(p,Q) systems with critical singular exponential nonlinearities in the Heisenberg group

1 Introduction

In this paper, we study the following system in the Heisenberg group ℍ n

where Q = 2 n + 2 is the homogeneous dimension of the Heisenberg group ℍ n , 1 < p < Q , 0 ≤ β < Q and g , h are nontrivial nonnegative functionals in the dual space H W − 1, Q ′ ( ℍ n ) of H W 1, Q ( ℍ n ) , which will be characterized later on. In other words, h , g ≠ 0 and 〈 h , u 〉 H W − 1 , Q ′ , H W 1 , Q ≥ 0 , 〈 g , u 〉 H W − 1 , Q ′ , H W 1 , Q ≥ 0 for all u ∈ H W 1, Q ( ℍ n ) , with u ≥ 0 in ℍ n . For brevity in what follows the duality pairing between H W 1, Q ( ℍ n ) and H W − 1, Q ′ ( ℍ n ) will be denoted simply by 〈 ⋅ , ⋅ 〉 . The function r in ( S ) is the Korányi norm in the Heisenberg group ℍ n , which is defined as

with ξ = ( z , t ) ∈ ℍ n , z = ( x , y ) ∈ ℝ n × ℝ n , t ∈ ℝ . From here on, ⋅ denotes the natural inner product in any Euclidean space ℝ d for any dimension d ≥ 1 and | ⋅ | the corresponding Euclidean norm.

We denote by D H φ the horizontal gradient of a regular function φ , that is,

where { X j , Y j } j = 1 n is the standard basis of the horizontal left invariant vector fields on ℍ n ,

for j = 1 , … , n . The operator Δ H , ℘ , with ℘ ∈ { p , Q } , appearing in system ( S ) is the well-known horizontal ℘ -Laplacian on the Heisenberg group, which is defined as

for all φ ∈ C 2 ( ℍ n ) , where | D H φ | H = ∑ j = 1 n [ ( X j φ ) 2 + ( Y j φ ) 2 ] .

Taking inspiration from [1], we assume that the functions F u , F v are partial derivatives of a Carathéodory function F, of exponential type, and are assumed to satisfy

(F ₁) F ( ξ , ⋅ , ⋅ ) ∈ C 1 ( ℝ 2 ) for a.e. ξ ∈ ℍ n , F u ( ξ , u , v ) = 0 for all u ≤ 0 and v ∈ ℝ , F v ( ξ , u , v ) = 0 for all u ∈ ℝ and v ≤ 0 , F u ( ξ , u , 0 ) = 0 for all u ∈ ℝ , and F v ( ξ , 0 , v ) = 0 for all v ∈ ℝ . Furthermore, there is α 0 > 0 with the property that for all ε > 0 there exists κ ε > 0 such that

for a.e. x ∈ ℍ n and all ( u , v ) ∈ ℝ 0 + × ℝ 0 + , where ℝ 0 + = [ 0 , ∞ ) , ϱ = u 2 + v 2 , ∇ F = ( F u , F v ) ,

(F₂) there exists ν > Q such that 0 < ν F ( ξ , u , v ) ≤ ∇ F ( ξ , u , v ) ⋅ ( u , v ) for a.e. ξ ∈ ℍ n and for any pair ( u , v ) ∈ ℝ + × ℝ + , where ℝ + = ( 0 , ∞ ) .

A function satisfying ( F 1 ) – ( F 2 ) is given for example by F ( u , v ) = u + Q e v + Q ′ − S Q − 1 ( 1 , v + ) for all ( u , v ) ∈ ℝ × ℝ . Indeed, F u ( u , v ) = Q u + Q − 1 e v + Q ′ − S Q − 1 ( 1 , v + ) , F v ( u , v ) = Q ′ v + Q ′ − 1 u + Q e v + Q ′ − S Q − 2 ( 1 , v + ) for all ( u , v ) ∈ ℝ × ℝ and it is easy to check that condition ( F 1 ) is satisfied for any α 0 > 1 . It is not hard to see that also ( F 2 ) is verified with ν = Q ( 1 + Q ′ ) > Q . Of course, it is possible to produce an entire class of functions satisfying ( F 1 ) – ( F 2 ) by taking F ( ξ , u , v ) = a ( ξ ) Φ ( u , v ) for all ( ξ , u , v ) ∈ ℍ n × ℝ × ℝ , where a is a positive measurable function, a ∈ L ∞ ( ℍ n ) and ess inf ξ ∈ ℍ n a ( ξ ) > 0 , while Φ ( u , v ) = u + Q e v + Q ′ − S Q − 1 ( 1 , v + ) , ( u , v ) ∈ ℝ × ℝ .

Let us also introduce two alternative assumptions under which it is possible to construct a solution of ( S ) with different components

(H₁) if g ≢ h , assume that F u ( ξ , u , u ) ≡ F v ( ξ , u , u ) for a.e. x ∈ ℍ n and all u ∈ ℝ ,

(H₂) if g ≡ h , assume that F u ( ξ , u , u ) ≠ F v ( ξ , u , u ) for a.e. ξ ∈ ℍ n and all u ∈ ℝ + .

Clearly, the example F ( u , v ) = u + Q e v + Q ′ − S Q − 1 ( 1 , v + ) , ( u , v ) ∈ ℝ × ℝ , verifies ( H 2 ) .

The natural space where finding solutions of ( S ) is

endowed with the norm

where ∥ u ∥ H W 1 , ℘ = ∥ u ∥ ℘ ℘ + ∥ D H u ∥ ℘ ℘ 1 / ℘ for all u ∈ H W 1, ℘ ( ℍ n ) and ∥ ⋅ ∥ ℘ denotes the canonical L ℘ ( ℍ n ) norm for any ℘ ≥ 1 . We say that a pair ( u , v ) is nonnegative if both components are nonnegative in ℍ n .

Existence and multiplicity of nontrivial nonnegative solutions for equations and systems in the entire Heisenberg group ℍ n , involving elliptic operators with standard Q-growth and critical Trudinger-Moser nonlinearities, have been proved in a series of papers. We refer to [2,3,4,5,6] and to references therein.

On the other hand, in the literature there are few contributions devoted to the study of coupled systems involving both exponential nonlinearities and nonstandard growth conditions in the Heinsenberg context. In the Euclidean setting, a similar problem has been studied in [1], where the authors consider coupled systems involving exponential nonlinearities and ( p , N ) growth. Other references, again in the Euclidean setting, are given by [7,8,9,10,11]. We also refer to the recent paper [12], which contains the proof of a non-singular version of the Moser-Trudinger inequality in the Cartesian product of Sobolev spaces.

In the Heisenberg context, existence of solutions for the equation corresponding to system ( S ) has been established in [13]. Let us cite [14,15,16] for related problems.

In this paper, we solve for the first time in the literature a coupled exponential system in the Heisenberg group ℍ n driven by a ( p , Q ) operator and establish the existence of nonnegative solutions for system ( S ), which have both components nontrivial and under further reasonable assumptions different.

Since the solution ( u g , h , v g , h ) , constructed in Theorem 1.1, has both components nontrivial and different, it is evident that it solves an actual system, which does not reduce into an equation.

The essential tool when dealing with exponential nonlinearities is the celebrated Trudinger-Moser inequality. The Trudinger-Moser inequality in bounded domains Ω of the Heisenberg group was first established in [17] by Cohn and Lu, using a sharp representation formula for functions of class C c ∞ ( ℍ n ) in terms of the horizontal gradient. The authors in [17] adapted Adams’ idea in [18] to avoid considering the horizontal gradient of the rearrangement function, which is not available in the Heisenberg setting. The situation is more involved when concerning the Trudinger-Moser-type inequalities for unbounded domains of ℍ n , since Adams’ approach does not work any longer. However, Lam, Lu and Tang obtained in [3], see also [19], a sharp Trudinger-Moser inequality on the whole Heisenberg group ℍ n , which is subcritical in the sense clarified in Theorem 2.2. This inequality is crucial in the proof of Theorem 1.1.

The proof of Theorem 1.1 is obtained via an application of the Ekeland variational principle and the Trudinger-Moser inequality on the whole Heisenberg group ℍ n . Even if the argument follows somehow the strategies in [1,13,14,15,20] and relies on standard variational methods, the extension to this more general context is pretty involved and leads to new difficulties, arising from the non-Euclidean and vectorial nature of the problem. In particular, a delicate step is the proof of the fact that both components of the constructed solution are nontrivial and different. A similar process to show that solutions of systems have both components nontrivial and different first appears, under different assumptions, in [1,15,21].

Theorem 1.1 improves in several directions previous results, not only from the Euclidean to the Heisenberg setting but also for the presence of the coupled exponential nonlinearities and the ( p , Q ) growth. In particular, Theorem 1.1 extends Theorem 1.1 of [1] because of the non-Euclidean context and the presence of the singularity at zero in the right hand side of ( S ), and also Theorem 1.1 of [13] from the scalar to the vectorial case.

The paper is organized as follows. In Section 2, we recall some basic definitions and backgrounds related to the Heisenberg group ℍ n , as well as useful properties of the solution space W and some technical lemmas on the Trudinger-Moser inequality in the Heisenberg group. In Section 3, we prove Theorem 1.1, using a minimization argument based on the Ekeland variational principle.

2 Preliminaries

In this section, we briefly recall some useful notations and preliminaries on the Heisenberg group. For a complete treatment we refer to [22,23,24,25,26].

Let ℍ n be the Heisenberg group of topological dimension 2 n + 1 , that is, the Lie group which has ℝ 2 n + 1 as a background manifold and whose group structure is given by the non-Abelian law

for all ξ , ξ ′ ∈ ℍ n , with

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

We can consider the family of dilations ( δ R ) R , where δ R : ℍ n → ℍ n is defined for any R ∈ ℝ by

It is easy to check that the Jacobian determinant of dilatations δ R is constant and equal to R 2 n + 2 , where the natural number Q = 2 n + 2 is the homogeneous dimension of ℍ n .

The anisotropic dilation structure on ℍ n induces the Korányi norm, which is given by

Consequently, the Korányi norm is homogeneous of degree 1, with respect to the dilations δ R , R > 0 , that is,

for all ξ = ( z , t ) ∈ ℍ n . The corresponding distance, the so-called Korányi distance, is

We denote by B R ( ξ 0 ) = { ξ ∈ ℍ n : d K ( ξ , ξ 0 ) < R } the Korányi open ball of radius R centered at ξ 0 . For simplicity we put B R = B R ( O ) , where O = ( 0 , 0 ) is the natural origin of ℍ n .

For any measurable set U ⊂ ℍ n let | U | be the Haar measure of U, which coincides with the (2 n + 1) -dimensional Lebesgue measure and is invariant under left translations and Q-homogeneous with respect to dilations.

The real Lie algebra of ℍ n is generated by the left-invariant vector fields on ℍ n

for j = 1 , … , n . This basis satisfies the Heisenberg canonical commutation relations

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

We define the horizontal gradient of a C 1 function u : ℍ n → ℝ by

Clearly, D H u is an element of the span of { X j , Y j } j = 1 n , denoted by span { X j , Y j } j = 1 n . In span { X j , Y j } j = 1 n ≃ ℝ 2 n , we consider the natural inner product given by

for 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 X.

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 ) , we define the horizontal divergence of X 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 ≤ ℘ < ∞ and Ω is an open set in ℍ n . Denote by H W 1, ℘ ( Ω ) the horizontal Sobolev space consisting of the functions u ∈ L ℘ ( Ω ) such that D H u exists in the sense of distributions and | D H u | H ∈ L ℘ ( Ω ) , endowed with the natural norm

When Ω = ℍ n the notation will be simplified into ∥ ⋅ ∥ ℘ = ∥ ⋅ ∥ L ℘ ( ℍ n ) and ∥ ⋅ ∥ H W 1 , ℘ = ∥ ⋅ ∥ H W 1 , ℘ ( ℍ n ) . For a complete treatment on the topic we refer to [27,28,29] and we just recall that, if 1 ≤ ℘ < Q , then the embedding H W 1 , ℘ ( Ω ) ↪ L q ( Ω ) is continuous for all q ∈ [ ℘ , ℘ ⁎ ] , being ℘ ⁎ = ℘ Q / ( Q − ℘ ) , while H W 1 , Q ( Ω ) ↪ L q ( Ω ) is continuous for all q ∈ [ Q , ∞ ) . As a consequence, we get a simple lemma for the real separable reflexive Banach space W, which is the solution space for ( S ) defined in Section 1. Before stating this result, let us introduce some useful remarks concerning the vectorial Sobolev norms.

If ( u , v ) ∈ [ H W 1 , Q ( ℍ n ) ] 2 , then ϱ = | ( u , v ) | = u 2 + v 2 ∈ H W 1 , Q ( ℍ n ) and

The above inequality is trivial when ϱ ≡ 0 . Otherwise, in the case in which ϱ ( ξ ) ≠ 0 at a point ξ ∈ ℍ n , we have

This implies that | D H ϱ | ≤ | D H u | H + | D H v | H in ℍ n . Consequently,

Both estimates give (2.3).

Combining the classical results in the Sobolev space theory, we easily get the next lemma, where we endow [ L ℘ ( ℍ n ) ] 2 , with the norm ∥ ( u , v ) ∥ ℘ = ∥ u ∥ ℘ + ∥ v ∥ ℘ , whenever 1 ≤ ℘ < ∞ .

Clearly, when in particular Q / 2 ≤ p < Q , the embedding (2.4) is continuous for all q ∈ [ p , ∞ ) .

Throughout the paper, H W − 1, Q ′ ( ℍ n ) denotes the dual space of H W 1, Q ( ℍ n ) . It is well known, see for example [30], that

where the pairing between a function u ∈ H W 1 , Q ( ℍ n ) and a distribution h = h 0 + ∑ j = 1 n ( h j 1 X j + h j 2 Y j ) is given as usual by

The corresponding norm is

Let us now report the next result obtained by Lam et al. in [3], see also [19].

The Trudinger-Moser inequality in Theorem 2.2 is subcritical, since α cannot reach the sharp threshold α Q , β . For later purposes, let us introduce for all q ∈ [ Q , ∞ ) , the singular eigenvalue defined by

see also [3,4,6]. By the continuity of the Sobolev embedding and by the Hölder inequality, we get that λ q > 0 for any q ∈ [ Q , ∞ ) , see [13]. Let us now state Lemma 4.1 of [4] and prove in Lemma 2.4 an extension of this result, which is crucial in the proof of Theorem 1.1.

3 Existence of solutions

This section is devoted to the proof of the main result. From now on we assume, without further mentioning, that the structural assumptions required in Theorem 1.1 hold. From here on we adopt the notation

for all ( u , φ ) ∈ W .

We say that the couple ( u , v ) ∈ W is a (weak) solution of system ( S ) if

for any ( φ , ψ ) ∈ W . Evidently, system ( S ) has a variational structure. Indeed, the functional I : W → ℝ , defined by

for all ( u , v ) ∈ W is the Euler-Lagrange functional associated with ( S ). Clearly, I is well defined and of class C 1 ( W ) by ( F 1 ) and the assumptions on g and h, and the (weak) solutions of ( S ) are exactly the critical points of I. Here and in the following we denote by W ′ the dual space of W and by 〈 ⋅ , ⋅ 〉 W ′ , W the dual pairing between W ′ and W.

We first prove in Lemma 3.1 the geometric properties of the functional I, necessary to apply a minimization argument. The next lemma is an extension of Lemma 3.1 of [1].