SYMMETRY AND ASYMPTOTIC BEHAVIOR OF GROUND STATE SOLUTIONS FOR SCHR¨ODINGER SYSTEMS WITH LINEAR INTERACTION

. We study symmetry and asymptotic behavior of ground state solutions for the doubly coupled nonlinear Schr¨odinger elliptic system ) , where N ≤ 3 , Ω ⊆ R N is a smooth domain. First we establish the symmetry of ground state solutions, that is, when Ω is radially symmetric, we show that ground state solution is foliated Schwarz symmetric with respect to the same point. Moreover, ground state solutions must be radially symmetric under the condition that Ω is a ball or the whole space R N . Next we investigate the asymptotic behavior of positive ground state solution as κ → 0 − , which shows that the limiting proﬁle is exactly a minimizer for c 0 (the minimized energy constrained on Nehari manifold corresponds to the limit systems). Finally for a system with three equations, we prove the existence of ground state solutions whose all components

where N ≤ 3, Ω ⊆ R N is a smooth domain. First we establish the symmetry of ground state solutions, that is, when Ω is radially symmetric, we show that ground state solution is foliated Schwarz symmetric with respect to the same point. Moreover, ground state solutions must be radially symmetric under the condition that Ω is a ball or the whole space R N . Next we investigate the asymptotic behavior of positive ground state solution as κ → 0 − , which shows that the limiting profile is exactly a minimizer for c 0 (the minimized energy constrained on Nehari manifold corresponds to the limit systems). Finally for a system with three equations, we prove the existence of ground state solutions whose all components are nonzero.
1. Introduction. In this paper, we are mainly concerned with the doubly coupled nonlinear Schrödinger system (see [10]) where N ≤ 3, Ω ⊆ R N is a smooth domain, µ 1 , µ 2 are positive constants, κ, β are linear and nonlinear coupling constants respectively. System (1) models naturally many physical problems, especially in nonlinear optics. Physically, the solutions Φ and Ψ denote the first and second component of the beam in Kerr-like photorefractive media (cf. [1]). The positive constant µ j is for self-focusing in the j-th component of the beam, j = 1, 2. The nonlinear coupling constant β is the interaction between the two components of the beam. As β > 0, the interaction is attractive, but the interaction is repulsive if β < 0. The linear coupling is generated either by a twist applied to the fiber in the case of circular polarization, or by an elliptic deformation of the fibers core in the case of circular polarizations. Problem (1) also arises in the Hartree-Fock theory for Bose-Einstein condensates, in which all the above parameters have the specific physical meaning, for more details we refer the reader to [10,13,19,21,25].
To obtain solitary wave solutions, we set Φ = e iλ1t u(x) and Ψ = e iλ2t v(x), system (1) is reduced to the following elliptic system for u, v with Dirichlet boundary conditions: in Ω, u = v = 0 on ∂Ω (or u, v ∈ H 1 (R N ) as Ω = R N ). (2) Here the coefficients µ 1 , µ 2 , β and κ are each twice the corresponding coefficients in (1). Besides, we assume that λ 1 , λ 2 > 0, β ∈ R. The nonlinear elliptic system (2) has attracted considerable attention in the last ten years. When κ = 0, there are many interesting works devoting to study the quantitative and qualitative properties of solutions to the system (2); see [2,3,4,5,16,17,18,22] for the existence of ground state or bound state solutions, [23,27,28] for the symmetry of least energy solutions or bound state solutions, and [20] about the limits of bound state solutions as β → −∞. However, when κ = 0, β = 0, that is, linear coupling terms and nonlinear coupling terms both exist, only a few interesting results have been obtained in [6,11,15,24]. To be more specific, the existence of bound state and ground state solutions have been investigated by the topological and variational methods in [6,15], while the authors in [11,24] study the bifurcation of synchronized solutions with parameter κ and β respectively. With regard to the other properties (such as symmetry, asymptotic limit and so on) of solutions to system (2), there are no results. This may led us to characterize the symmetry and asymptotic behavior of ground state solutions for (2). In addition, as it has been seen in the case that the linear coupling term doesn't exist, there is a sharking contract between two components and more than two components for system (2). Taking the fact into account, we will also study the existence of ground state solutions to a doubly coupled system with three components (for simplicity), which can be seen as an extension result compared with that of [15].
We begin with some notations. In the following we always assume that |κ| < √ λ 1 λ 2 . Set Define the inner product on H as follows: is the corresponding norm, which is equivalent to the standard product norm on the product space H.

SYMMETRY AND ASYMPTOTIC BEHAVIOR OF GROUND STATE SOLUTIONS 789
We observe that the solutions of (2) correspond to critical points of the energy functional where Since we suppose that N ≤ 3, the Sobolev embedding implies that I κ (u, v) is welldefined and of class C 2 . Introduce the Nehari manifold by then we consider the following minimizing problem thus an equivalent characterization of c κ is as follows: First, as to the existence of ground state solutions to system (2), let us recall from [15] the following theorem.
, and that Ω is a smooth bounded domain in R N or Ω = R N , N ≤ 3. Then system (2) has a ground state solution (u, v). Moreover,
(2) According to the proof of theorem 1.2(cf. [15]), any minimizer (u κ , v κ ) for c κ is a solution of system (2). Note that (2) has no semi-trivial solution which assumes the form of (u, 0) with u = 0 or (0, v) with v = 0, thus any minimizer for c κ is a ground state solution. However, there may exist semi-trivial solutions for a system with more than two equations, which shows that to find its ground state solutions(cf. Definition 1.7) may be more involved.
We are in position to give our main results.
1.1. Symmetry results. When β ≥ 0, κ = 0 and the underlying domain Ω is radially symmetric, the authors in [27] proved the partial symmetry of the least energy solutions to system (2). For κ = 0, we can obtain a similar result. Before the following theorem, let us recall that a positive function u defined on a radially symmetric domain Ω is said to be foliated Schwarz symmetric with respect to e ∈ S N −1 if u depends only on (r, θ) = (|x|, arccos(x·e)/|x|) and is non-increasing in θ. Thus for a negative function u, we can call u foliated Schwarz symmetric with respect to e ∈ S N −1 if u(x) depends only on (r, θ) = (|x|, arccos(x · e)/|x|) and is non-decreasing in θ. Let B r (0) := {x ∈ R N , |x| < r}, r > 0. Now we state our theorems.
and that Ω is a bounded radial domain in R N (i.e., Ω = B r (0), r > 0 or Ω = B r1 (0) \ B r2 (0), r 1 > r 2 > 0) or the whole space R N , let (u, v) be a ground state solution to system (2). Then there exists e ∈ S N −1 such that u and v are foliated Schwarz symmetric with respect to e.
Moreover, when Ω is a ball or the whole space R N , we can give a further result on the symmetry of ground state solutions.
, and that Ω is a ball B r (0), r > 0 or the whole space R N . Then any ground state solution of system (2) must be radially symmetric.
Fixing µ 1 , µ 2 > 0 we may assume without loss of generality that µ 1 ≤ µ 2 . Consider the limiting system of (6) Let By direct computations, we have Since µ 1 ≤ µ 2 , then I 0 (w, 0) ≥ I 0 (0, w), whence Besides, if µ 1 < µ 2 , system (7) has a synchronized solution of the form But if µ 1 = µ 2 = µ, system (7) has a synchronized solution of the form consist of a family of solutions to system (7). With regard to the asymptotic behavior of positive ground state solutions to (6), we obtain the following two theorems.
be a positive ground state solution to system (6), then for any sequence κ n → 0 − , (i) for β < 0, there exist a subsequence(still denoted by κ n ) and {x n } ⊆ R N such that , there exists a subsequence(still denoted by κ n ) such that (iii) for β ∈ (µ 2 , +∞), there exists a subsequence(still denoted by κ n ) such that be a positive ground state solution to system (6), then for any sequence κ n → 0 − , (i) for β < 0, there exist a subsequence(still denoted by κ n ) and {x n } ⊆ R N such that (iii) for β = µ, there exists a subsequence(still denoted by κ n ) such that (iv) for β ∈ (µ, +∞), there exists a subsequence(still denoted by κ n ) such that
(2) When Ω is a ball in R N , N = 2 or N = 3, we have similar results which may be easier to prove. For the sake of simplicity, we omit it here.
1.3. Ground states to the system with three equations. In this subsection, we briefly discuss the systems with more than two equations. To simplify notation, we focus on a system of three equations like We assume that K is positive definite, then we can define an inner product on H as follows. For where represents for vector transpose. Thus, is the corresponding norm.
Remark 4. Since K is positive definite, denoting the smallest and biggest eigenvalues by σ min and σ max respectively, we have σ max ≥ σ min > 0. This implies that (u 1 , u 2 , u 3 ) K is equivalent to the standard product norm on the product space H.
Then the corresponding energy functional is and the Nehari manifold is We consider the following minimizing problem Definition 1.7. We call a solution u = (u 1 , u 2 , u 3 ) ∈ H a ground state solution of (11) if u achieves inf{I K ( u)|I K ( u) = 0, u i = 0, i = 1, 2, 3}.
The following theorem gives the existence of ground state solutions to system (11).
Then there is a ground state solution u = (u 1 , u 2 , u 3 ) to system (11).
Moreover, when K is positive definite and κ 12 < 0, κ 13 < 0, κ 23 < 0, we have This paper is organized as follows. In Section 2, we devote to state some preliminaries. The symmetry results Theorems 1.3-1.4 will be proved in Section 3. In Section 4, we give a detailed analysis on the asymptotic behavior of the positive ground state solutions to (2). The proof of existence of ground state solution to system with three equations will be done in the final section.

2.
Preliminaries. In this section, some preliminaries are given by the following lemmata.
We define the sets For each H ∈ H 0 we denote by σ H : R N → R N the reflection in R N with respect to the hyperplane ∂H, and define the polarization of a function u : Ω → R with respect to H by  [7]). Let f, g be nonnegative and in L 2 (R N ). Suppose that f * and g * are the Schwarz symmetrized function of f and g respectively, then Lemma 2.4 (cf. [31]). Let r > 0 and 2 ≤ q < 2 3. Proof of Theorems 1.3-1.4. This section is devoted to the symmetry results of ground state solutions to system (2). Firstly, we give a proof concerned with a partial symmetry result.

ZHITAO ZHANG AND HAIJUN LUO
Proof of Theorem 1.3. First, it can be easily seen that system (2) is invariant under the following transformation: Using the σ-invariance of system (2), we can see that (u, v) is a ground state solution of the system (2) with coupling coefficients κ and β if and only if (u, −v) is a ground state solution of the system (2) with coupling coefficients −κ and β. Thus, we only suffice to deal with the case β ≥ 0, κ ∈ (− √ λ 1 λ 2 , 0). By Theorem 1.2, we know that u > 0, v > 0 or u < 0, v < 0. Up to a sign, we may assume that u > 0, v > 0.
Next, let us divide the proof into three steps.
Step 1: We claim that (u H , v H ) (defined as (13)) is also a minimizer for c κ .
Since u > 0, v > 0, we have By Proposition 31.7 of [32], we have Furthermore, in virtue of Lemma 3.1 in [30], we have Besides, observe that κ < 0, together with (14) and (15), we can infer that t H ∈ (0, 1]. Therefore, we have where the last equality follows from Remark 1 and the fact that (u, v) is a minimizer for c κ . This means that all the previous inequalities are indeed equalities, and in particular, which implies that t H = 1 and I κ (u H , v H ) = c κ . Thus, (u H , v H ) ∈ N κ is also a minimizer for c κ .
Step 2: Let us show that u and v are foliated Schwarz symmetric with respect to e 1 and e 2 respectively. By Step 1, (u H , v H ) is a minimizer for c κ and therefore a solution of system (2)(cf.(2) of Remark 2). Hence (u H , v H ) satisfies the following system   x ∈ H ∩ Ω, Since u H − u ≥ 0 in H ∩ Ω, by the maximum principle, we have For H ∈ H 0 (e 1 ), we have u H (x) = u(x) for all x ∈ H ∩ Ω because u H (y 1 ) = u(y 1 ). By Lemma 2.2, u is foliated Schwarz symmetric with respect to e 1 .
According to the analogus argument as above, we can also show that v is foliated Schwarz symmetric with respect to e 2 .
If y 2 = y 1 , i.e. e 2 = e 1 , set e = e 2 = e 1 , then u and v are foliated Schwarz symmetric with respect to the same point e. If y 2 = y 1 , by Step 2, we can imply that v is foliated Schwarz symmetric with respect to both e 1 and e 2 . Therefore, let e = e 1 , then the desired result also follows.
If the underlying domain Ω is a ball or the whole space R N , we can improve the above result, as is said in Theorem 1.4. Next, we commerce with the proof of the theorem.
Proof of Theorem 1.4. We divide the proof into two cases according to the range of the parameter κ.
Case 1: κ ∈ (− √ λ 1 λ 2 , 0). For this case, we know that up to a sign the ground state solutions are positive. In addition, system (2) can also be written into the following form Since β ≥ 0, κ < 0, we can infer that this is a cooperative system. The positive solutions must be radially symmetric and strictly decreasing following from [26] as Ω is a ball B r (0), r > 0, while from [9] as Ω = R N .

ZHITAO ZHANG AND HAIJUN LUO
Moreover, we have Note that v * = −(−v) * and κ > 0, together with (17), we get Thus, if (u, v) with u > 0, v < 0 is a ground state solution, then Introduce the function ϕ(t) := I κ (tu * , tv * ), then by a direct computation, we get that is a maximal point of ϕ, which implies that (t 0 u * , t 0 v * ) ∈ N κ . Furthermore, we have which means that all the previous inequalities are indeed equalities. Therefore, we have I κ (u * , v * ) = c κ and Since (u * , v * ) is also a minimizer for c κ , then we have By the elliptic regularity theory, we get that u * , v * ∈ C 2 (Ω). By the strong maximum principle, we have u * > 0, −v * = (−v) * > 0 in Ω. We claim that where | · | denotes the N -dimension Lebesgue measure. In fact, by the very definition of u * , we know that u * is a non increasing radially symmetric function in Ω, thus (u * ) (s) ≤ 0 in (0, r) as Ω = B r (0) or in (0, ∞) as Ω = R N . Note that u * is C 2 , we can suppose, in views of contradiction, that there exist 0 < a < b such that This implies that u * (x) ≡ C > 0 for some constant C in the annulus a < |x| < b. Nextly we show that the situation doesn't occur and hence obtain a contradiction. Note that u * (x) = 0, |x| = r as Ω = B r (0) or lim |x|→∞ u * (x) = 0 as Ω = R N , we can infer that b < r or b < +∞.
If Ω = B r (0), the function C − u * satisfies the following equations in the annulus b < |x| < r : For a < |x| < b, u * (x) ≡ C, together with the system (19), we have In particular, take x = x 0 with |x 0 | = (a + b)/2, we obtain Since u * , −v * are positive radially non-increasing functions in B r (0), we can imply that for every b < |x| < r. Since C − u * ≡ 0, by the Hopf's lemma (see [12,Lemma 3.4]), we obtain that ∂u * ∂ n (x) = − ∂(C−u * ) ∂ n (x) > 0 for |x| = b, where n denotes the unit outward normal to ∂{b < |x| < r}. However, observe that u * is C 2 (B r (0)) and identically equals to the positive constant C in a < |x| < b, this leads to a contradiction.
If Ω = R N , choose a constant R > b such that u * (x) = C/2 on |x| = R. By a similar argument in b < |x| < R, we can also obtain a contradiction.
In this way, we prove that Analogous argument holds for v * . According to the above claim, together with (18) and by Theorem 1.1 of Brothers and Ziemer [8], we obtain u * = u, v * = v a.e. in Ω.
Remark 5. For the second case in the proof of Theorem 1.4, in virtue of the σinvariance of system (2) in the proof of Theorem 1.3, we can infer that the ground state solutions to system (2) are radially symmetric following from the first case. However, the method to prove the second case has its own interest. 4. Asymptotic behavior as κ → 0 − . We study the following elliptic systems: where N = 2, 3, κ ∈ (−1, 0), µ 1 , µ 2 > 0, β ∈ R. Fixing µ 1 , µ 2 > 0, we may assume without loss of generality that µ 1 ≤ µ 2 .
is the unique positive solutions to system (20), where (u β κ , v β κ ) is defined as follows: where w is given by (8).
Proof. Since the proof of the above proposition can be carried out line by line following from [29,Theorem 4.2], for the sake of simplicity, we omit it here.
With the above preparations, we now commence to study the asymptotic behavior of positive ground state solution (u κ , v κ ) to system (20).
Proof of Theorem 1.5 and 1.6. For clarity, we divide our proof into three steps.