Multiple solutions to a weakly coupled purely critical elliptic system in bounded domains

We study the weakly coupled critical elliptic system \begin{equation*} \begin{cases} -\Delta u=\mu_{1}|u|^{2^{*}-2}u+\lambda\alpha |u|^{\alpha-2}|v|^{\beta}u&\text{in }\Omega,\\ -\Delta v=\mu_{2}|v|^{2^{*}-2}v+\lambda\beta |u|^{\alpha}|v|^{\beta-2}v&\text{in }\Omega,\\ u=v=0&\text{on }\partial\Omega, \end{cases} \end{equation*} where $\Omega$ is a bounded smooth domain in $\mathbb{R}^{N}$, $N\geq 3$, $2^{*}:=\frac{2N}{N-2}$ is the critical Sobolev exponent, $\mu_{1},\mu_{2}>0$, $\alpha, \beta>1$, $\alpha+\beta =2^{*}$ and $\lambda\in\mathbb{R}$. We establish the existence of a prescribed number of fully nontrivial solutions to this system under suitable symmetry assumptions on $\Omega$, which allow domains with finite symmetries, and we show that the positive least energy symmetric solution exhibits phase separation as $\lambda\to -\infty$. We also obtain existence of infinitely many solutions to this system in $\Omega=\mathbb{R}^N$.

This type of systems arises, e.g., in the Hartree-Fock theory for double condensates, that is, Bose-Einstein condensates of two different hyperfine states which overlap in space; see [14]. The sign of µ i reflects the interaction of the particles within each single state. This interaction is attractive if µ i > 0. The sign of λ reflects the interaction of particles in different states. It is attractive if λ > 0 and it is repulsive if λ < 0. If the condensates repel, they separate spatially. This phenomenon is called phase separation and has been described in [23]. The system (1.1) is called cooperative if λ > 0 and it is called competitive if λ < 0.
Weakly coupled elliptic systems have attracted considerable attention in recent years, and there is an extensive literature on subcritical systems, specially on the cubic system (where α = β = 2 and 2 * is replaced by 4) in dimensions N ≤ 3; we refer to [21] for a detailed account of the achievements in the subcritical case. In contrast, there are still few results for critical systems.
When λ = 0 the system (1.1) reduces to the problem − ∆w = |w| 2 * −2 w, w ∈ D 1,2 0 (Ω), (1.2) which has been extensively studied in the last decades. W.Y. Ding showed in [13] that it has infinitely many solutions if Ω = R N . It is also well known that it does not have a nontrivial solution if Ω is strictly starshaped and Ω = R N . A remarkable result by Bahri and Coron [1] establishes the existence of a positive solution to (1.2) in every bounded smooth domain with nontrivial Z 2 -homology. The existence of multiple solutions to (1.2) in a bounded domain is, to a great extent, an open problem. It is shown in [15] that, in a bounded punctured domain, the number of sign-changing solutions to the problem (1.2) becomes arbitrarily large, as the hole schrinks. Multiplicity in bounded symmetric domains was studied in [6], where it is shown that the number of sign-changing solutions to (1.2) increases, as the cardinality of the orbits increases.
Note that, if w solves (1.2), then (µ w) solve the system (1.1) for every λ. Solutions of this type are called semitrivial. We are interested in fully nontrivial solutions to (1.1), i.e., solutions where both components, u and v are nontrivial. A solution is said to be synchronized if it is of the form (sw, tw) with s, t ∈ R, and it is called positive if u ≥ 0 and v ≥ 0.
Our aim is to present some results regarding the existence and multiplicity of fully nontrivial solutions to the critical system (1.1).
With this assumption, every nontrivial solution to the problem (1.2) gives rise to a fully nontrivial synchronized solution of the system (1.1) with λ > 0; see Section 4. In particular, we obtain the following version of the Bahri-Coron theorem. Here H * ( · ; Z 2 ) stands for reduced singular homology with Z 2 -coefficients. The special case of punctured domains was treated in [18].
In the competitive case the situation is quite different. In fact, there exists λ * < 0 such that the system (1.1) does not have a synchronized solution if λ < λ * ; see [9,Proposition 2.3].
It is an open question whether Theorem 1.1 is true or not for λ < 0. Answering this question is not easy. The only results that we know of for the system (1.1) with λ < 0 in a bounded domain are those recently obtained by Pistoia and Soave in [19], where they established existence of multiple fully nontrivial solutions in bounded punctured domains of dimension 3 or 4 using the Lyapunov-Schmidt reduction method. This method cannot be applied in higher dimensions, due to the low regularity of the interaction term.
In this paper we shall consider symmetric domains, and we will obtain results in every dimension.
Let O(N ) denote, as usual, the group of linear isometries of R N . If G is a closed subgroup of O(N ), we write Gx := {gx : g ∈ G} for the G-orbit of x ∈ R N and #Gx for its cardinality. A subset X of R N is said to be G-invariant if Gx ⊂ X for every x ∈ X, and a function u : Fix a closed subgroup Γ of O(N ) and a nonempty Γ-invariant bounded smooth domain Θ in R N such that the Γ-orbit of every point x ∈ Θ has positive dimension. Then, we prove the following result. Theorem 1.2. Fix Γ and Θ as above, and assume (A). Then, for any given n ∈ N, there exists ℓ n > 0, depending on Γ and Θ, such that, if Ω is a bounded smooth domain in R N which contains Θ, Ω is G-invariant for some closed subgroup G of Γ, and min x∈Ω #Gx > ℓ n , then, for each λ = 0, the system (1.1) has at least n nonequivalent fully nontrivial G-invariant solutions (u 1 , v 1 ), . . . , (u n , v n ) which satisfy where µ 0 := max{µ 1 , µ 2 } and S is the best Sobolev constant for the embedding D 1,2 (R N ) ֒→ L 2 * (R N ). Moreover, (u 1 , v 1 ) is positive and has least energy among all G-invariant solutions.
If every G-orbit in Ω is infinite, Theorem 1.2 yields the following result.
If Ω is a G-invariant bounded smooth domain in R N and the G-orbit of every point x ∈ Ω has positive dimension, then, for each λ = 0, the system (1.1) has infinitely many fully nontrivial G-invariant solutions.
Furthermore, Theorem 1.2 yields multiple solutions even in domains which have finite G-orbits. Let us give an example. If 2k ≤ N we write R N ≡ C k × R N −2k and the points in R N as (z, x) with z ∈ C k , x ∈ R N −2k . The group S 1 := {e iϑ : ϑ ∈ [0, 2π)} of unit complex numbers acts on R N by e iϑ (z, x) := (e iϑ z, x). The S 1 -orbit of (z, x) is homeomorphic to S 1 iff z = 0. For each m ≥ 2, let Z m be the cyclic subgroup of S 1 generated by e 2πi/m . For these group actions we obtain the following result.
Moreover, for any given n there exists ℓ n > 0 such that, if m > ℓ n , then the system (1.1) has n nonequivalent Z m -invariant fully nontrivial solutions in every Z m -invariant bounded smooth domain Ω which contains Θ and does not intersect {0} × R N −2k .
In particular, if N = 2k, we may take Θ to be an annulus. Then, Corollary 1.4 yields examples of annular domains, with a hole of arbitrary size, in which the system (1.1) has a prescribed number of solutions for any λ = 0. A similar statement for the single equation (1.2) was proved in [8].
Our next result says that the G-invariant least energy solutions given by Theorem 1.2 exhibit phase separation as λ → −∞. Theorem 1.5. Let Ω be a G-invariant bounded smooth domain. Assume that, for some sequence (λ k ) with λ k → −∞, there exists a positive fully nontrivial least energy G-invariant solution (u k , v k ) to the system (1.1) with λ = λ k , such that where µ 0 := max{µ 1 , µ 2 } and S is the best Sobolev constant for the embedding Then, after passing to a subsequence, we have that u k → u ∞ and v k → v ∞ strongly in D 1,2 0 (Ω), the functions u ∞ and v ∞ are continuous and G-invariant, and v ∞ solves the problem where Note that, if Ω is an annulus and the solutions (u k , v k ) given by Theorem 1.5 are radial, then the limiting domains Ω 1 and Ω 2 must be annuli.
Phase separation for weakly coupled subcritical systems in a bounded domain was established in [10,11] via minimization on a suitable constraint. Critical Brezis-Nirenberg type systems, obtained by adding a linear term to both equations in (1.1), have been recently treated in [3,4,17,20]. For these systems, phase separation occurs in dimensions N ≥ 6; see [4].
Our final result concerns the case when the domain is the whole space R N . For the system (1.1) in R N , with α = β, it is shown in [3,4] that there exists a positive fully nontrivial solution for all λ > 0 if N ≥ 5 and for a wide range of λ > 0 if N = 4. For λ < 0 a positive fully nontrivial solution was exhibited in [9], and infinitely many fully nontrivial solutions were obtained in [9] when µ 1 = µ 2 , α = β and λ ≤ µ1 α . These results are contained the following one. Theorem 1.6. Assume (A). If Ω = R N , then, for each λ = 0, the system (1.1) has infinitely many fully nontrivial solutions, which are not conformally equivalent, one of which is positive.
As in [9], we use the conformal invariance of the system (1.1) in R N to prove Theorem 1.6. A different approach was used in [16] to establish the existence of positive multipeak solutions for λ < 0 in dimension N = 3.
The positive entire solutions given by Theorem 1.6 also exhibit phase separation as λ → −∞. This was shown in [9, Theorem 1.2]. Moreover, a precise description of the limit domains Ω 1 and Ω 2 is given in [9].
To obtain our results, we use variational methods. As in the case of the single equation (1.2), the main difficulty is the lack of compactness of the functional associated to the system (1.1). We prove a representation theorem for G-invariant Palais-Smale sequences for this functional, which provides a full description of the loss of compactness in the presence of symmetries for every λ ∈ R; see Theorem 3.1 below. This paper is organized as follows. In Section 2 we introduce the variational setting. Section 3 is devoted to the study of the loss of compactness in the presence of symmetries. Our main results are proved in Section 4 in the cooperative case, and in Section 5 in the competitive case. Finally, in the Appendix we derive some Brezis-Lieb type results for the interaction term.
Let Ω be a smooth domain in R N , N ≥ 3. As usual, D 1,2 0 (Ω) denotes the closure of C ∞ c (Ω) in the space Since α, β > 1, this functional is of class C 1 and its derivative is given by The fully nontrivial solutions to the system (1.1) belong to the set Proposition 2.1. If λ < 0, then the following statements hold true: (a) For every (u, v) ∈ N (Ω), one has that where S is the best constant for the embedding D 1,2 (R N ) ֒→ L 2 * (R N ). Proof. The proof follows as in [9, Proposition 2.1], with minor modifications.
and this value is never attained by E on N (Ω).
Proof. The argument used in [9, Proposition 2.2] to prove this statement for R N can be easily adapted to a general domain Ω.
As for the single equation (1.2), one has the following nonexistence result. It is stated in [18] for µ 1 = 1 = µ 2 and λ = 1 2 * , but the proof carries over to the general case. We give it here for the sake of completeness.
If Ω = R N and Ω is strictly starshaped, then the system (1.1) does not have a nontrivial solution.
Proof. Without loss of generality, we assume that Ω is strictly starshaped with respect to the origin. Let (u, v) be a solution to the system (1.1). Multiplying the first equation in (1.1) by ∇u · x and the second one by ∇v · x we get Note that So multiplying equations (2.1) and (2.2) by 1 N , adding them up and integrating, we obtain the identity where ν = ν(s) is the outer unit normal to ∂Ω at s. As (u, v) solves the system (1.1), this identity reduces to Since Ω is strictly starshaped, this implies that ∂u ∂ν = 0 = ∂v ∂ν on ∂Ω so, extending u and v by zero outside of Ω, we obtain a solution to the system (1.1) in the whole of R N which vanishes in an open subset of R N . By the unique continuation principle, u = 0 and v = 0 in Ω, as claimed.
Propositions 2.2 and 2.3 showcase the lack of compactness of the functional E. Symmetries help restore compactness.

Symmetries and compactness
Let G be a closed subgroup of O(N ). We will assume, from now on, that Ω is a G-invariant bounded smooth domain and we will look for G-invariant solutions to the system (1.1), i.e., solutions (u, v) such that both components, u and v, is the isotropy group of x. In particular, #Gx = |G/G x |, where |G/K| denotes, as usual, the index of the subgroup K in G.
We will prove the following result.
Then, after passing to a subsequence, there exist a solution (u, v) to the system (1.1), an integer m ≥ 0 and, for each with the following properties: ∈ K j and each j = 1, ..., m.
The rest of the section is devoted to the proof of this theorem. The following lemma will allow us to choose a G-orbit of concentration in a convenient way.
and a closed subgroup K of G such that, after passing to a subsequence, the following statements hold true: We will also need the following lemma.
where S is the best constant for the embedding D 1,2 (R N ) ֒→ L 2 * (R N ). Note that t β ≤ 1 + t 2 * . Therefore, for every t ≥ 0. This completes the proof.
The main step in the proof of Theorem 3.1 is given by the following result.
. Then, after passing to a subsequence, there exist a closed , and a sequence ((w k , z k )) with the following properties: Proof. Passing to a subsequence, we may assume that (u k , v k ) = (0, 0) for all k. Since for k large enough, the sequence ((u k , v k )) is bounded in D(Ω) and, thus, Then, we have that As δ ∈ (0, N c 2 ), there exist bounded sequences (ε k ) in (0, ∞) and (ζ k ) in R N such that, after passing to a subsequence, For these sequences we choose K and (ξ k ) as in Lemma 3.2. Then, G ξ k = K and there exist g k ∈ G and a positive constant C 1 such that where C 0 := C 1 + 1. This implies, in particular, that We claim that |G/K| < ∞. Otherwise, according to Lemma 3.2(d), for any n ∈ N there exist g 1 , ..., g n ∈ G such that ε −1 k |g i ξ k − g j ξ k | → ∞ for every i = j and, hence, for k large enough, This is a contradiction.
Set Ω k := {y ∈ R N : ε k y + ξ k ∈ Ω} and, for y ∈ Ω k , set Note also that, as u k and v k are G-invariant and G ξ k = K, we have that u k and (Ω) and, hence, and similarly for v k .
In order to show that ( u, v) = (0, 0) we argue by contradiction. Assume that and a similar expression for v k . Adding both identities, and using Hölder's inequality and inequalities (3.4) and (3.2), we obtain This implies that (ϕ u k , ϕ v k ) = o(1) and, hence, that |ϕ u k | 2 * = o(1) and |ϕ v k | 2 * = o(1) for every ϕ ∈ C ∞ c (B 1 (y)) and every y ∈ R N . Therefore, u k → 0 and v k → 0 in L 2 * loc (R N ), contradicting (3.4). This proves that ( u, v) = (0, 0). Passing to a subsequence, we have that ξ k → ξ and ε k → ε ∈ [0, ∞). Moreover, ε = 0, otherwise, as u k ⇀ 0 and v k ⇀ 0 weakly in D 1,2 0 (Ω), we would get that u = 0 = v, which is a contradiction. Also, passing to a subsequence, Arguing by contradiction, assume that d ∈ [0, ∞). Then, as ε k → 0, we have that ξ ∈ ∂Ω. If a subsequence of (ξ k ) is contained inΩ we setd := −d, otherwise we setd := d. We define where ν is the inner unit normal to ∂Ω at ξ. It is easy to see that, if X is compact and X ⊂ H, then X ⊂ Ω k for k large enough and, if X is compact and X ⊂ R N H , then X ⊂ R N Ω k for k large enough. As u k → u and v k → v a.e. in R N , this implies, in particular, that u = 0 = v a.e. in R N H. (1). Passing to the limit as k → ∞, we conclude that ( u, v) solves the system (1.1) in H, contradicting Proposition 2.3. This proves that d = ∞ and, by (3.3), we have that ξ k ∈ Ω. Moreover, every compact subset X of R N is contained in Ω k for k large enough. So, arguing as above, we conclude that ( u, v) solves the limit system (3.1) Choose a radially symmetric cut-off function Since u and v are K-invariant and G ξ k = K for all k ∈ N, we have that w k and z k are G-invariant. Clearly, (w k , z k ) ⇀ (0, 0) weakly in D(Ω). Now, for each j = 1, ..., n, we define As r k ε −1 k → ∞, an easy computation shows that and similarly for v k . This proves that (w k , z k ) satisfies (iii).
We rescale w j k and use the G-invariance of u k to obtain Since u k ⇀ u weakly in D 1,2 (R N ) and ε −1 k |g j ξ k − g i ξ k | → ∞ for every i = j, we have that Therefore, where the second equality is given by the change of variable x = ε k y + g j ξ k . Iterating the identity (3.7) and using (3.6) we obtain u k 2 = w k 2 + n u 2 + o(1).

(3.8)
Adding this last identity with the similar one for v k gives A similar argument, using Lemma A.2 yields and From (3.9), (3.10) and (3.11) we obtain

This proves (iv).
In a similar way, using Lemma A.4, we get that in (D(R N )) ′ . This completes the proof of (v). The proof of Lemma 3.4 is now complete.
Proof of Theorem 3.1. The sequence ((u k , v k )) is bounded in D(Ω). So, after passing to a subsequence, we have that u k ⇀ u and v k ⇀ v weakly in D 1,2 (Ω), u k → u and v k → v in L 2 loc (Ω), and u k → u and v k → v a.e. in Ω. A standard argument shows that (u, v) solves the system (1.1). If (u k , v k ) → (u, v) strongly in D(Ω), the proof is finished. If not, we set u 1 k := u k − u and v 1 k := v k − v.

Cooperative systems
This section is devoted to the proof of Theorems 1.1, 1.2 and 1.6 in the cooperative case.

Competitive systems
This section is devoted to the proof of Theorems 1.2 and 1.6 in the competitive case. We will assume throughout that λ < 0. It was shown in [9, Proposition 2.3] that there exists λ * < 0 such that the system (1.1) does not have a fully nontrivial synchronized solution if λ < λ * . So the argument given in the previous section for the cooperative case does not work in the competitive case. We will use a C 1 -Ljusternik-Schnirelmann theorem due to A. Szulkin, stated below.
Let X be a real Banach space, M be a closed C 1 -submanifold of X and Φ ∈ C 1 (M, R). We denote by d x Φ the differential of Φ at a point x ∈ M , and write K c := {x ∈ M : Φ(x) = c and d x Φ = 0}.
Recall that Φ is said to satisfy the (P S) c -condition on M if every sequence (x k ) in M such that Φ(x k ) → c and d x Φ → 0 has a convergent subsequence. Let Z be a symmetric subset of X with 0 ∈ Z. Recall that Z is called symmetric if −Z = Z. If Z is nonempty, the genus of Z is the smallest integer j ≥ 1 such that there exists an odd continuous function Z → S j−1 into the unit sphere S j−1 in R j . We denote it by genus(Z). If no such j exists, we set genus(Z) := ∞. We define genus(∅) := 0. The properties of the genus may be found in [22,Proposition 2.3].
If M is symmetric and 0 ∈ M , we define Theorem 5.1 (Szulkin [22]). Let M be a closed symmetric C 1 -submanifold of X which does not contain the origin, and let Φ ∈ C 1 (M, R) be an even function which is bounded below. If Σ n = ∅ for some n ≥ 1 and Φ satisfies (P S) cj for j = 1, ..., n, then, for each j = 1, ..., n, one has that c j is a critical value of Φ and genus(K cj ) ≥ m + 1 if c j = · · · = c j+m for some m ≥ 0.
We shall use this theorem to prove Theorems 1.2 and 1.6 for λ < 0. It is easy to see that ∇E(u, v), ∇F 1 (u, v), ∇F 2 (u, v) ∈ D(Ω) G for every (u, v) ∈ D(Ω) G , where F 1 and F 2 are the functions defined in Proposition 2.1; see, e.g., [24,Theorem 1.28]. Therefore, N (Ω) G is a closed C 1 -submanifold of D(Ω) G and a natural constraint for E. The orthogonal projection of ∇E(u, v) onto the tangent space of N (Ω) G at the point (u, v) will be denoted by ∇ N (Ω) E(u, v).

Multiplicity in bounded domains
Proof. The same argument used in [9, Lemma 3.5] yields this statement.
Lemma 5.2 allows us to apply Theorem 3.1 to obtain the following compactness condition.
Proof. Let ((u k , v k )) be a sequence such that Arguing by contradiction, assume that the number m given by Theorem 3.1 is such that m ≥ 1, and let ( u 1 , v 1 ), ..., ( u m , v m ) be the nontrivial solutions to the limit problem (3.1) given by that theorem.
For (u, v) ∈ D(Ω) with u = 0, v = 0, let s u and t v be the unique positive numbers such that s u u 2 = Ω µ 1 |s u u| 2 * and t v v 2 = Ω µ 2 |t v v| 2 * , and set Let {e 1 , ..., e j } be the canonical basis of R j . The boundary of the convex hull of the set {±e 1 , ..., ±e j }, which is given by has genus j and the map ψ : Q → N (Ω) G , given by is odd and continuous. Hence, the set Z := ψ(Q) ⊂ N (Ω) G is symmetric and compact, and genus(Z) ≥ j; see [22,Proposition 2.3]. This completes the proof.
Proof of Theorem 1.2 for λ < 0. Let Γ be the subgroup of O(N ) and Θ be the Γ-invariant bounded smooth domain given in the statement of Theorem 1.2. Lemma 5.4 asserts that Σ G j (Θ) = ∅. Hence, c Γ j (Θ) ∈ R for every j ∈ N. Given n ∈ N, we define where µ 0 := max{µ 1 , µ 2 }. If G is a closed subgroup of Γ and Ω is a G-invariant bounded smooth domain in R N which contains Θ, then N (Θ) Γ ⊂ N (Ω) G and, hence, Σ Γ n (Θ) ⊂ Σ G n (Ω) and c G n (Ω) ≤ c Γ n (Θ). So, if min x∈Ω #Gx > ℓ n , then, for each 1 ≤ j ≤ n, we have that It follows from Lemma 5.3 that E satisfies the (P S) cj -condition on N (Ω) G for every 1 ≤ j ≤ n. Moreover, by Lemma 5.4, Σ G n (Ω) = ∅. Therefore, by Theorem 5.1, E has n critical points (u 1 , v 1 ), ..., and these points are nonequivalent in the sense defined in the introduction. If c G i (Ω) = · · · = c G i+m (Ω) =: c for some m ≥ 1, then genus(K c ) ≥ 2. This implies that #K c = ∞, so E has infinitely many nonequivalent critical points with critical value c.

Phase separation in bounded domains
In this subsection we prove Theorem 1.5.
Let G be a closed subgroup of O (N ) and Ω be a G-invariant smooth bounded domain. Consider the problem where w + = max{w, 0} and w − = min{w, 0}. Let be its energy functional and its Nehari manifold. The sign-changing solutions to problem (5.1) lie on the set It is easy to see that E G = ∅. We define To emphasize the dependence on λ, in the following we write N λ (Ω) G and E λ , instead of N (Ω) G and E, for the Nehari manifold and the energy funcional of the system (1.1). Notice that (w + , w − ) ∈ N G λ and J(w) = E λ (w + , w − ) for every w ∈ E G and each λ < 0. Therefore, Proof of Theorem 1.5. Let λ k < 0, λ k → −∞, and (u k , v k ) ∈ N λ k (Ω) G be such that u k ≥ 0, v k ≥ 0 and, for each k ∈ N, Hence, passing to a subsequence, there exist u ∞ , v ∞ ∈ D 1,2 0 (Ω) G such that a.e. in Ω.
In particular, and, from Fatou's lemma, we obtain Therefore, u ∞ v ∞ = 0. We claim that To prove this claim, we argue by contradiction. Assume that u ∞ = 0 and v ∞ = 0; the other cases can be treated in a similar way. Then, u k ⇀ 0 in D 1,2 0 (Ω) and u k 2 ≥ c 0 > 0 by Proposition 2.1(a). Following the argument in the first part of the proof of Lemma 3.4, one shows that there exists a closed subgroup K of finite index in G and sequences (ξ k ) in R N and (ε k ) in (0, ∞) such that G ξ k = K, ε −1 k |gξ k −gξ k | → ∞ for any g,g ∈ G with g −1g / ∈ K, and dist(ξ k ,Ω) → 0. Moreover, the rescaled functions Arguing as we did to prove equation (3.8), we obtain The last inequality holds true because Ω is smooth and dist(ξ k ,Ω) → 0. Define Set v k (y) := ε . Then, as (u k , v k ) solves (1.1), u k ≥ 0 and u ≥ 0, we have that Passing to the limit we obtain and, passing to the limit, we get As s u belongs to the Nehari manifold associated to the problem and tv ∞ belongs to the Nehari manifold associated to the problem we have that s u 2 ≥ µ We define arguing as above, we see that s, t ∈ (0, 1]. So, using (5.2), we get The argument given in [2,Lemma 2.6] shows that u ∞ − v ∞ is a critical point of J, i.e., u ∞ − v ∞ is a sign-changing G-invariant solution of (5.1). In particular, and v ∞ solves the problem as claimed.

Multiple entire solutions
Now we turn our attention to the competitive system (1.1) in R N . We shall consider symmetries given by conformal transformations. We give a brief account of the symmetric setting. Details may be found in [9,Section 3]. Let G be a closed subgroup of O(N + 1). Then, G acts isometrically on the standard sphere S N . Using the stereographic projection σ : S N → R N ∪ {∞}, we transfer this action to R N . Namely, for each g ∈ G we have a conformal transformation g := σ • g −1 • σ −1 : R N → R N , which is well defined except at a single point.
The space D 1,2 (R N ) is a G -Hilbert space with the action defined by gu := | det g ′ | 1/2 * u • g, g ∈ G , u ∈ D 1,2 (R N ), and D(R N ) is also a G -Hilbert space with the diagonal action g(u, v) := (gu, gv). We set It is easy to see that the functional E is G -invariant, and so are the functionals F 1 and F 2 defined in Proposition 2.1. Hence, is a closed C 1 -submanifold of D(R N ) G and a natural constraint for E. The advantage of taking this kind of actions is that O(N + 1) contains subgroups G such that the G -orbit of every point p ∈ S N satisfies 0 < dim(G p) < N . For example, the group G = O(m) × O(n) with m + n = N + 1, m, n ≥ 2, has this property, as, for this group, G p is homeomorphic to either S m−1 , or to S n−1 , or to S m−1 × S n−1 . This property plays a role in the following lemmas.  Proof. If dim(G p) < N for every p ∈ S N , then, for any given j ≥ 1, there exist 2j pairwise disjoint open G -invariant subsets of R N (take, for example, 2j distinct G -orbits and pairwise disjoint G -invariant neighborhoods of them). Now we may argue as in the proof of Lemma 5.4.
Proof of Theorem 1.6 for λ < 0. Let G be a closed subgroup of O(N + 1) such that 0 < dim(G p) < N for every p ∈ S N . Then, by Lemmas 5.5 and 5.6 and Theorem 5.1, we have that c G j (R N ) is a critical value of the restriction of E to N (R N ) G for every j ≥ 1.
Moreover, as E satisfies the (P S) c -condition on N (R N ) G , the critical sets K c are compact and, hence, genus(K c ) < ∞ for every c ∈ R. It follows from Theorem 5.1 that #{c G j (R N ) : j ≥ 1} = ∞, i.e., E has infinitely many critical values on N (R N ) G .
After replacing the minimizer (u 1 , v 1 ) of E on N (R N ) G with (|u 1 |, |v 1 |), we get a positive critical point. The proof is complete.
Proof. Throughout the proof C will denote a positive constant, not necessarily the same one. (a) : Passing to a subsequence we have that v k → v in L q (Θ) and a.e. in Θ. Using the mean value theorem we get that Hence, as v ∈ L ∞ (Θ), we obtain |v k | β − |v k − v| β − |v| β q ≤ C |v k − v| β−1 |v| + |v| β q ≤ C|v k − v| q + C a.e. in Θ.
Then, |w + k | ≤ C and, by the dominated convergence theorem, lim k→∞ Θ |v k | β − |v k − v| β − |v| β q ≤ lim k→∞ Θ w + k + lim k→∞ C Θ |v k − v| q = 0, as claimed. (b) : Passing to a subsequence we have that u k → u in L p (Θ) and a.e. in Θ. From the mean value theorem we obtain As u ∈ L ∞ (Θ), the statement follows immediately from the dominated convergence theorem if α ∈ (1, 2]. For α > 2 the argument is similar to the one we used to prove (a).
Set f (t) := |t| α−2 t and, for R > 0, let B R be the ball centered at 0 of radius R and B c R be its complement in R N . Let ϕ ∈ D 1,2 (R N ). Then if R is large enough, where | · | s stands for the norm in L s (R N ). Similarly, from (A.9) we obtain that for R large enough. Clearly, the same is true for the last integral in (A.10). Now, we fix R > 0 such that In B R we have that Now, we estimate the integrals on the RHS using Lemma A.3. For the first one, we fix q ∈ [1, 2 * ) such thatq := q β−1 > 1 and 1 − α−1 2 * − β−1 q ≥ 1 2 * . Then, for large enough k. The other integrals are estimated in a similar way. This shows that for large enough k, and finishes the proof of the lemma.