A COMPLETE CLASSIFICATION OF GROUND-STATES FOR A COUPLED NONLINEAR SCHR¨ODINGER SYSTEM

. In this paper, we establish the existence of nontrivial ground-state solutions for a coupled nonlinear Schr¨odinger system where n = 1 , 2 , 3 ,m ≥ 2 and b ij are positive constants satisfying b ij = b ji . By nontrivial we mean a solution that has all components non-zero. Due to possible systems collapsing it is important to classify ground state solutions. For m = 3, we get a complete picture that describes whether nontrivial ground-state solutions exist or not for all possible cases according to some algebraic conditions of the matrix B = ( b ij ). In particular, there is a nontrivial ground-state solution provided that all coupling constants b ij ,i (cid:54) = j are suﬃciently large as opposed to cases in which any ground-state solution has at least a zero component when b ij ,i (cid:54) = j are all suﬃciently small. Moreover, we prove that any ground-state solution is synchronized when matrix B = ( b ij ) is positive semi-deﬁnite. m ≥ 3. We investigate the classiﬁcation of non-trivial ground-state solutions of (1) for m = 3 in the present paper for all possible cases. The results established here describe a complete picture in which we know the type of ground-state solutions, non-trivial or semi-trivial, for


1.
Introduction. In this note, we are concerned with the existence of nontrivial ground-state solutions of the time-independent Schrödinger equations where n = 1, 2, 3, m ≥ 2, and λ j , b ij all are positive constants for i, j = 1, 2, · · · , m, which is closely related to the following time-dependent system of m-coupled nonlinear Schrödinger(CNLS) equations The relation between the two equations is that (u 1 , u 2 , · · · , u m ) satisfies (1) if and only if v j (x, t) = e iλj t u j (x), j = 1, 2, · · · , m solves (2) which arises physically under conditions when there are m-wave trains moving with nearly the same group velocities [13,17]. The CNLS system also models physical systems whose fields have more than one component; for example, in optical fibers and wave guides, the propagating electric field has two components that are transverse to the direction of propagation. This type of systems also arises from physical models in nonlinear optics and in Bose Einstein condensates for multi-species condensates (i.e., [11,14] and references therein). Readers are referred to the works [5,6,4,11,14,18,19] for the derivation as well as applications of this system. As the terminology about ground states are not quite uniformly used throughout the literature we first fix some definitions in the paper. A solution of (1) is called nontrivial if all components are non-zero. In contrast, semi-trivial solutions are the non-zero solutions with at least one zero component. In this case the system collapses to a system with fewer number of equations. A solution (u 1 , · · · , u m ) of (1) having all components u j > 0 is called a positive solution as opposed to a non-negative solution which satisfies u j ≥ 0 for all j and u j > 0 for at least one j. Solutions u = (u 1 , · · · , u m ) ∈ (H 1 (R n )) m correspond to critical points of the energy functional associated with (1) Since we suppose that n < 4, the Sobolev embedding implies that Ψ is well-defined and of class C 2 . A solution is called ground-state solution if it has the least energy among all non-zero solutions of (1). Note that a ground state solution may be a semi-trivial solution due to collapsing of the system. Lemma 3.1 in [9] implies that there exists a ground state solution which is a non-negative radial solution. We consider the Nehari manifold hereafter, X m = H 1 (R n )×H 1 (R n )×· · ·×H 1 (R n ) is the m-times Cartesian product of H 1 (R n ). Then a ground-state solution of (1) is equivalent to a minimizer of the following variational problem because N is a natural constraint of Ψ. Thus it is important to identify conditions of the matrix B so that the ground states are non-trivial or semi-trivial. For m = 2, Bartsch and Wang have investigated the following linear equations in [3] in order to study existence of the positive ground-state solutions of (1) which was well understood for the case of m = 2 and λ 1 = λ 2 > 0. See also [2,10] for more general cases. Some interesting progress has been made partially in [9] under some conditions on the structure of the matrix B for arbitrary m ≥ 3. We investigate the classification of non-trivial ground-state solutions of (1) for m = 3 in the present paper for all possible cases. The results established here describe a complete picture in which we know the type of ground-state solutions, non-trivial or semi-trivial, for each case under the condition that (3) has at least a solution with each component x i > 0. Such solutions are called positive solutions to (3). Similarly, a solution of (3) is called non-negative if each component is non-negative. Throughout the present paper, we suppose that the symmetric real matrix B satisfies that (5) Let w be the unique positive radial solution of the equation The main results in the present paper are the following theorems. For simplicity and matching notations used in the literatures we write Theorem 1.1. Let m = 3 and det(B) > 0. Suppose that (3) has at least one positive solution (t 1 , t 2 , t 3 ). Then, the following conclusions hold.
then each ground-state solution of (1) is nontrivial. In particular, then each ground-state solution of (1) has exactly one non-zero component.
If det(B) < 0, the theorem below says that every ground-state solution of (1) is semi-trivial.
Let p denote the number of non-zero components in any ground-state solution of (1). For instance, p = 3 if and only if the associated solution is nontrivial. The type of ground-state solutions determined in above theorems can be displayed explicitly in the following table.
Next we turn to more qualitative property of ground state solutions. When the components of a solution to (1) are proportional to one another, it is termed a synchronized solution. Under attractive couplings synchronized solutions have been studied in recent years ( [3,7,9,12,15,16] and references there in). We define another minimization problem. Let On synchronized ground-state solutions, we have the following result.
where v is a ground-state solution of (6) and (t 1 , t 2 , · · · , t m ) is a positive solution to (3).

Remark 1.
The existence of the minimizers of Ψ on N is guaranteed by Lemma 2.2 here when (3) has a positive solution or by the Theorem 2 in [7] for all b ij , i = j small (see [8]) and B is positive definite.

Remark 2.
For m = 2, the result in the above theorem is obtained in [16] in which for small coupling constant the existence and synchronization of a minimizer on N is proved. Our result here shows that, when B is positive semi-definite, the solution of m-coupled system (1) which has the least energy on N is unique provided the ground-state solution of the scaler equation (6) is unique (or unique up to translations) and the solution of linear equations (3) is unique.
The rest of the paper is organized as follows. Section 2 provides several lemmas that are important in the proof of the above main theorems about ground-state solutions of (1) which will be proved in Section 3. Furthermore, the final section is to deal with the synchronized ground-state solutions.

Preliminary.
Lemma 2.1. Let A = (a ij ) be a symmetric real matrix and suppose that s j ≥ 0, j = 1, 2, · · · , N, satisfy N j=1 a ij s j = 1, i = 1, 2, · · · , N. Let {c k,j } ∞ k=1 , j = 1, 2, · · · , N be any sequences of real numbers satisfying the inequality system Noticing that s j ≥ 0 and d k,j ≥ 0, then the desired result follows from N j=1 c k,j ≤ g(k) and lim k→∞ g(k) = 0. Remark 4. The above lemma has been proved in [9] for a special case in which the matrix A is invertible. Therefore the conclusion here is weaker than that of [9] there, c k,j → 0 as k → ∞ for all j = 1, 2, · · · , N . However, it is enough for the following application.
The following lemma says that the synchronized solution is a minimizer for α J , which has been proved in [9] for the special case when J = {1, 2, · · · , m} and B = (b ij ) is invertible. Following the idea in [9], one can prove it easily by Lemma 2.1. Therefore, we state only the lemma without more details.
Remark 5. Suppose that t j > 0 for all j = 1, 2, · · · , m. Then the above lemma guarantees that the existence of positive minimizers is a nontrivial result because in general the existence of minimizers of the variational problem needs some additional conditions. For instance, the results in [7] imply that if B is positive definite then the variational problem admits at least one minimizer and there is no positive minimizer when b ij are negative for all i = j.
The following lemma describes the least energy level of (1) with two equations. Let Λ 2 = R n w 4 dx where w is the solution of (6).
Proof. It follows from Theorem 0.5 in [3] that the above minimum problem admits a minimizer with one zero component for β ≤ max{µ 1 , µ 2 } as well as one positive minimizer for β > max{µ 1 , µ 2 }. We therefore have Then (10) and (11) follow from Lemma 2.2. Moreover, a straightforward calculation reveals that Then, it follows from (10) and (11) that (12). Consequently, the rest of the lemma follows from theorem 0.5 in [3].
Before discussing the classification of ground-state solutions of (1) for m = 3, it is helpful to understand solutions of the linear equations (3) have the same sign, and (2) if there is one number of (13) being zero, then det(B) ≤ 0.
(ii) If rank(B) = 2, then there exists at most one number in (13) which equals zero. In this case, if more β 2 3 = µ 1 µ 2 and suppose that (3) has at least one solution, then and all solutions of (3) can be formulated as follows for all i, j, k, permutation of 1, 2, 3, and suppose that (3) has at least one solution, then rank(B) = 1.

Proof. A straightforward calculation reveals
for any i, j, k, permutation of 1, 2, 3. It implies that part (1), (2) and (3). To prove (i) of part (4), notice that rank(B) = 1, then we therefore assume that there exist three positive constants c 1 , c 2 , c 3 > 0 such that So, it follows from (3) having a solution that c 1 = c 2 = c 3 . Then, we obtain that (14) due to the symmetry of matrix B.
Proof. To prove the corollary, it suffices to verify that all assumptions of part (1) in theorem 1.1 hold here. Indeed, we have in current case det(B) can be rewritten as and the unique solution of (3) It implies that det(B) and det(B) · t i , the product of det(B) and t i , i = 1, 2, 3 are all polynomials with respect to variable β with positive leading coefficient because of |k 1 − k 2 | < k 3 . Therefore, they are all positive as β > 0 sufficiently large. Moreover, it is easy to see that β > max{µ 1 , µ 2 , µ 3 } min{k 1 , k 2 , k 3 } implies that (7) holds. Consequently, the corollary is proved.
Remark 8. It is interesting to consider three positive numbers k 1 , k 2 , k 3 as the lengths of the sides of a triangle due to the inequality |k 1 − k 2 | < k 3 . Therefore, the assumptions of the above corollary are equivalent to that three coupling constants are corresponding to the lengths of the sides of a triangle which is similar to another one with its sides having lengths k 1 , k 2 , k 3 and the ratio β > 0 is sufficiently large.
The following corollary can be proved in a similar way to that of corollary 1, so the proof is omitted.
The proof of theorem 1.2. It is equivalent to proving In order to show (29), we consider the following two cases: one case is that there exists some i, j, k, permutation of 1, 2, 3 such that β i > max{µ j , µ k }, and another case is that β 1 ≤ max{µ 2 , µ 3 }, β 2 ≤ max{µ 1 , µ 3 }, β 3 ≤ max{µ 1 , µ 2 }. For the former, a calculation similar to (24) reveals which implies that Therefore, the theorem in this case follows from Lemma 2.3. On the other hand, for the latter, we have Then, it follows from (3) that We therefore deduce that It is easy to see that if were true then it would cause rank(B) = 1. Consequently, the theorem in this case follows from The proof of theorem 1.3. It follows from lemma 2.4 that for all i, j = 1, 2, 3. It is easy to see that by lemma 2.2. Then, the theorem follows from the equality where t 1 + t 2 + t 3 = 1 µ1 and t j ≥ 0, j = 1, 2, 3.
Remark 10. In this case b ij = b 11 for all i, j = 1, 2, 3, all positive solutions to (1) are synchronized solutions, which has been proved in [15] in the case of m = 2 and space dimension n = 1. For a general case of m ≥ 3, the result can be proved by the same argument in [15].
Next, we discuss ground-state solutions for the case in which rank(B) = 2. Taking Lemma 2.4 into account, we see that all possible cases are contained in the three parts of Theorem 1.4.
The proof of Theorem 1.4. For part (a), the matrix B can be rewritten as follows due to lemma 2.4. Thus, all non-negative solutions of (3) can be formulated by Indeed, we observe that either β 1 > max{µ 2 , µ 3 } or β 1 < min{µ 2 , µ 3 } must hold because β 1 ∈ [min{µ 2 , µ 3 }, max{µ 2 , µ 3 }] would imply rank(B) = 1 by part (5) of Lemma 2.4, which is contradictory to rank(B) = 2. Then, the part (1) of (a) follows from the following equalities where Lemma 2.3 has been used. Similarly, part (2) of (a) follows from the following inequality thanks to β 1 < min{µ 2 , µ 3 }. For part (b), we suppose that We notice that the following three vectors are all solutions to the linear equations (3). Certainly, it is possible that some of their components are negative. By remark 3, the sums of all components for all solutions are equal to each other. That is, We can assume, by part (5) of lemma 2.4, that β 3 > max{µ 1 , µ 2 } without the generality. Thus, all non-negative solutions to (3) can be expressed as follow which is well defined because of (µ 2 β 2 − β 1 β 3 ) + (µ 1 β 1 − β 2 β 3 ) = µ 1 µ 2 − β 2 3 < 0, where the equality follows from the assumption that (3) has at least one solution. Notice that t 1 and t 2 in (31) must be non-negative because both t 1 and t 2 are linear functions with respect to the variable t 3 . Then, part (b) follows from due to β 3 > max{µ 1 , µ 2 } and β 1 > √ µ 2 µ 3 .
Let (t 1 , t 2 , t 3 ) be a non-negative solution to (3). It follows from (32) and lemma 2.3 that there is no ground state solution having exactly one trivial component. Moreover, an argument similar to that of theorem 1.2 reveals t 1 + t 2 + t 3 > 1 max{µ 1 , µ 2 , µ 3 } , which implies that part (c) holds.