On entire solutions of an elliptic system modeling phase separations

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Advances in Mathematics 243 (2013) 102–126 www.elsevier.com/locate/aim

On entire solutions of an elliptic system modeling phase separations Henri Berestycki a , Susanna Terracini b,∗ , Kelei Wang c , Juncheng Wei d a Ecole ´ des hautes e´ tudes en sciences sociales, CAMS, 54, boulevard Raspail, F - 75006 - Paris, France b Dipartimento di Matematica “G. Peano”, Universit`a di Torino, Via Carlo Alberto 10, 10123 Torino, Italy c Wuhan Institute of Physics and Mathematics, The Chinese Academy of Sciences, Wuhan 430071, China d Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong

Received 3 April 2012; accepted 16 April 2013

Communicated by O. Savin

Abstract We study the qualitative properties of a limiting elliptic system arising in phase separation for Bose–Einstein condensates with multiple states:  ∆u = uv 2 in Rn , ∆v = vu 2 in Rn ,  u, v > 0 in Rn . When n = 1, we prove uniqueness of the one-dimensional profile. In dimension 2, we prove that stable solutions with linear growth must be one-dimensional. Then we construct entire solutions in R2 with polynomial growth | x |d for any positive integer d ≥ 1. For d ≥ 2, these solutions are not one-dimensional. The construction is also extended to multi-component elliptic systems. c 2013 Elsevier Inc. All rights reserved. ⃝ MSC: 35B45 Keywords: Stable solutions; Elliptic systems; Phase separations; Almgren monotonicity formulas

∗ Corresponding author.

E-mail addresses: [email protected] (H. Berestycki), [email protected], [email protected] (S. Terracini), [email protected] (K. Wang), [email protected] (J. Wei). c 2013 Elsevier Inc. All rights reserved. 0001-8708/$ - see front matter ⃝ http://dx.doi.org/10.1016/j.aim.2013.04.012

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1. Introduction and main results Consider the following two-component Gross–Pitaevskˇıi system −∆u + αu 3 + Λv 2 u = λ1 u

in Ω ,

(1.1)

−∆v + βv 3 + Λu 2 v = λ2 v

in Ω ,

(1.2)

u > 0,

v>0

in Ω ,

u = 0, v = 0 on ∂Ω ,   2 u = N1 , v 2 = N2 , Ω

(1.3) (1.4) (1.5)

Ω

where α, β, Λ > 0 and Ω is a bounded smooth domain in Rn . Such systems are associated with mixtures of Bose–Einstein condensates cfr, [3,24,25]. Solutions of (1.1)–(1.5) can be regarded as critical points of the energy functional    α β Λ E Λ (u, v) = |∇u|2 + |∇v|2 + u 4 + v 4 + u 2 v 2 , (1.6) 2 2 2 Ω on the space (u, v) ∈ H01 (Ω ) × H01 (Ω ) with constraints   2 u d x = N1 , v 2 d x = N2 . Ω

(1.7)

Ω

The eigenvalues λ j ’s are Lagrange multipliers with respect to (1.7). Both eigenvalues λ j = λ j,Λ , j = 1, 2, and eigenfunctions u = u Λ , v = vΛ depend on the parameter Λ. As the parameter Λ tends to infinity, the two components tend to separate their supports. In order to investigate the basic rules of phase separations in this system one needs to understand the asymptotic behavior of (u Λ , vΛ ) as Λ → +∞. We shall assume that the solutions (u Λ , vΛ ) of (1.1)–(1.5) are such that the associated eigenvalues λ j,Λ ’s are uniformly bounded, together with their energies E Λ (u Λ , vΛ ). Then, as Λ → +∞, there is weak convergence (up to a subsequence) to a limiting profile (u ∞ , v∞ ) which formally satisfies  −∆u ∞ + αu 3∞ = λ1,∞ u ∞ in Ωu , (1.8) 3 −∆v∞ + βv∞ = λ2,∞ v∞ in Ωv , where Ωu = {x ∈ Ω : u ∞ (x) > 0} and Ωv = {x ∈ Ω : v∞ (x) > 0} are positivity domains composed of finitely disjoint components with positive Lebesgue measure, and each λ j,∞ is the limit of λ j,Λ ’s as Λ → ∞ (up to a subsequence). There is a large literature about this type of questions. Effective numerical simulations for (1.8) can be found in [5,6,14]. Chang–Lin–Lin–Lin [14] proved pointwise convergence of (u Λ , vΛ ) away from the interface Γ ≡ {x ∈ Ω : u ∞ (x) = v∞ (x) = 0}. In Wei–Weth [32] the uniform equicontinuity of (u Λ , vΛ ) is established, while Noris–Tavares–Terracini–Verzini [26] proved the uniform-in-Λ H¨older continuity of (u Λ , vΛ ); afterwards, Wang [31] proved localized uniform H¨older estimates. The regularity of the nodal set of the limiting profile has been investigated in [12,30] and in [17]: it turns out that the limiting pair (u ∞ (x), v∞ (x)) is the positive and the negative pair (w+ , w− ) of a solution of the equation −∆w + α(w+ )3 − β(w − )3 = λ1,∞ w + − λ2,∞ w− .

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To derive the asymptotic behavior of (u Λ , vΛ ) near the interface Γ = {x ∈ Ω : u ∞ (x) = v∞ (x) = 0}, one is led to consider the points xΛ ∈ Ω such that u Λ (xΛ ) = vΛ (xΛ ) = m Λ → 0 and xΛ → x∞ ∈ γ ⊂ Ω as Λ → +∞ (up to a subsequence). Let us first consider a special, yet fundamental, case: the one-dimensional one, that we will discuss in detail in Section 2. Energy and symmetry considerations (see Theorem 1.1 in [8]) yield m 4Λ Λ → C0 > 0

(1.9)

(by a simple scaling argument we may assume that C0 = 1). Turning to higher dimensions, it seems now reasonable to assume the validity of (1.9): then, by blowing up, we find the following limiting nonlinear elliptic system ∆u = uv 2 ,

∆v = vu 2 ,

u, v > 0 in Rn .

(1.10)

This paper focuses on classification of entire solutions to this class of system. Of course, throughout this paper, solution means classical (analytic) solution. As already mentioned, problem (1.10) has been studied in Berestycki–Lin–Wei–Zhao [8], and Noris–Tavares–Terracini–Verzini [26]. It has been proved in [8] that, in the one-dimensional case, (1.9) always holds. In addition, the authors showed the existence, symmetry and nondegeneracy of the solution to one-dimensional limiting system u ′′ = uv 2 ,

v ′′ = vu 2 ,

u, v > 0 in R.

(1.11)

In particular they showed that entire solutions are reflectionally symmetric, i.e., there exists x0 such that u(x − x0 ) = v(x0 − x). They also established a two-dimensional version of the De Giorgi Conjecture in this framework. Namely, under the growth condition u(x) + v(x) ≤ C(1 + |x|),

(1.12)

all monotone solutions are one dimensional. By the way, one-dimensional entire solutions satisfy a global Lipschitz bound, as easily seen from the energy conservation law, and are exactly linear at infinity. On the other hand, in [26], it was proved that the linear growth is the lowest possible for solutions to (1.10). In other words, if there exists α ∈ (0, 1) such that u(x) + v(x) ≤ C(1 + |x|)α ,

(1.13)

then one component is constant and the other vanishes identically. In this paper we address three problems left open in [8]. First, we prove the uniqueness of solutions to (1.11), up to translations (u(x), v(x)) → (u(x + t), v(x + t)), t ∈ R and scalings (u(x), v(x)) → (λu(λx), λv(λx)), λ > 0. This answers the question stated in Remark 1.4 of [8]. Second, we prove that the De Giorgi conjecture still holds in the two dimensional case, when we replace the monotonicity assumption by the stability condition. A third open question of (1.10) is whether all solutions to (1.10) necessarily satisfy the growth bound (1.12). We shall answer this question negatively in this paper. Theorem 1.1. The solution to (1.11) is unique, up to translations and scaling. Next, we want to classify the stable solutions in R2 . We recall that a stable solution (u, v) to (1.10) is such that the linearization is weakly positive definite. That is, it satisfies  [|∇ϕ|2 + |∇ψ|2 + v 2 ϕ 2 + u 2 ψ 2 + 4uvϕψ] ≥ 0, ∀ϕ, ψ ∈ C0∞ (Rn ). Rn

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In addition, in [8], it was proved that the one-dimensional solution is stable in Rn . Our first result states that the only stable solution in R2 , among those growing at most linearly, is the one-dimensional family. Theorem 1.2. Let (u, v) be a stable solution to (1.10) in R2 satisfying the growth bound (1.12) holds. Then (u, v) is one-dimensional, i.e., there exists a ∈ R2 , |a| = 1 such that (u, v) = (U (a · x), V (a · x)) where (U, V ) are functions of one variable and satisfies (1.11). This result has been recently extended to more general nonlinear operators and nonlinearities of gradient form (see [21,22]). Our third result shows that there are solutions to (1.10) with polynomial growth |x|d that are not one dimensional. The construction depends on the following harmonic polynomial Φ of degree d: Φ := Re(z d ). Note that Φ has some dihedral symmetry; indeed, let us take its d nodal lines L 1 , . . . , L d and denote the corresponding reflection with respect to these lines by T1 , . . . , Td . Then there holds Φ(Ti z) = −Φ(z).

(1.14)

The third result of this paper is the following one. Theorem 1.3. For each positive integer d ≥ 1, there exists a solution (u, v) to problem (1.10), satisfying 1. u − v > 0 in {Φ > 0} and u − v < 0 in {Φ < 0}; 2. u ≥ Φ + and v ≥ Φ − ; 3. ∀i = 1, . . . , d, u(Ti z) = v(z); 4. ∀r > 0, the Almgren frequency function satisfies  r Br (0) |∇u|2 + |∇v|2 + u 2 v 2  N (r ) := ≤ d; (1.15) 2 2 ∂ Br (0) u + v 5. lim N (r ) = d.

r →+∞

(1.16)

6. Furthermore, there exists b ∈ (0, +∞) such that:  k  1 lim 1+2d u i2 = b. r →∞ r ∂ Br (0) 1 Note that the one-dimensional solution constructed in [8] can be viewed as corresponding to the case d = 1. Thanks to property 6, such solutions have polynomial growth at infinity. For d ≥ 2, the solutions of Theorem 1.3 will be obtained by a minimization argument under symmetric variations (ϕ, ψ) (i.e. satisfying ϕ ◦ Ti = ψ for every reflection Ti ). The first four claims will be derived from the construction. See Theorem 4.1. Regarding property 5, a remarkable fact is that the Almgren frequency function N (r ) is increasing in r : this is given by the Almgren monotonicity formula (see Proposition 5.2). Therefore limr →+∞ N (r ) exists. To understand the asymptotics at infinity of an entire solution, at infinity we consider the normalized blow-down sequence defined by:   1 1 (u R (x), v R (x)) := u(Rx) v(Rx) , L(R) L(R)

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where L(R) is chosen so that  u 2R + v 2R = 1. ∂ B1 (0)

In Section 6, we will prove the following. Theorem 1.4. Let (u, v) be a solution of (1.10) such that d := lim N (r ) < +∞. r →+∞

Then d is a positive integer. As R → ∞, (u R , v R ) defined above (up to a subsequence) converges to (Ψ + , Ψ − ) uniformly on any compact set of R N where Ψ is a homogeneous harmonic polynomial of degree d. When d = 1 (linear growth) this implies Reifenberg flatness of the coincidence set {x : u(x) = v(x)}. By applying a new sharp Alt–Caffarelli–Friedman type (see, e.g. [11,13]) inequality for symmetric solutions (Theorem 5.6 and its Corollary) to the solutions found by Theorem 1.3, Theorem 1.4 can be improved, thus yielding the following property. Corollary 1.5. Let (u, v) be a solution of (1.10) given by Theorem 1.3. Then   1 1 (u R (x), v R (x)) := u(Rx) v(Rx) Rd Rd converges uniformly on compact subsets of R2 to a multiple of (Φ + , Φ − ), where Φ := Re(z d ). Theorem 1.4 roughly says that (u, v) is asymptotic to (Ψ + , Ψ − ) at infinity to some homogeneous harmonic polynomial. The extra information we have in the setting of Theorem 1.3 is a sharp growth estimate and, moreover, the information that Ψ ≡ Φ = Re(z d ). This can be inferred from the symmetries of the solution (property 3 in Theorem 1.3). For another elliptic system with a similar form,  ∆u = uv, u > 0 in Rn , (1.17) ∆v = vu, v > 0 in Rn the same result has been proved by Conti–Terracini–Verzini in [16]. In fact, their result holds for any dimension n ≥ 1 and any harmonic polynomial function on Rn . Note however that the problem here differs in nature from (1.17). Indeed, Eq. (1.17) can be reduced to a single equation: indeed, the difference u − v is a harmonic function (∆(u − v) = 0) and thus we can write v = u − Φ where Φ is a harmonic function. By restricting to certain symmetry classes, (1.17) can be solved by the sub–super solution method. However, this reduction does not work for system (1.10) that we study here. For the proof of Theorem 1.3, we first construct solutions to (1.10) in any bounded ball B R (0) satisfying appropriate boundary conditions:  ∆u = uv 2 , in B R (0), (1.18) ∆v = vu 2 , in B R (0),  + − u=Φ , v=Φ on ∂ B R (0). This is done by variational method and using heat flow. The next natural step is to let R → +∞ and obtain some convergence result. This requires some uniform (in R) upper bound

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for solutions to (1.18). In order to prove this, we will exploit a new monotonicity formula for symmetric functions (Proposition 5.7). We also need to rule out a possible degeneracy, that is the vanishing of the limit, or the lowering f its degree. To this end, we will give some lower bound using the symmetric Almgren monotonicity formula. Finally, we observe that a similar construction works also for a system with many components. Theorem 1.6. Let d be an integer or a half-integer and 2d = hk be a multiple of the number of components k, and G denote the rotation of order 2d. In this way we prove the following result. There exists a positive solution to the system  k  ∆u = u  u 2 , in C = R2 , i = 1, . . . , k, i i j (1.19) j̸=i, j=1   u i > 0, i = 1, . . . , k, having the following symmetries (here z is the complex conjugate of z) u i (z) = u i+1 (Gz), u i+k (z) = u i (z),

on C, i = 1, . . . , k, on C, i = 1, . . . , k,

u k+1−i (z) = u i (z),

(1.20)

on C, i = 1, . . . , k.

Furthermore, we have r lim

r →∞



Br (0)

k 

|∇u i |2 +

i< j

1



∂ Br (0)



k  1

u i2 u 2j = d,

u i2

and there exists b ∈ (0, +∞) such that:  k  1 lim 1+2d u i2 = b. r →∞ r ∂ Br (0) 1 Note that Theorem 1.6 includes Theorem 1.3 as a special case, when k = 2, d = k. The problem of the full classification of solutions to (1.10) is largely open. In view of our results, one can formulate several open questions. Open Problem 1. We recall from [8] that it is still an open problem to know in which dimension it is true that all monotone solutions are one-dimensional. A partial result in this direction has been recently obtained by Wang [31] and seems to indicate that, in any dimension, minimal solutions having at most linear growth are one-dimensional. A similar open question is in which dimension it is true that all stable solutions are one-dimensional. We refer the reader to [2,23,20, 27,28] for similar results for the Allen–Cahn equation. Open Problem 2. Let us recall that, in the one dimensional case, there exists a unique solution to (1.11) (up to translations and scalings). Such solution has linear growth at infinity and, in the Almgren monotonicity formula, satisfies lim N (r ) = 1.

r →+∞

(1.21)

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It is natural to conjecture that, in any space dimension, a solution of (1.10) satisfying (1.21) is actually one dimensional, that is, there is a unit vector a such that (u(x), v(x)) = (U (a · x), V (a · x)) for x ∈ Rn , where (U, V ) solves (1.11). However this result seems to be difficult to obtain at this stage. Open Problem 3. A further step would be to prove uniqueness of the (family of) solutions having polynomial asymptotics given by Theorem 1.3 in two space dimension. An extremely difficult and challenging question is to classify all solutions with lim N (r ) = d.

(1.22)

r →+∞

Open Problem 4. For the Allen–Cahn equation ∆u + u − u 3 = 0 in R2 , symmetric solutions similar to those provided by Theorem 1.3 were first constructed in [18] in two dimensions and in [1] for higher dimensions; let us recall that, contrarily to the present case, all solutions to the Allen–Cahn equation are bounded. On the other hand, it was also proved in [19] that the Allen–Cahn equation in R2 admits solutions with multiple fronts. An open question is whether a similar result holds for (1.10). Namely, are there solutions to (1.10) such that the set {u = v} contains disjoint multiple curves? Let us mention that solutions with this property having exponential growth has been recently achieved by Soave–Zilio in [29]. Open Problem 5. This question is related to the possible extension of Theorem 1.3 to higher dimensions. We recall that, for the Allen–Cahn equation ∆u + u − u 3 = 0 in R2m with m ≥ 2, saddle-like solutions were constructed in [10] by suitably employing properties of Simons cone. Stable solutions to the Allen–Cahn equation in R8 with a non planar level set were found in [27], using minimal cones. We conjecture that all these results should have analogues for (1.10). 2. Uniqueness of solutions in R: proof of Theorem 1.1 In this section we prove Theorem 1.1. Without loss of generality, we assume that lim u(x) = +∞,

x→+∞

lim v(x) = 0.

x→+∞

(2.1)

The existence of such entire solutions has been proved in [8]. By the symmetry property of solutions to (1.11) (Theorem 1.3 of [8]), we may consider the following problem  u ′′ = uv 2 , v ′′ = vu 2 , u, v > 0 in R, (2.2) ′ lim u (x) = − lim v ′ (x) = a x→+∞

x→−∞

where a > 0 is a constant. We now prove that there exists a unique solution (u, v) to (2.2), up to translations. We will prove it using the method of moving planes. At first, note that both u and v are convex positive functions. Were both increasing in one halfline, they would become unbounded in a finite time, as a simple comparison argument shows. Thus, one is decreasing (say u) while the other is increasing (say v), globally on R. Next, it is immediate to check that they have to vanish at the end they are unbounded and that the decay at infinity is exponential Hence, we infer that for any solution (u, v) of (2.2), u ′′ and v ′′ decay exponentially at infinity. Integration shows that as x → +∞, |u ′ (x) − a| decays exponentially (see also [8]). This implies the existence of a positive constant A such that |u(x) − ax + | + |v(x) − ax − | ≤ A.

(2.3)

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Moreover, the limits lim (u(x) − ax + ),

lim (v(x) − ax − )

x→+∞

x→−∞

exist. Now assume (u 1 , v1 ) and (u 2 , v2 ) are two solutions of (2.2). For t > 0, denote u 1,t (x) := u 1 (x + t),

v1,t (x) := v1 (x + t).

We want to prove that there exists an optimal t0 such that for all t ≥ t0 , u 1,t (x) ≥ u 2 (x),

v1,t (x) ≤ v2 (x) in R.

(2.4)

Then we will show that when t = t0 these inequalities are identities. This will imply the uniqueness result. Without loss of generality, assume (u 1 , v1 ) and (u 2 , v2 ) satisfy the estimate (2.3) with the same constant A. Step 1. For t ≥ 16A a (A as in (2.3)), (2.4) holds. First, in the region {x ≥ −t + 2A a }, by (2.3) we have u 1,t (x) ≥ a(x + t) − A ≥ ax + + A ≥ u 2 (x); while in the region {x ≤ −t +

2A a },

(2.5)

we have

v1,t (x) ≤ a(x + t)− + A ≤ ax − − A ≤ v2 (x).

(2.6)

On the interval {x < −t + 2A a }, we have  ′′ 2 u 1,t = u 1,t v1,t ≤ u 1,t v22 , ′′ 2 u 2 = u 2 v2 . With the right boundary conditions     2A 2A ≥ u 2 −t + , u 1,t −t + a a

(2.7)

lim u 1,t (x) = lim u 2 (x) = 0,

x→−∞

x→−∞

a direct application of the maximum principle implies inf

(u 1,t − u 2 ) ≥ 0.

{x<−t+ 2A a }

A similar argument also shows that sup

(v1,t − v2 ) ≤ 0.

{x>−t+ 2A a }

Therefore, we have shown that for t ≥ 16A a , u 1,t ≥ u 2 and v1,t ≤ v2 . Step 2. We now decrease t to an optimal value when (2.4) holds t0 = inf{t ′ | such that (2.4) holds for all t ≥ t ′ }. 2 (u ′′ Thus t0 is well defined by Step 1. Since −(u 1,t0 − u 2 )′′ + v1,t 1,t0 − u 2 ) ≥ 0, −(v2 − v1,t0 ) + 0 u 21,t0 (v2 − v1,t0 ) ≥ 0, by the strong maximum principle, either

u 1,t0 (x) ≡ u 2 (x),

v1,t0 (x) ≡ v2 (x) in R,

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or u 1,t0 (x) > u 2 (x),

v1,t0 (x) < v2 (x)

in R.

(2.8)

Let us argue by contradiction that (2.8) holds. By the definition of t0 , there exists a sequence of tk < t0 such that limk→+∞ tk = t0 and either inf(u 1,tk − u 2 ) < 0,

(2.9)

R

or sup(v1,tk − v2 ) > 0. R

Let us only consider the first case. Define w1,k := u 1,tk − u 2 and w2,k := v2 − v1,tk . Direct calculations show that they satisfy  ′′ 2 −w1,k + v1,t w = u 2 (v2 + v1,tk )w2,k in R, k 1,k (2.10) ′′ −w2,k + u 21,tk w2,k = v2 (u 2 + u 1,tk )w1,k in R. We use the auxiliary function g(x) = log(|x| + 3) as in [15]. Note that g ≥ 1,

g ′′ < 0

in {x ̸= 0}.

Define w 1,k := w1,k /g and w 2,k := w2,k /g. For x ̸= 0 we have    g′ ′ g′ ′′ 2   − w − 2 w  + v − w 1,k = u 2 (v2 + v1,tk ) w2,k ,  1,k 1,tk g 1,k g    g′ ′ g′  ′′ − w2,k −2 w 2,k + u 21,tk − w 2,k = v2 (u 2 + u 1,tk ) w1,k , g g

in R, (2.11) in R.

By definition, w1,k and w2,k are bounded in R, and hence w 1,k , w 2,k → 0

as |x| → ∞.

In particular, in view of (2.9), we know that infR ( w1,k ) < 0 is attained at some point xk,1 . Note that |xk,1 | must be unbounded, for if xk,1 → x∞ , tk → t0 , then w1,k (xk,1 ) → u 1,t0 (x∞ ) − u 2 (x∞ ) = 0. But this violates assumption (2.8). Since |xk,1 | is unbounded, at x = xk,1 there holds ′′ w 1,k ≥0

and

′ w 1,k = 0.

Substituting this into the first equation of (2.11), we get   g ′′ (xk,1 ) v1,tk (xk,1 )2 − w 1,k (xk,1 ) ≥ u 2 (xk,1 )(v2 (xk,1 ) + v1,tk (xk,1 )) w2,k (xk,1 ) (2.12) g(xk,1 ) which implies that w 2,k (xk,1 ) < 0. Thus we also have infR w 2,k < 0. Assume it is attained at xk,2 . The same argument as before shows that |xk,2 | must also be unbounded. Similar to (2.12), we have   g ′′ (xk,2 ) 2 u 1,tk (xk,2 ) − w 2,k (xk,2 ) ≥ v2 (xk,2 )(u 2 (xk,2 ) + u 1,tk (xk,2 )) w1,k (xk,2 ). (2.13) g(xk,2 )

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Observe that w 2,k (xk,2 ) = inf w 2,k ≤ w 2,k (xk,1 ), R

w 1,k (xk,1 ) = inf w 1,k ≤ w 1,k (xk,2 ). R

Substituting these into (2.12) and (2.13), we obtain w 1,k (xk,1 ) ≥

u 2 (xk,1 )[v2 (xk,1 ) + v1,tk (xk,1 )] v1,tk (xk,1 )2 − ×

g ′′ (xk,1 ) g(xk,1 )

v2 (xk,2 )[u 2 (xk,2 ) + u 1,tk (xk,2 )] u 1,tk (xk,2 )2 −

g ′′ (xk,2 ) g(xk,2 )

w 1,k (xk,1 ).

(2.14)

Since w˜ 1,k (xk,1 ) < 0, we conclude from (2.14) that u 2 (xk,1 )[v2 (xk,1 ) + v1,tk (xk,1 )] v2 (xk,2 )[u 2 (xk,2 ) + u 1,tk (xk,2 )] v1,tk (xk,1 )2 −

g ′′ (xk,1 ) g(xk,1 )

u 1,tk (xk,2 )2 −

g ′′ (xk,2 ) g(xk,2 )

≥1

(2.15)

(x) 1 as where |xk,1 | → +∞, |xk,2 | → +∞. This is impossible since gg(x) ∼ − |x|2 log(|x|+3) |x| → +∞, and we also use the decaying as well as the linear growth properties of u and v at ∞. We have thus reached a contradiction, and the proof of Theorem 1.1 is thereby completed. ′′

3. Stable solutions: proof of Theorem 1.2 In this section, we prove Theorem 1.2. The proof follows an idea from Berestycki–Caffarelli– Nirenberg [7]; see also Ambrosio–Cabr´e [2] and Ghoussoub–Gui [23]. First, by the stability, we have the following. Lemma 3.1. There exist a constant λ ≥ 0 and two functions ϕ > 0 and ψ < 0, smoothly defined in R2 such that  ∆ϕ = v 2 ϕ + 2uvψ − λϕ, (3.1) ∆ψ = 2uvϕ + v 2 ψ − λψ. Proof. For any R < +∞ the stability assumption reads  2 2 2 2 2 2 B R (0) |∇ϕ| + |∇ψ| + v ϕ + u ψ + 4uvϕψ  λ(R) := min ≥ 0; 2 2 ϕ,ψ∈H01 (B R (0))\{0} B R (0) ϕ + ψ it is well known that the corresponding minimizer is the first eigenfunction. That is, let (ϕ R , ψ R ) realizing λ(R), then  ∆ϕ R = v 2 ϕ R + 2uvψ R − λ(R)ϕ R , in B R (0), (3.2) ∆ψ R = 2uvϕ R + v 2 ψ R − λ(R)ψ R , in B R (0),  ϕ R = ψ R = 0 on ∂ B R (0). By possibly replacing (ϕ R , ψ R ) with (|ϕ R |, −|ψ R |), we can assume ϕ R ≥ 0 and ψ R ≤ 0. After a normalization, we also assume |ϕ R (0)| + |ψ R (0)| = 1.

(3.3)

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λ(R) is decreasing in R, thus uniformly bounded as R → +∞. Let λ := lim λ(R). R→+∞

The equations for ϕ R and −ψ R (both of them are nonnegative functions) form a cooperative system; thus by the Harnack inequality ([4] or [9]), ϕ R and ψ R are uniformly bounded on any compact set of R2 . By letting R → +∞, we can obtain a converging subsequence and the limit (ϕ, ψ) satisfies (3.1). We also have ϕ ≥ 0 and ψ ≤ 0 by passing to the limit. Hence −∆ϕ + (v 2 − λ)ϕ ≥ 0. Applying the strong maximum principle, either ϕ > 0 strictly or ϕ ≡ 0. If ϕ ≡ 0, substituting this into the first equation in (3.1), we obtain ψ ≡ 0. This contradicts the normalization condition (3.3). Thus, it holds true that ϕ > 0 and similarly ψ < 0.  Fix a unit vector ξ . Differentiating Eq. (1.10) yields the following equation for (u ξ , vξ )  ∆u ξ = v 2 u ξ + 2uvvξ , (3.4) ∆vξ = 2uvu ξ + v 2 vξ . Let w1 =

uξ , ϕ

w2 =

vξ . ψ

Direct calculations using (3.1) and (3.4) show  div(ϕ 2 ∇w1 ) = 2uvϕψ(w2 − w1 ) + λϕ 2 w1 , div(ϕ 2 ∇w2 ) = 2uvϕψ(w1 − w2 ) + λψ 2 w2 . For any η ∈ C0∞ (R2 ), testing these two equations with w1 η2 and w2 η2 respectively, we obtain    2 2 2 2  − ϕ |∇w1 | η − 2ϕ w1 η∇w1 ∇η = 2uvϕψ(w2 − w1 )w1 η2 + λϕ 2 w1 η2 ,    − ψ 2 |∇w2 |2 η2 − 2ψ 2 w2 η∇w2 ∇η = 2uvϕψ(w1 − w2 )w2 η2 + λψ 2 w2 η2 . Adding these two and applying the Cauchy–Schwarz inequality, we infer that   ϕ 2 |∇w1 |2 η2 + ψ 2 |∇w2 |2 η2 ≤ 16 ϕ 2 w12 |∇η|2 + ψ 2 w22 |∇η|2  ≤ 16 (u 2ξ + vξ2 )|∇η|2 .

(3.5)

Here we have taken away the positive term on the right hand side and used the fact that 2uvϕψ(w2 − w1 )w1 η2 + 2uvϕψ(w1 − w2 )w2 η2 = −2uvϕψ(w1 − w2 )2 η2 ≥ 0, because ϕ > 0 and ψ < 0. On the other hand, testing the equation ∆u ≥ 0 with uη2 (η as above) and integrating by parts, we get   |∇u|2 η2 ≤ 16 u 2 |∇η|2 .

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The same estimate also holds for v. For any r > 0, take η ≡ 1 in Br (0), η ≡ 0 outside B2r (0) and |∇η| ≤ 2/r . By the linear growth of u and v, we obtain a constant C such that  |∇u|2 + |∇v|2 ≤ Cr 2 . (3.6) Br (0)

Now for any R > 0, in (3.5), we take η to be  x ∈ B R (0),  1,  0, x ∈ B R 2 (0)c , η(z) = log(|z|/R)   x ∈ B R 2 (0) \ B R (0). 1 − log R With this η, we infer from (3.5) that  ϕ 2 |∇w1 |2 + ψ 2 |∇w2 |2 ≤ B R (0)

≤ =

C (log R)2 C (log R)2 C (log R)2



1 (|∇u|2 + |∇v|2 ) 2 B R 2 (0)\B R (0) |z|    R2 |∇u|2 + |∇v|2 dr r −2 ∂ Br (0)

R R2



r −2 R



≤

d dr

 Br (0)

 |∇u|2 + |∇v|2 dr

 R 2  |∇u| + |∇v|  ∂ Br (0) R  



C r −2 = (log R)2   R2 + 2 r −3 R



Br (0)

2

2

|∇u|2 + |∇v|2 dr

C . log R

By letting R → +∞, we see ∇w1 ≡ 0 and ∇w2 ≡ 0 in R2 . Thus, there is a constant c such that (u ξ , vξ ) = c(ϕ, ψ). Because ξ is an arbitrary unit vector, from this we actually know that after changing the coordinates suitably, u y ≡ 0,

vy ≡ 0

in R2 .

That is, u and v depend on x only and they are one dimensional. 4. Existence in bounded balls In this section we first construct a solution (u, v) to the problem  ∆u = uv 2 in B R (0), ∆v = vu 2 in B R (0),

(4.1)

satisfying the boundary condition u = Φ+,

v = Φ−

on ∂ B R (0) ⊂ R2 .

(4.2)

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More precisely, we prove the following. Theorem 4.1. There exists a solution (u R , v R ) to problem (4.1), satisfying 1. 2. 3. 4.

u R − v R > 0 in {Φ > 0} and u R − v R < 0 in {Φ < 0}; u R ≥ Φ + and v R ≥ Φ − ; ∀i = 1, . . . , d, u R (Ti z) = v R (z); ∀r ∈ (0, R),  r Br (0) |∇u R |2 + |∇v R |2 + u 2R v 2R  ≤ d. N (r ; u R , v R ) := 2 2 ∂ Br (0) u R + v R

Proof. Let us denote U ⊂ H 1 (B R (0))2 the set of pairs of non negative functions (u, v) satisfying the boundary condition (4.2), together with conditions 1 and 3 of the statement of the Theorem (with the strict inequality < replaced by ≤, and so now U is a closed set). The desired solution will be a minimizer of the energy functional  |∇u|2 + |∇v|2 + u 2 v 2 E R (u, v) := B R (0)

over U. The existence of at least one minimizer follows easily from the direct method of the Calculus of Variations. To prove that the minimizer also satisfies Eq. (4.1), we use the heat flow method. More precisely, we consider the following parabolic problem  Ut − ∆U = −U V 2 , in [0, +∞) × B R (0), (4.3) Vt − ∆V = −V U 2 , in [0, +∞) × B R (0), with the boundary conditions U = Φ + and V = Φ − on (0, +∞)×∂ B R (0) and initial conditions in U. By the standard parabolic theory, there exists a unique local solution (U, V ) in spaces of H¨older continuous functions. The solutions are actually analytic in the interior of the ball. We note that the solution’s components are (globally in time) bounded above (by the harmonic extension of Φ + and Φ − resp.) and below (by zero). Hence, they can be globally continued in time. By noting the energy inequality       ∂U 2  ∂ V 2 d     E R (U (t), V (t)) = − (4.4)   +  ∂t  dt B R (0) ∂t and the fact that E R ≥ 0, standard parabolic theory implies that for any sequence ti → +∞, there exists a subsequence of ti such that (U (ti ), V (ti )) converges to a solution (u, v) of (4.1). Next we show that U is positively invariant by the parabolic flow. First of all, by the symmetry of initial and boundary data, (V (t, Ti z), U (t, Ti z)) is also a solution to the problem (4.3). By the uniqueness of solutions to the parabolic system (4.3), (U, V ) inherits the symmetry of (Φ + , Φ − ). That is, for all t ∈ [0, +∞) and i = 1, . . . , d, U (t, z) = V (t, Ti z). This implies U −V =0

on {Φ = 0}.

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Thus, in the open set D R := B R (0) ∩ {Φ > 0}, we have, for any initial datum (u 0 , v0 ) ∈ U,  (U − V )t − ∆(U − V ) = U V (U − V ), in [0, +∞) × D R (0), U − V ≥ 0, on [0, +∞) × ∂ D R (0), (4.5)  U − V ≥ 0, on {0} × D R (0). The strong maximum principle implies U − V > 0 in (0, +∞) × D R (0). In addition we have  (U − V )t − ∆(U − V ) ≥ 0, in D R (0), (4.6) U − V = Φ + , on ∂ D R (0). Comparing with Φ + on D R (0), we obtain U − V > Φ + > 0,

in D R (0).

(4.7)

By the parabolic maximum principle, we find that U − V ≥ Φ on D R (0), and therefore U − V > Φ+. By letting t → +∞, we obtain that the limit satisfies u − v ≥ Φ+

in D R (0).

(4.8)

(u, v) also has the symmetry, ∀i = 1, . . . , d u(Ti z) = v(z). Because u > 0 and v > 0 in B R (0), we in fact have u > Φ+,

in B R (0).

(4.9)

In conclusion, (u, v) satisfies conditions (1, 2, 3) in the statement of the theorem. Let (u R , v R ) be a minimizer of E R over U. Now we consider the parabolic equation (4.3) with the initial condition U (x, t) = u R (x),

V (x, t) = v R (x).

(4.10)

By (4.4), we deduce that E R (u R , v R ) ≤ E R (U, V ) ≤ E R (u R , v R ) and hence (U (x, t), V (x, t)) ≡ (u R (x), v R (x)) for all t ≥ 0. By the arguments above, we see that (u R , v R ) satisfies (4.1) and conditions (1, 2, 3) in the statement of the theorem. In order to prove (4), we first note that, as (u R , v R ) minimizes the energy and (Φ + , Φ − ) ∈ U, there holds   2 2 2 2 |∇u R | + |∇v R | + u R v R ≤ |∇Φ|2 . B R (0)

B R (0)

Now by the Almgren monotonicity formula (Proposition 5.2) and the boundary conditions, ∀r ∈ (0, R), we derive  R B R (0) |∇Φ|2 N (r ; u R , v R ) ≤ N (R; u R , v R ) ≤  = d. 2 ∂ B R (0) |Φ| This completes the proof of Theorem 4.1.



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Let us now turn to the system with many components. In a similar way we shall prove the existence on bounded sets. Let d be an integer or a half-integer and 2d = hk be a multiple of the number of components k, and G denote the rotation of order 2d. Take the fundamental domain F of the rotations group of degree 2d, that is F = {z ∈ C : θ = arg(z) ∈ (−π/2d, π/2d)}.  d h−1 ik r cos(dθ ) if z ∈ ∪i=0 G (F), Ψ (z) = (4.11) 0 otherwise in C. Note that Ψ (z) is positive whenever it is not zero. Next we construct a solution (u 1 , . . . , u k ) to the system ∆u i = u i

k 

u 2j ,

in B R (0), i = 1, . . . , k

(4.12)

j̸=i, j=1

satisfying the symmetry and boundary condition (here z is the complex conjugate of z)  u i (z) = u i+1 (Gz), on B R (0), i = 1, . . . , k, u k+1−i (z) = u i (z), on B R (0), i = 1, . . . , k,  u i+k (z) = u i (z), on B R (0), i = 1, . . . , k, u i+1 (z) = Ψ (G i (z)),

on ∂ B R (0), i = 0, . . . , k − 1.

(4.13) (4.14)

More precisely, we prove the following. Theorem 4.2. For every R > 0, there exists a solution (u 1,R , . . . , u k,R ) to the system (4.12) with symmetries (4.13) and boundary conditions (4.14), satisfying, r N (r ) :=



Br (0)

k 

|∇u i,R |2 +

 i< j

1



∂ Br (0)

k  1

2 u2 u i,R j,R

≤ d,

∀r ∈ (0, R).

2 u i,R

Proof. Let us denote by U ⊂ H 1 (B R (0))k the set of pairs satisfying the symmetry and boundary condition (4.13) and (4.14). The desired solution will be the minimizer of the energy functional  k   2 |∇u i,R |2 + u i,R u 2j,R Br (0) 1

i< j

over U. Once more, to deal with the constraints, we may take advantage of the positive invariance of the associated heat flow:   ∂Ui − ∆Ui = −Ui U 2j , in [0, +∞) × B R (0), (4.15) ∂t j̸=i

which can be solved under conditions (1.20), (4.14) and initial conditions in U. Thus, the minimizer of the energy (u 1,R , . . . , u k,R ) solves the differential system. In addition, using the test function (Ψ1 , . . . , Ψk ), where Ψi = Ψ ◦ G i−1 , i = 1, . . . , k, we have   k   2 |∇u i,R |2 + u i,R u 2j,R ≤ k |∇Ψ |2 . B R (0) 1

i< j

B R (0)

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Now by the Almgren monotonicity formula below (Proposition 5.2) and the boundary conditions, we get  R B R (0) |∇Ψ |2 = d, ∀r ∈ (0, R).  N (r ) ≤ N (R) ≤  2 ∂ B R (0) |Ψ | In order to conclude the proof of Theorems 4.1 and 4.2, we need to find upper and lower bounds for the solutions, uniform with respect to R on bounded subsets of C. That is, we will prove that for any r > 0, there exist positive constants 0 < c(r ) < C(r ) (independent of R) such that c(r ) < sup u R ≤ C(r ).

(4.16)

Br (0)

Once we have this estimate, then by letting R → +∞, a subsequence of (u R , v R ) will converge to a solution (u, v) of problem (1.10), uniformly on any compact set of R2 . It is easily seen that properties (1)–(4) in Theorem 4.1 can be derived by passing to the limit, and we obtain the main results stated in Theorems 1.3 and 1.6. It then remains to establish the bound (4.16). In the next section, we shall obtain this estimate by using the monotonicity formula. 5. Monotonicity formulæ Let us start by stating some monotonicity formulas for solutions to (1.10), for any dimension n ≥ 2. The first two are well-known and we include them here for completeness. But we will also require some refinements. Proposition 5.1. For r > 0 and x ∈ Rn ,  k   E(r ) = r 2−n |∇u i |2 + u i2 u 2j Br (x) 1

i< j

is nondecreasing in r . Proof. We have d E(r ) = (2 − n)r 1−n dr + r 2−n



k 



Br (x) 1 k 

∂ Br (x) 1

|∇u i |2 +

|∇u i |2 +



u i2 u 2j

i< j



u i2 u 2j .

i< j

The Pohoˇzaev identity then yields (see also the proof of the monotonicity formula in [26])   d u 2v2.  (5.1) E(r ) = 2r 2−n [u r2 + vr2 ] + 2r 1−n dr Br ∂ Br The next statement is an Almgren-type monotonicity formula with remainder. Proposition 5.2. For r > 0 and x ∈ Rn , let us define  k  H (r ) = r 1−n u i2 . ∂ Br (x) 1

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Then E(r ) H (r ) is nondecreasing in r . In addition, for every r > 0 there holds N (r ; x) :=

 0

r

2

 Bs



k  i< j

∂ Bs

u i2 u 2j

k  1

ds ≤ N (r ).

(5.2)

u i2

Proof. For simplicity, take x to be the origin 0 and let k = 2. Then, direct calculations show that  d 1−n H (r ) = 2r |∇u|2 + |∇v|2 + 2u 2 v 2 . (5.3) dr Br From (5.1) and (5.3), we infer that        H 2r 2−n ∂ Br (u r2 + vr2 ) + 2r 1−n Br u 2 v 2 − E 2r 1−n ∂ Br uu r + vvr d E (r ) = dr H H2  2   2r 3−2n ∂ Br (u 2 + v 2 ) ∂ Br (u r2 + vr2 ) − 2r 3−2n ∂ Br uu r + vrr ≥ 2   H2 2 1−n 2 2 1−n 2r 2r Br u v Br u v + ≥ . H H Here we have used the following inequality   2 2 2 2 E(r ) ≤ |∇u| + |∇v| + 2u v = uu r + vrr . Br

∂ Br

Hence this yields monotonicity of the Almgren quotient. In addition, by integrating the above inequality we obtain  r 2  u 2v2  Bs ds ≤ N (r ).  2 2 r0 ∂ Bs u + v If x = 0, we simply denote N (r ; x) as N (r ). Assuming an upper bound on N (r ), we establish a doubling property by the Almgren monotonicity formula. Proposition 5.3. Let R > 1 and let (u 1 , . . . , u k ) be a solution of (1.19) on B R . If N (R) ≤ d, then for any 1 < r1 ≤ r2 ≤ R r 2d H (r2 ) ≤ ed 22d . H (r1 ) r1

(5.4)

Proof. For simplicity of notation, we expose the proof for the case of two components. By direct calculation using (5.3), we obtain     2 Br |∇u|2 + |∇v|2 + 2u 2 v 2 d 1−n 2 2  log r u +v = 2 2 dr ∂ Br (0) ∂ Br (0) u + v

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119

2 Br u 2 v 2 2N (r )  ≤ + 2 2 r ∂ B (0) u + v  r 2 2 2 Br u v 2d . ≤ + 2 2 r ∂ Br (0) u + v 

Thanks to (5.2), by integrating, we find that, if r1 ≤ r2 ≤ 2r0 then r 2d H (r2 ) ≤ ed 22d . H (r1 ) r1



(5.5)

An immediate consequence of Proposition 5.3 is the lower bound on bounded sets for the solutions found in Theorems 4.1 and 4.2. Proposition 5.4. Let (u 1,R , . . . , u k,R ) be a family of solutions to (1.19) such that N (R) ≤ d and H (R) = C R 2d . Then, for every fixed r < R, there holds H (r ) ≥ Ce−d r 2d . Another byproduct of the monotonicity formula with the remainder (5.2) is the existence of the limit of H (r )/r 2d . Corollary 5.5. Let (u 1 , . . . , u k ) be a solution of (1.19) on C such that limr →+∞ N (r ) ≤ d, then there exists lim

r →+∞

H (r ) := b < +∞. r 2d

(5.6)

Now we prove the optimal lower bound on the growth of the solution. To this aim, we need a fine estimate on the asymptotics of the lowest eigenvalue as the competition term diverges. The following result is an extension of Theorem 1.6 in [8], where the estimate was proved in the case of two components. Theorem 5.6. Let k be the number of components, h any fixed integer and let 2d = hk; consider     k   2π  L(d, Λ) = min |u i′ |2  0  i     2π  k    u i2 = 1, u i (x) = u i+1 (x + π/d),  k     i + Λ u i2 u 2j  0 .  u k+1−i (x) = u i (−x),   i< j   u i+ jk = u i , i = 1, . . . , k, j = 0, . . . , h − 1

(5.7)

Then, there exists a constant C such that for all Λ > 1 we have d 2 − CΛ−1/4 ≤ L(d, Λ) ≤ d 2 . Proof. At first, it is easy to see that, as Λ → +∞, there holds L(d, Λ) → L(d, ∞),

L(d, Λ) < L(d, ∞),

(5.8)

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where   2π  k    2  u i = 1, u i (x) = u i+1 (x + π/d), k   2π  ′ 2 0 i L(d, ∞) = min |u i |   u k+1−i (x) = u i (−x), u i+ jk = u i   0   i     u i (x)u i+1 (x) = 0, i = 1, . . . , k, j = 0, . . . , h − 1     

and that L(d, ∞) = d 2 . We wish to catch the exact convergence rate with respect to Λ. To this aim, we follow the argument in the proof of Theorem 1.1 in [8]. Any minimizer (u 1,Λ , . . . , u k,Λ ) solves the system of ordinary differential equations  u i′′ = Λu i u 2j − λu i , i = 1, . . . , k, (5.9) j̸=i

together with the associated energy conservation law k 

(u i′ )2 + λu i2 − Λ

k 

u i2 u 2j = e.

(5.10)

i< j

1

Note that the Lagrange multiplier satisfies  2π   k k  ′ 2 2 2 λ= |u i | + 2Λ u i u j = L(d, Λ) + 0

i

0

i< j

2π

Λ

k 

u i2 u 2j .

i< j

As Λ → ∞, we see convergence of the eigenvalues λ ≃ L(d, Λ) → d 2 , together with the energies e → d 2 /2π . Moreover, according to [8], the solutions remain bounded in the Lipschitz norm and converge in Sobolev and H¨older spaces. Now, let us focus on an interval I = (a, a + π/d) where the i-th component is active (this means a = (i − 1 + k j)π/d, for some j = 0, . . . , h). The symmetry constraints imply u i−1 (a) = u i (a),

′ u i−1 (a) = −u i′ (a),

u i+1 (a + π/d) = u i (a + π/d),

′ u i+1 (a + π/d) = −u i′ (a + π/d).

We observe that there is interaction only with the two prime neighboring components, while the others are exponentially small (in Λ) on I . Close to the endpoint a, the component u i is increasing and convex, while u i−1 is decreasing and again convex. By the very same blow-up argument used in the proof of Theorem 1.1 in [8], we infer that u i (a) = u i−1 (a) ≃ K Λ−1/4 ,

′ u i′ (a) = −u i−1 (a) ≃ H = (e + K )/2.

(5.11)

′′ Hence, in a right neighborhood of a, there holds u i (x) ≥ u i (a), and therefore, as u i−1 ≥ Λu i2 (a)u i−1 , from the initial value problem (5.11) we infer that 1/2 u

u i−1 (x) ≤ Cu i (a)e−Λ

i (a)(x−a)

,

∀x ∈ [a, b].

On the other hand, on the same interval we have u i (x) ≤ u i (a) + C(x − a),

∀x ∈ [a, b]

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(here and below C denotes a constant independent of Λ). Consequently, there holds  3 2 u i + u i−1 u i2 ≤ CΛ−1/2 u i (a)−1 ≃ CΛ−1/4 . Λ u i−1 u i2 + u i−1

(5.12)

I

In particular, this yields L(d, Λ) ≥ λ − CΛ−1/4 .

(5.13)

In order to estimate λ, let us consider  ui =



ui −

and u ui ∈ i (a + 2π/d) = u i+1 (a + 2π/d),  u i − j=i±1 u j with  u i on I we find   ′ 2 | u i | ≤ λ | u i |2 + CΛ−1/4 , I



H01 (I ).

j=i±1 u j

+

. Then, as u i (a) = u i−1 (a)

By testing the differential equation for

I

where in the last term we have majorized all the integrals of mixed fourth order monomials with (5.12). As |I | = π/d, using the Poincar´e inequality and (5.13) we obtain the desired estimate on L(d, Λ).  We are now ready to apply the estimate from below on L to derive a lower bound on the energy growth. We recall that there holds   k k    ∂u i 2 2 2  ui |∇u i | + 2 ui u j = E(r ) := . ∂r ∂ Br (x) 1 Br (x) 1 i< j Proposition 5.7. Let (u 1,R , . . . , u k,R ) be a solution of (1.19) having the symmetries (1.20) on B R . There exists a constant C (independent of R) such that for all 1 ≤ r1 ≤ r2 ≤ R there holds  2) r 2d E(r ≥ C 22d .  1) E(r r

(5.14)

1

Proof. Let us compute,    d  ) = − 2d + log r −2d E(r dr r

k 

∂ Br (x)

|∇u i |2 + 2

∂ Br (x)

k  1

2 k   ∂u i ∂r

1

i u i ∂u ∂r

 + 

2 k   2π  ∂u i 2d =− + r

0

1

u i2 u 2j

 i< j

1



 2d =− + r

∂ Br (x)

∂r

1 r2

1 r2

 2π 0

∂θ

1

∂ Br (x)

 +

2 k   ∂u i k  1

k  1

 i< j

u i2 u 2j

i u i ∂u ∂r

2 k   ∂u i 1

 + 2r 2

∂θ

i u i ∂u ∂r

 + 2r 2

 i< j

u i2 u 2j .

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Now we use Theorem 5.6 and we continue the chain of inequalities: 2 k  k  2π   2π  2 ∂u i + L(d,2rr 2 H (r )) 0 u i2 0 ∂r   2d d 1 1  ) ≥− log r −2d E(r + k  2π  dr r i u i ∂u 0 ∂r 1

 2d − 2 L(d, 2r 2 H (r )) C C ≥− ≥− ≥ − 3/2 , √ 4 3/2 r r r H (r )

(5.15)

where in the last line we have used the H¨older inequality and the lower bound for H given Proposition 5.4. By integration we easily obtain the assertion.  A direct consequence of the above inequalities is the non vanishing of the quotient E/r 2d . Corollary 5.8. Let R > 1 and let (u 1 , . . . , u k ) be a solution of (1.19) on C satisfying (1.20); then there exists lim

r →+∞

 ) E(r = b ∈ (0, +∞]. r 2d

(5.16)

If, in addition, limr →+∞ N (r ) ≤ d, then we have that b < +∞ and lim N (r ) = d,

r →+∞

and

lim

r →+∞

E(r ) = b. r 2d

(5.17)

Proof. Note that (5.16) is a straightforward consequence of the monotonicity formula (5.15). To prove (5.17), we first notice that lim

r →+∞

E(r ) H (r ) = lim N (r ) 2d . r →+∞ r 2d r

So the limit of E(r )/r 2d exists finite. Now we use (5.2) 

+∞

2

0

 Bs



k  i< j

u i2 u 2j

k 

∂ Bs

1

ds < +∞ u i2

and we infer that r

 Br

lim inf r →+∞



k  i< j

∂ Br

u i2 u 2j

k  1

= 0. u i2

Next, using Corollary 5.5 we can compute  Br

lim inf r →+∞

k 

u i2 u 2j

i< j r 2d

 Br

= lim inf r →+∞

k  i< j

u i2 u 2j

H (r )

H (r ) = 0, r 2d

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123

and finally  ) − E(r ) E(r = 0. r →+∞ r 2d Was the limit of N (r ) strictly less that d, the growth of H (r ) would be in contradiction with that of E(r ).  lim inf

Now we can combine the upper and lower estimates to obtain convergence of the approximating solutions on compact sets and complete the proof of Theorem 1.6. Proof of Theorems 1.3 and 1.6. Let (u 1,R , . . . , u k,R ) be a family of solutions to (1.19) such that N R (R) ≤ d and H R (R) = C R 2d . Since H R (R) = C R 2d , then, by Proposition 5.3 we deduce that, for every fixed 1 < r < R, there holds H R (r ) ≥ Ce−d r 2d . All the constants below are independent of R. Assume first that there holds a uniform bound for some r > 1, H R (r ) ≤ C.

(5.18)

Then H R (r ) and E R (r ) are uniformly bounded on R. This implies a uniform bound on the H 1 (Br ) norm, for any positive r . From the differential system, using standard elliptic estimates we easily pass from local H 1 to C 2 bounds on Br/2 , which are independent on R. Note that, by Proposition 5.4, H R (r ) is bounded away from zero, so the limit cannot be zero. By the doubling Proposition 5.3 the uniform bound on H R (r2 ) ≤ Cr22d holds for every r2 ∈ R larger than r . Thus, a diagonal procedure yields the existence of a nontrivial limit solution of the differential system, defined on the whole of C. It is worthwhile noticing that this solution inherits all the symmetries of the approximating solutions together with the upper bound on the Almgren quotient. Finally, from Corollaries 5.5 and 5.8 we infer the limit H (r ) 1 b E(r ) = lim = ∈ (0, +∞). , lim (5.19) 2d 2d r →+∞ r →+∞ N (r ) d r r Let us now show that H R (r ) is uniformly bounded with respect to R for fixed r . We argue by contradiction and assume that, for a sequence Rn → +∞, there holds lim

r →+∞

lim H Rn (r ) = +∞.

(5.20)

n→+∞

Denote u i,n = u i,Rn and Hn , E n , Nn the corresponding functions. Note that, owing to Proposition 5.7, E n is bounded; therefore we must have Nn (r ) → 0. For each n, let λn ∈ (0, r ) such that λ2n Hn (λn ) = 1 (such λn exist right because of (5.20)) and scale u˜ i,n (z) = λn u i,n (λn z),

|z| < Rn /λn .

Note that the (u˜ i,n )i still solve system (1.19) on the disk B(0, Rn /λn ) and enjoy all the symmetries (1.20). Let us denote H˜ n , E˜ n , N˜ n the corresponding quantities. We have H˜ n (1) = λ2n Hn (λn ) = 1, E˜ n (1) = λ2n E n (λn ) → 0 N˜ n (1) = Nn (λn ) → 0.

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In addition there holds N˜ n (s) ≤ d for s < Rn /λn . By the compactness argument exposed above, we can extract a subsequence converging in the compact-open topology of C 2 to a nontrivial symmetric solution of (1.19) with the Almgren quotient vanishing constantly. Thus, such solution should be a nonzero constant in each component, but constant solutions are not compatible with the system of PDE’s (1.19).  6. Asymptotics at infinity We now come to the proof of Theorem 1.4. Note that by Proposition 5.3, the condition on N (r ) implies that u and v have a polynomial growth. (In fact, with more effort we can show the reverse also holds. Namely, if u and v have polynomial growth, then N (r ) approaches a positive integer as r → +∞. We leave out the proof.) Recall the blow-down sequence is defined by   1 1 u(Rx), v(Rx) , (u R (x), v R (x)) := L(R) L(R) where L(R) is chosen so that   2 2 u R + vR = ∂ B1 (0)

∂ B1 (0)

Φ2.

(6.1)

For the solutions in Theorem 1.3, by (5.19), we have L(R) ∼ R d .

(6.2)

We will now analyze the limit of (u R , v R ) as R → +∞. Because for any r ∈ (0, +∞), N (r ) ≤ d, (u, v) satisfies Proposition 5.3 for any r ∈ (1, +∞). After rescaling, we see that Proposition 5.3 holds for (u R , v R ) as well. Hence, there exists a constant C > 0, such that for any R and r ∈ (1, +∞),  u 2R + v 2R ≤ Ced r d . (6.3) ∂ Br (0)

Next, (u R , v R ) satisfies the equation  2 2 2 ∆u R = L(R) R u R v R , ∆v = L(R)2 R 2 v R u 2R ,  R u R , vR > 0 in R2 .

(6.4)

Here we need to observe that, by (6.2), lim L(R)2 R 2 = +∞.

R→+∞

By (6.3), as R → +∞, u R and v R are uniformly bounded on any compact set of R2 . Then by the main result in [17,26,30,31], there is a harmonic function Ψ defined in R2 , such that (a subsequence of) (u R , v R ) → (Ψ + , Ψ − ) in H 1 and in H¨older spaces on any compact set of R2 . By (6.1),   2 Ψ = Φ2, ∂ B1 (0)

∂ B1 (0)

H. Berestycki et al. / Advances in Mathematics 243 (2013) 102–126

125

so Ψ is nonzero. Because L(R) → +∞, u R (0) and v R (0) go to 0; hence Ψ (0) = 0.

(6.5)

After rescaling in Proposition 5.2, we obtain a corresponding monotonicity formula for (u R , v R ),  r Br (0) |∇u R |2 + |∇v R |2 + L(R)2 R 2 u 2R v 2R  = N (Rr ) N (r ; u R , v R ) := 2 2 ∂ Br (0) u R + v R is nondecreasing in r . By (4) in Theorem 1.3 and from Corollary 5.8, N (r ; u R , v R ) ≤ d = lim N (r ; u R , v R ), r →+∞

∀r ∈ (0, +∞).

(6.6)

1 and for any r < +∞, In [17], it is also proved that (u R , v R ) → (Ψ + , Ψ − ) in Hloc  lim L(R)2 R 2 u 2R v 2R = 0. R→+∞ Br (0)

After letting R → +∞ in (6.6), we get  r Br (0) |∇Ψ |2 N (r ; Ψ ) :=  = lim N (r ; u R , v R ) = lim N (Rr ) = d. 2 R→+∞ R→+∞ ∂ Br (0) Ψ

(6.7)

In particular, N (r ; Ψ ) is a constant for all r ∈ (0, +∞). This implies that Ψ is a homogeneous function of degree d: this can be easily deduced from the proof of the monotonicity formula, Proposition 5.2. Hence, Φ is a homogeneous harmonic polynomial. Actually the number d is the vanishing order of Ψ at 0, which must therefore be a positive integer. Now it remains to prove that Ψ ≡ Φ: this is easily done by exploiting the symmetry conditions on Ψ (point (3) of Theorem 1.3). Acknowledgments Part of this work was carried out while Henri Berestycki was visiting the Department of Mathematics at the University of Chicago. He was supported by an NSF FRG grant DMS1065979 and by the French “Agence Nationale de la Recherche” within the project PREFERED (ANR 08-BLAN-0313). Juncheng Wei was supported by a GRF grant from RGC of Hong Kong. Susanna Terracini was partially supported by the Italian PRIN2009 grant “Critical Point Theory and Perturbative Methods for Nonlinear Differential Equations”. Kelei Wang was supported by the Australian Research Council. References [1] F. Alessio, A. Calamai, P. Montecchiari, Saddle-type solutions for a class of semilinear elliptic equations, Adv. Differential Equations 12 (2007) 361–380. [2] L. Ambrosio, X. Cabr´e, Entire solutions of semilinear elliptic equations in R3 and a conjecture of De Giorgi, J. Amer. Math. Soc. 13 (4) (2000) 725–739. [3] P. Ao, S.T. Chui, Binary Bose–Einstein condensate mixtures in weakly and strongly segregated phases, Phys. Rev. A 58 (1998) 4836–4840. [4] A. Arapostathis, M. Ghosh, S. Marcus, Harnack’s inequality for cooperative weakly coupled elliptic systems, Comm. Partial Differential Equations 24 (9–10) (1999) 1555–1571.

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