Infinitely many segregated solutions for coupled nonlinear Schrödinger systems

In this paper, we consider the following coupled nonlinear Schrodinger system \begin{document}$ \left\{\begin{array}{ll} -\Delta u+(1+\delta a(x))u = \mu_1 u^3+\beta uv^2 &\mbox{in }\mathbb{R}^3,\\ -\Delta v+(1+\delta b(x))v = \mu_2 v^3+\beta u^2v &\mbox{in }\mathbb{R}^3,\\ u\to 0,\quad v\to 0, &\mbox{as } |x|\to\infty \end{array}\right. $\end{document} where \begin{document}$ \mu_1>0 $\end{document} , \begin{document}$ \mu_2>0 $\end{document} , \begin{document}$ \beta\in\mathbb{R} $\end{document} , \begin{document}$ \delta\in\mathbb{R} $\end{document} , and \begin{document}$ a(x) $\end{document} and \begin{document}$ b(x) $\end{document} are two \begin{document}$ C^\alpha $\end{document} potentials with \begin{document}$ 0 , satisfying some slow decay assumptions, but do not need to fulfill any symmetry property. Using the Lyapunov–Schmidt reduction method and some variational techniques, we show that there exist \begin{document}$ 0 and \begin{document}$ 0 such that the above system has infinitely many positive segregated solutions for any \begin{document}$ 0 and \begin{document}$ 0 .

Nonlinear Schrödinger system arises in the Hartree-Fock theory for a double condensate, i.e., a binary mixture of Bose-Einstein condensates in two different hyperfine states |1 and |2 (see [8]). Physically, u and v are the corresponding condensate amplitudes. µ j and β are the intraspecies and interspecies scattering lengths. The sign of scattering length β determines whether the interactions of states |1 and |2 are repulsive or attractive. When β > 0, the interactions of states |1 and |2 are attractive, the components tend to go along with each other, leading to synchronization. In contrast, when β < 0, the interactions of states |1 and |2 are repulsive, the components tend to repel each other, leading to segregation.
Nonlinear Schrödinger systems have been studied extensively in Mathematics, please see [4], [7], [10], [13], [17], [14], [18], [19], and the references therein. The first work we want to refer the readers to is by Dancer, Wei, and Weth, see [7], where they studied the set of solutions of the following Dirchlet problem    −∆u + λ 1 u = µ 1 u 3 + βuv 2 in Ω, where Ω ⊂ R N is smooth and bounded with N ≤ 3. They showed that the value β = − √ µ 1 µ 2 is critical in the sense that the solutions of (2) are priori bounded for β > − √ µ 1 µ 2 , and in contrast, (2) admits an unbounded sequence of solutions when When λ 1 = λ 2 , (2) was also studied by Bartsch, Dancer, and Wang. In [4], they established a new Liouville type theorem which provides priori bounds of solution's branches, and then they used this theorem and spectral analysis to investigate local and global bifurcations of (2) in terms of the parameter β. If the domain is radial, possibly unbounded, they also showed that the nodal structure of a certain weighted difference of the components of solutions along the bifurcating branches can also be controlled.
The proof of Theorem 1.1 in [1] is mainly based on the Lyapunov-Schmidt reduction method and some variational techniques. The strategy can be stated as following: First, the authors defined the configuration spaces and constructed the approximate solutions for the synchronized case of (1). Second, the existence and uniqueness of solutions for the projection of (1) were proved by a priori estimate of solutions, the Lax-Milgram theorem and the Banach fixed point theorem. Third, to study the maximization of the energy function over the configuration space, a error estimation was obtained by a secondary reduction method, and the error estimation was also used to show that the maximization of the energy function can be attained by some finite points in the configuration space by the induction method. Last, the authors showed that the maximization of the energy function could be attained by some interior points in configuration space, and the existence of synchronized solutions was proved by solving a linear algebra system. Since the estimations were independent of the number of spikes, then infinitely many synchronized solutions were obtained by changing the number of spikes.
However, the segregated case has not been considered yet, and in this paper, we are going to follow the main strategy in [1] by Ao and Wei to fill the gap for small β > 0. Slightly different from the conditions (H 1 )-(H 3 ) given by Ao and Wei in [1], we assume that a(x) and b(x) satisfy (A1) a(x) and b(x) are C α in R 3 for 0 < α < 1; (A2) a(x), b(x) → 0 as |x| → ∞, and there exists 0 < η < 1 such that lim |x|→∞ a(x)e η|x| = lim |x|→∞ b(x)e η|x| = +∞. Remark 1. The condition (H 2 ) can be implied by the condition (A 2 ), and so we omit it in this paper. As we know, the condition like (H 3 ) or (A 2 ) also can be found in the work by Cerami, Passaseo, and Solimini, see [6], where they considered a scalar equation and obtained the existence of infinitely many positive solutions by purely variational methods.
Remark 2. Note that a(x) and b(x) do not satisfy any symmetry property. When the potentials are radial and polynomial decay, we want to refer the readers to the work by Peng and Wang, see [15], where they studied a class of nonlinear Schrödinger systems involving potential functions, and constructed an unbounded sequence of non-radial positive segregated solutions in the repulsive case, and synchronized solutions in the attractive case.
Our main result in this paper is stated as following.

Remark 3.
In what follows, a secondary reduction method in [1] by Ao and Wei will be used to prove the existence of infinitely many segregated solutions for small β > 0, and we will find that Theorem 1.2 also holds for the coupled nonlinear Schrödinger systems (1) defined in R N for N = 1, 2. However, we don't know how to deal with the auxiliary term in (1) for β < 0, for more details, please see Proposition 3. Therefore, the existence of segregated solutions to (1) for β < 0 is still an open problem.
In order to outline the main idea and approach in the proof of Theorem 1.2, we introduce some notations and formulate a new version of Theorem 1.2. Let be the Hilbert space endowed with the inner product and the induced norm The functional corresponding to (1) is defined by for any (u, v) ∈ H, Let ω be the unique solution of . From Gidas, Ni, and Nirenberg [9], we can obtain The functional corresponding to (3) is defined by for any u ∈ H 1 (R 3 ) In this paper, we always assume that 0 < δ < 1 is sufficiently small, P k = (P 1 , · · · , P k ) ∈ R 3k , and Q l = (Q 1 , · · · , Q l ) ∈ R 3l for k ≥ 2 and l ≥ 2. In order to construct segregated solutions of (1) for small β > 0 by Lyapunov-Schmidt reduction method, the configuration spaces can be defined as following: Remark 4. These configuration spaces are used for proving that the maximization of the energy function can be attained by some finite points in the configuration space by induction method, see the proof of Lemma 5.1. Different from the synchronized case, see Ao, Wei [1], we assume that |P i − Q j | ≥ | ln δ|/4 in the configuration spaces for segregated case. In fact, from Lemma A.2, it is easy to find that Therefore, this term is so small such that the contradiction can not be obtained in Proposition 3 if |P k − Q l | = | ln δ|/2. For more details, please see the proof of Proposition 3.
We use the main ideas of Ao, Wei [1] to prove theorem 1.3 and the proof can be divided into four steps. In Step 1, we prove the existence of solutions to a linear problem by a priori estimate of solutions and the Fredholm alternative theorem. In Step 2, we use the Banach fixed point theorem to prove the existence of solutions to a nonlinear problem. In Step 3, we prove that the maximization of the energy function over the configuration space Λ k,l can be attained by some interior points of Λ k,l . In Step 4, we finish our proof by solving a linear algebra system and applying the strong maximum principle of elliptic equations.
To do these four steps, some modifications and improvements in techniques are necessary for the segregated case for small β > 0. First, we use two norms · * and · * * to do estimations. Second, a general error estimation is built up to prove that the maximization of the energy function can be attained by some finite points in our configuration space by a secondary reduction, see the proof of Lemma 5.2 in Section 5, and the parameter β in this general error estimation is kept for the reason that the distance of P i and Q j is too small in our configuration space, see Lemma 4.1. As a corollary of the general error estimation, a simpler error estimation, see Corollary 1, is built up to show that the maximization of the energy function can be attained at the interior point of the configuration space. Last, we use the characteristic of the configuration space Λ k,l to do some estimations, see (43) in Proposition 3.
This paper is organized as follows. We study a linear problem in Section 2 and a nonlinear problem in Section 3. In section 4, we built up a general error estimation of solutions to the nonlinear problem which is essential in our paper. In Section 5, we study the maximization of the energy function over the configuration space. In Section 6, we complete the proof of Theorem 1.2 by solving a linear algebra system. Finally, some technical computations are carried out in Appendix A.
Throughout this paper, unless otherwise stated, suppf denotes the support set of f . B R (x) and B(x, R) denote the ball centered at x with radius R. dist(P, Q) and |P − Q| denote the distance of P and Q. β 0 and δ 0 denote different positive parameters. C denotes different positive constants that are independent of δ, k, and l.
For (P, Q) ∈ Λ kl , we denote Now, we consider the following linear problem where The first lemma gives a priori estimation of solutions to (4).
Proof. It is sufficient to show the case for h = (h 1 , h 2 ) satisfying h * * < +∞ and for all i, s, and j, for the reason that the coefficients a ij and b sj in (4) can be replaced by a ij + First, we claim that for i = 1, 2, · · · , k, s = 1, 2, · · · , l, and j = 1, 2, 3, we have for some C > 0 that is independent of δ, k, and l. It is sufficient to show the first inequality in (7). For simplicity, we denote U ij = ∂U P i ∂xj and χ i = χ Pi . If we multiply the first equation in (4) against X ij and integrate over R 3 on both sides, we can obtain Since Now, we estimate each term on the right hand side of the above equality. By Lemmas A.1 and A.2, we find easily that Then it follows that Therefore, (7) follows from (8), (9), and Next, we claim that if (u, v) * = 1, then we have where ρ 1 > 0 is sufficiently large and C > 0 is independent of δ, ρ 1 , k, and l.

LUSHUN WANG, MINBO YANG AND YU ZHENG
Thus there is a subsequence (Q n,sn ) such that (u n (x − Q n,sn ), v n (x − Q n,sn )) converges on each compact set to (u, v) a solution of It follows that (u, v) = (0, 0), contradicts to (15). Therefore, combining the above two cases, we deduce that (5) holds and finish the proof of this lemma. Proposition 1. Let β, δ 0 , and C be as in Lemma 2.1, then for any 0 < δ < δ 0 Then the Riesz Lemma yields that for any (ψ, φ) ∈ H, From the fact that U P k and V Q l are bounded and (u, v) ∈ H, we can rewrite the problem (4) by the Riesz lemma as following where for any (ψ, φ) ∈ H, Let P be the orthogonal projection from H to H, then it follows from the problem By Hölder's inequality, we see that . Then as n → ∞, Since the projection P is bounded, then P K is also a compact operator in H. By using Lemma 2.1 and the Fredholm alternative theorem, we see that (17) has a unique solution in H. Therefore, (4) has a unique solution for any h with h * * norm bounded.

3.
A nonlinear problem. In this section, we consider the following nonlinear problem where i = 1, · · · , k, s = 1, · · · , l, j = 1, 2, 3, From a simple computation, it follows that where Let's denote A(h) be the solution to (4) corresponding to the vector function h. Then (20) can be rewritten as then solving the equation (20) is equivalent to finding a fixed point of T in some suitable space.
Before solving (18) by the Banach fixed point theorem, we give a lemma as follows.

LUSHUN WANG, MINBO YANG AND YU ZHENG
So there exists 0 < ξ < 1 such that Therefore, we find that S 1 (U P k , V Q l ) * * ≤ Cδ 1+ξ 6 , and so we finish this lemma. Now, we give the main result in this section.
Proof. Define where r > 0 will be determined later. Then solving the equation (20) is equivalent to finding a fixed point of T onto B. First, it is sufficient to show that T is a contract mapping onto B for some r > 0 and δ > 0, since by using Lemma A.4, for sufficiently small δ > 0, there exists r > 0 such that Therefore, T has a unique fixed point (u P k ,Q l , v P k ,Q l ) by the Banach fixed point theorem, and so (20) has a unique solution (u P k ,Q l , v P k ,Q l ) satisfying (21).
Next, we prove the solution (u P k ,Q l , v P k ,Q l ) of (20) is C 1 in (P, Q). Indeed, we define a mapping from Λ k,l × H × R 3(k+l) to H × R 3(k+l) as follows For any (P k , Q l ) ∈ Λ k,l , (20) has a solution (u P k ,Q l , v P k ,Q l , a P k ,Q l , b P k ,Q l ), i.e.
After a simple computation, By using Lemma 2.1, we can obtain easily which implies that (ψ, φ) = (0, 0). Therefore, ∂H(P k ,Q l ,u P k ,Q l ,v P k ,Q l ,a P k ,Q l ,b P k ,Q l ) ∂(u,v,a,b) is invertible. Since ∂H(P k ,Q l ,u,v,a,b) and ∂H(P k ,Q l ,u,v,a,b) are both continuous, then by the implicit function theorem, (u P k ,Q l , v P k ,Q l ) is C 1 in (P k , Q l ).

A general error estimation.
In the sequence, we always assume that β < min{µ 1 , µ 2 } and 0 < δ < δ 0 without special statement, where δ 0 is as in Proposition 2. For any (P, Q) ∈ Λ k,l , let's denote where (u P k ,Q l , v P k ,Q l ) is the solution to (18). We also write From Proposition 2, it follows that u k+s,l+t * ≤ Cδ We shall use a secondary reduction method to prove the following lemma which describes the norms of (u k+s,l+t , v k+s,l+t ) in Lemma 4.1. Let k, l, s,t ≥ 1. There exists a constant C > 0 independent of δ, k, l, s, and t such that Proof. We denote From Lemma 13.5 in [20] by Wei and Winter, we see that 0 is the second eigenvalue of L 0 and L 0 . We further assume that λ 1 and λ 1 are the first eigenvalues of L 0 and L 0 with eigenfunctions φ 0 and φ 0 respectively. Precisely, From the works [1] by Ao and Wei and [2] by Ao, Wei, and Zeng, we also know that φ 0 and φ 0 are radial, positive, exponential decay.
Multiplying ( φ i , φ j ) on both sides of (32) and integrating over R 3 , we have Similarly,

Then a simple computation yields
(33)

LUSHUN WANG, MINBO YANG AND YU ZHENG
For the above estimation (33), we have Since, by Hölder's inequality and Sobolev inequality, Then it follows that Similarly, we also deduce that Since, by Hölder's inequality, (u k+s,l+t , v k+s,l+t ) , (u k+s,l+t , v k+s,l+t ) .
Then (31) follows from the above argument. Now we are in the position to estimate the norm ( ψ, ψ) . Indeed, multiplying ( ψ, ψ) on both sides of the equation (32) and integrating over R 3 , we have (34) Claim 3. There exists C > 0 independent of δ, k, l, s, and t such that In fact, since W and W are exponential decay, then by Hölder's inequality, we have
Let us continue to estimating the norm ( ψ, ψ) . Since, by Hölder's inequality, Then, by (34) and (35), we have Finally, we estimate (u k+1 , v k+1 ) to finish this lemma. As we see, from (28), we have Therefore, this lemma follows from Claim 1 and the following

A supremum problem. Recall that
is the solution to (18). Let us define an energy function from Λ kl to R as follows and consider the following problem The following lemma will be required when we prove that C k,l can be attained by some interior points in Λ k,l .
Lemma 5.1. Suppose that the conditions (A1) and (A2) hold, there exists δ 0 > 0 such that for any 0 < δ < δ 0 , if C k,l can be attained by some points in Λ k,l , then for k ≥ 0 and l ≥ 0 one has C k+1,l > C k,l + I 1 (U ), where C 0,0 = 0 and for i = 1, 2, Proof. We prove this lemma by the following three cases.

So Case 1 holds directly.
Case 2. Consider the relations between C k,1 and C k,0 , C 1,l and C 0,l for k ≥ 1 and l ≥ 1.
To prove Case 2, we assume that C k,0 is attained by P k , i.e., C k,0 = J 1 (u P k ), and take Q ∈ R 3 such that |P i − Q| > 4|Q| 5 > 10| ln δ| for i = 1, · · · , k. Let u k,1 = u P k ,Q − u P k and v k,1 = v P k ,Q − V Q . From Corollary 1, we see that for some C > 0 independent of δ. After a simple expansion, we have From the definition of C k,1 , it follows that From the argument in the proof of Lemma 2.5 in [1], we find that where M > 0 is sufficiently large and δ is sufficiently small. Since for η ≥ 2 3 , << δe −η|Q| .
Then, combining the above inequalities, it follows that Case 3. Consider the relations between C k+1,l and C k,l , C k,l+1 and C k,l for k ≥ 1 and l ≥ 1.
To prove Case 3, we assume that C k,l is attained by (P k , Q l ), i.e. C k,l = J(u P k ,Q l , v P k ,Q l ). For i = 1, · · · , k and j = 1, · · · , l, let P k+1 ∈ R 3 be a point such that As before, let us denote It follows from Corollary 1 that By (3) and (18), a decomposition shows that For I, since η ≥ 2 3 and |a ij | ≤ Cδ for any i and j, then it follows that For II, by a similar argument in Case 2, we find that From Case 2, it is easy to see that for M > 0 large enough. Then, combining the above inequalities, one has C k+1,l > C k,l + I 1 (U ).
The following lemma shows that C k,l can be attained at some point (P k , Q l ) ∈ Λ k,l .
Proof. For k = 0, it is easy to see that this lemma holds by the argument in Ao and Wei [1]. So we just consider the case k ≥ 1. In what follows, we prove this lemma for fixed k ≥ 1 by induction on l ≥ 0.
For l = 0, the lemma also holds by Ao and Wei [1]. Now let us consider the case l = m + 1 under the assumption that the lemma holds for l ≤ m.
We show the case by contradiction. Assume that M (P n k , Q n m+1 ) → C k,m+1 for some (P n k , Q n m+1 ) satisfying |(P n k , Q n m+1 )| → ∞ as n → ∞. Without loss of generality, we may assume that (P n k , Q n m+1 ) = (P n k−s , P n k−s+1 , · · · , P n k , Q n m+1−t , Q m+1−t+1 , · · · , Q m+1 ) satisfying that Let us denote From (3) and (18), we have the following decomposition +O( (u k,m+1 , v k,m+1 ) 2 ). Since, as n → +∞, Then, it follows that Note that Then, we have So there exist β 0 and δ 0 such that for any 0 < β < β 0 and 0 < δ < δ 0 , we have Let n → +∞, it follows that However, by Lemma 5.1, we can obtain easily This is a contradiction, and thus the proof of this lemma is finished.
To finish this section, we prove that the attained points are located in the interior part of Λ k,l . Proposition 3. There exist β 0 > 0 and δ 0 > 0 such that for any 0 < β < β 0 and 0 < δ < δ 0 , C k,l = max Λ k,l M (P k , Q l ) can be attained at some (P, Q) ∈ intΛ k,l , the interior part of Λ k,l .
Proof. Suppose on contrary that C k,l is achieved at (P k , Q l ) ∈ ∂Λ k,l . We may assume that Let us denote A decomposition shows that (41) Take η ≥ 2 3 (see Lemma 3.1), from Lemma 4.1, we see that where we use In particular, (u k,l , v k,l ) ≤ Cδ 1 4 | ln δ| . Since, by Hölder's inequality and Sobolev inequality, then it follows from (41) that Note that ). (43) By the definition of Λ k,l , it is easy to see that there exists at most 6 3 points Q i such that Q i ∈ B(P k , | ln δ| 2 ). For the sake of simplicity, we may assume that |P k − Q i | ≥ | ln δ| 2 for i = 1, · · · , l − 2, | ln δ| 4 ≤ |P k − Q i | < | ln δ| 2 for i = l − 1, l.
Then, it follows that Then by (42), (43), and (44), we have C k,l ≤ C k−1,l + I 1 (U ) − µ 1 6. The proof of Theorem 1.2. In this section, we follow the argument in [11] to prove Theorem 1.3 and then finish the paper by proving Theorem 1.2.
Proof of Theorem 1.3. By Proposition 2, for each (P k , Q l ) ∈ Λ kl , there exists (u P k ,Q l , v P k ,Q l ) such that Let (P k , Q l ) be the maximum point of Λ kl and denote (u P k ,Q l , v P k ,Q l ) = (U P k , V Q l ) + (u P k ,Q l , v P k ,Q l ).
From Proposition 3, it follows that for each i = 1, · · · , k, s = 1, · · · , l, and j = 1, 2, 3, ∂M (P k , Q l ) ∂P ij = 0, ∂M (P k , Q l ) ∂Q sj = 0. Precisely, ∇u P k ,Q l ∇ ∂u P k ,Q l ∂P ij + (1 + δa(x))u P k ,Q l ∂u P k ,Q l ∂P ij By the equation (45), we obtain the following linear algebra system It is easy to see that (48) is diagonally dominant, then from the knowledge of linear algebra system it follows that a ij = 0, b sj = 0 for all i, s, j. Then (u P k , v Q l ) = (U P k + u P k ,Q l , V Q l + v P k ,Q l ) is indeed a solution to (1). Furthermore, similar to the argument in Section 6 of [11], we see that u P k ,Q l and v P k ,Q l are positive. Moreover, u P k ,Q l and v P k ,Q l have k and l local maximum points respectively.
Proof of Theorem 1.2. By changeing variables, Theorem 1.2 is a direct result of Theorem 1.3.