A simple variational approach to weakly coupled competitive elliptic systems

The main purpose of this paper is to exhibit a simple variational setting for finding fully nontrivial solutions to the weakly coupled elliptic system (1.1). We show that such solutions correspond to critical points of a $\mathcal{C}^1$-functional $\Psi:\mathcal{U}\to\mathbb{R}$ defined in an open subset $\mathcal{U}$ of the product $\mathcal{T}:=S_1\times\cdots\times S_M$ of unit spheres $S_i$ in an appropriate Sobolev space. We use our abstract setting to extend and complement some known results for the system (1.1).

The cubic system (1.1) in R 3 with α ij = β ij = 2 arises as a model in many physical phenomena, for example, in the study of standing waves for a mixture of Bose-Einstein condensates of M -hyperfine states which overlap in space. The sign of µ i reflects the interaction of the particles within each single state, whereas that of λ ij reflects the interaction between particles in two different states. The interaction is attractive if the sign is positive, and it is repulsive if the sign is negative. The system is called competitive if, as we are assuming here, all of the λ ij 's are negative.
A solution u i to the equation gives rise to a solution of the system (1.1) whose i-th component is u i and all other components are trivial, i.e., u j = 0 if j = i. A solution with at least one trivial and one nontrivial component is called semitrivial. We are interested in finding solutions all of whose components are nontrivial. These are called fully nontrivial solutions. A fully nontrivial solution is said to be positive if every component u i is nonnegative.
The main purpose of this paper is to exhibit a simple variational setting for finding fully nontrivial solutions to the system (1.1). Our approach is inspired by the ideas introduced by Szulkin and Weth in [19,20].
We will show that the fully nontrivial solutions to (1.1) correspond to the critical points of a C 1 -functional Ψ : U → R defined in an open subset U of the product T := S 1 × · · · × S M of unit spheres S i in H. The functional Ψ tends to infinity at the boundary of U in T , thus allowing the application of the usual descending gradient flow techniques to obtain existence and multiplicity of critical points.
This variational setting can be easily extended to systems whose coefficients κ i , µ i , λ ij are functions defined in Ω and satisfying suitable assumptions. It may also be extended, with some care, to systems having more general nonlinearities. We chose to treat only the constant coefficient system (1.1) in order to make the ideas more transparent.
Our abstract results (Theorems 3.3 and 3.4) apply to many interesting types of systems. Here we consider the following three.
Firstly, we consider the subcritical system with κ i , µ i > 0, λ ij = λ ji < 0, α ij , β ij > 1, α ij = β ji , and α ij +β ij = p ∈ (2, 2 * ), in an exterior domain Ω of R N (i.e., R N Ω is bounded, possibly empty), N ≥ 3. We assume that Ω is invariant under the action of a closed subgroup G of the group O(N ) of linear isometries of R N , and look for G-invariant solutions, i.e., solutions whose components are G-invariant.
Let Gx := {gx : g ∈ G} denote the G-orbit of x ∈ R N . We prove the following result.
and Ω is a G-invariant exterior domain in R N , then the system (1.2) has an unbounded sequence of Ginvariant fully nontrivial solutions. One of them is positive and has least energy among all G-invariant fully nontrivial solutions.
There is an extensive literature on subcritical systems in bounded domains and in the whole of R 3 . We refer to [17] for a detailed account. Theorem 1.1 seems to be the first existence result for the system (1.2) in an exterior domain. A cubic system of two equations with variable coefficients in an expanding exterior domain was recently considered in [10].
Our second application concerns the critical system We look for solutions which are invariant under the conformal action of the group Γ := O(m) × O(n) on R N , with m + n = N + 1 and n, m ≥ 2, which is induced by the isometric action of Γ on the standard N -dimensional sphere, by means of the stereographic projection. We prove the following result.
Theorem 1.2. The system (1.3) has an unbounded sequence of Γ-invariant fully nontrivial solutions. One of them is positive and has least energy among all Γ-invariant fully nontrivial solutions. Theorem 1.2 extends some earlier results obtained in [5,6] for a system of two equations; see also [9]. Existence and multiplicity results for the purely critical system in a bounded domain may be found in [5,13,14]. Supercritical systems were recently considered in [4].
Note that there is no condition on α ij , β ij , other than α ij , β ij > 1 and α ij + β ij = 2 * , if N ≥ 6. Theorem 1.3 extends some earlier results obtained in [2,3] for a system of two equations. Multiple positive solutions were constructed in [15] when N = 4, and the existence of infinitely many sign-changing solutions was established in [11] when N ≥ 7 and α ij = β ij = 2 * 2 ; see also [12]. Our variational approach is based on some elementary properties of a certain function in M variables, which are established in Section 2. In Section 3 we introduce our variational setting and we derive some abstract results concerning the existence and multiplicity of fully nontrivial solutions to the system (1.1). Section 4 is devoted to the proof of Theorems 1.1, 1.2 and 1.3.

On a function in M variables
Let J : (0, ∞) M → R be the function given by In particular, J attains its maximum on (0, ∞) M .
Proof. Fix R > r > 0 such that, for all i = 1, . . . , M , and Let s = (s 1 , . . . , s M ) ∈ (0, ∞) M . If s i ≥ R and s i = max{s 1 , . . . , s M }, we have that Therefore (2.2) holds true. Proof. Assume first that (1, . . . , 1) is a critical point of J. Then, from (2.1) we get that Arguing by contradiction, assume that s = (1, . . . , 1). We consider two cases. Suppose first that s i > 1 for some i. We may assume without loss of generality that s i ≥ s j for all j. Then, the left-hand side in (2.6) is negative whereas the right-hand side is ≥ 0. This is a contradiction. Now suppose that s i < 1 for some i. Again, we may assume that s i ≤ s j for all j. Now the left-hand side in (2.6) is positive while the right-hand side is not, a contradiction again. Hence (1, . . . , 1) is the only critical point of J in (0, ∞) M . The inequalities (2.5) allow us to apply Lemma 2.1 to conclude that (1, , and the conclusion follows from the special case considered above. has a unique critical point s 0 in (0, ∞) M which is a global maximum and satisfies | s 0 − s 0 | < ε.

The variational setting
The results of this section also apply to the case N = 1 or 2 and p ∈ (2, ∞).
Let H be either Since, by assumption, the operators −∆ + κ i are well defined and coercive in H, we have that · i is a norm in H, equivalent to the standard one.
, and let J : H → R be given by This function is of class C 1 and, since λ ij = λ ji and β ij = α ji , So the critical points of J are the solutions to the system (1.1). The fully nontrivial ones belong to the set This Nehari-type set was introduced in [7], and has been used in many works. Note that Given u = (u 1 , . . . , u M ) ∈ H and s = (s 1 , . . . , s M ) ∈ (0, ∞) M , we write If u i = 0 for all i = 1, . . . , M , then, as Then, Let S i := {v ∈ H : v i = 1}, T := S 1 × · · · × S M , U := U ∩ T , and let m : U → N be the restriction of m to U. We write ∂U for the boundary of U in T .
(c) m : U → N is continuous, and m : U → N is a homeomorphism.
Thus, N is a closed subset of H.
Proof. (a) : Let u = (u 1 , . . . , u M ) ∈ T be such that u i and u j have disjoint supports if i = j. Then, d u,ij = 0 for every i = j, and, setting We assume without loss of generality that i = 1 and Then, as α ij + β ij = p and λ ij < 0 for all i, j, we have that Since λ 12 = λ 21 and the right-hand sides above must be positive, we get that which is impossible if −λ 12 ≥ max{ µ1 β12 , µ2 β21 }. So, if this last inequality holds true, then (c) : If (u n ) is a sequence in U and u n → u ∈ U , then, for each i, j = 1, . . . , M with i = j, we have that a un,i → a u,i , b un,i → b u and d un,ij → d u,ij . So, from Lemma 2.3 we get that s un,i → s u,i . Hence, m : U → N is continuous.
The inverse of m : U → N is given by which is, obviously, continuous. (e) : Let (u n ) be a sequence in U such that u n → u ∈ ∂U. If the sequence (s un,i ) were bounded for every i = 1, . . . , M , then, after passing to a subsequence, s un,i → s i . Since N is closed, we would have that (s 1 u 1 , . . . , s M u M ) ∈ N and, therefore, u ∈ U. This is impossible because u ∈ ∂U and U is open in T .
A fully nontrivial solution u to (1.1) will be called synchronized if u i = t i v and u j = t j v for some i = j and t i , t j ∈ R. Let Ψ : U → R be given by Ψ(u) := J ( m(u)), and let Ψ be the restriction of Ψ to U. Then, If u ∈ U and the derivative Ψ ′ (u) of Ψ at u exists, then i.e., · * is the norm in the cotangent space T * u (T ) to T at u. A sequence (u n ) in U is called a (P S) c -sequence for Ψ if Ψ(u n ) → c and Ψ ′ (u n ) * → 0, and Ψ is said to satisfy the (P S) c -condition if every such sequence has a convergent subsequence.
As usual, a (P S) c -sequence for J is a sequence (u n ) in H such that J (u n ) → 0 and J ′ (u n ) H −1 → 0, and J satisfies the (P S) c -condition if any such sequence has a convergent subsequence.
is a (P S) c -sequence for J and u n ∈ N for all n ∈ N, then (m −1 (u n )) is a (P S) c -sequence for Ψ. (iv) If (u n ) is a sequence in U such that u n → u ∈ ∂U, then Ψ(u n ) → ∞.
Proof. We adapt the arguments of Proposition 9 and Corollary 10 in [20]. (i) : Let u ∈ U and v ∈ H. As s u is the maximum of J u , using the mean value theorem we obtain for |t| small enough and some τ 1 ∈ (0, 1). Similarly, for some τ 2 ∈ (0, 1). From the continuity of s u and these two inequalities we obtain The right-hand side is linear in v and continuous in v and u. Therefore Ψ is of class C 1 . If u ∈ U and v ∈ T u (T ), then m(u) = m(u), and the statement is proved.
(ii) : Note that H = T u (T ) ⊕ (Ru 1 , . . . , Ru M ) for each u ∈ U. Since m(u) ∈ N , we have that J ′ (m(u))w = 0 if w ∈ (Ru 1 , . . . , Ru M ). So, from (i) we get If (Ψ(u n )) converges, then (s un ) is bounded in R M by (3.4). Moreover, by Proposition 3.1(d), this sequence is bounded away from 0. Therefore, (m(u n )) is a (P S) c -sequence for J iff (u n ) is a (P S) c -sequence for Ψ, as claimed. Let Z be a subset of T such that −u ∈ Z iff u ∈ Z. If Z = ∅, the genus of Z is the smallest integer k ≥ 1 such that there exists an odd continuous function Z → S k−1 into the unit sphere S k−1 in R k . We denote it by genus(Z). If no such k exists, we define genus(Z) := ∞. We set genus(∅) := 0.
As usual, we write The previous theorem yields the following one. (b) If Ψ : U → R satisfies the (P S) c -condition for every c ≤ a, then the system (1.1) has, either an infinite (in fact, uncountable) set of fully nontrivial solutions with the same norm, or it has at least genus(Ψ ≤a ) fully nontrivial solutions with pairwise different norms.
(c) If Ψ : U → R satisfies the (P S) c -condition for every c ∈ R and genus(U) = ∞, then the system (1.1) has an unbounded sequence of fully nontrivial solutions.
Proof. Theorem 3.3(iii) states that u is a critical point of Ψ iff m(u) is a fully nontrivial critical point of J . Note that Ψ(u) = p−2 2p m(u) 2 , by (3.1). If inf N J = J (u) and u ∈ N , then m −1 (u) ∈ U and Ψ(m −1 (u)) = inf U Ψ. So u is a fully nontrivial critical point of J . As |u| ∈ N and J (|u|) = J (u) the same is true for |u|. This proves (a). Theorem 3.3(iv) implies that U is positively invariant under the negative pseudogradient flow of Ψ, so the usual deformation lemma holds true for Ψ; see, e.g., [18,Section II.3] or [21,Section 5.3]. Set Standard arguments show that, under the assumptions of (b), c j is a critical value of Ψ for every j = 1, . . . , genus(Ψ ≤a ). Moreover, if some of these values coincide, say c := c j = · · · = c j+k , then genus(K c ) ≥ k + 1 ≥ 2. Hence, K c is an infinite set; see, e.g., [18,Lemma II.5.6]. On the other hand, under the assumptions of (c), c j is a critical value for every j ∈ N, and a well known argument (see, e.g., [16,Proposition 9.33]) shows that c j → ∞ as j → ∞. This completes the proof.

Some applications 4.1 Subcritical systems in exterior domains
Consider the subcritical system (1.2) in an exterior domain Ω. First, we show that this system cannot be solved by minimization. Set Proposition 4.1. We have that and this infimum is not attained by J on N .
Proof. We consider H 1 0 (Ω) to be a subspace of H 1 (R N ), via trivial extension. If (u 1 , . . . , u M ) ∈ N then, as λ ij < 0 for every i = j, we have that u i 2 i ≤ |u i | p p,i for all i = 1, . . . , M . Hence, To prove the opposite inequality, set B r (x) := {y ∈ R N : |y − x| < r}, and let w i,R be a least energy solution to the problem It is easy to verify that lim R→∞ w i,R This completes the proof of (4.1).
To show that the infimum is not attained, we argue by contradiction. Assume To obtain multiple solutions to the system (1.2) we introduce some symmetries.
Let G be a closed subgroup of O(N ) and Gx := {gx : g ∈ G}. Set S N −1 := {x ∈ R N : |x| = 1}. We start with the following lemma.
Lemma 4.2. If dim(Gx) > 0 for every x ∈ R N {0}, then, for each k ∈ N, there exists d k > 0 such that, for every x ∈ S N −1 , there exist g 1 , . . . , g k ∈ G with min i =j Proof. Arguing by contradiction, assume that for some k ∈ N and every n ∈ N there exists x n ∈ S N −1 such that min i =j |g i x n − g j x n | < 1 n for any k elements g 1 , . . . , g k ∈ G.
After passing to a subsequence, we have that x n → x in S N −1 . Since dim(Gx) > 0, there existḡ 1 , . . . ,ḡ k ∈ G such thatḡ i x =ḡ j x if i = j. Fix i = j such that, after passing to a subsequence, |ḡ i x n −ḡ j x n | = min i =j |ḡ i x n −ḡ j x n | for every n ∈ N. Then, This is a contradiction.
We assume that Ω is G-invariant and define Recall that Ω is called G-invariant if Gx ⊂ Ω for all x ∈ Ω, and a function v : Proof. Let (w n ) be a bounded sequence in H 1 0 (Ω) G . Then, after passing to a subsequence, w n ⇀ w weakly in H 1 0 (Ω) G . Set v n := w n − w. A subsequence of (v n ) satisfies v n ⇀ 0 weakly in H 1 0 (Ω) G , v n → 0 in L 2 loc (Ω) and v n (x) → 0 a.e. in Ω. We claim that (4.2) sup To prove this claim, let ε > 0, and let C > 0 be such that v n 2 ≤ C for all n ∈ N, where · is the standard norm in H 1 0 (Ω). We choose k ∈ N such that C < εk and d k > 0 as in Lemma 4.2, and we fix R k > 2/d k . We consider two cases.
Assume first that |x| ≥ R k . By Lemma 4.2, there exist g 1 , . . . , g k ∈ G such that |g i x − g j x| ≥ |x|d k for all i = j.
Proof of Theorem 1.1. The functional J is G-invariant, so, by the principle of symmetric criticality, the critical points of the restriction of J to H G are the G-invariant critical points of J ; see, e.g., [21,Theorem 1.28].
It is readily seen that the results of Section 3 are also true for H := H 1 0 (Ω) G . Theorem 3.3(ii) and Lemma 4.4 imply that Ψ satisfies the (P S) c -condition for every c ∈ R. This, together with Lemma 4.5 and Theorem 3.4, yields Theorem 1.1.

Entire solutions to critical systems
Next, we consider the Yamabe system (1.3).
As usual, we denote where w 2 := R N |∇w| 2 and |w| 2 * 2 * := R N |w| 2 * . The next result says that the system (1.3) cannot be solved by minimization. To obtain multiple solutions to the system (1.3) we consider a conformal action on R N , as in [6,8].
Let Γ = O(m) × O(n) with m + n = N + 1 and m, n ≥ 2 act on R N +1 ≡ R m × R n in the obvious way. Then, Γ acts isometrically on the unit sphere S N := {x ∈ R N +1 : |x| = 1}. The stereographic projection σ : S N → R N ∪ {∞}, which maps the north pole (0, . . . , 0, 1) to ∞, induces a conformal action of Γ on R N , given by Note that the map γ is well defined except at a single point. The group Γ acts on the Sobolev space D 1,2 (R N ) by linear isometries as follows: for any γ ∈ Γ and w ∈ D 1,2 (R N ); see [6,Section 3]. We shall say that w is Γ-invariant if γw = w for all γ ∈ Γ, and that (u 1 , . . . , u M ) is Γ-invariant if each u i is Γ-invariant. We set One has the following results.
For each I ⊂ {1, . . . , M }, let (S I ) be the system of M − |I| equations obtained by replacing κ i , µ i , λ ij , λ ji with 0 if i ∈ I, where |I| is the cardinality of I, i.e.,  Proof. Note that inf v∈N J (v) = inf v∈U Ψ(v). So, by Ekeland's variational principle [21,Theorem 8.5] and Theorem 3.3, there exists a sequence (u n ) in N such that J (u n ) → c 0 and J ′ (u n ) → 0. It follows from (3.1) that (u n ) is bounded in H := (D 1,2 0 (Ω)) M . So, after passing to a subsequence, u n ⇀ u weakly in H, u n → u strongly in L 2 (Ω) and u n → u a.e. in Ω. A standard argument shows that u is a solution to the system (1.4). We claim that u is fully nontrivial.