POSITIVE SOLUTIONS OF DOUBLY COUPLED MULTICOMPONENT NONLINEAR SCHR¨ODINGER SYSTEMS

. In this paper, we study the following doubly coupled multicomponent system where Ω ⊂ R N and N = 2 , 3; λ j ,γ jk = γ kj ,µ j ,β jk = β kj are constants, j,k = 1 , 2 ,...,n , n ≥ 2. We prove some existence and nonexistence results for positive solutions of this system. If the system is fully symmetric, i.e. λ j ≡ λ,γ jk ≡ γ,µ j ≡ µ,β jk ≡ β , we study the multiplicity and bifurcation phenomena of positive solution.


1.
Introduction. Consider the following doubly coupled multicomponent nonlinear Schrödinger system where Ω ⊂ R N is a smooth domain, N = 2, 3; λ j > 0, γ jk = γ kj , µ j > 0, β jk = β kj are constants, j, k = 1, 2, ..., n and n ≥ 2. System (1) describes the standing wave solutions of doubly coupled Schrödinger equations, where linear and nonlinear coupling terms are both in presence. Due to the important applications in physics, particularly in the study of nonlinear optics and Bose-Einstein condensates, Schrödinger systems have been studied extensively in recent years. We refer to [12,21,22] for more background in physics, and [1]- [10], [14]- [19], [23]- [27], [29] and references therein for mathematical studies. Most of these papers dealt with nonlinear coupling terms only. But the linearly couplings also carry important physical information, see [11] for more interpretation in this respect. Roughly speaking, the linearly coupling coefficients affect the eigenvalues of the linear operator corresponding to the linearized system of (1), and bring a different kind of influence on the existence, multiplicity and bifurcations of positive solutions to system (1).
We define some notations first, then present our main results.  (1), By Sobolev embeddings, J is a C 2 functional, whose critical points are weak solutions of (1). Two Nehari type manifolds associated to J are considered in this paper: the one-constraint Nehari manifold where u j has only one nonzero component located at the j-th position. Clearly, N 1 contains all nontrivial solutions of (1). We call a nontrivial solution a ground state solution, if it achieves the minimum value of J on N 1 . Let λ * = min{λ j }, λ * = max{λ j } and γ * = max{|γ jk |}. If 0 ≤ γ * < λ * n−1 , we can define a norm on H u H :=   n j=1 |∇u j | 2 2 + λ j |u j | 2 2 + 2 1≤j<k≤n γ jk Direct calculation shows On the other hand, thus · H is equivalent to the standard product norm on H. Since I (u), u = 2 u 2 H − 4 u 2 H = −2 u 2 H < 0, for any u ∈ N 1 , N 1 is smooth and is of co-dimension one.
Remark 2. System (1) possesses a coefficient-component symmetry, which means if u = (u 1 , u 2 , ..., u n ) is solution corresponding to a set of linear coupling coefficients {γ jk } 1≤j<k≤n , thenũ ∈ H, which only differs from u by changing the sign of the j-th component (1 ≤ j ≤ n), is also a solution of (1) with all related coefficients γ jk replaced by −γ jk , k = j. For instance, in the case n = 2, if (γ, u, v) is a solution, then (−γ, −u, v) and (−γ, u, −v) are also solutions. Therefore, Theorem 1.1 has 2 n−1 equivalent versions.
The next theorem concerns the existence of positive solutions of (1). Similar problem has been obtained in [14,27] for n = 2. Here we shall extend such results to multicomponent cases. If Ω = R N and the following symmetric conditions hold the existence of positive solutions can be verified for a larger range of coupling parameters. Indeed, there exists a synchronized solution branch whose components are positive constant multiples of the non-degenerate positive radial solution of Precisely, we have for all 1 ≤ j = k ≤ n, then system (1) has a positive ground state solution for any β jk = β kj ∈ R.
Let u 1 , u 2 ∈ H. If there exists an integer j such that u 2 = σ j u 1 , then we say u 1 and u 2 belong to the same Z n -orbit.  (1). Some related work can be found in [4] and [6], where the nonlinearly coupled case, i.e. γ = 0, were studied. In both papers, sequences of β j 's were found and proved to be bifurcation parameters, where the attractive self-interaction was studied in [4], and the repulsive and mixed self-interactions were considered in [6]. More recently, when both linear and nonlinear couplings are in presence, [27,9] studied the bifurcation phenomena of (1) for n = 2.
To discuss γ-bifurcations, a dilation transformation will be used, as it is suggested by the expression of T ω . Therefore we only consider this type of bifurcation on Ω = R N . The γ-bifurcation on bounded domains is also an interesting problem, but new technique seems to be required to overcome the difficulty of losing scaling invariance. In contrast, β-bifurcation have been studied on both bounded and unbounded (radial) domains. We refer to [4,5,25,26] for more details.
(ii) For each integer k ≥ 2 there exists a connected set S k ⊂ R × H of positive solution of (1), such that Secondly, about γ-bifurcations, we have At last, we discuss the two-parameter bifurcations with respect to T ω . A few notations are needed.
Then there exist a sequence of bifurcation surfaces C k ⊂ O and C k ∩ Γ γ0 = ∅, whose dimension at each point is at least 2. Moreover, each C k satisfies one of the following properties: We call u a partially synchronized solution subjected to P, if u j (x)/u k (x) is a constant as long as j, k ∈ P k for some 1 ≤ k ≤ m. One can also obtain bifurcation results for partially synchronized solutions using the arguments in [4].
The paper is organized as follows. In Section 2, we establish the existence and nonexistence results using variational methods. In Section 3, we use Z n -index theory to find multiple positive solutions and give proof of Theorem 1.3. In Section 4, we find and verify the γ-bifurcations of positive solutions to (1) in linear coupling parameter γ. Some calculations are contained in the Appendix.
2. Nonexistence and existence of nonnegative solution. In this section, we apply a Liouville type theorem [10, Theorem 2.3] to prove Theorem 1.1. Then we use variational methods and some elementary calculations to establish Theorem 1.2. Similar problem has been studied in [27,14] for n = 2.
Proof of Theorem 1.1. Let u = (u 1 , ..., u n ) be a solution of (1) whose components have the same sign. Without loss of generality, assume u is nonnegative. Moreover, by standard regularity argument, u ∈ [C 2 (R N )] n .
In order to prove Theorem 1.2, we need a few estimates on functional J. For any u ∈ N 1 , H therefore there exists a constant C 0 > 0, which only depends on λ j , γ jk , µ j , β jk such that u ≥ C 0 , for all u ∈ N 1 . On the other hand, The following lemma indicates an important influence exerted by the linear coupling terms, which helps us to rule out nonnegative semi-trivial solutions.
Then a subsequence of {u m } converges to a positive solution u of (1).
Proof. Clearly, c ≥ C 0 . Since J is coercive on N 1 , {u m } is bounded in H. Therefore, there exists u ∈ H and a subsequence of {u m } (still denoted by {u m } for simplicity) such that u m u in H, as m → ∞. If Ω ⊂ R N is bounded, by compact Sobolev embedding u is a nonnegative solution of (1).
If Ω is unbounded, by applying the Concentration Compactness Principle as in [14], one can find a new PS sequence with positive components on N 1 which strongly converges to u. Precisely, we claim: there exists at least one component (u m ) j such that which contradicts with inf N1 J > 0. Assume the claim holds for the first component, then up to a subsequence, there exist {x m } ⊂ Ω such that Let v m (·) = u n (· − x m ). It is easy to check that {v m } is also a positive PS sequence of J. Let u be the weak limit of If u 0 has zero component, for example u n = 0, then the last equation of (1) gives γ 1n u 1 + γ 2n u 2 + ... + γ n−1,n u n−1 = 0 therefore u j = 0, j = 1, ..., n. But, on the other hand, the lower semi-continuity of norm implies u 4 H ≥ 4C 0 > 0 A contradiction. Thus u is a positive solution of (1). Then u ∈ N and has at least one nontrivial component. By the same argument as above, we deduce that u is a positive solution of (1). By Ekeland's variational principle, {u m } is a PS sequence of J| N1 . Since γ < 0, there holds For any fixed u ∈ N 1 , let t 0 > 0 be a constant such that t 0 |u| ∈ N 1 . Then Therefore we may assume that all components of u m are positive, i.e. J| N1 has a positive PS sequence. By Lemma 2.2, {u m } is also a positive sequence of J. At last, apply Lemma 2.1, {u n } converges to a positive solution u of system (1).
(ii) Assume Ω = R N . Let w be the unique least energy positive solution of (4). In order to obtain positive solutions, we set u j (x) = A j w(tx), where t > 0, A 1 > 0, ..., A n > 0 are unknown parameters to be determined, then (1) becomes for j = 1, 2, ..., n. According to the uniqueness of w that solves scalar equation (4), a system of equations about A j , t, β and γ can be derived. From the coefficient of w, we get From the coefficient of w 3 , we get Set A j ≡ A for j = 1, ..., n, then A and t satisfy for any fixed γ > − λ n−1 and β > − µ n−1 . Solve the above equations for A and t, then we obtain the synchronized solution branch T ω as defined in (5).

Remark 4.
If Ω = R N , then the existence of positive solution claimed by part (i) of Theorem 1.2 is easier to be obtained by applying compact embedding between radial spaces.

Remark 5.
It is easy to see that other synchronized solutions besides T ω can be derived from equations (8) and (9) . Two of the special cases are given below for example: (a) Set γ = 0 and β = µ in (8), (9) respectively. In this case, system (1) has infinitely many solution of the form (A 1 w, A 2 w, ..., A n w), as long as the constant coefficients satisfying n k=1 A 2 k = 1. See also [7] for the case n = 2. (b) For any γ > − λ n−1 and β > − µ n−1 , if n k=1 A k = 0 and A 2 k = A 2 l , ∀1 ≤ j < k ≤ n, then we must have n = 2m for some m ∈ N, and the deduced synchronized solution must have m positive components and m negative components.
In this paper, we are interested in positive solutions, thus other types of synchronized solution branches will not be studied further. But it is worth to point out that the bifurcation results can also be extended to these types of solutions.
3. Multiplicity of positive solutions. In this section, we study the multiplicity of positive solutions to (1) under symmetric assumption (3). The proof of Theorem 1.3 is similar to [24] and we shall be sketchy here. First, in order to get positive solution of (1), we consider the nontrivial solution of the following modified system Lemma 3.1. Let γ ∈ (− λ n−1 , 0) and β < 0. Any nontrivial solution of (11) is a positive solution of (1).
Proof. Multiply the j-th equation of system (11) by u − j and integrate over Ω, then add the n equations together, u is a positive solution of (1).
Let N = p t1 1 p t2 2 · · · p ts s be the prime factorization of N , where 1 < p 1 < · · · < p s are prime numbers and t 1 , · · · , t s are positive integers. Let 1 = q 0 < q 1 < q 2 < · · · < q a < n, for some integer a ≥ 0 be all the distinct factors of n. Correspondingly, define n = n 0 > n 1 > n 2 > · · · > n a > 1 by n b = n/q b for 0 ≤ b ≤ a. In this section, we use the codimension n manifold N n . Define the least energy of J on the sets of fixed points of σ q b , Proof. If β ≤ − µ n b −1 and σ q b (u) = u, i.e. u = (u 1 , · · · , u q b , · · · , u 1 , · · · , u q b ), system (11) reduces to q b equations. Multiply the j-th equation by u j and integrate over Ω, Then add these q b equalities together, we find Thus, u H,λ = 0, i.e. u = 0. On the other hand, 0 / ∈ N n , so σ q b has no fixed point on N n . By definition, c q b (β) = ∞.
If − µ N b −1 < β < 0 and σ q b (u) = u, then similar calculation as above yields Recall the Z n -index defined in [24]. The properties of Z n -index can be found in [24,Lemma 3.2]. Now, we construct a sequence of Liusternik-Schnirelman type levels on N n using this Z n -index.
Denote by K the set of critical points of J. We give an estimate of the index γ near the critical levels.
Comparing with the case n = 2 considered in [10,14], we need more technical arguments to estimate c k . Recall the continuous map ψ : S 2m−1 → N n constructed in [24], ψ(e i 2π N U ) = σψ(U ), for all U ∈ S 2m−1 . Let S 2m−1 be the unit sphere in C m . The constructions given in [24,Proposition 4.1] and [24,Proposition 4.2] both work for the doubly coupled system (11).
Lemma 3.5. Let {c k } be given by (12). Now assume that n is not a prime number. If u is a solution satisfying σ q b (u) = u, then u is also a solution of system Then we see that the reduction does not change the structure of the system in the sense: • the coefficient β only depends on the number of equations, i.e.β depends on q b in the same way as β depends on n. • γ ∈ (− λ n−1 , 0) is equivalent toγ ∈ (−λ q b −1 , 0) Therefore the arguments in the proof of [24, Theorem 1.1] can be applied here with simple changes of the linear coupling coefficients. 4. Linearized system and bifurcation parameters. In this section, symmetric condition (3) is always assumed. Without loss of generality, assume λ = µ = 1.
In [4], the β-bifurcations for γ = 0 and n ≥ 3 were studied. As an important difference with the doubly coupled system, when γ = 0, the existence of T ω does not require µ j ≡ µ. With similar arguments, we can generalize the β-bifurcations to all γ ∈ (− 1 n−1 , 0). Proof of Theorem 1.4 The proof is similar to [5] and we omit the details here.
From now on, we shall focus on γ-bifurcations. We first investigate the linearized system of (1) along T ω .
Proof. By (14), D(γ) and C(β) can be diagonalized with the same matrix P and its inverse P −1 In fact, the linearized system (13) can be written as where A is defined in (10). Then (15) is equivalent to Also note where t is defined in (10), then ξ satisfies −∆ξ +ξ = 3ω 2 ξ. The non-degeneracy of w in radial space H r (R) implies ξ = 0, i.e. n k=1 φ k = 0. On the other hand, the remaining n − 1 equations of (15) are for j = 2, .., n. Clearly, all φ j 's satisfy the same equation Therefore, we conclude that: 1. Every solution φ of system (13), satisfies n k=1 φ k = 0.
2. Every component φ k of φ can be found by solving the scalar equation (16). Denote e i the n-vector whose i-th component equals to 1 and all the other components equals to 0. Assume that ψ is a solution of (16), then φ j = ψ(e j + e n ), j = 1, ..., n − 1, give the base of the solution set to (13).
Therefore, if equation (16) has l nontrivial solutions, for a fixed set of γ, β and n, then the kernel space of (13) has dimension l(n − 1).
In [27,Lemma 3.3], Proposition 2 is proved for n = 2, but the calculation of ∂H ∂γ there contains a gap. We shall provide a modified proof in the Appendix.
Proof of Theorem 1.5. By Lemma 4.1 and Proposition 2, the assumption of [20,Theorem 8.9] is satisfied. Thus the Theorem 1.5 holds.

4.2.
Two dimensional bifurcation branches. In this subsection we discuss the global bifurcations of (1). Define operator F : R × H → H, where By the compact embedding H 1 r (R N ) → L p (R N ), 2 < p < 2 * , the map A : To get positive solutions, we confine the problem in the nonnegative cone P = {(u 1 , u 2 ) ∈ H r u 1 ≥ 0, u 2 ≥ 0}, and let r : H r → P be a retraction. According to the positive definiteness of the operator, it is easy to see that A(β, γ, P) ⊂ P.
Clearly, G is a completely continuous mapping satisfying G(b, 0) = 0 whenever (b, 0) ∈ O. We refer to I 2 × {0} ∩O as the trivial solutions. Moreover, Notice that h is continuously differentiable and has 0 as a regular value. To simplify notations, we identify Γ with Γ × {0} ⊂ O. For any constant It is easy to see that b and b lie in the same component of Γ γ0 , neither b nor b are Γ bifurcation points of G := I − G. It is easy to get that

5.
Appendix. In this section, we first prove a lemma regarding the monotonicity and continuity of λ j (γ), j = 1, 2, ... which are defined in (17). Note that the continuity has been observed in [27], but the original proof contained a gap, which is then fixed in [9]. For completeness, we provide a revised proof below. Then we prove Proposition 2. Let Q(γ) = 1−γ 1+(n−1)γ , then Q is strictly decreasing on (− 1 n−1 , 0). Define Recall the variational characterization of λ j (γ), where E j denotes a j-dimensional subspace of H 1 r (R N ), E ⊥ j denotes the orthogonal space of E j . It is easy to see that Λ(φ, ·) is a continuous and decreasing function of γ for fixed φ.
Proof of Proposition 2. We first calculate the derivative of H, which is defined by (18), with respect to γ, then evaluate at γ j ∂H ∂γ γ=γj where y = 1 + (n − 1)γ j x, ψ j (y) = φ j (x). It is easy to see that −∆ψ j + g(γ j )ψ j = f (β)w 2 ψ j and in radial coordinates i.e. the Hessian of J is strictly increasing at γ j . Hence the Morse index changes at each γ j . By [20,Theorem 8.9], γ j is a γ-bifurcation parameter. The proof is completed.