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

Abstract: This paper deals with the existence of entire nontrivial solutions for critical quasilinear systems (S) in the Heisenberg group Hn, driven by general (p, q) elliptic operators of Marcellini types. The study of (S) requires relevant topics of nonlinear functional analysis because of the lack of compactness. The key step in the existence proof is the concentration–compactness principle of Lions, here proved for the rst time in the vectorial Heisenberg context. Finally, the constructed solution has both components nontrivial and the results extend to the Heisenberg group previous theorems on quasilinear (p, q) systems.


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 H n , n = , , . . . , 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 H n , involving (p, q) operators, which generalize the ones introduced by Marcellini in [12]. In particular, we consider the system in H n where λ is a positive real parameter, Q = n + is the homogeneous dimension of the Heisenberg group H n , α > and β > are two exponents such that α + β = ℘ * and ℘ * is a critical exponent associated to ℘, with < ℘ < Q, that is denotes the horizontal gradient of u, where {X j , Y j } n j= is the basis of the horizontal left invariant vector elds on H n , that is for j = , . . . , 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 rst 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.
(A) A is a strictly positive and strictly increasing function of class C (R + ), (B) B ∈ C(R + ) is a strictly positive function and t → tB(t) is strictly increasing in R + , with tB(t) → as t → + .
For simplicity, we introduce the functions A and B, which are 0 at 0 and which are obtained by integration from A (t) = tA(t), B (t) = tB(t) for all t ∈ R + .
Notice that (A) implies that tA(t) → as t → + , and so tA(t) and tB(t) are de ned to be 0 at 0. We furthermore assume (C ) there exist constants a , a , b , b strictly positive, with a ≤ , a , a , b , b nonnegative, with the property that a > implies b > , a > and b > , and there are exponents p and q, with < p < q < ℘ * , where < ℘ < Q, ℘ = p if a = and ℘ = q if a > , such that for all t ∈ R + a t p− where 1 U is the characteristic function of a Lebesgue measurable subset U of R. Assumption (C ) was introduced by Figueiredo in [15]. Moreover, we assume (C ) there exist constants θ and ϑ, with ℘ ≤ min{θ, ϑ} < ℘ * , such that θA(t) ≥ tA (t), ϑB(t) ≥ tB (t) for all t ∈ R + holds.
Several general systems verify all the assumptions (A), (B), (C ) and (C ), and we refer to [13] for the main prototypes of the potentials A and B covered.
The functions Hu and Hv in (S) are partial derivatives of a function H of class C (R ), satisfying the condition (H) H > in R + × R + , Hu(u, ) = for all u ∈ R and Hv( , v) = for all v ∈ R. Furthermore, there exist m, m, σ such that ℘ < m < m < ℘ * , max{θ, ϑ} < σ < ℘ * and for every ε > there exists Cε > for which the inequality where |(u, v)| = √ u + v , ∇H = (Hu , Hv), and also the inequalities hold, where θ, ϑ are given in (C ).
Throughout the paper, · denotes the Euclidean inner product and | · | the corresponding Euclidean norm in any space R m , m = , , . . . . Since ℘ = p if a = , while ℘ = q if a > , the natural space where nding solutions of (S) is W = HW ,p (H n ) ∩ HW ,℘ (H n ) × HW ,p (H n ) ∩ HW ,℘ (H n ) , endowed with the norm (u, v) = u HW ,p + v HW ,p + 1 R + (a ) u HW ,q + v HW ,q for all u ∈ HW ,p (H n ), where HW ,p (H n ) is the horizontal Sobolev space de ned 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 (C ) allows us to treat simultaneously when either a = or a > , the most interesting case is the latter, in which ℘ = q and so the couple (p, q) appears in its importance. Indeed, when a > in (C ), 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 = HW ,p (Ω) ∩ HW ,q (Ω) = HW ,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 di culties intrinsic to (S). Because of the lack of compactness, the main diculty 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 ,℘ (H n ) × S ,℘ (H n ), where S ,℘ (H n ), < ℘ < Q, is the Folland-Stein space, that is the completion of C ∞ c (H n ) with respect to the norm and δ ξ j is the Dirac function at the point ξ j of H n .
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 di culties 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.
Then, u k → in L p (H n ) as k → ∞ for all p between p and ℘ * .
The paper is organized as follows. In Section 2, we recall some fundamental de nitions and properties related to the Heisenberg group H n . 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 nally Theorem 1.1, adapting the strategy of [13] and extending the results there to the Heisenberg group setting.

Preliminaries
In this section we present the basic properties of H n 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 di erences with the Euclidean case is that the homogeneous dimension Q = n + 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 H n be the Heisenberg Lie group of topological dimension n + , that is the Lie group which has R n+ as a background manifold, endowed with the non-Abelian group law for all ξ , ξ ∈ H n , with ξ = (z, t) = (x , . . . , xn , y , . . . , yn , t) and ξ = (z , t ) = (x , . . . , x n , y , . . . , y n , t ).
The inverse is given by ξ − = −ξ and so (ξ The vector elds for j = , . . . , n constitute a basis for the real Lie algebra of left-invariant vector elds on H n . This basis satis es the Heisenberg canonical commutation relations A left invariant vector eld X, which is in the span of {X j , Y j } n j= , is called horizontal. We de ne the horizontal gradient of a C function u : H n → R by In the span of {X j , Y j } n j= R n we consider the natural inner product given by If furthermore u ∈ C (H n ), then the Leibnitz formula continues to be valid, that is Similarly, if u ∈ C (H n ), then the Kohn-Spencer Laplacian, or equivalently the horizontal Laplacian in H n , of u is de ned as follows According to the celebrated Theorem 1.1 due to Hörmander in [28], the operator ∆ H is hypoelliptic. In particular, ∆ H u = div H D H u for each u ∈ C (H n ). A well known generalization of the Kohn-Spencer Laplacian is the horizontal p-Laplacian on the Heisenberg group, p ∈ ( , ∞), de ned by 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, R t), R > , since r(δ R (ξ )) = r(Rz, R t) = (|Rz| + R t ) / = R r(ξ ) for all ξ = (z, t) ∈ H n .
Let B R (ξ ) = {ξ ∈ H n : d K (ξ , ξ ) < R} be the Korányi open ball of radius R centered at ξ . For simplicity B R denotes the ball of radius R centered at ξ = O, where O = ( , ) is the natural origin of H n .
It is easy to verify that the Jacobian determinant of dilatations δ R : H n → H n is constant and equal to R n+ . This is why the natural number Q = n + is called homogeneous dimension of H n .
We recall also the de nition of Carnot-Carathéodory distance on H n and for further details we refer to [7,25]. A piecewise smooth curve γ : [ , ] → H n is called a horizontal curve ifγ(t) belongs to the span of {X j , Y j } n j= a.e. in [ , ]. The horizontal length of γ is de ned as Now, given two arbitrary points ξ , η ∈ H n , by the Chow-Rashevsky theorem there is a horizontal curve between them in H n , see [29,30]. Therefore, the Carnot-Carathéodory distance of two points ξ and η of H n is well-de ned as Clearly, d CC is a left invariant metric on H n , and for all ξ , η ∈ H n , see [7]. Moreover, the Carnot-Carathéodory distance is homogeneous of degree with respect to dilatations δ R , that is In the case of the Heisenberg group, it is easy to check that the Lebesgue measure on R n+ is invariant under left translations. Thus, from here on, we denote by dξ the Haar measure on H n that coincides with the ( n + )-Lebesgue measure, since the Haar measures on Lie groups are unique up to constant multipliers. We also denote by |U| the ( n + )-dimensional Lebesgue measure of any measurable set U ⊂ H n . Furthermore, the Haar measure on H n is Q-homogeneous with respect to dilations δ R . Consequently, In particular |B R | = |B |R Q . As usual, for any measurable set U ⊂ H n and for any general exponent p, with ≤ p ≤ ∞, we denote by L p (U) the canonical Banach space, endowed with the norm When U = H n 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, Let us now review some classical facts about the rst-order Sobolev spaces on the Heisenberg group H n . We restrict ourselves to the special case in which ≤ p < ∞ and Ω is an open set in H n . Denote by HW ,p (Ω) the horizontal Sobolev space consisting of the functions u ∈ L p (Ω) such that D H u exists in the sense of distributions and |D H u| H ∈ L p (Ω), endowed with the natural norm It is easy to check that the distributional horizontal gradient of a function u ∈ HW ,p (Ω) is uniquely de ned 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 H u 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 ∈ L (H n ) and v ∈ L p (H n ), with ≤ p ≤ ∞, then for a.e. ξ ∈ H n the function is called convolution of u and v. By the analog of the Young theorem u * v belongs to L p (H n ) and If p = ∞, then u * v is well de ned and uniformly continuous in H n . 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 H n . In particular, it is possible to generate a sequence of molli ers (ρ k ) k on H n with the usual properties, see the Appendix of [33]. Moreover, Proposition A.1.2 of [33] yield that if φ ∈ C ∞ c (H n ) and u ∈ L loc (H n ) then u * φ ∈ C ∞ (H n ) and in the sense of distributions. In particular, if q = p, then u * v ∈ HW ,p (H n ).
Proof. Let u and v be as in the statement and divide the proof into two cases.
Then, by the de nition of distributional derivative, by the fact thatǔ and φ are C ∞ c (H n ) and by (2.4),(2.5) and (2.6) we get Similarly, for j = , . . . , n, which is exactly the assertion for X j . We derive the result for Y j like so. This completes the proof.
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.
Proof. Let u ∈ HW ,p (H n ). Consider the sequence of molli ers (ρ k ) k on H n . Thus, ρ k * u ∈ C ∞ (H n ) by (2.4), for any ξ ∈ H n . Then, de ne the sequence of cut-o functions for all j = , . . . , n and k. Therefore, direct calculations show that X j ζ k = X j ζ (δ /k (ξ )) k, so that where C is a positive constant depending only on ζ . Repeating the argument when Y j replaces X j , we get the same property, so that we conclude that D H u k → D H u in L p (H n , R n ). Consequently, u k → u in HW ,p (H n ) as k → ∞, and this completes the proof.
The basic embedding theorems for the Sobolev space HW ,p (H n ), rst 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 = n + . In particular, if p is an exponent, with < p < Q, then the embedding For a complete treatment on the compactness of the embeddings HW ,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 de nition is taken from [36].
An open set Ω of H n is said to be a . This de nition is purely metric. There is a large number of PS domains in H n , 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]. is compact for all q, with ≤ q < p * , where Q is the homogeneous dimension of the Heisenberg group and p * is the Sobolev exponent related to p.
Combining Theorem 2.3, with the fact that the Carnot-Carathéodory distance and the Korányi distance are equivalent on H n , we get (i) when Ω is any Korányi ball B R (ξ ) centered at ξ ∈ H n , with radius R > .

The concentration compactness principle for critical systems on H n
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 < ℘ < Q, and that α > and β > 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 ,℘ (H n ) holds. Then, the following best constant is well de ned where S = S ,℘ (H n ) × S ,℘ (H n ). Indeed, the Hölder inequality and the Folland-Stein inequality (3.1) give for all (u, v) ∈ S, since α, β > and α + β = ℘ * . Therefore, (3.3) and the Young inequality yield for all (u, v) ∈ S. Hence I ≥ /C ℘ ℘ * > . 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.
Proof. Fix a sequence {(u k , v k )} k in S, as in the statement. Put I = [ , ] and consider the functions de ned for all (ξ , t) ∈ H n × I. Clearly, f k u ∈ L (H n × I) and g k v ∈ L (H n × I) by Fubini's theorem. Then, Tonelli's theorem gives
Finally, we are ready to prove Theorem 1.2.
Proof of Theorem . . Let {(u k , v k )} k and (u, v) be as in the statement and divide the proof into two cases.
Moreover, the elementary ℘-norm inequality in R n yields Both (u k ) k and (v k ) k converge to in L ℘ loc (H n ) by the Rellich Theorem 2.3, so that the right hand side of (3.8) goes to as k → ∞. Now, (3.2) and (3.8) give and then, letting k → ∞ in the above inequality, we get from (3.7) and (3.8) Finally, applying Lemma 1.4.6 of [7], we conclude the proof in Case 1.

The system (S)
The aim of this section is to prove the existence of nontrivial solutions for (S). From now on we assume that the structural assumptions required in Theorem 1.1 hold.
The couple (u, v) is called a (weak) solution of system (S) if The solutions of (S) are exactly the critical points of the Euler-Lagrange functional I = I λ : W → R, given by for all (u, v) ∈ W, where the functions A and B are the potentials, de ned in the Introduction.
From the main properties summarised in Section 2 we easily get the next result.
where Cp depends on p, Q, p and ℘. If p ∈ [ , ℘ * ), then for all R > the embedding is compact.
The next lemma shows that every nontrivial solution of (S) has both components non trivial, that is it solves the actual system (S), which does not reduce into an equation.

Lemma 4.2. Every nontrivial solution (u, v) ∈ W of (S) has both components nontrivial, that is u ≠ and v ≠ in H n .
Since the proof of Lemma 4.2 is not so much di erent from that of Lemma 2.3 in [13], we omit it here. For simplicity in notation, let us introduce which gives a key bound from below on A.

Lemma 4.3. Under assumptions (A) and (C ) we have
for all X and Y in the span of {X j , Y j } n j= .
The proof of Lemma 4.3, with obvious changes in notation, proceeds exactly as in Lemma 2.1 of [13]. We leave it out. The structural assumptions of Theorem 1.1 lead that the functional I possesses the geometric features of the mountain pass theorem of Ambrosetti and Rabinowitz at special levels. Now we introduce an asymptotic property of the levels c λ as λ → ∞, which is crucial in the proof of the key Lemma 4.6. This result was observed in the Euclidean vectorial case in [13], cf. Lemma 2.5, and also in the Euclidean scalar case in [15], cf. Lemma 2.2 and Remark 2.3. The proof of Lemma 4.5 follows directly from that of Lemma 2.5 of [13] and it is not reported here. Taking inspiration from [13], we prove a crucial result, observing rst some properties obtained straight from the structural assumptions. Indeed, (A), (B) and (C ) imply that for all t ∈ R + . Moreover, for any ε > , condition (H) gives the existence of a number Cε > such that holds. Clearly, (3.3) yields at once that for all since α, β > are such that α + β = ℘ * . For simplicity, in what follows we put while a = max{a , b , a , b } for all cases a ≥ . Clearly, < a ≤ a by (C ). We are going now to prove essential properties of the Palais-Smale sequences of I at the special level c λ . In particular, the next lemma, which is relevant in the proof of the main theorem, is a special case of Lemma 2.6 in [13], when H is independent of ξ and the Hardy terms are not considered. In any case it is worth to produce the proof since it relies on delicate arguments in the Heisenberg context, as Theorems 1.2 and 2.2. Lemma 4.6. Let {(u k , v k )} k ⊂ W be a Palais-Smale sequence of I at the level c λ for all λ > . Then, (ii) there exists λ * > such that the weak limit (u λ , v λ ) is a solution of (S) for all λ ≥ λ * , (iii) the set {(u λ , v λ )} λ≥λ * satis es the asymptotic property (1.2).

Moreover, by (C ) and the Hölder inequality
and similarly in v component. Therefore, by (4.9) (4.27) Again by (C ) and the Hölder inequality which yields by (4.9), also in v component, (4.28) Likewise, by (H), the Hölder inequality, (4.5) with ε = and (4.11) where Similarly, again (C ) and (4.11) give a.e. in B R as k → ∞. Therefore, the Hölder inequality gives as k → ∞, since (D H u k ) k is bounded in L ℘ (H n , R n ) and (4.35) holds. Consequently, from (4.35) and (4.36), we get the existence of a sequence (ε k ) k , independent of η, such that ε k → as k → ∞ and for all k ∈ N. Clearly, for all η ∈ H n , we have |u k |p ∈ L (B R (η)). Furthermore, the Hölder inequality yields for all η ∈ H n . Consequently, |u k |p ∈ HW , (B R (η)). Fix r ∈ ( , Q/(Q − )). Then the embedding Theorem 2.3, yields the existence of a constant C R , independent of η, such that where ε k is introduced in (4.37). Moreover, (u k ) k is bounded in Lp(H n ) and in L (p− )℘ (H n ) by the interpolation theorem, since (u k ) k is bounded in L p (H n ) and in L ∞ (H n ). Therefore, since (D H u k ) k is bounded in L ℘ (H n , R n ), the Hölder inequality gives where c is a number independent of k. Now, from Lemma 2.3 in [46], there exists a sequence (η j ) j ⊂ H n such that H n = ∞ j= B R (η j ) and each ξ ∈ H n is covered by at most Q balls B R (η j ). Hence, from (4.38) and (4.39), we have for any r ∈ ( , Q/(Q − )) and anyp, with p <p and p < (p − )℘ < ∞. Fix now p between p and ℘ * and r ∈ ( , Q/(Q − )). In the case p < ℘ * , we can choosep su ciently big so thatpr > ℘ * . Then, by the interpolation theorem applied to p, p, andpr, since p < p < ℘ * <pr, we get for a suitable τ ∈ ( , ) u k p ≤ u k τ p u k −τ pr = o( ) as k → ∞, since (u k ) k is bounded in L p (H n ) and (4.40) holds. Similarly, in the case p > ℘ * , we choosep su ciently big so thatpr > p and we apply the interpolation theorem to ℘ * , p, andpr. Thus, we obtain for a suitable τ ∈ ( , ) u k p ≤ u k τ ℘ * u k −τ pr = o( ) as k → ∞, since (4.40) holds, ℘ * < p < p <pr and (u k ) k is bounded in L ℘ * (H n ) by the Folland-Stein inequality (3.1). In conclusion, in all the cases, u k → in L p (H n ) as k → ∞ for all p between p and ℘ * , and this concludes the proof of Case 1.
Case 2. General case. Fix N ∈ N and put v k = min{|u k |, N} for all k ∈ N. Clearly, (v k ) k is a bounded sequence in L ∞ (H n ). Then, from Case 1, it results v k → in L p (H n ) (4.41) for all p between p and ℘ * . Fix now p and q between p and ℘ * , with q > p. By the interpolation theorem, (u k ) k is bounded in L q (H n ), since (u k ) k is bounded in L p (H n ) and also in L ℘ * (H n ) by the Folland-Stein inequality. Then, by the de nition of v k , where C is a nonnegative constant independent of k. Consequently, from (4.41) we get lim sup k→∞ H n |u k | p dξ ≤ C N q −p for all N ∈ N. |u k | p + |v k | p dξ > , which gives the required contradiction. Hence, (η k ) k is not bounded in H n as stated.
Finally, thanks to Proposition 4.7, we are ready to prove the existence of nontrivial solutions for system (S).
Proof of Theorem . . First, thanks to Lemmas 4.4 and 4.6, for any λ > the functional I has the geometry of the mountain pass theorem, and then I admits a Palais-Smale sequence {(u k , v k )} k at level c λ which, up to a subsequence, still denoted by {(u k , v k )} k , weakly converges to some limit (u λ , v λ ) ∈ W. Moreover, as asserted in Lemma 4.6, part (ii), there exists a threshold λ * > and the weak limit (u λ , v λ ) is a critical point of I for all λ ≥ λ * , namely a weak solution of (S). Furthermore, as stated in Lemma 4.6, part (iii), the solution has the asymptotic property (1.2). It remains to show that the constructed solution (u λ , v λ ) is nontrivial. Assume by contradiction that (u λ , v λ ) = ( , ). Clearly {(u k , v k )} k cannot converge strongly to ( , ) in W, since otherwise I (u λ , v λ ) = and = I(u λ , v λ ) = c λ > by Lemma 4.4. Therefore, by Proposition 4.7 there exist R > and a sequence (η k ) k ∈ H n such that lim sup k→∞ B R (η k ) |u k | p + |v k | p dξ > . (4.43) Now, de ne a new sequence {( u k , v k )} k , where u k (ξ ) = u k (ξ • η k ), v k (ξ ) = v k (ξ • η k ), for all ξ ∈ H n , where • is the product in H n de ned in (2.1). Therefore, I( u k , v k ) = I(u k , v k ) by the left invariance of the horizontal gradient and of the Haar measure. Moreover, for all (φ, ψ) ∈ W, with (φ, ψ) = , putting φ k (ξ ) = φ(ξ • η − k ) and ψ k (ξ ) = ψ(ξ • η − k ), ξ ∈ H n , by the change of variable ξ = ξ • η k we have Hence, ( u λ , v λ ) ≠ ( , ). Finally, Lemma 4.2 gives that both components of ( u λ , v λ ) are nontrivial, and this concludes the proof.