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

Abstract The paper deals with the existence of solutions for ( p , Q ) (p,Q) coupled elliptic systems in the Heisenberg group, with critical exponential growth at infinity and singular behavior at the origin. We derive existence of nonnegative solutions with both components nontrivial and different, that is solving an actual system, which does not reduce into an equation. The main features and novelties of the paper are the presence of a general coupled critical exponential term of the Trudinger-Moser type and the fact that the system is set in ℍ n {{\mathbb{H}}}^{n} .


Introduction
In this paper, we study the following system in the Heisenberg group n  with = ( ) ∈ ξ z t , n , = ( ) ∈ × z x y , n n , ∈ t . From here on, ⋅ denotes the natural inner product in any Euclidean space d for any dimension ≥ d 1 and |⋅| the corresponding Euclidean norm.
We denote by D φ H the horizontal gradient of a regular function φ, that is, 1 is the standard basis of the horizontal left invariant vector fields on n , 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 1 for a.e. ∈ x n and all ( ) ∈ × j  α  t  1 and , , 0, ; , , , for a.e. ∈ ξ n and for any pair and it is easy to check that condition ( ) F 1 is satisfied for any > α 1 0 . It is not hard to see that also ( ) . Of course, it is possible to produce an entire class of functions satisfying ( ) , . Let us also introduce two alternative assumptions under which it is possible to construct a solution of ( ) with different components for a.e. ∈ ξ n and all ∈ + u .
Clearly, the example ( ) = ( − ( )) Existence and multiplicity of nontrivial nonnegative solutions for equations and systems in the entire Heisenberg group 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 ( ) 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 n driven by a ( ) p Q , operator and establish the existence of nonnegative solutions for system ( ), which have both components nontrivial and under further reasonable assumptions different. . Moreover, has the property that , 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 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 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 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 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 ( ), 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 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.

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 n be the Heisenberg group of topological dimension + n 2 1, that is, the Lie group which has as a background manifold and whose group structure is given by the non-Abelian law , , , , , , and , , , , , , , . The inverse is given by It is easy to check that the Jacobian determinant of dilatations δ R is constant and equal to is the homogeneous dimension of n . The anisotropic dilation structure on n induces the Korányi norm, which is given by f o r a l l , . 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 , n . The corresponding distance, the so-called Korányi distance, is f o r a l l , .
the Korányi open ball of radius R centered at ξ 0 . For simplicity we put 0, 0 is the natural origin of n . For any measurable set ⊂ U n let | | U be the Haar measure of U, which coincides with the ( + ) n 2 1dimensional Lebesgue measure and is invariant under left translations and Q-homogeneous with respect to dilations.
The real Lie algebra of n is generated by the left-invariant vector fields on n

This basis satisfies the Heisenberg canonical commutation relations
j j n n 1 2 , we consider the natural inner product given by , H produces the Hilbertian norm for the horizontal vector field X. For any horizontal vector field function = ( ) Let us now review some classical facts about the first-order Sobolev spaces on the Heisenberg group n . We just consider the special case in which ≤ ℘ < ∞ . For a complete treatment on the topic we refer to [27][28][29] and we just recall that, if ≤ ℘ < Q 1 , then the embedding . As a consequence, we get a simple lemma for the real separable reflexive Banach space W, which is the solution space for ( ) defined in Section 1. Before stating this result, let us introduce some useful remarks concerning the vectorial Sobolev norms.
The above inequality is trivial when ≡ ϱ 0. Otherwise, in the case in which ( ) ≠ ξ ϱ 0at a point ∈ ξ n , we have Both estimates give (2.3).
Combining the classical results in the Sobolev space theory, we easily get the next lemma, where we endow , whenever ≤ ℘ < ∞ 1 .
where C q depends on q, p and Q. . It is well known, see for example [30], that The corresponding norm is Let us now report the next result obtained by Lam et al. in [3], see also [19].
, and for any α, such that the inequality 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

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 We say that the couple ( is the Euler-Lagrange functional associated with ( ). Clearly, I is well defined and of class ( ) C W 1 by ( ) F 1 and the assumptions on g and h, and the (weak) solutions of ( ) 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.
By (2.8) and (3.7) and taking ∈ ( ] δ 0, 1 sufficiently small to apply Lemma 2.4, we get for all ( Consider now the function , we get This completes the proof of the geometry of the functional I.
nontrivial and nonnegative such that , and a pair of functions ( An application of the Ekeland variational principle in B ρ yields the existence of a sequence {( 1 and , , 1 , ,     a.e. in n . Hence, ( is a nonnegative pair in n . Consequently, without loss of generality, we can assume that ( ) = ( ) in W. This completes the proof of (3.2). □ Taking inspiration from [1], we are now ready to prove the main result of the paper. Take > R 0 and ∈ ( ) in W. Similarly, we can obtain (3.10) also in the v variable, that is, as