EQUATIONS IN Rn, n ≤ 3
TAI-CHIA LIN AND JUNCHENG WEI
Abstract. We establish some general theorems for the existence and nonexistence of ground state solutions of a steady-state N coupled nonlinear Schr¨odinger equations. The sign of coupling constants βij’s is crucial for the existence of ground state solutions. When all βij’s are positive and the matrix Σ is positively definite, there exists a ground state solution which is radially symmetric. However, if all βij’s are negative, or one of βij’s is negative and the matrix Σ is positively definite, there is no ground state solution. Furthermore, we find a bound state solution which is non-radially symmetric when N = 3.
1. Introduction
In this paper, we study solitary wave solutions of time-dependent N coupled nonlinear Schr¨odinger equations given by
−i∂ ∂tΦj = ∆Φj + µj|Φj|2Φj+ X i6=j βij|Φi|2Φj for y ∈ Rn, t > 0, Φj = Φj(y, t) ∈ C , j = 1, . . . , N, Φj(y, t) → 0 as |y| → +∞, t > 0, (1.1)
where µj > 0’s are positive constants, n ≤ 3, and βij’s are coupling constants. The
system (1.1) has applications in many physical problems, especially in nonlinear optics. Physically, the solution Φj denotes the j-th component of the beam in Kerr-like
photore-fractive media(cf. [1]). The positive constant µj is for self-focusing in the j-th component
of the beam. The coupling constant βij is the interaction between the i-th and the j-th
component of the beam. As βij > 0, the interaction is attractive, but the interaction is
repulsive if βij < 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 Schr¨odinger equations(cf. [8], [17], [18], [19]).
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
Key words and phrases. Coupled nonlinear Schr¨odinger equations, ground state solutions, bound states.
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 Φj(y, t) = eiλjtuj(y)
and we may transform the system (1.1) to steady-state N coupled nonlinear Schr¨odinger equations given by ∆uj − λjuj+ µju3j + X i6=j βiju2iuj = 0 in Rn, uj > 0 in Rn, j = 1, . . . , N, uj(y) → 0 as |y| → +∞, (1.2)
where λj, µj > 0 are positive constants, n ≤ 3, and βij’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 sate solutions may depend on coupling constants βij’s. When all βij’s are positive and the matrix Σ(defined in (1.9)) is positively definite,
there exists a ground state solution which is radially symmetric i.e. uj(y) = uj(|y|), j =
1, · · · , N . Such a radially symmetric solution may support the existence of N circular self-trapped beams. However, if all βij’s are negative, or one of βij’s is negative and the
matrix Σ 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∈H1(Rn) R Rn| 5 u|2+ λ1 R Rnu2 (RRnu4) 1 2 . (1.3)
An equivalent formulation, called Nehari’s manifold approach (see [6] and [7]), is to con-sider the following minimization problem
inf u1∈N1 E[u1] where N1 = ½ u ∈ H1(Rn) : u 6≡ 0 , Z Rn | 5 u|2+ λ 1 Z Rn u2 = µ 1 Z Rn u4 ¾ . (1.4) 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) 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 = (u1, . . . , uN) ∈ ¡ H1(Rn)¢N : u j ≥ 0, uj 6≡ 0 , (1.5) Z Rn | 5 uj|2+ λj Z Rn u2 j = µj Z Rn u4 j + X i6=j βij Z Rn u2 iu2j, j = 1, . . . , N )
c = inf
u∈NE[u] , (1.6)
where the associated energy functional is given by E[u] = N X j=1 Ã 1 2 Z Rn | 5 uj|2+ λj 2 Z Rn u2j − µj 4 Z Rn u4j ! (1.7) −1 4 N X i,j=1, i6=j βij Z Rn u2iu2j for u = (u1, . . . , uN) ∈ (H1(Rn))N. (1.8)
Since n ≤ 3, by Sobolev embedding, E[u] is well-defined. A minimizer u0 = (u0
1, . . . , u0N)
of (1.6), if exists, is called a ground state solution of (1.2), and it may have the following properties:
(1) u0
j > 0 , ∀j , and u0 satisfies (1.2);
(2) E[u0
1, . . . , u0N] ≤ E[v1, . . . , vN] for any other solution (v1, . . . , vN) of (1.2).
It is natural to ask when the ground state solution exists. As N = 1, the existence of ground state solution is trivial (see [6]). However, the existence of ground state solution with multi-components is quite complicated.
For general N ≥ 2, we introduce the following auxiliary matrix X
= (|βij|), where we set βii= µi. (1.9)
Our first theorem concerns all repulsive case:
Theorem 1.1. If βij < 0, ∀i 6= j, then ground state solution doesn’t exist, i.e. c defined
at (1.6) can not be attained.
Our second theorem concerns all attractive case. Theorem 1.2. If βij > 0, ∀i 6= j, and the matrix
P
(defined at (1.9)) is positively definite, then there exists a ground state solution (u0
1, . . . , u0N). All u0j must be positive,
radially symmetric and strictly decreasing.
When attraction and repulsion coexist i.e. some of βij’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 ground state solution doesn’t exist.
Theorem 1.3. If there exists an i0 such that
βi0j < 0, ∀j 6= i0, and βij > 0, ∀i 6= i0, j 6∈ {i, i0} (1.10)
and assume that the matrix Σ is positively definite, then 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, λ1 = λ2 = λ3 = µ1 = µ2 = µ3 = 1. (1.11)
Theorem 1.4. Assume that N = 3 and
β12= δ ˆβ12= β13= δ ˆβ13 > 0, β23=
√
δ ˆβ23< 0. (1.12)
Then for δ sufficiently small, problem (1.2) admits a non-radial solution uδ =¡uδ 1, uδ2, uδ3
¢ with the following properties:
uδ
1(y) ∼ w(y), uδ2(y) ∼ w(y − Rδe1), uδ3(y) ∼ w(y + Rδe1) ,
where
Rδ ∼ log1
δ, e1 = (1, 0, . . . , 0)
T ,
and w is the unique solution of the following problem ∆w − w + w3 = 0 in Rn w > 0 in Rn, w(0) = max y∈Rnw(y) w(y) → 0 as |y| → +∞. (1.13) Graphically, we have
Note that under condition (1.12), there is also a radially symmetric solution ur of the
following form: ur= (ur 1, ur2, ur3), urj = p ξjw(y), j = 1, 2, 3 where ξj satisfies ξj+ X i6=j βijξi = 1, j = 1, 2, 3. (1.14) Then we have
Corollary 1.5. Assume that N = 3 and (1.12) holds. Then for δ sufficiently small, we have
E[uδ] < E[ur], (1.15)
where uδ is constructed in Theorem 1.3. As a consequence, if the ground state solution
It is known that (1.2) admits many radially symmetric bound states(see [17] and [18]). Theorem 1.4 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 1.1-1.3 to simple cases N = 2 and N = 3.
For the case N = 2, we have Corollary 1.6. If N = 2, then
(1) for β12< 0, ground state solution doesn’t exist,
(2) for 0 < β12 <√µ1µ2, ground state solution exists.
For the case N = 3, the matrix Σ becomes
Σ = Ã µ 1 |β12| |β13| |β12| µ2 |β23| |β13| |β23| µ3 ! .
Assume that βij 6= 0. Then we may divide into four cases given by
Case I: all repulsive: β12< 0, β13 < 0, β23< 0,
Case II: all attractive: β12 > 0, β13> 0, β23 > 0,
Case III: two repulsive and one attractive: β12 < 0, β13< 0, β23> 0
Case IV: one repulsive and two attractive: β12> 0, β13 > 0, β23 < 0.
For case I–III, we have a complete picture Corollary 1.7. If N = 3, then
(1) for case I, ground state solution doesn’t exist,
(2) for case II and assume Σ is positively definite, ground state solution exists, (3) for case III and assume Σ is positively definite, ground state solution doesn’t exists. It then remains only to consider case IV. Due to the existence of non-radial bound states in Theorem 1.4 and non-radial property of ground states in Corollary 1.5, 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 Schr¨odinger equations with trap potentials:
∆uj − Vj(x)uj+ µju3j + X i6=j βiju2iuj = 0, x ∈ Rn, uj > 0 in Rn, j = 1, . . . , N, uj(x) → 0 as |x| → +∞. (1.16)
The main idea in proving Theorem 1.1-1.3 is by Nehari’s manifold approach and Schwartz symmetrization technique. Theorem 1.4 is proved by Liapunov-Schmidt re-duction method combined with variational method. The organization of the paper is as follows: In Section 2, we collect some properties of the function w-solution of (1.13) and Schwartz symmetrization. In Section 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 Σ is positively definite. The proofs of Theorems 1.1, 1.2, 1.3, 1.4 are given in Sections 4, 5, 6, 7, respectively. Section 8 contains the proof of Corollary 1.5.
Acknowledgments: The research of the first author is partially supported by a research Grant from NSC of Taiwan. He wants to express his sincere thanks to Prof. J. Wei and Z. Xin for their kindly hospitality during his visit of IMS in HongKong. The research of the second author is partially supported by an Earmarked Grant from RGC of Hong Kong. The support from IMS under the program “Geometric Analysis and Nonlinear PDE” is greatly appreciated.
2. Some Preliminaries
In this section, we analyze some problems in Rn. 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
w0(|y|) < 0 for |y| > 0
and w(|y|) = Anr− n−1 2 e−r µ 1 + O µ 1 r ¶¶ , as r = |y| → +∞, (2.1) w0(|y|) = −Anr− n−1 2 e−r µ 1 + O µ 1 r ¶¶ , as r = |y| → +∞. (2.2) We denote the energy of w as
I[w] = 1 2 Z Rn | 5 w|2+1 2 Z Rn w2− 1 4 Z Rn w4. (2.3)
Let wλ,µ be the unique solution to the following problem
∆wλ,µ− λwλ,µ+ µwλ,µ3 = 0 in Rn, wλ,µ > 0, wλ,µ(0) = max y∈Rnwλ,µ(y), wλ,µ(y) → 0 as |y| → +∞.
It is easy to see that
wλ,µ(y) = s λ µw ³√ λ|y| ´ , (2.4) and 1 2 Z Rn | 5 wλ,µ|2+ λ 2 Z Rn w2 λ,µ− µ 4 Z Rn w4 λ,µ= λ 4−n 2 µ−1I[w]. (2.5)
We now collect some of the properties of wλ,µ.
Lemma 2.1. (1) w(|y|) is the unique solution to the following minimization problem inf u∈H1(Rn), u≥0 R Rn| 5 u|2+ R Rnu2 (RRnu4) 1 2 . (2.6)
(2) The following eigenvalue problem ½
∆φ − λφ + 3µw2
λ,µφ = βφ
admits the following set of eigenvalues
β1 > 0 = β2 = . . . = βn+1> βn+2 ≥ . . . ,
where the eigenfunctions corresponding to the zero eigenvalue are spanned by K0 := span ½ ∂wλ,µ ∂yj , j = 1, . . . , n ¾ = C0. (2.8)
As a result, the following map
Lλ,µφ := ∆φ − λφ + 3µw2λ,µφ is invertible from K⊥ 0 → C0⊥ where K⊥ 0 = ( u ∈ H2(Rn) ¯ ¯ ¯ ¯ ¯ Z Rn u∂wλ,µ ∂yj = 0 , j = 1, · · · , n ) , (2.9) C⊥ 0 = ( u ∈ L2(Rn) ¯ ¯ ¯ ¯ ¯ Z Rn u∂wλ,µ ∂yj = 0 , j = 1, · · · , n ) . (2.10)
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 that
Iλ,µ[u] = 1 2 Z Rn | 5 u|2+ λ 2 Z Rn u2−µ 4 Z Rn u4. (2.11) We then have Lemma 2.2. inf u∈Nλ,µ
Iλ,µ[u] is attained only by wλ,µ,
where Nλ,µ= ( u ∈ H1(Rn) ¯ ¯ ¯ ¯ ¯ Z Rn | 5 u|2+ λ Z Rn u2 = µ Z Rn u4 ) . (2.12)
Proof. It is easy to see that inf
u∈Nλ,µ Iλ,µ[u] is equivalent to inf u≥0, u∈H1(Rn) R Rn| 5 u|2+ λ R Rnu2 (RRnu4) 1 2 .
The rest follows from (1) of Lemma 2.1. ¤
The next lemma is not so trivial. Lemma 2.3. inf
u∈N0 λ,µ
Iλ,µ[u] is also attained only by wλ,µ,
where N0 λ,µ= ( u ∈ H1(Rn) ¯ ¯ ¯ ¯ ¯ Z Rn | 5 u|2+ λ Z Rn u2 ≤ µ Z Rn u4 ) . (2.13)
Proof. Let uk be a minimizing sequence and u∗k be its Schwartz symmetrization. Then by
the property of symmetrization Z Rn | 5 u∗ k|2+ λ Z Rn (u∗ k)2 ≤ Z Rn | 5 uk|2+ λ Z Rn u2 k≤ µ Z Rn u4 k = µ Z Rn (u∗ k)4, (2.14) and Iλ,µ[u∗k] ≤ Iλ,µ[uk]. (2.15)
Hence, we may assume that uk is radially symmetric and decreasing. Since uk∈ H1(Rn),
and uk is strictly decreasing, it is well-known that
uk(r) ≤ Cr− N −1
2 kukkH1. (2.16)
So uk → u0(up to a subsequence) in L4(Rn), where u0 is also radially symmetric and
decreasing. Moreover, by Fatou’s Lemma, u0 ∈ Nλ,µ0 . Hence infu∈N0 λ,µ
Iλ,µ[u] can be attained
by u0.
We then claim that Z
Rn | 5 u0|2+ λ Z Rn u2 0 = µ Z Rn u4 0. (2.17)
Suppose not. That is Z Rn | 5 u0|2+ λ Z Rn u2 0 < µ Z Rn u4 0.
Then u0 ∈ (Nλ,µ0 )0 - the interior of Nλ,µ0 . By standard elliptic theory, u0 is a critical point
of Iλ,µ[u], i.e.
5 Iλ,µ[u0] = 0 (2.18)
where “5” means the derivative. Multiplying (2.18) by u0, we have R Rn| 5 u0|2 + λ R Rnu20 = µ R Rnu40 a contradiction. Hence u0 ∈ Nλ,µ. By Lemma 2.2, u0 = q λ µw( √ λ|y|) = wλ,µ(y). ¤
We present another characterization of wλ,µ:
Lemma 2.4.
inf
u∈Nλ,µ
Iλ,µ[u] = u≥0,inf u∈H1(Rn)
sup
t>0Iλ,µ[tu].
Proof. This follows from a simple scaling. ¤
Finally, we recall the following well-known result, whose proof can be found in Theorem 3.4 of [21].
Lemma 2.5. Let u ≥ 0, v ≥ 0, u, v ∈ H1(Rn) and u∗, v∗ be their Schwartz
Symmetriza-tion. Then Z Rn uv ≤ Z Rn u∗v∗ Our last lemma concerns some integrals.
Lemma 2.6. Let y1 6= y2 ∈ Rn. Then as |y1− y2| → +∞, we have for λ1 < λ2, Z Rn w2 λ1,µ1(y − y1)w 2 λ2,µ2(y − y2) ∼ w 2 λ1,µ1(y1− y2) Z Rn w2 λ2,µ2(z)e 2√λ1 D z,|y1−y2|y1−y2 E dz. (2.19) If λ1 = λ2, then wλ2+σ1,µ1(y1− y2) ≤ Z Rn wλ21,µ1(y − y1)wλ22,µ2(y − y2) ≤ w 2−σ λ1,µ1(y1− y2) (2.20) for any 0 < σ < 1.
Proof. Let y = y2+ z. Then from (2.1), we have
w2 λ1,µ1(y − y1)w 2 λ2,µ2(y − y2) = w2λ1,µ1(y2− y1+ z)w2λ2,µ2(y − y2) = w2 λ1,µ1(y2− y1)e
2√λ1(|y2−y1|−|y2−y1+z|)(1 + o(1))w2
λ2,µ2(y − y2) = w2 λ1,µ1(y1− y2)(1 + o(1))e 2√λ1 D z,|y1−y2|y1−y2 E w2 λ2,µ2(z) .
Hence by Lebesgue Dominated Convergence Theorem gives (2.19). The proof of (2.20) is similar.
¤
3. Nehari’s Manifold Approach
In this section, we consider the relation between two minimization problems Problem 1: c = inf u∈NE[u] (3.1) where N = ( u ∈ (H1(Rn))N ¯ ¯ ¯ ¯ ¯ Z Rn |5uj|2+λj Z Rn u2j = µj Z Rn u4j+X i6=j βij Z Rn u2iu2j, j = 1, . . . , N ) . Problem 2: m = inf u≥0t1,...,tsupN>0E[ √ t1u1, . . . , √ tNuN]. (3.2) We have
Theorem 3.1. Suppose either βij < 0, ∀i 6= j, or the matrix Σ defined by
Σ = (|βij|) with βii= µi
is positively definite. Then c = inf
u∈NE[u] = m = infu≥0t1,...,tsupN>0E[
√
t1u1, . . . ,
√ tNuN].
Proof. We consider the following function β(t1, . . . , tN) = E[ √ t1u1, . . . , √ tNuN]. First we assume u ∈ N.
Claim 1: β(t1, . . . , tN) attains its global maximum at t1 = . . . = tN = 1. In fact,
β(t1, . . . , tN) = N X j= 1 tj "Z Rn | 5 uj|2+ λju2j # −Q[t1, . . . , tN] where Q[t1, . . . , tN] = 1 4 N X j= 1 µjt2j Z Rn u4 j + 1 4 N X i,j=1, i6=j βijtitj Z Rn u2 iu2j = 1 4t TΣ0t where t = (t1, ..., tN)T and Σ0 = Ã βij Z Rn u2 iu2j ! . (3.3)
If βij < 0, ∀i 6= j, then since u ∈ N, we have
µj Z Rn u4 j + X i6=j βij Z Rn u2 iu2j = Z Rn | 5 uj|2 + λj Z Rn u2 j > 0 .
Moreover, we see that the matrix Σ0 is diagonally dominant and hence Σ0 is positively
definite.
If βij > 0 for all i 6= j, then for tj > 0, j = 1, . . . , N
Q[t1, . . . , tN] = 1 4 " X i,j βijtitj Z Rn u2 iu2j # ≥ 1 4 " N X j=1 µjt2j Z Rn u4j # −1 4 N X i,j=1, i6=j |βij| ÃZ Rn u4i !1 2ÃZ Rn u4j !1 2 titj > 0.
Again, Q[t1, . . . , tN] is positively-definite. Thus β(t1, . . . , tN) 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 inf
u∈NE[u] ≥ infu≥0t1sup,...,tNE[
√
t1u1, . . . ,
√
tNuN]. (3.4)
On the other hand, suppose that sup
t1,...,tN
E[√t1u1, . . . ,
√
where u = (u1, · · · , uN) ≥ 0. Certainly, (t01, . . . , t0N) is a critical point of β(t1, . . . , tN) and hence (u0 1, . . . , u0N) ≡ ³p t0 1u1, . . . , p t0 NuN ´ ∈ N. So E[u01, . . . , u0N] = β(t01, . . . , t0N) ≥ inf u∈NE[u] which proves c = inf
u∈NE[u] ≤ m = infu≥0t1sup,...,tNE[
√
t1u1, . . . ,
√
tNuN]. (3.5)
Combining (3.4) and (3.5), we obtain Theorem 3.1.
¤
4. Proof of Theorem 1.1 In this section, we prove Theorem 1.1.
First by Theorem 3.1, c = inf u≥0t1sup,...,tNE[ √ t1u1, . . . , √ tNuN]. Now we choose uj(y) := wλj,µj(y − jRe1), j = 1, . . . , N (4.1)
where R >> 1 is a large number and e1 = (1, 0, . . . , 0)T.
By choosing R large enough and applying Lemma 2.5, we obtain that Z Rn u2 iu2j = Z Rn w2 λi,µi(y − iRe1)w 2 λj,µj(y − jRe1) = Z Rn w2 λi,µi(y)w 2 λj,µj(y + (i − j)Re1)dy → 0 as R → +∞. Let (tR
1, . . . , tRN) be the critical point of β(t1, . . . , tN). Then we have
Z Rn | 5 uj|2+ λj Z Rn u2 j = µjtRj Z Rn u4 j + X i6=j βijtRi Z Rn u2 iu2j
since the matrix µ βij Z Rn u2iu2j ¶
is positively definite (similar to arguments in Section 3), by implicit function theorem
tR j = 1 + o(1). Thus c ≤ lim R→+∞β(t R 1, . . . , tRN) = N X j= 1 " 1 2 µZ Rn | 5 wj|2+ λjw2j ¶ − µj 4 Z Rn w4j # . (4.2) Next we claim that
c ≥ N X j= 1 " 1 2 µZ Rn | 5 wj|2+ λjwj2 ¶ − µj 4 Z Rn w4 j # . (4.3)
In fact, let (u1, . . . , uN) ∈ N, then since βij < 0, ∀i 6= j E[u1, . . . , uN] ≥ n X j= 1 " 1 2 ÃZ Rn | 5 uj|2+ λju2j ! −µj 4 Z Rn u4j # (4.4) = n X j= 1 Iλj,µj[uj] , and Z Rn | 5 uj|2+ λju2j ≤ µj Z Rn u4 j. (4.5) By Lemma 2.2, E[u1, . . . , uN] ≥ n X j= 1 Iλj,µj[uj] (4.6) ≥ n X j= 1 inf w∈N0 λj ,µj Iλj,µj[w] = N X j= 1 Iλj,µj[wλj,µj]
which proves (4.3). Hence
c = N X j= 1 Iλj,µj[wλj,µj]. (4.7) If c is attained by some (u0
1, . . . , u0N). Then= (u01, . . . , u0N) ∈ N and u0j is a solution of
(1.2). By the Maximum Principle, u0
j > 0, j = 1, . . . , N. Then we have c = E[u0 1, . . . , u0N] > N X j= 1 Iλj,µj[u 0 j] ≥ N X j= 1 Iλj,µj[wλj,µj] (4.8)
which contradicts to (4.7), and we may complete the proof of Theorem 1.1.
5. Proof of Theorem 1.2
Now we prove Theorem 1.2 in this section. Our main idea is by Schwartz symmetriza-tion.
For uj ≥ 0, uj ∈ H1(Rn), we denote u∗j as its Schwartz symmetrization. By Lemma
2.6, for i 6= j Z Rn u2iu2j ≤ Z Rn (u∗i)2(u∗j)2. (5.1) Hence E[u∗ 1, . . . , u∗N] ≤ E[u1, . . . , uN]. (5.2)
The new function u∗ = (u∗
1, . . . , u∗N) will satisfy the following inequalities:
Z Rn | 5 u∗ j|2+ λ1 Z Rn (u∗ j)2− X i6=j βij Z Rn (u∗ i)2(u∗j)2 (5.3) ≤ µj Z Rn (u∗ j)4
(by (5.1) and the fact that βij > 0).
Therefore, we have
c = inf
u∈NE[u] ≥ infu∈N0E[u] := c 0 where N0 = ( u ∈ (H1(Rn))N ¯ ¯ ¯ ¯ ¯ Z Rn | 5 uj|2+ λju2j ≤ µj Z Rn u4 j+ X i6=j βij Z Rn u2 iu2j, j = 1, . . . , N ) (5.4) We first study c0 and then we show that c0 = c.
By the previous argument, we may assume any minimizing sequence (u1, . . . , uN) of c0
must be radially symmetric and decreasing. We follow the proof of Lemma 2.2 to conclude that a minimizer for c0 exists and must be radially symmetric and decreasing. Moreover,
we have Z Rn | 5 uj|2+ λju2j ≤ µj Z Rn u4 j + X i6=j βij Z Rn u2 iu2j, j = 1, . . . , N. (5.5)
If all the inequalities of (5.5) are strict, then as for the proof of Lemma 2.2, 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
Gj[u] := Z Rn | 5 uj|2+ λj Z Rn u2 j − µj Z Rn u4 j − X i6=j βij Z Rn u2 iu2j = 0, j = 1, . . . , k < N (5.6) Then we have 5 E[u1, . . . , uN] + k X j= 1 Λj 5 Gj[u1, . . . , uN] = 0 (5.7)
where Gj is defined at (5.6). We assume that Λk+1 = . . . = ΛN = 0 and we write (5.7)
as 5 E[u1, . . . , uN] + N X j= 1 Λj 5 Gj[u1, . . . , uN] = 0 (5.8) From (5.6), we obtain N X j= 1 Λjh5Gj, uji = 0
which is equivalent to N X j= 1 Ã βij Z Rn u2iu2j ! Λj = 0
since the matrix Σ0 is positively define, the matrix
à βij Z Rn u2iu2j ! is non-singular
and hence Λj = 0, j = 1, . . . , N . As for the proof of Lemma 2.1, u ∈ N. Hence c = c0
and c can be achieved by radially symmetric pairs (u0
1, . . . , u0N). Hence (u01, . . . , u0N) must
satisfy (1.2).
By the maximum principle, u0
j > 0, since u0j satisfies ∆u0 j − λju0j + µj(u0j)3+ X i6=j βij(u0i)2u0j = 0, βij > 0
by the moving plane method for cooperative systems (cf. [27]), u0
j must be radially
sym-metric and strictly decreasing. Therefore we may complete the proof of Theorem 1.2.
6. Proof of Theorem 1.3
In this section, we prove Theorem 1.3. The proof combines those of Theorem 1.1 and Theorem 1.2.
Assume u = (u1, . . . , uN) ∈ N. Without loss of generality, we may assume that i0 = 1.
Then
β1j < 0, ∀j > 0, and βij > 0, ∀i > 1, j 6∈ {1, i}.
We may divide the energy E[u1, . . . , uN] into two parts
E[u1, . . . , uN] = 1 2 Z Rn |5u1|2+ λ1 2 Z Rn u2 1− µ1 4 Z Rn u4 1− 1 2 N X j=2 β1j Z Rn u2 1u2j+E0[u2, . . . , uN] (6.1) where E0[u 2, . . . , uN] = N X j= 2 Ã 1 2 Z Rn | 5 uj|2+ λj 2 Z Rn u2 j − µj 4 Z Rn u4 j ! (6.2) −1 4 N X i,j=2, i6=j βij Z Rn u2 iu2j. Since β1j < 0, for j > 1, E[u1, . . . , uN] ≥ Iλ1,µ1[u1] + E 0[u 2, . . . , uN]. (6.3)
On the other hand, u1 satisfies Z Rn | 5 u1|2+ λ1 Z Rn u21− µ1 Z Rn u41 = N X j=2 β1j Z Rn u21u2j ≤ 0 (6.4) and uj, j = 2, . . . , N satisfies Z Rn | 5 uj|2+ λ1 Z Rn u2 j ≤ µj Z Rn u4 j + N X i=2, i6=j βij Z Rn u2 iu2j. (6.5)
Here we have used the system (1.2) and the fact that β1j < 0, for j > 1. By the proof of
Theorem 1.2, E0[u2, . . . , uN] ≥ inf (u2,...,uN)∈N1 E0[u2, . . . , uN] = c1 (6.6) where N1 = ( u0 = (u 2, . . . , uN) ¯ ¯ ¯ ¯ ¯ Z Rn | 5 uj|2+ λj Z Rn u2 j = µj Z Rn u4 j + N X i=2, i6=j βij Z Rn u2 iu2j ) .
On the other hand, by Lemma 2.3,
Iλ1,µ1[u1] ≥ Iλ1,µ1[wλ1,µ1]. (6.7)
Hence
inf
u∈NE[u] ≥ Iλ1,µ1[wλ1,µ1] + c1. (6.8)
Now we claim that
inf
u∈NE[u] = Iλ1,µ1[wλ1,µ1] + c1. (6.9)
In fact, by Theorem 3.1, c = inf
u∈NE[u] = infu≥0t1,...,tsupN≥0E[
√ t1u1, . . . , √ tNuN]. Now we choose u1 = wλ1,µ1(y − Re1)
and uj = u0j for j ≥ 2, where (u02, . . . , u0N) is a minimizer of c1 at (6.6). Then
Z Rn u21(u0j)2 → 0 as R → +∞ , ∀j > 1 . Thus if we set β(tR 1, . . . , tRN) = sup t1,...,tN≥0 E[√t1u1, . . . , √ tNuN] , then tR j = 1 + o(1) and c ≤ lim R→+∞β(t R 1, . . . , tRN) = Iλ1,µ1[wλ1,µ1] + c1. (6.10)
This, combined with (6.8), proves that
Finally, we show that c is not attained. In fact, if c is attained by some (u0 1, . . . , u0N), u0 j > 0, then c = E[u0 1, . . . , u0N] > Iλ1,µ1[u 0 1] + E0[u02, . . . , u0N] ≥ Iλ1,µ1[wλ1,µ1] + c1. A contradiction!
Remark 1. Theorem 1.3 also holds if βij satisfies
βij < 0, for i = i1, . . . , ik, j 6= i1, . . . , ik
and
βij > 0, for i 6∈ {i1, . . . , ik}, j 6= i.
7. Proof of Theorem 1.4
In this section, we construct non-radial bound state of (1.2) in the following case: N = 3, λ1 = λ2 = λ3 = µ1 = µ2 = µ3 = 1, (7.1)
β23=
√
δ ˆβ23< 0 , (7.2)
β12 = δ ˆβ12= β13= δ ˆβ13> 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 u2 and u3 can be exchanged.
We shall make use of the so-called Liapunov-Schmidt reduction process and varia-tional approach. The Liapunov-Schmidt reduction method was first used in nonlinear Schr¨odinger 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], [3], [4], [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
Sj[u] = ∆uj− uj + u3j + X i6=j βiju2iuj, (7.4) S[u] = S1[u] ... SN[u] ,
X = (H2(Rn) ∩ {u | u(x1, x0) = u(x1, |x0|)})3 ∩{(u1, u2, u2) | u2(x1, x0) = u3(−x1, x0)}, Y = (L2(Rn) ∩ {u | u(x 1, x0) = u(x1, |x0|)})3 ∩{(u1, u2, u2) | u2(x1, x0) = u3(−x1, x0)}, wRj(y) = w (y − R je1) , (7.5) X = (H2(Rn) ∩ {u | u(x 1, x0) = u(x1, |x0|)})3, X0 = X ∩ {(u1, u2, u3) | u2(x1, x0) = u3(−x1, x0) , u1(x1, x0) = u1(−x1, x0)}, Y = (L2(Rn) ∩ {u | u(x 1, x0) = u(x1, |x0|)})3, Y0 = Y ∩ {(u1, u2, u3) | u2(x1, x0) = u3(−x1, x0) , u1(x1, x0) = u1(−x1, x0)}.
Note that S[u] is invariant under the map
T : (u1(x1, x0), u2(x1, x0), u3(x1, x0)) (7.6)
→ (u1(−x1, x0), u3(−x1, x0), u2(−x1, x0)).
Thus S is map from X0 to Y0.
Fix R ∈ Λδ, where
Λδ = {R | w(R) < δ
1
4−σ}. (7.7)
Here we may choose
σ = 1 1000. We define uR := (w(y), w(y − Re 1), w(y + Re1))T (7.8) = (w, wR, w−R)T. We begin with
Lemma 7.1. The map
L0Φ = Ã ∆φ1− φ1+ 3w2φ1 ∆φ2− φ2+ 3(wR)2φ2 ∆φ3− φ3+ 3(w−R)2φ3 ! : X0 → Y0 (7.9)
has its kernel
K0 = span (Ã 0,∂wR ∂y1 , −∂w−R ∂y1 !T) (7.10) and cokernel C0 = span (Ã 0,∂w R ∂y1 , −∂w −R ∂y1 !T) . (7.11)
Proof. In fact, L0Φ = 0. Then we have by Lemma 2.1 (2) φ1 = n X j= 1 c1,j ∂w ∂yj , φ2 = n X j= 1 c2,j ∂wR ∂yj , φ3 = n X j= 1 c3,j ∂w−R ∂yj . (7.12)
Since (φ1, φ2, φ3)T ∈ X0, we have φ1(x1, |x0|) = φ1(−x1, |x0|) = φ1(x1, x0). This forces
φ1 = 0. Similarly, we have c2,2 = ... = c2,n = 0, c3,2 = ... = c3,n = 0. On the other
hand, φ2(x1, x 0
) = φ3(−x1, x 0
). So we have c2,1 = −c3,1. This proves (7.10). Since L0 is a
self-adjoint operator, (7.11) follows from (7.10).
¤ From Lemma 7.1, we deduce that
Lemma 7.2. The map
L := S0[uR] (7.13)
is uniformly invertible from
L := K⊥0 → C⊥0. (7.14)
Proof. We may write
L = L0+
√
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 7.2, we derive the following proposition:
Proposition 7.3. For δ sufficiently small, and R ∈ Λδ, there exists a unique solution
vR = (vR 1, v2R, vR3) such that S1[uR+ vR] = 0, (7.16) S2[uR+ vR] = cR ∂wR ∂y1 , (7.17) S3[uR+ vR] = −cR ∂w−R ∂y1 , (7.18)
for some constant cR. Moreover, vR is of C1 in R and we have
kvRk
H2(Rn) ≤ cδ1−2σ. (7.19)
Proof. Let R1 = 0, R2 = R, R2 = −R and
wRj = w(y − R je1).
We choose v ∈ B, where
B = {v ∈ X | kvkH2 < δ1−2σ} (7.20)
and then expand
S1[uR+ v] = ∆v1 − v1+ 3(wR1)2v1+ [(wR1 + v1)3− (wR1)3− 3(wR1)2v1]
+δ[ ˆβ12(wR2 + v2)2+ ˆβ13(wR3 + v3)2](wR1 + v1)
= L1v1+ H1[v1] + E1
where
E1 = δ[ ˆβ12(wR2 + v2)2+ ˆβ13(wR3 + v3)2](wR1 + v1), and H1[v1] = [(wR1 + v1)3− (wR1)3− 3(wR1)2v1] = O(|v1|2). Here we have E1 = O(δ)(wR2wR1 + wR3wR1) (7.21) = O(δ)(w(|R1− R2|) + w(|R1− R3|) = O(δ54−σ). Similarly, S2[uR+ v] = L2v2+ H2[v2] + E2, S3[uR+ v] = L3v3+ H3[v3] + E3, where L2v2 = ∆v2− v2+ 3(wR2)2v2, (7.22) L3v3 = ∆v3− v3+ 3(wR3)2v3, E2 = O(1)[δ ˆβ12(wR1)2+ √ δ ˆβ23(wR3)2]wR2, = O(δ54−σ+ δ1−σ) = O(δ1−σ), E3 = O(1)[δ ˆβ13(wR1)2+ √ δ ˆβ23(wR2)2]wR3 = O(δ1−σ), H2[v2] = [(wR2 + v2)3− (wR2)3− 3(wR2)2v2] = O(|v2|2), H3[v3] = [(wR3 + v3)3− (wR3)3− 3(wR3)2v3] = O(|v3|2). Since L : K⊥
0 → C⊥0 is invertible, solving (7.16)-(7.18) is equivalent to solving
Π ◦ [Lv + H[v] + E] = 0 , v ∈ K⊥
0 , (7.23)
where Π is the orthogonal projection on C⊥
0 and v = (v1, v2, v3)T, H[v] = (H1, H2, H3)T, E =
(E1, E2, E3)T. The equation (7.23) can be written in the following form
v = G[v] := (Π ◦ L ◦ Π0)−1[−H[v] − E] (7.24)
where Π0 is the orthogonal projection on K⊥ 0.
Since H[v] = O(|v|2) and E = O(δ1−σ), it is easy to see that the map G defined at
(7.24) is a contraction map from B to B. By contraction mapping theorem, (7.23) has a unique solution vR= (vR
1, v2R, vR3) ∈ K⊥0 with the property that
kvRkH2(Rn) ≤ CkEk1−σL2(Rn) (7.25)
≤ C(δ(1−σ)(1−σ)) ≤ Cδ1−2σ.
The C1 property of vR follows from the uniqueness of vR. See similar proof in Lemma
3.5 of [15]. ¤
Now we let
M[R] = E[uR+ vR] : Λ
δ → R1
Lemma 7.4. For R ∈ Λδ and δ sufficiently small, we have M[R] = 3I[w] (7.26) −1 2 " √ δ ˆβ23 Z Rn (wR)2(w−R)2+ 2δ ˆβ 12 Z Rn w2(wR)2 # +O Ã δ32+ σ 2 ! . Proof. We may calculate that
M[R] = E[uR+ vR] = 3 X j= 1 ( 1 2 "Z Rn | 5 (wRj + vR j )|2+ Z Rn (wRj+ vR j )2 # −1 4 Z Rn (wRj+ vR j )4 ) −1 4 X i,j i6=j βij Z Rn (wRi+ vR i )2(wRj + vRj )2 = E[uR] + 3 X j= 1 1 2 "Z Rn (| 5 vR j |2+ (vRj )2) − 3(wRj)2(vjR)2 # −1 2 X i,j i6=j βij Z Rn " wRivR i (wRj)2 + wRjvjR(wRi)2 # +O(δ2−4σ) = E[uR] + O(δ2−4σ) − " β23 ÃZ Rn vR 2wR2(wR3)2+ Z Rn vR 3(wR3)(wR2)2 !# = E[uR] + O(δ32+σ2).
Here we have used the assumption (1.12), the equation (1.13), and Proposition 7.3. Since E[uR] = 3I[w]−1 2 " √ δ ˆβ23 Z Rn (wR2)2(wR3)2+δ ˆβ 12 Z Rn (wR1)2(wR2)2+δ ˆβ 13 Z Rn (wR1)2(wR3)2 # , and β12= β13, R2 = −R3, we obtain (7.26). ¤ Next we have Lemma 7.5. If Rδ ∈ (Λ
δ)0 – the interior of Λδ is a critical point of M[R], then the
corresponding solution uδ = uRδ
+ vRδ
is a critical point of E[u]. Proof. Since Rδ ∈ (Λ
δ)0 – the interior of Λδ is a critical point of M[R], we then have
d dRM[R] ¯ ¯ ¯ ¯ ¯ R=Rδ = 0
which is equivalent to < ∇E[uR+ vR], d dR(u R+ vR) > ¯ ¯ ¯ ¯ ¯ R=Rδ = 0. Using proposition 7.3, we obtain
cR Z Rn ∂wR ∂y1 d dR(w R+ vR 2) − cR Z Rn ∂w−R ∂y1 d dR(w −R+ vR 3) = 0 (7.27)
for R = Rδ. Note that since v ∈ K⊥
0, we have Z Rn " ∂wR ∂y1 vR2 −∂w −R ∂y1 v3R # = 0. (7.28)
Differentiating (7.28) with respect to R, we obtain that Z Rn " ∂wR ∂y1 d dRv R 2 − ∂w−R ∂y1 d dRv R 3 # = − Z Rn " ∂2wR ∂R∂y1 vR 2 − ∂2w−R ∂R∂y1 vR 3 # = O(δ1−2σ). (7.29)
On the other hand, we see that Z Rn " ∂wR ∂y1 d dR(w R) − ∂w−R ∂y1 d dR(w −R) # = −2 Z Rn (∂w ∂y1 )2. (7.30) From (7.27), (7.29) and (7.30), we deduce that
cR= 0, for R = Rδ, (7.31)
which then implies that the corresponding solution uδ = uRδ
+ vRδ
is a critical point of E[u].
¤ Finally, we prove Theorem 1.4.
Proof of Theorem 1.4. We consider the following minimization problem M0 = min
R∈¯Λδ
M[R] (7.32)
since M[R] is continuous and ¯Λδ is closed, M[R] attains its minimum at a Rδ∈ ¯Λδ.
We claim that Rδ 6∈ ∂ ¯Λ
δ. Suppose not. That is Rδ ∈ ∂ ¯Λδ. Then we have w(Rδ) = δ
1 4−σ. Let ρ(R) = Z Rn w2(y)w2(y − Re 1). (7.33)
Then from Lemma 7.4, we have M[R] = 3I[w] − 1 2 √ δ ˆβ23ρ(2R) − δ ˆβ12ρ(R) + O(δ 3 2+ σ 2). (7.34)
By Lemma 2.6, ρ(2R) ≥ (ρ(R))2+σ4 and ρ(R) ≥ w2(R), we have for R = Rδ, −√δ ˆβ23ρ(2Rδ) − 2δ ˆβ12ρ(Rδ) ≥ − √ δ ˆβ23(ρ(Rδ))2+ σ 4 − 2δ ˆβ12ρ(Rδ) (7.35) ≥ ρ(Rδ) " √ δ(− ˆβ23)ρ1+ σ 4(Rδ) − 2δ ˆβ12 # ≥ ρ(Rδ) " √ δδ(1 2−2σ)(1+σ4)(− ˆβ23) − 2δ ˆβ12 # > 2ρ(Rδ)δ1−σ and thus by (7.34) M[Rδ] > 3I[w] + ρ(Rδ)δ1−σ (7.36) On the other hand, by choosing ¯R ∈ Λδ such that
√ δ ˆβ23ρ(2 ¯R) + δ ˆβ13ρ( ¯R) = 0 , (7.37) then we have M[Rδ] ≤ M[ ¯R] ≤ 3I[w] − δ ˆβ 12ρ( ¯R) + O(δ 3 2+σ2) ≤ 3I[w]. (7.38) A contradiction to (7.36).
It remains to show that (7.37) is possible since from (7.37) we have w2( ¯R) ≤ ρ( ¯R) ≤ Cδ12(1+σ4)
−1
< δ12−2σ
and hence it is possible to have ¯R ∈ Λδ, where C is a positive constant depending only
on ˆβ13 and ˆβ23. Here we have used the fact that
ρ(2R) ≥ (ρ(R))2+σ
4 and ρ(R) ≥ w2(R) .
This proves that Rδ ∈ (¯Λ
δ)0. So Rδ is a critical point of M[R]. By Lemma 7.5, uδ =
uRδ
+ vRδ
is a critical point of E[u] and hence a bound state of (1.2).
8. Proof of Corollary 1.5 In this section, we prove Corollary 1.5.
First, substituting uj =
p
ξjw into the equation (1.2), we obtain the following algebraic
equation
ξj +
X
i6=j
βijξi = 1, j = 1, 2, 3 (8.1)
Since by our assumption |βij| << 1, we see that solution to (8.1) exists and moreover, we
have
ξj = 1 −
X
i6=j
Now we compute E[ur] = " 3 X j=1 (ξj 2 − ξ2 j 4) − 1 4 X i,j i6=j βijξiξj #Z Rn w4 ≥ 3 4 Z Rn w4−1 4 X i,j i6=j βij Z Rn w4+ O(δ) = 3 4 Z Rn w4+1 2 √ δ| ˆβ23| Z Rn w4+ O(δ) . (8.3)
On the other hand, by (7.38), we have E[uδ] < 3
4 Z
Rn
w4. (8.4)
From (8.3) and (8.4), we arrive at the following
E[uδ] < E[ur]. (8.5)
Now if we have a ground state solution uδ which is radially symmetric, we have to
show that for δ small, uδ = ur. In fact, since uδ is a ground state solution, we have
that uδ is uniformly bounded. Letting δ → 0, we see that uδ → u0 = (w, w, w)T. Thus
uδ− ur= o(1) as δ → 0.
To show that uδ = ur, we let uδ= ur+ vδ. Then it is easy to see that vδ satisfies
∆vδ,j− vδ,j+ 3w2vδ,j + O(
√
δw|vδ| + |vδ|2) = 0, j = 1, 2, 3 (8.6)
Since the operator L1,1 is uniformly invertible in radially symmetric function class (by
Lemma 2.1) and vδ = o(1), we see that
vδ,j = L−11,1◦ O(
√
δw|vδ| + |vδ|2) = O(
√
δw|vδ| + |vδ|2), j = 1, 2, 3
and hence vδ = 0 for δ small.
This proves the Corollary 1.5.
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Department of Mathematics, Chung Cheng University, Minghsiung, Chia Yi, Taiwan E-mail address: [email protected]
Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong E-mail address: [email protected]
