Multiplicity of positive solutions for two coupled nonlinear Schrödinger equations

Multiplicity of positive solutions for two coupled

nonlinear Schrodinger equations

Tsung-fang Wu

Department of Applied Mathematics

National University of Kaohsiung, Kaohsiung, Taiwan

Outline

• The History of BEC • Motivation

• Nehari manifold

• Palais-Smale sequences • Our main result

2

Motivation

We consider a two-component system of nonlinear Schr¨odinger equations with optical lattice potentials:

       ~2 2m4Φj − ˜Vj(x)Φj + P2 i=1 β˜ij|Φi| 2 _Φ j = −i~ ∂tΦj for x ∈ Ω , Φj(x, t) = 0 for x ∈ ∂Ω , t > 0 , j = 1, 2, (2.1) where Ω ⊆ RN , N = 2, 3 is the region for condensate dwelling, Φj’s

Here each N_j ≥ 1 is a fixed number of atoms in the hyperfine state |ji, and U_ij = 4π ~2

2maij, where ajj’s and a12 are the intraspecies

and interspecies scattering lengths. ˜Vj is the optical lattice

potential for the j-th species and is a periodic function of spatial variables written as (see [Jaksch etc., Phys. Rev. Lett., 1998])

˜ Vj(x) = N X k=1 ˜ νj,k sin2(l xk) for x = (x1, · · · , xN) ∈ Ω, j = 1, 2 , (2.2) where ˜νj,k ≥ 0 is the associated potential depth, l = 2π/L, and L is

Here, we study steady state bright solitons with the form

Φj = ei˜λj t/~ uj(x), j = 1, 2 called bound states of the system (2.1),

where ˜λj’s are positive constants and uj’s satisfy

       ~2 2m4uj(x) − ˜Vj(x)uj(x) + P2 i=1 β˜iju 2 i(x)uj(x) = ˜λjuj in Ω , uj > 0 in Ω , u_j = 0 on ∂Ω. (2.3) Physically, uj’s are the associated condensate amplitudes of the

bound states Φj’s, and ˜λj’s are chemical potentials. Due to

By rescaling and some simple assumptions, the problem (2.3) with very large ˜βjj’s, ˜λj’s and ˜νj,k’s is equivalent to the following

singularly perturbed problem:        24uj − ˆVj(x)uj + P 2 i=1 βiju 2 iuj = 0 in Ω, uj > 0 in Ω , uj = 0 on ∂Ω , j = 1, 2, (2.4) where 0 <   1 is a small parameter, and ˆV_j’s are defined by

ˆ Vj(x) = λj + N X k=1 νj,k sin2(l xk), for x = (x1, · · · , xN) ∈ Ω . (2.5)

Problems

To see how the potentials act on the existence and number of steady state bright solitons (i.e. ground states or bound states). i.e. we study the existence and multiplicity of positive solutions of problem (2.4).

Without loss of generality, problem (2.4) can be generalized to the following problem:        ε2∆u − V1 (x) u + µ1u3 + βuv2 = 0 in RN, ε2∆v − V2 (x) v + µ2v3 + βu2v = 0 in RN, u > 0, v > 0 _{in R}N, (Eε)

where N = 1, 2, 3, and the potentials V₁, V_{2 ∈ C R}N
satisfy (I) V1 and V2 are bounded functions on RN and

Λl = sup_x∈RN |V_l (x) | < ∞ , l = 1, 2.

(II) The global minima of potentials Vl, l = 1, 2 only occur at κ

points a1, a2, . . . , aκ _{∈ R}N if l = 1, and m points b1, b2, . . . , bm _{∈ R}N if l = 2, respectively.

Moreover,

V1 ai
= λ1 ≡ min V1 (x) : x ∈ RN > 0 , i = 1, 2, . . . , κ ,

and

3

Nehari manifold

Let (u, v) ∈ H be a solution of problem (Eε). Then

Z RN ε2∇u∇ϕ₁+ Z RN V1uϕ1+ Z RN ε2∇v∇ϕ₂+ Z RN V2vϕ2− Z RN µ1u3ϕ1 − Z RN µ2v3ϕ2 − β Z RN uv2ϕ1 − β Z RN u2vϕ2 = 0 , ∀ (ϕ1, ϕ2) ∈ H ,

where H = H1 _RN
× H1 _RN
is a Sobolev space with the norm k · k_H given by k(φ, ψ)k2_H = kφk2_V 1 + kψk 2 V₂ = Z RN  ε2 |∇φ|2 + V1φ2  + Z RN  ε2 |∇ψ|2 + V2ψ2  .

It is well known that the solution (u, v) ∈ H of problem (Eε) is a

critical point of the energy functional Jε ∈ C1 (H, R) defined by

Jε (φ, ψ) = 1 2 k(φ, ψ)k 2 H − 1 4 Z RN µ1φ4 + Z RN µ2ψ4  − β 2 Z RN φ2ψ2 (3.1) for all (φ, ψ) ∈ H .

Note that the energy functional Jε is not bounded below on H, it is

The type of Nehari manifold A. codimension one

e

Nε = {(u, v) ∈ H\ {(0, 0)} : hJ_ε0 (u, v) , (u, v)i = 0}

where

hJ_ε0 (u, v) , (u, v)i = k(u, v)k2_H−  µ1 Z RN u4 + µ2 Z RN v4  −2β Z RN u2v2. • trivial solution (0, 0) /∈ eNε

• semitrivial solutions (u, 0), (0, v) ∈ eNε

B. codimension two Nε =    (u, v) ∈ T : kuk 2 V1 = µ1 R RN u 4 _{+ β} R RN u 2_v2 kvk2_V 2 = µ2 R RN v 4 _{+ β} R RN u 2_v2    ,

where T = {(u, v) ∈ H : u 6≡ 0 and v 6≡ 0} . • trivial solution (0, 0) /∈ N_ε

Example: Let Vl ≡ λl = 1 = µl for l = 1, 2 and β ∈ R. Define

αλl,µl is the minimizing energy of the functional Iλl,µl over the

Nehari manifold Mλl,µl defined by

α_λl,µl = inf {I_λl,µl (u) | u ∈ M_λl,µl} , Iλl,µl (u) = 1 2 Z RN  |∇u|2 + λlu2  − µl 4 Z RN u4 , ∀ : u ∈ H1 _RN
, Mλl,µl = u ∈ H 1 RN
\ {0} | Iλ0 l,µl (u) , u = 0 , for l = 1, 2.

Thus we should choose the Nehari manifold with codimension two. Lemma 1. Let (u0, v0) be a constrained critical point of Jε on

Nε with µ1µ2 Z RN u4₀ Z RN v₀4 − β2 Z RN u2₀v₀2 2 > 0. Then ∇Jε (u0, v0) = 0 on H∗. proof: Define f (u, v) = kuk2_V 1 − µ1 Z RN u4 − β Z RN u2v2, g (u, v) = kvk2_V 2 − µ2 Z RN v4 − β Z RN u2v2.

Since (u0, v0) is a constrained critical point of Jε on Nε, by the

Then we have θ1  ku₀k2_V 1 − 2µ1 Z RN u4₀ − 2β Z RN u2₀v₀2  − θ₂  β Z RN u2₀v₀2  = 0, −θ₁  β Z RN u2₀v₀2  + θ2  kv₀k2_V 2 − 2µ2 Z RN v₀4 − 2β Z RN u2₀v₀2  = 0 or Σ   θ1 θ₂   =   0 0   where Σ =   µ1 R RN u 4 0 β R RN u 2 0v02 β R RN u 2 0v02 µ2 R RN v 4 0  . Then

Now we state results which are useful to prove our main result as follows:

Lemma 2. Let β < 0, ε, σ > 0 and (u, v) ∈ N_ε. If R RN u 2_v2 _{≥ σε}N_{, then} ε−NJε (u, v) > αλ1,µ1 + αλ2,µ2 + δ (3.2) where δ = |β| σ 2 min  1, 1 2 min  S₁2 µ1 (4αλ₁,µ₁ + |β| σ) , S 2 2 µ2 (4αλ₂,µ₂ + |β| σ)  , (3.3) and Si = inf u∈H1_(RN_)\{0} R RN |∇u| 2 + λ_iu2 R u4
1/2 > 0 , i = 1, 2 . (3.4)

Furthermore, if 0 < σ < 4 √ α_λ1,µ1α_λ2,µ2 |β| , then " µ₁µ₂ Z RN u4 Z RN v4 − β2 Z RN u2v2 2# > ε2N 16αλ1,µ1αλ2,µ2 − β 2 σ2
> 0 for all (u, v) ∈ Nε and ε−NJε (u, v) ≤ αλ1,µ1 + αλ2,µ2 + δ.

When the potentials Vl’s satisfy

0 < λl = inf

x∈RN Vl(x) < lim_|x|→∞ Vl(x) ≤ ∞ for l = 1, 2 , (3.5)

Lin and Wei study the minimization of the functional Jε over the

manifold Nε. As β < 0 and ε > 0 sufficiently small, there exists a

least energy solution (uε,1, uε,2) of problem (Eε) such that

ε−N Jε(uε,1, uε,2) → αλ1,µ1 + αλ2,µ2 , (3.6)

Remark: The our conditions (I) and (II) may allow the case that lim inf

|x|→∞ Vl(x) < lim sup_|x|→∞ Vl(x) for some l = 1, 2.

Hence the argument of Lin-Wei may not be applicable to problem (Eε) with the conditions (I) and (II).

4

Palais-Smale sequences

To find Palais-Smale (PS) sequences of the functional Jε, we may

use the conditions (I) and (II) of potentials V_l’s and a generalized barycenter map given by Φ : L2 _RN
\ {0} → RN a continuous map satisfying

Φ (ξ ∗ u) = ξ + Φ (u) and Φ u ◦ A−1
= AΦ (u) , (4.1) and

Φ (u ◦ ε) = ε−1Φ (u) for all u ∈ L2 _RN
\ {0} and ε > 0. (4.2) for every ξ ∈ RN, every orthogonal N × N matrix A, every ε > 0 and every u ∈ L2 _RN
\ {0}, where (ξ ∗ u)(x) = u(x − ξ) and

We may use the map Φ to decompose the Nehari manifold Nε into

κ × m submanifolds Ni,j (ε)’s as follows:

Ni,j (ε) = (u, v) ∈ Nε : Φ (u) ∈ Cl ai

and Φ (v) ∈ Cl bj
,

with the boundary

O_i,j (ε) = (u, v) ∈ N_ε : Φ (u) ∈ ∂C_l ai
or Φ (v) ∈ ∂C_l bj
, for i = 1, 2, . . . , κ and j = 1, 2, . . . , m. Hereafter, Cl (x)’s are cubes

defined by Cl (x) = N

Π

By the conditions (I) and (II), there exists l > 0 such that {C_l ai
}κ_i=1 and {Cl bj
}m_j=1 are collections of disjoint cubes

satisfying V₁ (x) > V₁ ai
for x ∈ ∂C_l ai
and i = 1, 2, . . . , κ , V2 (x) > V2 bj
for x ∈ ∂Cl bj
and j = 1, 2, . . . , m , Cl ai
∩ Cl bj
= ∅ if ai 6= bj , Cl ai
= Cl bj
if ai = bj .

Now we consider the minimization of the functional Jε over Ni,j (ε)

and Oi,j (ε), respectively, and denote the corresponding minima as

γi,j (ε) = inf

(u,v)∈Ni,j(ε)

Jε (u, v) and γ_ei,j (ε) = inf

(u,v)∈Oi,j(ε)

Jε (u, v) .

Existence of Palais-Smale sequences

Proposition. For each i ∈ {1, 2, . . . , κ} , j ∈ {1, 2, . . . , m}, there exists a sequence of functions {(un, vn)}

∞

n=1 ⊂ Ni,j (ε) such that

un, vn ≥ 0 in RN for n ∈ N,

Jε ui,j_n , v_ni,j
→ γi,j (ε) in H as n → ∞

and

Palais-Smale conditions

Proposition. Let i ∈ {1, . . . , κ} and j ∈ {1, . . . , m}. Assume 

ui,j_n , v_ni,j
∞_n=1 is a sequence in Ni,j (ε) satisfying ui,j_n , v_ni,j ≥ 0 in

RN for n ∈ N ,

(i) Jε ui,j_n , v_ni,j
→ γi,j (ε) as n → ∞;

(ii) J_ε0 ui,j_n , v_ni,j
→ 0 strongly in H∗ as n → ∞.

Then there exists a convergent subsequence denoted also as 

ui,j_n , v_ni,j
∞_n=1 for notation convenience such that as n → ∞,

5

Our main results

Theorem A. Assume µ1, µ2 > 0 , β < 0 and the potentials

V1, V2 ∈ C RN
satisfy the conditions (I) and (II). Then

(i) There exists ε₀ > 0 such that for 0 < ε < ε₀, problem (E_ε) has κ × m solutions  ui,j_ε , v_εi,j
: i = 1, · · · , κ , j = 1, · · · , m satisfying

ε−NJε ui,j_ε , v_εi,j
< αλ1,µ1 + αλ2,µ2 + δ0 ,

for i = 1, · · · , κ and j = 1, · · · , m, where δ0 is a positive constant

(ii) Fix i ∈ {1, 2, . . . , κ} and j ∈ {1, 2, . . . , m} arbitrarily. Then ui,j_ε has a concentration point P_1,i,jε and v_εi,j has another concentration point P_2,i,jε such that

ε−NJε ui,j_ε , v_εi,j
→ αλ₁,µ₁ + αλ₂,µ₂ , (5.4)

P_1,i,jε → ai, P_2,i,jε → bj and |P_1,i,jε − P_2,i,jε |/ε → +∞ as ε goes to zero (up to a subsequence).

Theorem B. Assume µ₁, µ₂ > 0 , β < 0, the potentials

V1, V2 ∈ C RN
satisfy the conditions (I) and (II) and there exists

1 ≤ k0 ≤ min {κ, m} such that ai = bi for all i = 1, 2, . . . , k0. Then

there exists ε∗ > 0 such that for ε < ε∗, the Problem (Eε) has at

Theorem C. Assume µ1, µ2 > 0 , β < 0, the potentials

V1, V2 ∈ C RN
satisfy the conditions (I) and (II). Then

(i) if lim inf|x|→∞ Vl(x) ≡ Vl,∞ > λl for all l = 1, 2, then there

exists ε∗∗ > 0 such that for every ε < ε∗∗, we can find at least

one least energy solution in the these solutions of Theorems A, B;

(ii) if lim_|x|→∞ Vl(x) = λl for all l = 1, 2, then the all solutions of