardy-Sobolev constant H p = H(p, Q) is given by Let us introduce some notation. Define with associated norm Much interest has grown on systems involving Hardy terms and critical exponents. We refer to the recent papers [12,13,17] for a large bibliography on the topic in the Euclidean setting, and to [5] in the Heisenberg frame. holds.
The proof of Theorem 1.1 somehow follows [13], but there are some technical difficulties due to the more general setting considered in this paper, as well as to the presence of the Hardy terms and the critical nonlinearities. The main compactness result, Theorem 2.2, is proved following the argument of Lemma 2.2 in [7] given for the Euclidean case. We refer also to Theorem 2.1 in [27] for further details and comments. In any case, the new key result, of independent interest, is given by the crucial Lemma 3.4.
Taking inspiration from [13], we treat also the sublinear case, that is when the exponent s ∈ (1, p), and when H is of the special separated form H(q, u, v) = h(q)f (u, v). Hence, we deal with the new system in H n where V satisfies (V), σ > 0 and f verifies (f 1 ) f ∈ C 1 (R 2 , R + ) and there exist C > 0 and s ∈ (1, p) such that and f u , f v denote the partial derivatives of f with respect to the first and second variable; (f 2 ) there exist a 0 > 0, δ > 0 and s 1 ∈ (1, p) such that f (w) ≥ a 0 |w| s1 for all w ∈ R 2 , with |w| ≤ δ.
Concerning the function h in (1.6), we assume from now on that h verifies Clearly, conditions (h) simply require that h is not trivial.
In order to cover the more interesting case when σ > 0 in (1.6), we need a further assumption on h. Fix γ < H p and set is such that We are now able to state the existence result for (1.6).
If σ > 0 and h, depending on γ + , satisfies where s is the exponent in (f 1 ), then there exists a threshold σ * = σ * (γ, h) > 0 such that system (1.6) admits at least one nontrivial solution (u γ,σ , v γ,σ ) in W for all σ ∈ (0, σ * ). Theorem 1.2 was recently established in the Euclidean setting in Theorem 1.3 of [13] and in Theorem 1.2 of [27] when γ = 0, that is without the Hardy terms. Again, the Heisenberg setting makes Theorem 1.2 more difficult to handle than in [13,27]. As far as we know, Theorems 1.1 and 1.2 are new even when p = 2. Indeed, Theorem 1.2 generalizes the existence results obtained in [8] in several directions, as well as in the papers cited in [13].
The paper is organized as follows. In Section 2 we recall the main notations and definitions related to the Heisenberg group and for the natural solution space W of (1.1) and (1.6) we prove the key compactness theorems, particularly helpful for the next sections. In Section 3, using the mountain pass of Ambrosetti and Rabinowitz, we obtain the existence of nontrivial solutions for (1.1), that is we prove Theorem 1.1. Section 4 is devoted to the proof of Theorem 1.2 via the Ekeland variational principle.
2. Notations and premilinaries. We briefly recall the relevant definitions and notations related to the Heisenberg group functional setting. For a complete treatment, we refer to [9,18,23,24].
In H n the natural origin is denoted by O = (0, 0). Define where | · | stands for the Euclidean norm in R 2n . The Korányi norm is homogeneous of degree 1, with respect to the dilations δ R : Hence, the Korányi distance, is and the Korányi open ball of radius R centered at q 0 is For simplicity B R denotes the ball of radius R centered at q 0 = O.
The Jacobian determinant of δ R is R 2n+2 . The natural number Q = 2n + 2, which is the so-called homogeneous dimension of H n , plays a role analogous to the topological dimension in the Euclidean context, see [24] and the references therein.
The Haar measure on H n coincides with the Lebesgue measure on R 2n+1 . It is invariant under left translations and Q-homogeneous with respect to dilations. Hence, as noted in [23], the topological dimension 2n + 1 of H n is strictly less than its Hausdorff dimension Q = 2n + 2. We denote by |E| the (2n + 1)-dimensional Lebesgue measure of any measurable set E ⊂ H n . Then, The vector fields for j = 1, . . . , n constitute a basis for the real Lie algebra of left-invariant vector fields on H n . This basis satisfies the Heisenberg canonical commutation relations for position and momentum [X j , Y k ] = −4δ jk ∂/∂t, all other commutators being zero. A vector field in the span of {X j , Y j } n j=1 will be called horizontal.
The natural inner product in the span of If furthermore g ∈ C 1 (R), then the Leibnitz formula holds, namely , then the Kohn-Spencer Laplacian, or equivalently the horizontal Laplacian in H n , of u is defined as follows and ∆ H n is hypoelliptic according to the celebrated Theorem 1.1 due to Hörmander in [21]. In particular, The main geometrical function ψ in (1.1) is defined by and 0 ≤ ψ ≤ 1, ψ(0, t) ≡ 0, ψ(z, 0) ≡ 1. Furthermore, ψ 2 is the density function, which is homogeneous of degree 0, with respect to the dilatation δ R . Direct calculations show that for details we refer to Section 2.1 of [25]. A well known generalization of the Kohn-Spencer Laplacian is the horizontal p-Laplacian on the Heisenberg group, defined by Banach space. A proof of this fact is given in the Euclidean setting in Lemma 10 of [29] and can be extended, with obvious changes, in the Heisenberg context. Hence, by Theorem 1.12 of [1], the main solution space From now on HW 1,p (H n ) denotes the horizontal Sobolev space of the functions u ∈ L p (H n ) such that D H n u exists in the sense of distributions and |D H n u| H n is in L p (H n ), endowed with the natural norm The embedding is continuous for any ν ∈ [p, p * ] by (1.2) and the interpolation inequality. Furthermore, by [19,22,33] we know that, if Ω is a bounded Poincaré-Sobolev domain in H n , the embedding is compact provided that 1 ≤ ν < p * . This result holds for Carnot-Carathéodory balls, since by [16,22,33] such sets are Poincaré-Sobolev domains. In the proofs of the main compactness results below, we apply (2.5) to the Korányi balls B R . Indeed, it is well-known that the Carnot-Carathéodory distance and the Korányi distance are equivalent on H n , see [3,26,30]. The next result was established in Lemma 2.1 of [5] in the Heisenberg context, and is an adaptation of Lemma 4.1 of [6] and Lemma 1 of [29], where the Euclidean space R n is replaced by H n .
are continuous. In particular, are continuous and there exists a constant C ν such that where C ν depends on ν, n and p.
The proof of the next result relies on Lemma 2.2 in [7], see also Theorem 2.1 in [27], for the Euclidean case. The extension to the Heisenberg setting can be derived proceeding as in [7,27], with obvious changes. For a different proof in H n we refer also to Theorem 3.1 and Lemma 3.6 of [5], which extends to the Heisenberg case Theorem 2.1 in [29] and Lemma 4.4 in [6]. v) a.e. in H n as k → ∞. Lemma 3.5 of [5] is an extension to the Heisenberg setting of Proposition A.10 of [2].
Since α > 1 and β > 1 in (1.1) and (1.6) are such that α + β = p * , then the Hölder inequality and (1.2) yield for all (u, v) ∈ W . The next results is essential for the main proofs of Sections 3 and 4.
Proof. The first part of (2.8) can be obtained just adapting the proof of Lemma 2.1 of [20] to the Heisenberg group.
For the second part of (2.8), let us fix any measurable subset U ⊂ H n . Clearly, the sequences (u k ) k and (v k ) k are bounded in HW 1,p (H n ), so that the Hölder inequality gives and, similarly, Moreover, (u, v) ∈ L p * (H n ) × L p * (H n ) by Lemma 2.1. Thus, the sequences a.e. in H n by assumption. Hence, the Vitali convergence theorem yields (2.9) Obviously, (|u k |) k is bounded in L p * (H n ) and (|v k |) k is bounded in L p * (H n ) by Lemma 2.1. Therefore, the Hölder inequality, the fact that α + β = p * and (2.9) give as k → ∞ and similarly This completes the proof of (2.8).
3. Proof of Theorem 1.1. In this section, we assume, without further mentioning, that the assumptions required in Theorem 1.1 are satisfied. System (1.1) has a variational structure and to prove Theorem 1.1 we use the celebrated mountain pass theorem of Ambrosetti and Rabinowitz at a special critical level. The Euler-Lagrange functional, I = I γ,λ , associated to (1.1) is Clearly, the functional I is well-defined on W . Under condition (H), it is easy to see that I is of class C 1 (W ), and for (u, v) ∈ W for all (Φ, Ψ ) ∈ W . From here on ·, · simply denotes the dual pairing between W and its dual space W , that is ·, · = ·, · W ,W . Hence, the critical points of I in W are exactly the (weak) solutions of (1.1).
Before verifying that I satisfies the Palais-Smale condition at level c γ,λ let us prove an essential lemma, which is inspired by Lemma 3.8 of [2], see also Lemma 4.2 of [5]. Lemma 3.3. Let γ and λ be two fixed parameters and let {(u k , v k )} k be a bounded sequence in W . Consider the sequences (g k ) k and (h k ) k , defined for all k and all q ∈ H n by (3.10) For all compact set K of H n there exists C K > 0 such that Proof. Fix γ, λ and {(u k , v k )} k as in the statement. Let K be a compact set of H n . Concerning the first and the fifth term, by Hölder's inequality where c 1 = c 1 (K), x = p * /(p * − p + 1), since V ∈ C(H n ) and Lemma 2.1 can be applied, being 1 < p < p * . The second and the sixth term can be similarly evaluated, as and c 2 = c 2 (K) by (1.3). Indeed, ψ p r −p ∈ L 1 loc (H n ), since ψ = |ψ| ≤ 1, the Jacobian determinant is r Q and 1 < p < Q. Elementary inequalities and (H), with ε = 1, give where c 3 = c 3 (K), y = p * /(p * − µ + 1) > 1 andỹ = p * /(p * − s + 1) > 1, since p < µ ≤ s < p * by (H). Finally, the fourth and the eight term can be treated in the same way, since α > 1, β > 1 and α + β = p * , that is where c 4 = c 4 (K). This completes the proof.
The next crucial lemma relies on the proofs of Theorem 2.1 of [4], of Lemma 2 of [11] and of Step 1 of Theorem 4.4 of [2] in the Euclidean context. We also refer to proof of Lemma 4.3 of [5] for the Heisenberg setting.  v) a.e. in H n and I (u k , v k ) → 0 strongly in W . Assume furthermore that there exist two vector field functions Θ and Λ in H n of class L p (H n ; R 2n ), and such that weakly in L p (H n ; R 2n ). Then, up to a subsequence, if necessary, so that φ k and similarly ψ k are in HW 1,p (H n ) by Lemma 2.1. Taking Φ = φ k and Ψ = ψ k in (3.1), we get where (g k ) k and (h k ) k are the sequences associated to {(u k , v k )} k and defined in (3.10). Now, as k → ∞ as k → ∞, since I (u k , v k ) → 0 in W by assumption and (φ k , ψ k ) 0 in W as k → ∞ by construction.
In conclusion, the first five terms in the right hand side of (3.12) go to zero as k → ∞. Now, recalling that 0 ≤ ϕ R ≤ 1 in H n , we have since (g k ) k and (h k ) k are bounded in L 1 loc (H n ) by Lemma 3.3. By the definitions of ϕ R and η ε , Combining all these facts with (3.12), we find that lim sup Define the function e k = e k (q) by Clearly, e k is nonnegative a.e. in H n for all k. Moreover, (e k ) k is bounded in where C 0 is an appropriate constant, independent of k, since (D H n u k ) k and (D H n v k ) k are bounded in L p (H n ; R 2n ), and similarly (|D H n u k | p−2 H n D H n u k ) k and (|D H n v k | p−2 H n D H n v k ) k are bounded in L p (H n ; R 2n ), since (|D H n u k | p−2 H n D H n u k ) k and (|D H n v k | p−2 H n D H n v k ) k converge weakly in L p (H n ; R 2n ) by assumption.

PATRIZIA PUCCI
Letting ε tend to 0 + we find that e θ k → 0 in L 1 (B R ) and so, since R > 0 is arbitrary, we deduce that e k → 0 a.e. in H n up to a subsequence, if necessary. From Lemma 3 of [11] it follows the validity of (3.11) and this completes the proof. Now we are ready to prove the compactness property of I at the special level c γ,λ introduced in (3.6).