Existence for (p, q) critical systems in the Heisenberg group

1 Introduction

In recent years, great attention has been focused on the study of (p, q) systems, not only for their mathematical interest, but also for their relevant physical interpretation in applied sciences. It is also well known that the Heisenberg group ℍⁿ, n = 1, 2, …, appears in various areas, such as quantum theory (uncertainty principle, commutation relations) cf. [1, 2], signal theory cf. [3], theory of theta functions cf. [1, 4], and number theory. For additional physical interpretations we mention [5], while for general motivations in setting problems in the Heisenberg group context we refer to [6, 7, 8, 9, 10, 11] and the papers cited there.

Here we prove the existence of nontrivial solutions for quasilinear elliptic systems in the Heisenberg group ℍⁿ, involving (p, q) operators, which generalize the ones introduced by Marcellini in [12]. In particular, we consider the system in ℍⁿ

where λ is a positive real parameter, Q = 2n + 2 is the homogeneous dimension of the Heisenberg group ℍⁿ, α > 1 and β > 1 are two exponents such that α + β = ℘^* and ℘^* is a critical exponent associated to ℘, with 1 < ℘ < Q, that is

which is related to the (p, q) operator A in (𝓢). The vector

denotes the horizontal gradient of u, where {Xj,Yj}j=1n is the basis of the horizontal left invariant vector fields on ℍⁿ, that is

for j = 1, …, n.

The starting point is the paper [13], where the authors studied similar and more general systems in the Euclidean context. The main novelty of the paper is indeed to properly set (𝓢) in the Heisenberg context. In fact, several theorems have to be proved in the new framework for the first time. Indeed, the key existence argument relies on the celebrated Lemma I.1 of [14] as well on the concentration–compactness principle in the vectorial Heisenberg context, both due to Lions. Following [13], we require the structure conditions.

1. A is a strictly positive and strictly increasing function of class C¹(ℝ⁺),

2. B ∈ C(ℝ⁺) is a strictly positive function and t ↦ tB(t) is strictly increasing in ℝ⁺, with tB(t) → 0 as t → 0⁺.

For simplicity, we introduce the functions 𝓐 and 𝓑, which are 0 at 0 and which are obtained by integration from

Notice that (A) implies that tA(t) → 0 as t → 0⁺, and so tA(t) and tB(t) are defined to be 0 at 0. We furthermore assume

1. there exist constants a₀, 𝔞₀, b₀, 𝔟₀ strictly positive, with a₀ ≤ 1, a₁, 𝔞₁, b₁, 𝔟₁ nonnegative, <with the property that 𝔞₁ > 0 implies 𝔟₁ > 0, a₁ > 0 and b₁ > 0, and there are exponents p and q, with 1 < p < q < ℘^*, where 1 < ℘ < Q, ℘ = p if 𝔞₁ = 0 and ℘ = q if 𝔞₁ > 0, such that for all t ∈ R0+

   a0tp−1+1R+(a1)a1tq−1≤A′(t)≤a0tp−1+a1tq−1,b0tp−1+1R+(b1)b1tq−1≤B′(t)≤b0tp−1+b1tq−1,

   where 𝟙_U is the characteristic function of a Lebesgue measurable subset U of ℝ. Assumption (C₁) was introduced by Figueiredo in [15]. Moreover, we assume

2. there exist constants θ and ϑ, with ℘ ≤ min{θ, ϑ} < ℘^*, such that

   θA(t)≥tA′(t),ϑB(t)≥tB′(t)for  all  t∈R0+

   holds.

Several general systems verify all the assumptions (A), (B), (C₁) and (C₂), and we refer to [13] for the main prototypes of the potentials 𝓐 and 𝓑 covered.

The functions H_u and H_v in (𝓢) are partial derivatives of a function H of class C¹(ℝ²), satisfying the condition

(H) H > 0 in ℝ⁺ × ℝ⁺, H_u(u, 0) = 0 for all u ∈ ℝ and H_v(0, v) = 0 for all v ∈ ℝ. Furthermore, there exist 𝔪, m, σ such that ℘ < 𝔪 < m < ℘^*, max{θ, ϑ} < σ < ℘^* and for every ε > 0 there exists C_ε > 0 for which the inequality

where |(u,v)|=u2+v2, ∇ H = (H_u, H_v), and also the inequalities

hold, where θ, ϑ are given in (C₂).

Throughout the paper, ⋅ denotes the Euclidean inner product and |⋅| the corresponding Euclidean norm in any space ℝ^m, m = 1, 2, ….

Since ℘ = p if 𝔞₁ = 0, while ℘ = q if 𝔞₁ > 0, the natural space where finding solutions of (𝓢) is

endowed with the norm

for all u ∈ HW^1,𝔭(ℍⁿ), where HW^1,𝔭(ℍⁿ) is the horizontal Sobolev space defined in Section 2. We are now able to state the existence result for (𝓢).

Since the solution (u_λ, v_λ), constructed in Theorem 1.1, has both components non trivial, it is evident that it solves an actual system, which does not reduce into an equation. Moreover, Theorem 1.1 extends in several directions previous results, not only from the Euclidean to the Heisenberg setting, but also for the mild growth conditions on the main elliptic operator of (𝓢), cf. e.g. [16, 17, 18, 19].

Even if assumption (C₁) allows us to treat simultaneously when either 𝔞₁ = 0 or 𝔞₁ > 0, the most interesting case is the latter, in which ℘ = q and so the couple (p, q) appears in its importance. Indeed, when 𝔞₁ > 0 in (C₁), the main elliptic operator A has a (p, q) growth. Moreover, in this case, the solution space W has a strong dependence on (p, q), since we consider existence of entire solutions in the Heisenberg group. In fact, (p, q) problems are usually settled in bounded domains Ω, so that the natural solution space is W=HW01,p(Ω)∩HW01,q(Ω)=HW01,q(Ω). In this paper the situation is much more delicate, since the problem is in the entire group of Heisenberg.

The importance of studying problems involving operators with non standard growth conditions, or (p, q) operators, begins with the papers of Marcellini [12] and Zhikov [20]. Since then, the topic has been attracting increasing attention on existence and qualitative properties of solutions, but the vectorial case is much harder. Indeed, (𝓢) has a relevant physical interpretation in applied sciences as well as a mathematical challenge in overcoming the new difficulties intrinsic to (𝓢). Because of the lack of compactness, the main difficulty in treating (p, q) systems in our context relies on the proof of the key Lemma 4.6, dedicated on the crucial properties of the Palais–Smale sequences at special levels. To this aim, we prove a concentration compactness principle for systems in S = S^1,℘(ℍⁿ) × S^1,℘(ℍⁿ), where S^1,℘(ℍⁿ), 1 < ℘ < Q, is the Folland–Stein space, that is the completion of Cc∞ (ℍⁿ) with respect to the norm

To the best of our knowledge, the conclusions obtained in Theorem 1.2 are new in the Heisenberg context. The proof of this result follows somehow the arguments of [21] and [22, 23], but there are some technical difficulties due to the more general context, which we overcome.

Finally, the existence of solutions for problem (𝓢) rely on a readaptation of Proposition 2.8 of [13] in the Heisenberg group. Therefore, we have to prove an extension from the Euclidean to the Heisenberg context of the celebrated Lemma I.1 in [14] due to Lions, which is given in its general statement.

The paper is organized as follows. In Section 2, we recall some fundamental definitions and properties related to the Heisenberg group ℍⁿ. Section 3 is devoted to the proof of Theorem 1.2, while Section 4 deals with some lemmas useful to the study of system (𝓢). In particular, we prove Theorem 1.3 and finally Theorem 1.1, adapting the strategy of [13] and extending the results there to the Heisenberg group setting.

2 Preliminaries

In this section we present the basic properties of ℍⁿ as a Lie group. Analysis on the Heisenberg group is very interesting because this space is topologically Euclidean, but analytically non–Euclidean, and so some basic ideas of analysis, such as dilatations, must be developed again. One of the main differences with the Euclidean case is that the homogeneous dimension Q = 2n + 2 of the Heisenberg group plays a role analogous to the topological dimension in the Euclidean context. For a complete treatment, we refer to [24, 25, 26, 27].

Let ℍⁿ be the Heisenberg Lie group of topological dimension 2n + 1, that is the Lie group which has ℝ²ⁿ⁺¹ as a background manifold, endowed with the non–Abelian group law

for all ξ, ξ′ ∈ ℍⁿ, with

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

The vector fields for j = 1, …, n

constitute a basis for the real Lie algebra of left–invariant vector fields on ℍⁿ. This basis satisfies the Heisenberg canonical commutation relations

A left invariant vector field X, which is in the span of {Xj,Yj}j=1n , is called horizontal.

We define the horizontal gradient of a C¹ function u:ℍⁿ → ℝ by

Clearly, D_Hu is an element of the span of {Xj,Yj}j=1n . Furthermore, if f ∈ C¹(ℝ), then

In the span of {Xj,Yj}j=1n ≃ ℝ²ⁿ we consider the natural inner product given by

for X={xjXj+x~jYj}j=1n and Y={yjXj+y~jYj}j=1n. The inner product (⋅, ⋅)_H produces the Hilbertian norm

for the horizontal vector field X. Moreover, the Cauchy–Schwarz inequality

holds for any horizontal vector fields X and Y.

For any horizontal vector field function X = X(ξ), X={xjXj+x~jYj}j=1n , of class C¹(ℍⁿ, ℝ²ⁿ), we define the horizontal divergence of X by

If furthermore u ∈ C¹(ℍⁿ), then the Leibnitz formula continues to be valid, that is

Similarly, if u ∈ C²(ℍⁿ), then the Kohn–Spencer Laplacian, or equivalently the horizontal Laplacian in ℍⁿ, of u is defined as follows

According to the celebrated Theorem 1.1 due to Hörmander in [28], the operator Δ_H is hypoelliptic. In particular, Δ_Hu = div_HD_Hu for each u ∈ C²(ℍⁿ).

A well known generalization of the Kohn–Spencer Laplacian is the horizontal 𝔭–Laplacian on the Heisenberg group, 𝔭 ∈ (1, ∞), defined by

for all φ ∈ Cc∞ (ℍⁿ).

The Korányi norm is given by

The corresponding distance, the so called Korányi distance, is

This distance acts like the Euclidean distance in horizontal directions and behaves like the square root of the Euclidean distance in the missing direction. Consequently, the Korányi norm is homogeneous of degree 1, with respect to the dilations δ_R : (z, t) ↦ (Rz, R²t), R > 0, since

for all ξ = (z, t) ∈ ℍⁿ.

Let B_R(ξ₀) = {ξ ∈ ℍⁿ : d_K(ξ, ξ₀) < R} be the Korányi open ball of radius R centered at ξ₀. For simplicity B_R denotes the ball of radius R centered at ξ₀ = O, where O = (0, 0) is the natural origin of ℍⁿ.

It is easy to verify that the Jacobian determinant of dilatations δ_R : ℍⁿ → ℍⁿ is constant and equal to R²ⁿ⁺². This is why the natural number Q = 2n + 2 is called homogeneous dimension of ℍⁿ.

We recall also the definition of Carnot–Carathéodory distance on ℍⁿ and for further details we refer to [7, 25]. A piecewise smooth curve y : [0, 1] → ℍⁿ is called a horizontal curve if ÿ(t) belongs to the span of {Xj,Yj}j=1n a.e. in [0, 1]. The horizontal length of y is defined as

Now, given two arbitrary points ξ, η ∈ ℍⁿ, by the Chow–Rashevsky theorem there is a horizontal curve between them in ℍⁿ, see [29, 30]. Therefore, the Carnot–Carathéodory distance of two points ξ and η of ℍⁿ is well–defined as

Clearly, d_CC is a left invariant metric on ℍⁿ, and

for all ξ, η ∈ ℍⁿ, see [7]. Moreover, the Carnot–Carathéodory distance is homogeneous of degree 1 with respect to dilatations δ_R, that is

for all ξ, η ∈ ℍⁿ.

In the case of the Heisenberg group, it is easy to check that the Lebesgue measure on ℝ²ⁿ⁺¹ is invariant under left translations. Thus, from here on, we denote by dξ the Haar measure on ℍⁿ that coincides with the (2n+ 1)–Lebesgue measure, since the Haar measures on Lie groups are unique up to constant multipliers. We also denote by |U| the (2n + 1)–dimensional Lebesgue measure of any measurable set U ⊂ ℍⁿ. Furthermore, the Haar measure on ℍⁿ is Q–homogeneous with respect to dilations δ_R. Consequently,

In particular |B_R| = |B₁|R^Q.

As usual, for any measurable set U ⊂ ℍⁿ and for any general exponent 𝔭, with 1 ≤ 𝔭 ≤ ∞, we denote by L^𝔭(U) the canonical Banach space, endowed with the norm

while

When U = ℍⁿ or when there is not ambiguity about the set considered, for simplicity we denote the norm ∥⋅ ∥_𝔭. All the usual properties about the Lebesgue Banach spaces continue to be valid. In particular, L^𝔭(U) is a separable Banach space and C_c (U) is dense in it if 1 ≤ 𝔭 < ∞. Moreover, L^𝔭(U) is a reflexive Banach space if 1 < 𝔭 < ∞.

Let us now review some classical facts about the first–order Sobolev spaces on the Heisenberg group ℍⁿ. We restrict ourselves to the special case in which 1 ≤ 𝔭 < ∞ and Ω is an open set in ℍⁿ. Denote by HW^1,𝔭(Ω) the horizontal Sobolev space consisting of the functions u ∈ L^𝔭(Ω) such that D_Hu exists in the sense of distributions and |D_Hu|_H ∈ L^𝔭(Ω), endowed with the natural norm

It is easy to check that the distributional horizontal gradient of a function u ∈ HW^1,𝔭(Ω) is uniquely defined a.e. in Ω. Furthermore, if u is a smooth function, then its classical horizontal gradient is also the distributional horizontal gradient. For this reason, if u is a non smooth function, D_Hu is meant in the distributional sense.

For later purposes, let us introduce the convolution, which is useful also for density results, see [31, 32]. If u ∈ L¹(ℍⁿ) and v ∈ L^𝔭(ℍⁿ), with 1 ≤ 𝔭 ≤ ∞, then for a.e. ξ ∈ ℍⁿ the function

is in L¹(ℍⁿ). Moreover, u * v, defined a.e. on ℍⁿ by

is called convolution of u and v. By the analog of the Young theorem u * v belongs to L^𝔭(ℍⁿ) and

If 𝔭 = ∞, then u * v is well defined and uniformly continuous in ℍⁿ.

Using the convolution (2.2), the technique of regularization, originally introduced by Leray and Friedrichs in the Euclidean context, can be extended to the Heisenberg group ℍⁿ. In particular, it is possible to generate a sequence of mollifiers (ρ_k)_k on ℍⁿ with the usual properties, see the Appendix of [33]. Moreover, Proposition A.1.2 of [33] yield that if φ ∈ Cc∞ (ℍⁿ) and u ∈ Lloc1 (ℍⁿ) then u * φ ∈ C^∞(ℍⁿ) and

As in the Euclidean case, the density theorem for the horizontal Sobolev space continues to hold in the Heisenberg group. We present the proof for the sake of completeness and for later purposes, since this result is crucial to prove the main Lemma 4.6.

The basic embedding theorems for the Sobolev space HW^1,𝔭(ℍⁿ), first established in [34] by Folland and Stein in this type of generality, have a form similar to those in the Euclidean case, but the exponent governing the transition to the supercritical case is the homogeneous dimension Q = 2n + 2. In particular, if 𝔭 is an exponent, with 1 < 𝔭 < Q, then the embedding

is continuous for any 𝔮 ∈ [𝔭, 𝔭^*].

For a complete treatment on the compactness of the embeddings HW^1,𝔭(Ω) ↪ L^𝔮(Ω), when Ω is a well behaved domain, we refer to [25, 35, 36] and also to [37], as well as the references therein. The next definition is taken from [36].

An open set Ω of ℍⁿ is said to be a Poincaré–Sobolev domain, briefly PS domain, if there exist a bounded open subset U ⊂ ℍⁿ, with Ω ⊂ Ω ⊂ U, a covering {B}_B∈𝓕 of Ω by Carnot–Carathéodory balls B and numbers N > 0, α ≥ 1 and ν ≥ 1 such that

1. ∑_B∈𝓕 𝟙_(α+1)B ≤ N 𝟙_Ω in U,

2. there exists a (central) ball B₀ ∈ 𝓕 such that for all B ∈ 𝓕 there is a finite chain B₀, B₁, …, B_s(B), with B_i ∩ B_i+1 ≠ ∅ and |B_i ∩ B_i+1| ≥ max {|B_i|, |B_i+1|}/N, i = 0, 1, …, s(B) − 1 and moreover B ⊂ ν B_i for i = 0, 1, …, s(B).

This definition is purely metric. There is a large number of PS domains in ℍⁿ, as explained in details in [36].

For our purposes it is important also to recall a version of the Rellich–Kondrachov theorem in the Heisenberg context. The next result is a special case of Theorem 1.3.1 in [7].

Combining Theorem 2.3, with the fact that the Carnot–Carathéodory distance and the Korányi distance are equivalent on ℍⁿ, we get (i) when Ω is any Korányi ball B_R(ξ₀) centered at ξ₀ ∈ ℍⁿ, with radius R > 0.

3 The concentration compactness principle for critical systems on ℍⁿ

For the study of nonlinear elliptic problems, involving critical nonlinearities in the sense of the Sobolev inequality, the concentration compactness principle due to Lions has been being a fundamental tool for proving existence of solutions since its appearance. We just mention [13, 38, 39, 40, 41, 42, 43] and the references therein.

In this section, taking inspiration from [21] and following the basic ideas of the papers [22, 23] of Lions, we extend the vectorial concentration compactness principle to the Heisenberg group setting. This key result is one of the main tools to prove the existence Theorem 1.1. However, it is of independent interest and so we present it in a general setting, giving a detailed proof not included in the original work.

Throughout the section, we assume that ℘ is an exponent, with 1 < ℘ < Q, and that α > 1 and β > 1 are such that α + β = ℘^*, where ℘^* = ℘ Q / (Q − ℘). First, by [31] we know that there exists a positive constant C_℘^*, depending only on Q and ℘, such that for all u ∈ S^1,℘(ℍⁿ)

holds. Then, the following best constant is well defined

where S = S^1,℘(ℍⁿ) × S^1,℘(ℍⁿ). Indeed, the Hölder inequality and the Folland–Stein inequality (3.1) give

for all (u, v) ∈ S, since α, β > 1 and α + β = ℘^*. Therefore, (3.3) and the Young inequality yield

for all (u, v) ∈ S. Hence J≥1/C℘∗℘>0.

Before turning to Theorem 1.2 and its proof, let us show the next result, which is an extension and generalization of Lemma 2.1 in [44] to the Heisenberg setting.

Finally, we are ready to prove Theorem 1.2.