INFINITELY MANY NONRADIAL POSITIVE SOLUTIONS FOR MULTI-SPECIES NONLINEAR SCHR ODINGER SYSTEMS IN RN TUOXIN LI JUNCHENG WEI AND YUANZE WU

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INFINITELY MANY NONRADIAL POSITIVE SOLUTIONS FOR

MULTI-SPECIES NONLINEAR SCHR ¨

ODINGER SYSTEMS IN RN

TUOXIN LI, JUNCHENG WEI, AND YUANZE WU

Abstract. In this paper, we consider the multi-species nonlinear Schr¨odinger

systems in RN:











−∆uj+Vj(x)uj=µju3

i=1;i6=j

βi,j u2

iujin RN,

uj(x)>0 in RN,

uj(x)→0 as |x| → +∞, j = 1,2,··· , d,

where N= 2,3, µj>0 are constants, βi,j =βj,i 6= 0 are coupling parameters,

d≥2 and Vj(x) are potentials. By Ljapunov-Schmidt reduction arguments,

we construct inﬁnitely many nonradial positive solutions of the above system

under some mild assumptions on potentials Vj(x) and coupling parameters

{βi,j },without any symmetric assumptions on the limit case of the above

system. Our result, giving a positive answer to the conjecture in Pistoia and

Vaira [50] and extending the results in [50,52], reveals new phenomenon in the

case of N= 2 and d= 2 and is almost optimal for the coupling parameters

{βi,j }.

Keywords: nonlinear Schrodinger systems, inﬁnitely many positive solutions,

reduction method, min-max argument.

AMS Subject Classiﬁcation 2010: 35B09; 35B33; 35B40; 35J20.

1. Introduction

1.1. Backgrounds. In this paper, we consider the multi-species nonlinear Schr¨odinger

systems in RN:











−∆uj+Vj(x)uj=µju3

i=1;i6=j

βi,j u2

iujin RN,

uj(x)>0 in RN,

uj(x)→0 as |x| → +∞, j = 1,2,··· , d,

(1.1)

where N= 2,3, µj>0 are constants, βi,j =βj,i 6= 0 are coupling parameters,

d≥2 and Vj(x) are potentials.

arXiv:2210.03330v1 [math.AP] 7 Oct 2022

2 T. LI, J. WEI, AND Y. WU

It is well known that solutions of (1.1) are related to the bright solitons of the

Gross-Pitaevskii equations (cf. [40]),











ι∂Ψj

∂t = ∆Ψj−Vj(x)Ψj+µj|Ψj|2Ψj+

i=1;i6=j

βi,j |Ψi|2Ψj,

Ψj= Ψj(t, x)∈H1(RN+1;C), j = 1,2,··· , d, N = 2,3,

(1.2)

by the relation Ψj(t, x) = e−ιλjtuj(x), where ιis the imaginary unit. The Gross-

Pitaevskii equations (1.2) have applications in many physical models, such as in

nonlinear optics (cf. [1]) and in Bose-Einstein condensates for multi-species con-

densates (cf. [23,37,54]). In Bose-Einstein condensates for multi-species conden-

sates, µjand βi,j in (1.1) are the intraspecies and interspecies scattering lengths

respectively, while Vj(x) stands for the magnetic trap (cf. [70]) arising from the

chemical potentials. The sign of the scattering length βi,j determines whether the

interactions of states iiand jiare repulsive (βi,j <0) or attractive (βi,j >0).

In the autonomous case, i.e., the potentials Vjare positive constants for all

j= 1,2,··· , d, multi-species nonlinear Schr¨odinger systems (1.1) have been stud-

ied extensively in the pase two decades after the pioneer work [41]. By using

variational methods, Lyapunov-Schmidt reduction methods or bifurcation meth-

ods, various theorems, about the existence, multiplicity and qualitative properties

of nontrivial solutions of autonomous multi-species nonlinear Schr¨odinger systems

like (1.1), have been established in the literature under various assumptions on the

coupling parameters. Since it seems almost impossible for us to provide a complete

list of references, we refer the readers only to [2,3,6–10,16–18,22,23,25,26,31,32,34,

38,39,43–49,55,67,68,71–73,77] and the references therein for the two coupled case

d= 2, [24,28–30,42,50,57,59,62–64] and the references therein for the multi-coupled

case d≥3 with the purely repulsive couplings or the purely attractive couplings

and [11–15,21,27,33,41,51,56,58,60,61,65,74] and the references therein for the

multi-coupled case d≥3 with the mixed couplings. Here, we call the couplings

{βi,j }is purely repulsive if βi,j <0 for all i6=j, we call the couplings {βi,j }is

purely attractive if βi,j >0 for all i6=jand we call the couplings {βi,j }is mixed

if there exist (i1, j1) and (i2, j2) such that βi1,j1>0 and βi2,j2<0.

In the non-autonomous case, it is well known nowadays that the magnetic trap-

ping potentials will play important roles in constructing nontrivial solutions of

nonlinear Schr¨odinger equations or systems, see, for example, [5,19,20,35,50,52,53,

66,69,76] and the references therein. In particular, in [76], Wei and Yan constructed

inﬁnitely many nonradial positive solutions of the nonlinear Schr¨odinger equation,

(−∆u+V(x)u=|u|p−1uin RN,

u(x)→0 as |x| → +∞,(1.3)

where V(x) satisﬁes the assumption:

(V∗)V(x)>0 is continuous and radial with V(x) = 1 + δ

|x|ν+O1

|x|ν+εas

|x| → +∞, where δ∈Rand ν > 1 and ε > 0.

INFINITELY MANY POSITIVE SOLUTION 3

Let us brieﬂy sketch Wei and Yan’s construction in [76]. Let wjbe the unique (up

to translations) positive solution of the following equation:











−∆u+λju=µju3in RN,

u(x)>0 in RN,

u(x)→0 as |x| → +∞,

(1.4)

where j= 1,2,··· , d. Then it is well known that there exists AN,j >0, which only

depends on Nand j, such that

wj(x) = AN,j (1 + O(|x|−1))|x|1−N

2e−√λj|x|as |x| → +∞.(1.5)

For the sake of clarity, we denote wj=ewjin the partially symmetric case λj= 1

for all jand we denote wj=win the totally symmetric case λj= 1 and µj= 1 for

all j. Then by the assumption (V∗), (1.4) and (1.5), Pϑ

t=1 w(x−ηt) + v∗is very

close to a genuine solution of (1.3) if

min

t6=s|ηt−ηs| → +∞and min

t|ηt| → +∞

as ϑ→+∞by the Lyapunov-Schmidt reduction, where ϑ∈N,t= 1,2,··· , ϑ and

v∗is much smaller than the approximation Pϑ

t=1 w(x−ηt) in a suitable sense. To

prove Pϑ

t=1 w(x−ηt) + v∗is a genuine solution of (1.3), the adjustment of the

locations of {ηt}is needed. In [76], the key idea is to put {ηt}on a circle with a

large radius, which are invariant under the action of a discrete subgroup of SO(N),

to reduce the number of parameters in adjusting {ηt}and using ϑ, the number

of spikes, as a parameter in the construction of spiked solutions of (1.3). More

precisely, Wei and Yan choose ηt=ρϑξtwhere ρϑ∼ϑlog ϑis the radius and the

locations ξtsatisﬁes

ξt=cos 2(t−1)π

ϑ,sin 2(t−1)π

ϑ,0.(1.6)

Then the adjustment of the locations of {ηt}is reduced to ﬁnd a critical point of

the reduced energy functional of the parameter ϑwhich can be solved by taking

the maximum of this reduced energy functional of the parameter ϑover a suitable

set. We point out that generated by the fact that the building block shares the

same decaying property (1.5), the locations of {ηt}in [76] are invariant under the

rotation of the angle 2π

ϑand this invariance is crucial in the above construction.

Wei and Yan’s idea in [76] is applied by Peng and Wang in [52], where by

Ljapunov-Schmidt reduction arguments, Peng and Wang proved that for the two

coupled case d= 2, multi-species nonlinear Schr¨odinger systems (1.1) has inﬁnitely

many nonradial positive solutions in dimension three N= 3 under the following

assumptions on Vj(x),

(V0)Vj(x)>0 is continuous and radial with Vj(x) = 1 + δj

|x|νj+O1

|x|νj+ε

as |x| → +∞, where δj∈Rand νj>1 and ε > 0,

and some further assumptions on the parameters δjand β1,2. The solutions con-

structed by Peng and Wang in [52] either look like ϑcopies of synchronized spikes

ϑ

t=1

aw(x−ρϑξt),

t=1

bw(x−ρϑξt),

4 T. LI, J. WEI, AND Y. WU

where ξtis given by (1.6), or look like ϑcopies of segregated spikes

ϑ

t=1 ew1(x−ρϑξt),

t=1 ew2(x−ρϑξ0

t),

where

ξ0

t=cos (2t−1)π

ϑ,sin (2t−1)π

ϑ,0,

with ρϑ∼ϑlog ϑas ϑ→+∞and (a, b) being the unique solution of the following

algebraic equation: µ1a+β1,2b= 1,

µ2b+β1,2a= 1.

Again, we point out that generated by the fact that the building block shares the

same decaying property (1.5), the locations of spikes in [52] are still invariant under

the rotations of the angle 2π

ϑor π

ϑ, respectively, and these invariance is also crucial

in Peng and Wang’s construction in [52], since the adjustment of the locations of

spikes can still be reduced to ﬁnd a critical point of the reduced energy functional of

the parameter ϑwhich can be solved by taking the maximum of this reduced energy

functional of the parameter ϑover a suitable set. Moreover, to our best knowledge,

there is no results about the existence of inﬁnitely many nonradial positive solutions

of multi-species nonlinear Schr¨odinger systems (1.1) for the two coupled case d= 2

in the dimension two N= 2.

Peng and Wang’s results in [52] have been extended in a recent interesting pa-

per [50] by Pistoia and Vaira to the case d≥3 and N= 2,3. More precisely,

Pistoia and Vaira proved in [50], by Ljapunov-Schmidt reduction arguments too,

that there exists ϑ0>0 such that for every ϑ≥ϑ0, (1.1) has a positive solu-

tion (eu1,ϑ,eu2,ϑ,··· ,eud,ϑ) for d≥3 and N= 2,3 under the following symmetry

assumption

Vi(x) = Vj(x), µi=µj, βi,j=β, for all i, j (1.7)

where Vi(x) = Vj(x) satisﬁes the assumption (V∗) and βsatisﬁes the smallness

assumption

β∈(0, βϑ) for δ > 0or β∈(−βϑ,0) for δ < 0(1.8)

with βϑ→0 as ϑ→+∞. Moreover,

euj,ϑ ∼

t=1

w(x−ρϑe

ξt,j ) for all j= 1,2,··· , d,

and (eu1,ϑ,eu2,ϑ,··· ,eud,ϑ) is invariant under the rotation of the angle 2π

dϑ in R2, where

ρϑ∼ϑlog ϑas ϑ→+∞and

ξt,j =









cos 2(j−1)π

dϑ +2(t−1)π

ϑ,sin 2(j−1)π

dϑ +2(t−1)π

ϑ, N = 2,

cos 2(j−1)π

dϑ +2(t−1)π

ϑ,sin 2(j−1)π

dϑ +2(t−1)π

ϑ,0, N = 3.

The idea of Pistoia and Vairia in [50] is to combine the two symmetries of the

system (1.1) under the assumption (1.7): First of all, under the condition (1.7)

the system (1.1) is invariant under any permutation of (u1, ..., ud). Secondly, the

INFINITELY MANY POSITIVE SOLUTION 5

system (1.1) is rotationally invariant if the potentials Vj(x) are radial. Under these

two invariance, it is possible to arrange the (u1, ..., ud) ﬁrst in a sector, and then

use the rotational symmetry to extend.

1.2. Main Results. The main purpose of this paper is to extend the results in

[50,52] to obtain an almost optimal existence results of inﬁnitely many nonradial

positive solutions of (1.1) under some mild assumptions on the potentials Vj(x)

which are more general than (V0). In particular, we shall remove the symmetry

assumption (1.7) and the smallness assumption (1.8) in [50, Theorem 1]. We also

include the results in the case of N= 2, d = 2, not covered in [52].

To state our result precisely, let us ﬁrst introduce some necessary notations and

assumptions. We make the following assumptions on Vj(x):

(V1)Vj(x)>0 is continuous and Vj(x0, x00) = Vj(|x0|,|x00|) in RN, where

x0= (x1, x2)∈R2and x00 ∈RN−2;

(V2)Vj(x) = λj+δj

|x|νj+O1

|x|νj+εas |x| → +∞in the C1-sense, where

λj>0, δj∈Rand νj, ε > 0.

Remark 1.1. In the assumptions (V1)and (V2), we do not require that the poten-

tials Vj(x)are symmetric at inﬁnity, i.e. we do not assume that λj=λifor all

i6=j. Note that in [50,52] the symmetry assumption λi=λjat ∞plays key roles

in the construction.

By rearranging if necessary, we may assume that

λ1=··· =λn1< λn1+1 =··· =λn2<··· < λnk−1+1 =··· =λnk,(1.9)

where 1 ≤n1< n2<··· < nk=dand 1 ≤k≤d. For the sake of simplicity,

we denote n0= 0 and nτ={nτ−1+ 1, nτ−1+ 2,··· , nτ}for all τ= 1,2,··· , k.

By [4, Lemma 3.7] and (1.5),

ZRN|x|−νjw2

j(· − ξ)dx = (Bj+o(1))|ξ|−νj,(1.10)

ZRN

jwj(· − ξ)dx = (Cj+o(1))|ξ|1−N

2e−√λnτ|ξ|,(1.11)

ZRN

jw2

i(· − ξ)dx =





(Dτ+o(1))|ξ|−1

2e−2√λnτ|ξ|, N = 2,

(Dτ+o(1))|ξ|−2e−2√λnτ|ξ|log |ξ|, N = 3,

(1.12)

for all i, j ∈nτwith all τ= 1,2,··· , k,

ZRN

nτw2

nτ+1(· − ξ)dx = (D0

τ+o(1))|ξ|1−Ne−2√λnτ|ξ|(1.13)

for all τ= 1,2,··· , k −1 and

ZRN

nkw2

n1(· − ξ)dx = (D0

k+o(1))|ξ|1−Ne−2√λn1|ξ|,(1.14)

as |ξ| → +∞, where Bj,Cj,Dτand D0

τare positive constants. Moreover, it is also

well known that by (1.5), the spectrum of −∆ + λjin L2(RN;w2

i) is discrete for

all i, j. Let βi,j,∗be the ﬁrst eigenvalue of −∆ + λjin L2(RN;w2

i) and we denote

ν∗= min{νj}and m∗={j= 1,2,··· , d |νj=ν∗}. Then our main result can be

stated as follows.

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INFINITELYMANYNONRADIALPOSITIVESOLUTIONSFORMULTI-SPECIESNONLINEARSCHRODINGERSYSTEMSINRNTUOXINLI,JUNCHENGWEI,ANDYUANZEWUAbstract.Inthispaper,weconsiderthemulti-speciesnonlinearSchrodingersystemsinRN:8>>>>><>>>>>:uj+Vj(x)uj=ju3j+dXi=1;i6=ji;ju2iujinRN;uj(x)>0inRN;uj(x)!0asjxj!+1;j=1;2;;d;where...

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INFINITELY MANY NONRADIAL POSITIVE SOLUTIONS FOR MULTI-SPECIES NONLINEAR SCHR ODINGER SYSTEMS IN RN TUOXIN LI JUNCHENG WEI AND YUANZE WU

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