Some symmetry results for entire solutions of an elliptic system arising in phase separation

We study the one dimensional symmetry of entire solutions to an elliptic system arising in phase separation for Bose-Einstein condensates with multiple states. We prove that any monotone solution, with arbitrary algebraic growth at infinity, must be one dimensional in the case of two spatial variables. We also prove the one dimensional symmetry for half-monotone solutions, i.e., for solutions having only one monotone component.


Introduction and main results
We study smooth solutions of the elliptic system where the pair (u, v) has at most algebraic growth at infinity and N ≥ 2.
Problems and solutions of this type naturally arise in the study of phase separation phenomena for Bose-Einstein condensates with multiple states (cfr. [1], [2] and the references therein).
In order to motivate our study and to understand the difficulties that one has to face when dealing with system (1.1), we review the known results about the considered problem. The one dimensional case was studied in [1]. The Authors of [1] proved the existence, symmetry, monotonicity and the growth estimates for the solutions to (1.1). In particular they proved that Uniqueness (up to translations, scaling and reflection) of the solution to the one dimensional system (1.1) has been recently settled in [2]. Thus, the one dimensional case is well-understood. On the other hand, the higher dimensional case is more involved and much less is known in that situation.
In [5] it is proved that, in any dimension N ≥ 1, there are no solutions to (1.1) with sublinear growth, i.e., such that On the other hand, when N = 2, solutions with arbitrary integer algebraic growth at infinity has been recently constructed in [2]. In particular, these solutions are not one dimensional when the growth at infinity is superlinear, showing thus the great difference between the one dimensional case (where all the solutions have linear growth) and the higher dimensional case.
Inspired by a celebrated conjecture of E. De Giorgi [3] about monotone solutions to the Allen-Cahn equation (see also [4] for a recent review on the conjecture of De Giorgi and related topics) and motivated by the results in the one dimensional case, Berestycki, Lin, Wei and Zhao [1] raised the following Under what conditions is it true that all monotone solutions to (1.1), i.e., such that They gave a positive answer to the above question if Both the assumptions are crucial in their proof. In particular, in their approach it is not possible to replace the growth condition in (1.3) by the more general condition for some ǫ > 0.
Our first result states that, for N = 2, monotone solutions to (1.1) with at most arbitrary algebraic growth at infinity, must be one dimensional.
Then any monotone solution (u, v) to (1.1) with at most arbitrary algebraic growth at infinity, must be one dimensional.
Our proof uses a different strategy based on the Almgren frequency function We shall describe it in Section 3.
To state our second result we need the following Our second result states that, for N = 2, half-monotone solutions to (1.1) with at most arbitrary algebraic growth at infinity, must be one dimensional.
with at most arbitrary algebraic growth at infinity, must be one dimensional.
Theorem 1.1 is proved in Section 3, while Section 4 will be devoted to Theorem 1.2.

Some auxiliary results
In this section we prove some preliminary results which will be used in the course of the main theorems.
We first recall that the Almgren frequency function defined by (1.4) is nondecreasing in r (cfr. Proposition 5.2 of [2]) and then we prove the following result.
Lemma 2.1. Assume N ≥ 1 and let (u, v) be a solution to (1.1) with algebraic growth at infinity, i.e., satisfying where in the latter we have used the monotonicity of the Almgren frequency function N .
From (2.3) we infer that and thus, by (2.1), where c 1 is a positive constant depending only on the dimension N and on the constant C appearing in (2.1). From (2.5) we immediately get and the desired conclusion (2.2) follows from the monotonicity of the Almgren frequency function N .
Now we prove a Liouville-type theorem which will be useful in Section 4. and for some positive constant C independent of R.
We close the present section by recalling a result proved in [2] (cfr. Theorem 1.4. therein).
and there is a homogeneous harmonic polynomial of degree d, denoted by Ψ, such that the blow-down sequence defined by : where L(R) > 0 : converges (up to a subsequence) to (Ψ + , Ψ − ) uniformly on compact sets of R N . In addition, if N (∞) = 1 then (u, v) has linear growth at infinity.
Here, and in the sequel, we denote by w + the positive part of the function w and by w − the negative part of w.

Monotone solutions
Proof of Theorem 1.1. By Lemma 2.1 we get that N (∞) is finite. This enables us to use Theorem 2.3 with N = 2. The monotonicity assumption implies that ∂uR ∂x2 > 0 and ∂vR ∂x2 < 0 in R 2 , for every R > 0. Therefore, for every x = (x 1 , x 2 ) ∈ R 2 , every t > 0 and every R > 0 we have and an application of Theorem 2.3 immediately yields that Ψ + is nondecreasing with respect to x 2 , while Ψ − is nonincreasing with respect to x 2 . In particular we obtain that ∂Ψ ∂x2 ≥ 0 in R 2 .
To conclude the proof we invoke the subsequent Proposition 3.1, which tells us that Ψ must be a linear function. Hence, d = 1 in (2.11), which means that (u, v) has at most linear growth at infinity, that is (1.3) is satisfied. The desired result then follows from [1], as discussed in the Introduction. Now we turn to Proposition 3.1, which deals with entire monotone harmonic functions. Then where h is a harmonic function on R N −1 and γ ∈ R.
In particular, H must be an affine function when N = 2.
Proof of Proposition 3.1. By assumption we have that ∂H ∂xN is a nonnegative entire harmonic function. So, it must be constant by the classical Liouville Theorem, say ∂H ∂xN ≡ γ ∈ R. In particular, the function h := H − γx N satisfies ∂h ∂xN ≡ 0 and thus, it must be an entire harmonic function depending only on the variables x 1 , ..., x N −1 . This gives (3.2). When N = 2, h must be affine (since in this case h depends only on one variable). This concludes the proof. To conclude the section we prove a proposition which will be crucial in the proof of Theorem 1.2. Proposition 3.3. Assume N = 2 and let P be a homogeneous harmonic polynomial of degree d ≥ 1 such that P + is nondecreasing with respect to x 2 . Then P must be a linear function. Remark 1. In the above proposition the homogeneous harmonic polynomial P cannot be replaced by an arbitrary harmonic function as shown by H(x, y) = e y sin(x) in R 2 .
Proof of Proposition 3.3 . If P vanishes at a point z 0 = (x 0 , y 0 ) ∈ R 2 \ {0}, it must vanishes on the entire straight line passing through z 0 and the origin. To see this, we first observe that P vanishes on the half-line {tz 0 : t ≥ 0} by homogeneity. On the other hand, the restriction of P to the entire straight line passing through z 0 and the origin is a polynomial of one variable, which is identically zero on the half-line {tz 0 : t ≥ 0}. This clearly implies that P must vanish identically on the entire straight line containing {tz 0 : t ≥ 0}.
Since d ≥ 1, the polynomial P must vanish somewhere outside the origin. Suppose that P (z 0 ) = 0 for some z 0 = (x 0 , y 0 ) with x 0 = 0 and denote by S z0 the straight line passing through z 0 and the origin. By the monotonicity assumption on P + we get that P ≤ 0 on the open half-plane lying below S z0 . Indeed, if P (x, y) > 0, then P (x, s) > 0 for every s ≥ y, since P + is nondecreasing on R 2 . Now, since P is harmonic and nonconstant, the strong maximum principle implies that P < 0 everywhere on the open half-plane lying below S z0 . Thus, an application of Hopf's Lemma gives that |∇P | > 0 on the straight line S z0 . In particular |∇P (0)| > 0, which clearly implies d = 1 (by the homogeneity of P ) and P (x, y) = αx + βy, with β > 0.
Next we suppose that P (z 0 ) = 0 for some z 0 = (0, y 0 ) and y 0 = 0. In this case P vanishes on the straight line {(0, y) : y ∈ R} and the above argument tell us that P cannot vanish on R 2 \ {(0, y) : y ∈ R}. Hence, either P > 0 or P < 0 on the open half-plane {(x, y) ∈ R 2 : y > 0}. Applying once again Hopf's Lemma we get d = 1 and then P (x, y) = αx. This concludes the proof.

Half-monotone solutions
Proof of Theorem 1.2. Without loss generality we can suppose that ∂u ∂x2 > 0 in R 2 . If we prove that ∂v ∂x2 < 0 in R 2 we are done, since in this case the desired conclusion will follow from Theorem 1.1. To this end, we first prove that (u, v) has at most linear growth at infinity. By Lemma 2.1 we get that N (∞) is finite and so we can use Theorem 2.3 with N = 2. Since (u, v) is half-monotone we see that ∂uR ∂x2 > 0 in R 2 , for every R > 0. Therefore, for every x = (x 1 , x 2 ) ∈ R 2 , every t > 0 and every R > 0 we have and an application of Theorem 2.3 immediately yields that Ψ + is nondecreasing with respect to x 2 . The latter enables us to invoke Proposition 3.3 from which we infer that Ψ is linear. Hence d = 1 and (u, v) has at most linear growth at infinity. With this information in our hands we are ready to prove that ∂v ∂x2 < 0 in R 2 . The latter claim follows from the next result. This completes the proof of Theorem 1.2.