Ground State of N Coupled Nonlinear Schrödinger Equations in R n,n 3

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1 Commun. Math. Phys. 255, (2005) Digital Object Identifier (DOI) /s x Communications in Mathematical Physics Ground State of N Coupled Nonlinear Schrödinger Equations in R n,n 3 Tai-Chia Lin 1, Juncheng Wei 2 1 Department of Mathematics, National Taiwan University, Taipei, Taiwan, R.O.C. tclin@math.ntu.edu.tw 2 Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong, P.R. China. wei@math.cuhk.edu.hk Received: 5 August 2003 / Accepted: 12 November 2004 Published online: 2 March 2005 Springer-Verlag 2005 Abstract: We establish some general theorems for the existence and nonexistence of ground state solutions of steady-state N coupled nonlinear Schrödinger equations. The sign of coupling constants β ij s is crucial for the existence of ground state solutions. When all β ij s are positive and the matrix is positively definite, there exists a ground state solution which is radially symmetric. However, if all β ij s are negative, or one of β ij s is negative and the matrix is positively definite, there is no ground state solution. Furthermore, we find a bound state solution which is non-radially symmetric when N = Introduction In this paper, we study solitary wave solutions of time-dependent N coupled nonlinear Schrödinger equations given by i t j = j + µ j j 2 j + β ij i 2 j for y R n,t >0, (1.1) j = j (y, t) CI, j = 1,...,N, j (y, t) 0 as y +,t >0, where µ j > 0 s are positive constants, n 3, and β ij s are coupling constants. The system (1.1) has applications in many physical problems, especially in nonlinear optics. Physically, the solution j denotes the j th 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. The coupling constant β ij is the interaction between the i th and the j th component of the beam. As β ij > 0, the interaction is attractive, but the interaction is repulsive if β ij < 0. When the spatial dimension is one, i.e. n = 1, the system (1.1) is integrable, and there are many analytical and numerical results on solitary wave solutions of the general N coupled nonlinear Schrödinger equations(cf. [8, 17 19]).

2 630 T.-C. Lin, J. Wei From physical experiment(cf. [23]), two dimensional photorefractive screening solitons and a two dimensional self-trapped beam were observed. It is natural to believe that there are two dimensional N-component(N 2) solitons and self-trapped beams. However, until now, there is no general theorem for the existence of high dimensional N-component solitons. Moreover, some general principles like the interaction and the configuration of two and three dimensional N-component solitons are unknown either. This may lead us to study solitary wave solutions of the system (1.1) for n = 2, 3. Here we develop some general theorems for N-component solitary wave solutions of the system (1.1) in two and three spatial dimensions. To obtain solitary wave solutions of the system (1.1), we set j (y, t) = e iλ j t u j (y) and we may transform the system (1.1) to steady-state N coupled nonlinear Schrödinger equations given by u j λ j u j + µ j u 3 j + β ij u 2 i u j = 0in R n, u j > 0 in R n (1.2),j = 1,...,N, u j (y) 0 as y +, where λ j,µ j > 0 are positive constants, n 3, and β ij s are coupling constants. Here we want to study the existence and the configuration of ground state solutions of the system (1.2). The existence of ground state solutions may depend on coupling constants β ij s. When all β ij s are positive and the matrix (defined in (1.9)) is positively definite, there exists a ground state solution which is radially symmetric, i.e. u j (y) = u j ( y ), j = 1,,N. Such a radially symmetric solution may support the existence of N circular self-trapped beams. However, if all β ij s are negative, or one of β ij s is negative and the matrix is positively definite, there is no ground state solution. Furthermore, we find a bound state solution which is non-radially symmetric when N = 3. We will prove these results in the rest of this paper. Now we give the definition of ground state solutions as follows: In the one component case (N = 1), we may obtain a solution to (1.2) through the following minimization: inf u 0, u H 1 (R n ) R n u 2 + λ 1 ( R n u 4 ) 1 2 R n u 2. (1.3) An equivalent formulation, called Nehari s manifold approach (see [6] and [7]), is to consider the following minimization problem: inf u 1 N 1 E[u 1 ], where N 1 = { } u H 1 (R n ) : u 0, u 2 + λ 1 u 2 = µ 1 u 4. (1.4) R n R n R n It is easy to see that (1.3) and (1.4) are equivalent. A solution obtained through (1.4) is called a ground state solution in the following sense: (1) u>0 and satisfies (1.2), (2)

3 Ground State of N Coupled Nonlinear Schrödinger Equations in R n,n E[u] E[v] for any other solution v of (1.2). Hereafter, we extend the definition of ground state solutions to N-component case. To this end, we define first N = { u = (u 1,...,u N ) R n u j 2 + λ j ( H 1 (R n )) N : uj 0,u j 0, (1.5) R n u 2 j = µ j R n u 4 j + and consider the following minimization problem: } β ij u R 2 n i u2 j, j = 1,...,N c = inf E[u], (1.6) u N where the associated energy functional is given by for E[u] = ( 1 u j 2 + λ j u 2 2 R n 2 R n j µ j 4 j=1 1 4 β ij u R 2 n i u2 j i,j=1, R n u 4 j ) (1.7) u = (u 1,...,u N ) (H 1 (R n )) N. (1.8) Since n 3, by Sobolev embedding, E[u] is well-defined. A minimizer u 0 = (u 0 1,..., u 0 N ) of (1.6), if it exists, is called a ground state solution of (1.2), and it may have the following properties: 1. u 0 j > 0, j,and u0 satisfies (1.2); 2. E[u 0 1,...,u0 N ] E[v 1,...,v N ] for any other solution (v 1,...,v N ) of (1.2). It is natural to ask when the ground state solution exists. As N = 1, the existence of the ground state solution is trivial (see [6]). However, the existence of the ground state solution with multi-components is quite complicated. For general N 2, we introduce the following auxiliary matrix: = ( βij ), where we set β ii = µ i. (1.9) Our first theorem concerns the all repulsive case: Theorem 1. If β ij < 0, i j, then the ground state solution doesn t exist, i.e. c defined at (1.6) can not be attained. Our second theorem concerns the all attractive case. Theorem 2. If β ij > 0, i j, and the matrix (defined at (1.9)) is positively definite, then there exists a ground state solution (u 0 1,...,u0 N ). All u0 j must be positive, radially symmetric and strictly decreasing.

4 632 T.-C. Lin, J. Wei When attraction and repulsion coexist, i.e. some of β ij s are positive but some of them are negative, things become very complicated. Our third theorem shows that if one state is repulsive to all the other states, then the ground state solution doesn t exist. Theorem 3. If there exists an i 0 such that β i0 j < 0, j i 0, and β ij > 0, i i 0,j {i, i 0 } (1.10) and assume that the matrix is positively definite, then the ground state solution to (1.2) doesn t exist. Finally, we discuss the existence of bound states, that is, solutions of (1.2) with finite energy. We show that if repulsion is stronger than attraction, there may be non-radial bound states. To simplify our computations, we choose N = 3, λ 1 = λ 2 = λ 3 = µ 1 = µ 2 = µ 3 = 1. (1.11) Theorem 4. Assume that N = 3 and β 12 = δ ˆβ 12 = β 13 = δ ˆβ 13 > 0, β 23 = δ ˆβ 23 < 0. (1.12) Then for δ sufficiently small, problem (1.2) admits a non-radial solution u δ = ( u δ ) 1,uδ 2,uδ 3 with the following properties: where u δ 1 (y) w(y), uδ 2 (y) w(y Rδ e 1 ), u δ 3 (y) w(y + Rδ e 1 ), R δ log 1 δ, e 1 = (1, 0,...,0) T, and w is the unique solution of the following problem: w w + w 3 = 0 in R n w>0 in R n,w(0) = max y R w(y) n w(y) 0 as y +. (1.13) Graphically, we have

5 Ground State of N Coupled Nonlinear Schrödinger Equations in R n,n Note that under condition (1.12), there is also a radially symmetric solution u r of the following form: where ξ j satisfies u r = (u r 1,ur 2,ur 3 ), ur j = ξ j w(y), j = 1, 2, 3, ξ j + β ij ξ i = 1, j = 1, 2, 3. (1.14) Then we have Corollary 1. Assume that N = 3 and (1.12) holds. Then for δ sufficiently small, we have E[u δ ] <E[u r ], (1.15) where u δ is constructed in Theorem 3. As a consequence, if the ground state solution exists, it must be non-radially symmetric. It is known that (1.2) admits many radially symmetric bound states (see [17] and [18]). Theorem 4 suggests that there are many non-radially symmetric bound states which have lower energy than radially symmetric bound states. We consider the applications of Theorems 1 3 to simple cases N = 2 and N = 3. For the case N = 2, we have Corollary 2. If N = 2, then 1. for β 12 < 0, the ground state solution doesn t exist, 2. for 0 <β 12 < µ 1 µ 2, the ground state solution exists. For the case N = 3, the matrix becomes = µ 1 β 12 β 13 β 12 µ 2 β 23. β 13 β 23 µ 3 Assume that β ij 0. Then we may divide into four cases given by Case I: all repulsive: β 12 < 0,β 13 < 0,β 23 < 0, Case II: all attractive: β 12 > 0,β 13 > 0,β 23 > 0, Case III: two repulsive and one attractive: β 12 < 0,β 13 < 0,β 23 > 0, Case IV: one repulsive and two attractive: β 12 > 0,β 13 > 0,β 23 < 0. For Case I III, we have a complete picture Corollary 3. If N = 3, then 1. for Case I, the ground state solution doesn t exist, 2. for Case II and assume is positively definite, the ground state solution exists, 3. for Case III and assume is positively definite, the ground state solution doesn t exist.

6 634 T.-C. Lin, J. Wei It then remains only to consider Case IV. Due to the existence of non-radial bound states in Theorem 4 and non-radial property of ground states in Corollary 1, Case IV becomes very complicated. Our results here will be very useful in the study of (1.2) for bounded domains which relates to multispecies Bose-Einstein condensates, and in the study of solitary wave solutions of N coupled nonlinear Schrödinger equations with trap potentials: u j V j (x)u j + µ j u 3 j + β ij u 2 i u j = 0, x R n, u j > 0 in R n (1.16),j = 1,...,N, u j (x) 0 as x +. The main idea in proving Theorem 1 3 is by Nehari s manifold approach and Schwartz symmetrization technique. Theorem 4 is proved by the Liapunov-Schmidt reduction method combined with the variational method. The organization of the paper is as follows: In Sect. 2, we collect some properties of the function w-solution of (1.13) and Schwartz symmetrization. In Sect. 3, we state another equivalent approach of Nehari s method which is more useful in our proofs. It is here that we need that the matrix is positively definite. The proofs of Theorems 1, 2, 3, 4 are given in Sects. 4, 5, 6, 7, respectively. Section 8 contains the proof of Corollary Some Preliminaries In this section, we analyze some problems in R n. Recall that w is the unique solution of (1.13). By Gidas-Ni-Nirenberg s Theorem, [14], w is radially symmetric. By a theorem of Kwong [20], w is unique. Moreover, we have w ( y ) <0 for y > 0 and ( w( y ) = A n r n 1 2 e r 1 + O ( )) 1, as r = y +, (2.1) r ( w ( y ) = A n r n 1 2 e r 1 + O ( )) 1, as r = y +. (2.2) r We denote the energy of w as I[w] = 1 w w 2 1 w 4. (2.3) 2 R n 2 R n 4 R n Let w λ,µ be the unique solution to the following problem: w λ,µ λw λ,µ + µwλ,µ 3 = 0 in Rn, w λ,µ > 0,w λ,µ (0) = max y R w λ,µ(y), n w λ,µ (y) 0 as y +.

7 Ground State of N Coupled Nonlinear Schrödinger Equations in R n,n It is easy to see that w λ,µ (y) = λ µ w ( λ y ), (2.4) and 1 w λ,µ 2 + λ w 2 2 R n 2 R n λ,µ µ w 4 4 R n λ,µ = λ 4 n 2 µ 1 I[w]. (2.5) We now collect some of the properties of w λ,µ. Lemma 1.(1) w( y ) is the unique solution to the following minimization problem: R n u 2 + R n u 2 (. (2.6) R n u 4 ) 2 1 inf u H 1 (R n ), u 0 (2) The following eigenvalue problem: { φ λφ + 3µw 2 λ,µ φ = βφ φ H 2 (R n ) (2.7) admits the following set of eigenvalues: β 1 > 0 = β 2 =...= β n+1 >β n+2..., where the eigenfunctions corresponding to the zero eigenvalue are spanned by { } wλ,µ K 0 := span,j = 1,...,n = C 0. (2.8) y j As a result, the following map: L λ,µ φ := φ λφ + 3µw 2 λ,µ φ is invertible from K0 C 0 where K0 {u = H 2 (R n ) u w } λ,µ = 0,j = 1,,n, (2.9) R n y j C0 {u = L 2 (R n ) u w } λ,µ = 0,j = 1,,n. (2.10) R n y j Proof. (1) follows from the uniqueness of w(cf. [20]). (2) follows from Theorem 2.12 of [22] and Lemma 4.2 of [24]. Set also I λ,µ [u] = 1 u 2 + λ u 2 µ u 4. (2.11) 2 R n 2 R n 4 R n

8 636 T.-C. Lin, J. Wei We then have Lemma 2. inf I λ,µ [u] is attained only by w λ,µ, u N λ,µ where { N λ,µ = u H 1 (R n ) u 2 + λ u 2 = µ R n R n Proof. It is easy to see that inf I λ,µ [u] is equivalent to u N λ,µ inf u 0, u H 1 (R n ) The rest follows from (1) of Lemma 1. R n u 2 + λ R n u 2 (. R n u 4 ) 2 1 R n u 4 }. (2.12) The next lemma is not so trivial. Lemma 3. inf I λ,µ [u] is also attained only by w λ,µ, u N λ,µ where } N λ,µ {u = H 1 (R n ) u 2 + λ u 2 µ u 4. R n R n R n (2.13) Proof. Let u k be a minimizing sequence and u k be its Schwartz symmetrization. Then by the property of symmetrization u R n k 2 + λ (u R n k )2 u k 2 + λ u 2 R n R n k µ u 4 R n k = µ (u R n k )4, (2.14) and I λ,µ [u k ] I λ,µ[u k ]. (2.15) Hence, we may assume that u k is radially symmetric and decreasing. Since u k H 1 (R n ), and u k is strictly decreasing, it is well-known that u k (r) Cr N 1 2 u k H 1. (2.16) So u k u 0 (up to a subsequence) in L 4 (R n ), where u 0 is also radially symmetric and decreasing. Moreover, by Fatou s Lemma, u 0 N λ,µ. Hence inf I λ,µ [u] can be u N λ,µ attained by u 0. We then claim that u λ R n u 2 R n 0 = µ u 4 R n 0. (2.17) Suppose not. That is u λ R n u 2 R n 0 <µ u 4 R n 0.

9 Ground State of N Coupled Nonlinear Schrödinger Equations in R n,n Then u 0 (N λ,µ )0 - the interior of N λ,µ. By standard elliptic theory, u 0 is a critical point of I λ,µ [u], i.e. I λ,µ [u 0 ] = 0, (2.18) where " means the derivative. Multiplying (2.18) by u 0,wehave R n u 0 2 +λ R n u 2 0 = µ R n u 4 0 a contradiction. Hence u 0 N λ,µ. By Lemma 2, u 0 = λµ w( λ y ) = w λ,µ (y). We present another characterization of w λ,µ : Lemma 4. inf u N λ,µ I λ,µ [u] = Proof. This follows from a simple scaling. inf u 0, u H 1 (R n ) sup I λ,µ [tu]. t>0 Finally, we recall the following well-known result, whose proof can be found in Theorem 3.4 of [21]. Lemma 5. Let u 0,v 0,u,v H 1 (R n ) and u,v be their Schwartz Symmetrization. Then uv u v. R n R n Our last lemma concerns some integrals. Lemma 6. Let y 1 y 2 R n. Then as y 1 y 2 +, we have for λ 1 <λ 2, w 2 R n λ 1,µ 1 (y y 1 )wλ 2 2,µ 2 (y y 2 ) wλ 2 1,µ 1 (y 1 y 2 ) w 2 R n λ 2,µ 2 (z)e 2 If λ 1 = λ 2, then w 2+σ λ 1,µ 1 (y 1 y 2 ) for any 0 <σ <1. λ1 z, y 1 y 2 y 1 y 2 dz. (2.19) R n w 2 λ 1,µ 1 (y y 1 )w 2 λ 2,µ 2 (y y 2 ) w 2 σ λ 1,µ 1 (y 1 y 2 ) (2.20) Proof. Let y = y 2 + z. Then from (2.1), we have wλ 2 1,µ 1 (y y 1 )wλ 2 2,µ 2 (y y 2 ) = wλ 2 1,µ 1 (y 2 y 1 + z)wλ 2 2,µ 2 (y y 2 ) = wλ 2 1,µ 1 (y 2 y 1 )e 2 λ 1 ( y 2 y 1 y 2 y 1 +z ) (1 + o(1))wλ 2 2,µ 2 (y y 2 ) = wλ 2 1,µ 1 (y 1 y 2 )(1 + o(1))e 2 λ 1 z, y 1 y 2 y 1 y 2 wλ 2 2,µ 2 (z). Hence by Lebesgue Dominated Convergence Theorem gives (2.19). The proof of (2.20) is similar.

10 638 T.-C. Lin, J. Wei 3. Nehari s Manifold Approach In this section, we consider the relation between two minimization problems Problem 1. where c = inf E[u], (3.1) u N N = {u (H 1 (R n )) N u j 2 + λ j u 2 R n R n j = µ j + } β ij u R 2 n i u2 j,j = 1,...,N. R n u 4 j Problem 2. We have m = inf sup u 0 t 1,...,t N >0 E[ t 1 u 1,..., t N u N ]. (3.2) Theorem 5. Suppose either β ij < 0, i j, or the matrix defined by is positively definite. Then c = inf E[u] = m = inf u N u 0 = ( β ij ) with β ii = µ i Proof. We consider the following function sup t 1,...,t N >0 E[ t 1 u 1,..., t N u N ]. β(t 1,...,t N ) = E[ t 1 u 1,..., t N u N ]. First we assume u N. Claim 1. β(t 1,...,t N ) attains its global maximum at t 1 =...= t N = 1. In fact, where β(t 1,...,t N ) = ] t j u j [R 2 + λ j u 2 n j Q[t 1,...,t N ], Q[t 1,...,t N ] = 1 4 µ j tj 2 u 4 R n j = 1 4 tt t, β ij t i t j u R 2 n i u2 j i,j=1, where t = (t 1,..., t N ) T and = (β ij R n u 2 i u2 j ). (3.3)

11 Ground State of N Coupled Nonlinear Schrödinger Equations in R n,n If β ij < 0, i j, then since u N,wehave µ j R n u 4 j + β ij R n u 2 i u2 j = u j 2 + λ j u 2 R n R n j > 0. Moreover, we see that the matrix is diagonally dominant and hence is positively definite. If β ij > 0 for all i j, then for t j > 0,j = 1,...,N, [ ] Q[t 1,...,t N ] = 1 4 i,j 1 [ N µ j tj 2 4 > 0. j=1 β ij t i t j R n u 2 i u2 j R n u 4 j ] 1 4 i,j=1, ( ) 1 ( ) β ij u 4 2 R n i u R n j ti t j Again, Q[t 1,...,t N ] is positively-definite. Thus β(t 1,...,t N ) is concave and hence there exists a unique critical point. Since u N,(1...,1) is a critical point. So we complete the proof of Claim 1. From Claim 1, we deduce that On the other hand, suppose that inf E[u] inf sup E[ t 1 u 1,..., t N u N ]. (3.4) u N u 0 t 1,...,t N sup E[ t 1 u 1,..., t N u N ] = β(t1 0,...,t0 N )<+, t 1,...,t N where u = (u 1,,u N ) 0. Certainly, (t1 0,...,t0 N ) is a critical point of β(t 1,...,t N ) ) and hence (u 0 1,...,u0 N ( t ) 1 0u 1,..., tn 0 u N N. So which proves E[u 0 1,...,u0 N ] = β(t0 1,...,t0 N ) inf u N E[u] c = inf E[u] m = inf sup E[ t 1 u 1,..., t N u N ]. (3.5) u N u 0 t 1,...,t N Combining (3.4) and (3.5), we obtain Theorem 5.

12 640 T.-C. Lin, J. Wei 4. Proof of Theorem 1 In this section, we prove Theorem 1. First by Theorem 5, Now we choose c = inf sup E[ t 1 u 1,..., t N u N ]. u 0 t 1,...,t N u j (y) := w λj,µ j (y jre 1 ), j = 1,...,N, (4.1) where R>>1is a large number and e 1 = (1, 0,...,0) T. By choosing R large enough and applying Lemma 5, we obtain that u 2 R n i u2 j = w 2 R n λ i,µ i (y ire 1 )wλ 2 j,µ j (y jre 1 ) = w 2 R n λ i,µ i (y)wλ 2 j,µ j (y + (i j)re 1 )dy 0 as R +. Let (t1 R,...,tR N ) be the critical point of β(t 1,...,t N ). Then we have u j 2 + λ j u 2 R n R n j = µ j tj R u 4 R n j + β ij ti R since the matrix (β ij by implicit function theorem R n u 2 i u2 j R n u 2 i u2 j ) is positively definite (similar to arguments in Sect. 3), t R j = 1 + o(1). Thus c lim R + β(tr 1,...,tR N ) = Next we claim that c N [ ( ) 1 w j 2 + λ j w 2 2 R n j µ ] j w 4 4 R n j. (4.2) [ ( ) 1 w j 2 + λ j w 2 2 R n j µ ] j w 4 4 R n j. (4.3) In fact, let (u 1,...,u N ) N, then since β ij < 0, i j, E[u 1,...,u N ] = n [ ( ) 1 u j 2 + λ j u 2 2 R n j µ ] j u 4 4 R n j n I λj,µ j [u j ], (4.4)

13 Ground State of N Coupled Nonlinear Schrödinger Equations in R n,n and u j 2 + λ j u 2 R n j µ j u 4 R n j. (4.5) By Lemma 2, n E[u 1,...,u N ] I λj,µ j [u j ] (4.6) which proves (4.3). Hence n inf I λj w N λ,µ j [w] j,µ j = I λj,µ j [w λj,µ j ] c = I λj,µ j [w λj,µ j ]. (4.7) If c is attained by some (u 0 1,...,u0 N ), then= (u0 1,...,u0 N ) N and u0 j is a solution of (1.2). By the Maximum Principle, u 0 j > 0, j = 1,...,N. Then we have c = E[u 0 1,...,u0 N ] > N I λj,µ j [u 0 j ] N which contradicts (4.7), and we may complete the proof of Theorem 1. I λj,µ j [w λj,µ j ] (4.8) 5. Proof of Theorem 2 Now we prove Theorem 2 in this section. Our main idea is by Schwartz symmetrization. For u j 0,u j Lemma 6, for i j H 1 (R n ), we denote u j as its Schwartz symmetrization. By u 2 R n i u2 j (u R n i )2 (u j )2. (5.1) Hence E[u 1,...,u N ] E[u 1,...,u N ]. (5.2) The new function u = (u 1,...,u N ) will satisfy the following inequalities: u R n j 2 + λ 1 (u R n j )2 β ij (u R n i )2 (u j )2 µ j (5.3) R n (u j )4 (by (5.1) and the fact that β ij > 0).

14 642 T.-C. Lin, J. Wei Therefore, we have where N = c = inf E[u] inf E[u] := u N u N c, {u (H 1 (R n )) N µ j R n u 4 j + u j 2 + λ j u 2 R n j } β ij u R 2 n i u2 j,j = 1,...,N. (5.4) We first study c and then we show that c = c. By the previous argument, we may assume any minimizing sequence (u 1,...,u N ) of c must be radially symmetric and decreasing. We follow the proof of Lemma 2 to conclude that a minimizer for c exists and must be radially symmetric and decreasing. Moreover, we have u j 2 + λ j u 2 R n j µ j u 4 R n j + β ij u R 2 n i u2 j, j = 1,...,N. (5.5) If all the inequalities of (5.5) are strict, then as for the proof of Lemma 2, we may have a contradiction. So we may assume at least one of (5.5) is an equality. Without loss of generality, we may assume that G j [u] := u j 2 + λ j u 2 R n R n j µ j u 4 R n j β ij u R 2 n i u2 j = 0, j = 1,...,k <N. (5.6) Then we have E[u 1,...,u N ] + k j G j [u 1,...,u N ] = 0, (5.7) where G j is defined at (5.6). We assume that k+1 =... = N = 0 and we write (5.7) as From (5.6), we obtain which is equivalent to E[u 1,...,u N ] + j G j [u 1,...,u N ] = 0. (5.8) j Gj,u j = 0 (β ij R n u 2 i u2 j ) j = 0

15 Ground State of N Coupled Nonlinear Schrödinger Equations in R n,n since the matrix is positively define, the matrix ) (β ij is non-singular R n u 2 i u2 j and hence j = 0, j = 1,...,N. As for the proof of Lemma 1, u N. Hence c = c and c can be achieved by radially symmetric pairs (u 0 1,...,u0 N ). Hence (u0 1,...,u0 N ) must satisfy (1.2). By the maximum principle, u 0 j > 0, since u0 j satisfies u 0 j λ j u 0 j + µ j (u 0 j )3 + β ij (u 0 i )2 u 0 j = 0, β ij > 0 by the moving plane method for cooperative systems (cf. [27]), u 0 j must be radially symmetric and strictly decreasing. Therefore we may complete the proof of Theorem Proof of Theorem 3 In this section, we prove Theorem 3. The proof combines those of Theorem 1 and Theorem 2. Assume u = (u 1,...,u N ) N. Without loss of generality, we may assume that i 0 = 1. Then β 1j < 0, j >0, and β ij > 0, i >1,j {1,i}. We may divide the energy E[u 1,...,u N ] into two parts E[u 1,...,u N ] = 1 u λ 1 u 2 2 R n 2 R n 1 µ 1 u 4 4 R n 1 1 β 1j u 2 R 2 n 1 u2 j + E [u 2,...,u N ], (6.1) where E [u 2,...,u N ] = Since β 1j < 0, for j>1, j=2 ( 1 u j 2 + λ j u 2 2 R n 2 R n j µ j 4 j= On the other hand, u 1 satisfies β ij u R 2 n i u2 j. i,j=2, R n u 4 j ) (6.2) E[u 1,...,u N ] I λ1,µ 1 [u 1 ] + E [u 2,...,u N ]. (6.3) R n u λ 1 R n u 2 1 µ 1 R n u 4 1 = β 1j u R 2 n 1 u2 j 0 (6.4) j=2

16 644 T.-C. Lin, J. Wei and u j,j = 2,...,N satisfies R n u j 2 + λ 1 R n u 2 j µ j R n u 4 j + β ij u R 2 n i u2 j. (6.5) Here we have used the system (1.2) and the fact that β 1j < 0, for j>1. By the proof of Theorem 2, where Hence E [u 2,...,u N ] i=2, inf E [u 2,...,u N ] = c 1, (6.6) (u 2,...,u N ) N 1 { N 1 = u = (u 2,...,u N ) u j 2 R n +λ j u R 2 n j = µ j u 4 R n j + β ij u R 2 n i u2 j On the other hand, by Lemma 3, Now we claim that In fact, by Theorem 5, Now we choose c = inf E[u] = inf u N u 0 i=2, I λ1,µ 1 [u 1 ] I λ1,µ 1 [w λ1,µ 1 ]. (6.7) inf u N E[u] I λ 1,µ 1 [w λ1,µ 1 ] + c 1. (6.8) inf u N E[u] = I λ 1,µ 1 [w λ1,µ 1 ] + c 1. (6.9) sup t 1,...,t N 0 u 1 = w λ1,µ 1 (y Re 1 ) }. E[ t 1 u 1,..., t N u N ]. and u j = u 0 j for j 2, where (u0 2,...,u0 N ) is a minimizer of c 1 at (6.6). Then u 2 R n 1 (u0 j )2 0 asr +, j >1. Thus if we set β(t R 1,...,tR N ) = sup E[ t 1 u 1,..., t N u N ], t 1,...,t N 0 then t R j = 1 + o(1) and c lim R + β(tr 1,...,tR N ) = I λ 1,µ 1 [w λ1,µ 1 ] + c 1. (6.10)

17 Ground State of N Coupled Nonlinear Schrödinger Equations in R n,n This, combined with (6.8), proves that c = I λ1,µ 1 [w λ1,µ 1 ] + c 1. Finally, we show that c is not attained. In fact, if c is attained by some (u 0 1,...,u0 N ), u 0 j > 0, then A contradiction! c = E[u 0 1,...,u0 N ] >I λ 1,µ 1 [u 0 1 ] + E [u 0 2,...,u0 N ] I λ1,µ 1 [w λ1,µ 1 ] + c 1. Remark 1. Theorem 3 also holds if β ij satisfies and β ij < 0, for i = i 1,...,i k,j i 1,...,i k β ij > 0, for i {i 1,...,i k },j i. 7. Proof of Theorem 4 In this section, we construct non-radial bound state of (1.2) in the following case: N = 3, λ 1 = λ 2 = λ 3 = µ 1 = µ 2 = µ 3 = 1, (7.1) β 23 = δ ˆβ 23 < 0, (7.2) β 12 = δ ˆ β 12 = β 13 = δ ˆ β 13 > 0. (7.3) As we shall see, assumption (7.1) is not essential and it is just for simplification of our computations. The assumption (7.3) imposes some sort of symmetry which means that the role of u 2 and u 3 can be exchanged. We shall make use of the so-called Liapunov-Schmidt reduction process and variational approach. The Liapunov-Schmidt reduction method was first used in nonlinear Schrödinger equations by Floer and Weinstein [13] in one-dimension, later was extended to higher dimension by Oh [25, 26]. Later it was refined and used in a lot of papers. See [2 5, 15, 16, 25, 26, 28, 29] and the references therein. A combination of the Liapunov- Schmidt reduction method and the variational principle was used in [3, 10, 11, 15] and [16]. Here we follow the approach used in [15]. Let us first introduce some notations: let S j [u] = u j u j + u 3 j + β ij u 2 i u j, (7.4) S[u] = S 1 [u]. S N [u],

18 646 T.-C. Lin, J. Wei XI = (H 2 (R n ) {u u(x 1,x ) = u(x 1, x )}) 3 {(u 1,u 2,u 2 ) u 2 (x 1,x ) = u 3 ( x 1,x )}, YI = (L 2 (R n ) {u u(x 1,x ) = u(x 1, x )}) 3 {(u 1,u 2,u 2 ) u 2 (x 1,x ) = u 3 ( x 1,x )}, w R j (y) = w ( ) y R j e 1, (7.5) X = (H 2 (R n ) {u u(x 1,x ) = u(x 1, x )}) 3, X 0 = X {(u 1,u 2,u 3 ) u 2 (x 1,x ) = u 3 ( x 1,x ),u 1 (x 1,x ) = u 1 ( x 1,x )}, Y = (L 2 (R n ) {u u(x 1,x ) = u(x 1, x )}) 3, Y 0 = Y {(u 1,u 2,u 3 ) u 2 (x 1,x ) = u 3 ( x 1,x ),u 1 (x 1,x ) = u 1 ( x 1,x )}. Note that S[u] is invariant under the map Thus S is map from X 0 to Y 0. Fix R δ, where Here we may choose T : (u 1 (x 1,x ), u 2 (x 1,x ), u 3 (x 1,x )) (7.6) (u 1 ( x 1,x ), u 3 ( x 1,x ), u 2 ( x 1,x )). δ = {R w(r)<δ 1 4 σ }. (7.7) We define σ = u R := (w(y), w(y Re 1 ), w(y + Re 1 )) T (7.8) = (w, w R,w R ) T. We begin with Lemma 7. The map L 0 = φ 1 φ 1 + 3w 2 φ 1 φ 2 φ 2 + 3(w R ) 2 φ 2 : X 0 Y 0 (7.9) φ 3 φ 3 + 3(w R ) 2 φ 3 has its kernel ) T } K 0 = span {(0, wr, w R y 1 y 1 (7.10) and cokernel ) T } C 0 = span {(0, wr, w R. (7.11) y 1 y 1

19 Ground State of N Coupled Nonlinear Schrödinger Equations in R n,n Proof. In fact, L 0 = 0. Then we have by Lemma 1 (2), φ 1 = n c 1,j w y j, φ 2 = n c 2,j w R y j, φ 3 = n c 3,j w R y j. (7.12) Since (φ 1,φ 2,φ 3 ) T X 0,wehaveφ 1 (x 1, x ) = φ 1 ( x 1, x ) = φ 1 (x 1,x ). This forces φ 1 = 0. Similarly, we have c 2,2 = = c 2,n = 0,c 3,2 = = c 3,n = 0. On the other hand, φ 2 (x 1,x ) = φ 3 ( x 1,x ).Sowehavec 2,1 = c 3,1. This proves (7.10). Since L 0 is a self-adjoint operator, (7.11) follows from (7.10). From Lemma 7, we deduce that Lemma 8. The map is uniformly invertible from Proof. We may write L := S [u R ] (7.13) L := K 0 C 0. (7.14) L = L 0 + δb, (7.15) where B is a bounded and compact operator. Since L 1 0 exists, by standard perturbation theory, L is also invertible for δ sufficiently small. Using Lemma 8, we derive the following proposition: Proposition 1. For δ sufficiently small, and R δ, there exists a unique solution v R = (v1 R,vR 2,vR 3 ) such that S 1 [u R + v R ] = 0, (7.16) S 2 [u R + v R ] = c R w R y 1, (7.17) S 3 [u R + v R w R ] = c R, (7.18) y 1 for some constant c R. Moreover, v R is of C 1 in R and we have v R H 2 (R n ) cδ 1 2σ. (7.19)

20 648 T.-C. Lin, J. Wei Proof. Let R 1 = 0,R 2 = R,R 2 = R and We choose v B, where and then expand where w R j = w(y R j e 1 ). B = {v X v H 2 <δ 1 2σ } (7.20) S 1 [u R + v] = v 1 v 1 + 3(w R 1 ) 2 v 1 + [(w R 1 + v 1 ) 3 (w R 1 ) 3 3(w R 1 ) 2 v 1 ] +δ[ ˆβ 12 (w R 2 + v 2 ) 2 + ˆβ 13 (w R 3 + v 3 ) 2 ](w R 1 + v 1 ) = L 1 v 1 + H 1 [v 1 ] + E 1, L 1 v 1 = v 1 v 1 + 3(w R 1 ) 2 v 1, and Here we have Similarly, E 1 = δ[ ˆβ 12 (w R 2 + v 2 ) 2 + ˆβ 13 (w R 3 + v 3 ) 2 ](w R 1 + v 1 ), H 1 [v 1 ] = [(w R 1 + v 1 ) 3 (w R 1 ) 3 3(w R 1 ) 2 v 1 ] = O( v 1 2 ). E 1 = O(δ)(w R 2 w R 1 + w R 3 w R 1 ) (7.21) = O(δ)(w( R 1 R 2 ) + w( R 1 R 3 ) = O(δ 4 5 σ ). S 2 [u R + v] = L 2 v 2 + H 2 [v 2 ] + E 2, where S 3 [u R + v] = L 3 v 3 + H 3 [v 3 ] + E 3, L 2 v 2 = v 2 v 2 + 3(w R 2 ) 2 v 2, (7.22) L 3 v 3 = v 3 v 3 + 3(w R 3 ) 2 v 3, E 2 = O(1)[δ ˆβ 12 (w R 1 ) 2 + δ ˆβ 23 (w R 3 ) 2 ]w R 2, = O(δ 4 5 σ + δ 1 σ ) = O(δ 1 σ ), E 3 = O(1)[δ ˆβ 13 (w R 1 ) 2 + δ ˆβ 23 (w R 2 ) 2 ]w R 3 = O(δ 1 σ ), H 2 [v 2 ] = [(w R 2 + v 2 ) 3 (w R 2 ) 3 3(w R 2 ) 2 v 2 ] = O( v 2 2 ), H 3 [v 3 ] = [(w R 3 + v 3 ) 3 (w R 3 ) 3 3(w R 3 ) 2 v 3 ] = O( v 3 2 ). Since L : K 0 C 0 is invertible, solving (7.16) (7.18) is equivalent to solving [Lv + H[v] + E] = 0, v K 0, (7.23)

21 Ground State of N Coupled Nonlinear Schrödinger Equations in R n,n where is the orthogonal projection on C 0 and v=(v 1,v 2,v 3 ) T, H[v]=(H 1,H 2,H 3 ) T, E = (E 1,E 2,E 3 ) T. Equation (7.23) can be written in the following form: v = G[v] := ( L ) 1 [ H[v] E], (7.24) where is the orthogonal projection on K0. Since H[v] = O( v 2 ) and E = O(δ 1 σ ), it is easy to see that the map G defined at (7.24) is a contraction map from B to B. By the contraction mapping theorem, (7.23) has a unique solution v R = (v1 R,vR 2,vR 3 ) K 0 with the property that v R H 2 (R n ) C E 1 σ (7.25) L 2 (R n ) C(δ (1 σ)(1 σ) ) Cδ 1 2σ. The C 1 property of v R follows from the uniqueness of v R. See a similar proof in Lemma 3.5 of [15]. Now we let M[R] = E[u R + v R ]: δ R 1, where v R is given by Proposition 1. We have Lemma 9. For R δ and δ sufficiently small, we have M[R] = 3I[w] (7.26) 1 [ δ ] ˆβ 23 (w R ) 2 (w R ) 2 + 2δ ˆβ 12 w 2 (w R ) 2 2 R n R ( ) n +O δ σ 2. Proof. We may calculate that M[R] = E[u R + v R ] 3 { [ 1 = (w R j + v R 2 R n j ) 2 + (w R j + v R R n j )2 1 β ij (w R i + v R 4 R n i )2 (w R j + vj R )2 i,j 3 [ ] = E[u R 1 ] + ( v R 2 R n j 2 + (vj R )2 ) 3(w R j ) 2 (vj R )2 1 ] β ij [w R i v Ri 2 (wr j ) 2 + w R j v Rj (wr i ) 2 +O(δ 2 4σ ) i,j R n ] 1 } (w R j + v R 4 R n j )4 ( )] = E[u R ] + O(δ 2 4σ ) [β 23 v R R n 2 wr 2 (w R 3 ) 2 + v R R n 3 (wr 3 )(w R 2 ) 2 = E[u R ] + O(δ σ 2 ). Here we have used the assumption (1.12), Eq. (1.13), and Proposition 1.

22 650 T.-C. Lin, J. Wei Since E[u R ] = 3I[w] 1 [ δ ˆβ 23 (w R 2 ) 2 (w R 3 ) 2 2 R n +δ ˆβ 12 R n (w R 1 ) 2 (w R 2 ) 2 + δ ˆβ 13 R n (w R 1 ) 2 (w R 3 ) 2 and β 12 = β 13,R 2 = R 3, we obtain (7.26). Next we have Lemma 10. If R δ ( δ ) 0 the interior of δ is a critical point of M[R], then the corresponding solution u δ = u Rδ + v Rδ is a critical point of E[u]. Proof. Since R δ ( δ ) 0 the interior of δ is a critical point of M[R], we then have d dr M[R] 0 R=R δ= which is equivalent to < E[u R + v R ], d dr (ur + v R )> 0. R=R δ= Using Proposition 1, we obtain w R d Rn c R R n y 1 dr (wr + v2 R ) c w R d R y 1 dr (w R + v3 R ) = 0 (7.27) for R = R δ. Note that since v K0,wehave [ w R ] v R n 2 R y w R v3 R = 0. (7.28) 1 y 1 Differentiating (7.28) with respect to R, we obtain that R n [ w R y 1 On the other hand, we see that [ w R d dr vr 2 w R d y 1 dr vr 3 [ 2 w R = v R n 2 R R y 2 w R ] v3 R = O(δ 1 2σ ). (7.29) 1 R y 1 d R n y 1 dr (wr ) w R y 1 From (7.27), (7.29) and (7.30), we deduce that ] ] d dr (w R ) = 2 R n ], ( ) w 2. (7.30) y 1 c R = 0, for R = R δ, (7.31) which then implies that the corresponding solution u δ = u Rδ + v Rδ is a critical point of E[u].

23 Ground State of N Coupled Nonlinear Schrödinger Equations in R n,n Finally, we prove Theorem 4. Proof of Theorem 4. We consider the following minimization problem: M 0 = min M[R], R δ (7.32) since M[R] is continuous and δ is closed, M[R] attains its minimum at a R δ δ. We claim that R δ δ. Suppose not. That is R δ δ. Then we have w(r δ ) = δ 4 1 σ. Let ρ(r) = w 2 (y)w 2 (y Re 1 ). R n (7.33) Then from Lemma 9, we have M[R] = 3I[w] 1 2 δ ˆβ 23 ρ(2r) δ ˆβ 12 ρ(r) + O(δ σ 2 ). (7.34) By Lemma 6, ρ(2r) (ρ(r)) 2+ σ 4 and ρ(r) w 2 (R), wehaveforr = R δ, δ ˆβ 23 ρ(2r δ ) 2δ ˆβ 12 ρ(r δ ) δ ˆβ 23 (ρ(r δ )) 2+ σ 4 2δ ˆβ 12 ρ(r δ ) (7.35) [ ] δ( ρ(r δ ) ˆβ 23)ρ 1+ σ 4 (R δ ) 2δ ˆβ 12 and thus by (7.34) [ ] δδ ρ(r δ ( ) 2 1 2σ)(1+ σ 4 ) ( ˆβ 23 ) 2δ ˆβ 12 > 2ρ(R δ )δ 1 σ, M[R δ ] > 3I[w] + ρ(r δ )δ 1 σ. (7.36) On the other hand, by choosing R δ such that δ ˆβ 23 ρ(2 R) + δ ˆβ 13 ρ( R) = 0, (7.37) then we have M[R δ ] M[ R] 3I[w] δ ˆβ 12 ρ( R) + O(δ σ 2 ) 3I[w], (7.38) a contradiction to (7.36). It remains to show that (7.37) is possible since from (7.37) we have w 2 ( R) ρ( R) Cδ 1 2(1+ σ 4 ) 1 <δ 1 2 2σ and hence it is possible to have R δ, where C is a positive constant depending only on ˆβ 13 and ˆβ 23. Here we have used the fact that ρ(2r) (ρ(r)) 2+ σ 4 and ρ(r) w 2 (R). This proves that R δ ( δ ) 0.SoR δ is a critical point of M[R]. By Lemma 10, u δ = u Rδ + v Rδ is a critical point of E[u] and hence a bound state of (1.2).

24 652 T.-C. Lin, J. Wei 8. Proof of Corollary 1 In this section, we prove Corollary 1. First, substituting u j = ξ j w into Eq. (1.2), we obtain the following algebraic equation ξ j + β ij ξ i = 1, j = 1, 2, 3. (8.1) Since by our assumption β ij << 1, we see that solution to (8.1) exists and moreover, we have ξ j = 1 β ij + O(δ), j = 1, 2, 3. (8.2) Now we compute [ 3 E[u r ] = ( ξ j 2 ξ j 2 4 ) 1 ] β ij ξ i ξ j 4 j=1 i,j R n w 4 3 w 4 1 β ij w 4 R n 4 R 4 + O(δ) n i,j = 3 w δ ˆβ 23 w 4 + O(δ). (8.3) 4 R n 2 R n On the other hand, by (7.38), we have E[u δ ] < 3 w 4. (8.4) 4 R n From (8.3) and (8.4), we arrive at the following: E[u δ ] <E[u r ]. (8.5) Now if we have a ground state solution u δ which is radially symmetric, we have to show that for δ small, u δ = u r. In fact, since u δ is a ground state solution, we have that u δ is uniformly bounded. Letting δ 0, we see that u δ u 0 = (w,w,w) T. Thus u δ u r = o(1) as δ 0. To show that u δ = u r,weletu δ = u r + v δ. Then it is easy to see that v δ satisfies v δ,j v δ,j + 3w 2 v δ,j + O( δw v δ + v δ 2 ) = 0, j = 1, 2, 3. (8.6) Since the operator L 1,1 is uniformly invertible in radially symmetric function class (by Lemma 1) and v δ = o(1), we see that v δ,j = L 1 1,1 O( δw v δ + v δ 2 ) = O( δw v δ + v δ 2 ), j = 1, 2, 3, and hence v δ = 0 for δ small. This proves Corollary 1. Acknowledgement. The research of the first author is partially supported by a research Grant (NSC M-002) from NSC of Taiwan. The research of the second author is partially supported by RGC no and a Direct Grant from the Chinese University of Hong Kong.

25 Ground State of N Coupled Nonlinear Schrödinger Equations in R n,n References 1. Akhmediev, N., Ankiewicz, A.: Partially coherent solitons on a finite background. Phys. Rev. Lett. 82, 2661 (1999) 2. Alikakos, N., Kowalczyk, M.: Critical points of a singular perturbation problem via reduced energy and local linking. J. Diff. Eqns. 159, (1999) 3. Bates, P., Dancer, E. N., Shi, J.: Multi-spike stationary solutions of the Cahn-Hilliard equation in higher-dimension and instability. Adv. Differ. Eqs. 4, 1 69 (1999) 4. Bates, P., Fusco, G.: Equilibria with many nuclei for the Cahn-Hilliard equation. J. Diff. Eqns. 4, 1 69 (1999) 5. Bates, P., Shi, J.: Existence and instability of spike layer solutions to singular perturbation problems. J. Funct. Anal. 196(2), (2002) 6. Berestycki, H., Lions, P. L.: Nonlinear scalar field equaiton I. Existence of a ground state. Arch. Rat. Mech. Anal. 82(4), (1983) 7. Berestycki, H., Lions, P. L.: Nonlinear scalar field equaiton II. Existence of infinitely many solutions. Arch. Rat. Mech. Anal. 82(4), (1983) 8. Chow, K. W.: Periodic solutions for a system of four coupled nonlinear Schrodinger equations. Phys. Lett. A 285, 319 V326 (2001) 9. Christodoulides, D. N., Coskun, T. H., Mitchell, M., Segev, M.: Theory of incoherent self-focusing in biased photorefractive media. Phys. Rev. Lett. 78, 646 (1997) 10. Dancer, E. N., Yan, S.: Multipeak solutions for a singular perturbed Neumann problem. Pacific J. Math. 189, (1999) 11. Dancer, E. N., Yan, S.: Interior and boundary peak solutions for a mixed boundary value problem. Indiana Univ. Math. J. 48, (1999) 12. Esry, B. D., Greene, C. H., Burke, Jr. J. P., Bohn, J. L.: Hartree-Fock theory for double condensates. Phys. Rev. Lett. 78, (1997) 13. Floer, A., Weinstein, A.: Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential. J. Funct. Anal. 69, (1986) 14. Gidas, B., Ni, W.-M., Nirenberg, L.: Symmetry of positive solutions of nonlinear elliptic equations in R n, In: Mathematical Analysis and Applications, Part A, Sdv. Math. Suppl. Studies 7A, New York: Academic Press, 1981, pp Gui, C., Wei, J.: Multiple interior spike solutions for some singular perturbed Neumann problems. J. Diff. Eqs, 158, 1 27 (1999) 16. Gui, C., Wei, J.: On multiple mixed interior and boundary peak solutions for some singularly perturbed Neumann problems. Can. J. Math. 52, (2000) 17. Hioe, F. T.: Solitary Waves for N Coupled Nonlinear Schrödinger Equations. Phys. Rev. Lett. 82, (1999) 18. Hioe, F. T., Salter, T. S.: Special set and solutions of coupled nonlinear schrödinger equations. J. Phys. A: Math. Gen. 35, (2002) 19. Kanna, T., Lakshmanan, M.: Exact soliton solutions, shape changing collisions, and partially coherent solitons in coupled nonlinear Schrödinger equations. Phys. Rev. Lett. 86, 5043 (2001) 20. Kwong, M. K.: Uniqueness of positive solutions of u u + u p = 0inR n. Arch. Rational Mech. Anal. 105, (1989) 21. Lieb, E., Loss, M.: Analysis. Providence, RI: American Mathematical Society, Lin, C.-S., Ni, W.-M.: On the diffusion coefficient of a semilinear Neumann problem. Calculus of variations and partial differential equations (Trento, 1986), Lecture Notes in Math. 1340, Berlin- New York: Springer, 1988, pp Mitchell, M., Chen, Z., Shih, M., Segev, M.: Self-Trapping of partially spatially incoherent light. Phys. Rev. Lett. 77, (1996) 24. Ni, W.-M., Takagi, I.: Locating the peaks of least energy solutions to a semilinear Neumann problem. Duke Math. J. 70, (1993) 25. Oh,Y. G.: Existence of semi-classical bound states of nonlinear Schrödinger equations with potentials of the class (V ) a. Comm. PDE 13(12), (1988) 26. Oh, Y.G.: On positive multi-bump bound states of nonlinear Schrödinger equations under multiplewell potentials. Commun. Math. Phys. 131, (1990) 27. Troy, W. C.: Symmetry properties in systems of semilinear elliptic equations. J. Diff. Eq. 42(3), (1981) 28. Wei, J., Winter, M.: Stationary solutions for the Cahn-Hilliard equation. Ann. Inst. H. Poincaré Anal. Non Linéaire, 15, (1998) 29. Wei, J., Winter, M.: Multiple boundary spike solutions for a wide class of singular perturbation problems. J. London Math. Soc. 59, (1999) Communicated by P. Constantin