(毛宇) (吳行平) (唐春雷)

School of Mathematics and Statistics, Southwest University, Chongqing 400715, China

E-mail: 2531416750@qq.com; wuxp@swu.edu.cn; tangcl@swu.edu.cn

Abstract In this paper,we study normalized solutions of the Chern-Simons-Schr?dinger system with general nonlinearity and a potential in H1(R2).When the nonlinearity satisfies some general 3-superlinear conditions,we obtain the existence of ground state normalized solutions by using the minimax procedure proposed by Jeanjean in [L.Jeanjean,Existence of solutions with prescribed norm for semilinear elliptic equations,Nonlinear Anal.(1997)].

Key words Chern-Simons-Schr?dinger system;non-constant potential;Poho?aev identity;ground state normalized solution

1 Introduction and main results

In recent years,many scholars have paid attention to the planar nonlinear Chern-Simons-Schr?dinger system

where i denotes the imaginary unit,for (t,x1,x2)∈R1+2,φ:R1+2→C is the complex scalar field,Aj:R1+2→R is the gauge field,andDj=?j+iAjis the covariant derivative forj=0,1,2.The Chern-Simons-Schr?dinger system consists of Schr?dinger equations augmented by the gauge field,a situation that was first studied in[10,11].The Chern-Simons-Schr?dinger system describes the electromagnetic phenomena in a planar domain,which is related to the study of the high-temperature superconductor,Aharovnov-Bohm scattering and the fractional quantum Hall effect.Due to the physical motivations for studying system (1.1),many authors have investigated the initial value problem,wellposedness and blow-up of solutions,scattering and the uniqueness results for system (1.1);see [8,21,22].

In [3],Byeon,Huh and Seok first researched the standing wave solutions of the form

for system(1.1),whereλ>0 is a given frequency andu,k,hare real value functions on[0,+∞)withh(0)=0.Inserting ansatz (1.2) into system (1.1),we have the nonlocal semilinear elliptic equation

For equation (1.3) withf(u)=ω|u|p-2u,p>2 andω>0,the existence and nonexistence results of radial solutions were studied in [3,4,9,17,26].For when equation (1.3) has the general nonlinearityf,the existence and multiplicity of solutions were obtained in [13,14,18,25,28,30,35].Recently,the normalized solution of equation (1.3) has become a subject of increasing concern in the physical context.For whenf(u)=|u|p-2uandp ∈(2,4),the existence and multiplicity of normalized solutions to equation (1.3) were considered in [3,34].For equation(1.3)withf(u)=|u|2u,Li and Luo[16]researched the existence and nonexistence results of normalized solutions.In[16,34],the existence and multiplicity of normalized solutions to equation (1.3) were obtained for whenf(u)=|u|p-2uandp>4.Furthermore,Chen and Xie,in[5],investigated the existence and multiplicity of normalized solutions for equation(1.3)with the general nonlinearityf.For when equation (1.3) involves the harmonic potential|x|2,Luo[23]researched the existence and mass collapse behavior of normalized solutions in the case wheref(u)=|u|2u.He also investigated,in [24],the existence and multiplicity of normalized solutions in the case wheref(u)=|u|p-2uandp>4.

IfAj(t,x)=Aj(x),j=0,1,2 satisfies the Coulomb gauge condition?1A1+?2A2=0 andφ(t,x)=u(x)eiλt,u:R2→R,λ>0,then system (1.1) becomes

As is well known,the componentsA1,A2of the gauge field can be expressed by solving the elliptic equations

which give that

where?denotes the convolution in R2.We deduce from?2A0=-A1|u|2,?1A0=A2|u|2and?1A1+?2A2=0 that ?A0=?1(A2|u|2)-?2(A1|u|2),which gives the following representation ofA0:

For when system (1.4) has a non-constant potential;namely,for when

whereV ∈C1(R2,R) satisfies that

Wan and Tan [32]assumedf(u)=|u|p-2uwithp>4,and they investigated the existence of nontrivial solutions for system (1.5).Moreover,the authors of [31]studied the existence and concentration of semiclassical solutions for system(1.5)withf(u)=|u|p-2u,p>6 under some suitable conditions ofV.For system (1.5) with a coercive potential,Li and Yang [19]obtained a nontrivial solution forf(u)=|u|p-2u,p>4 and two nontrivial solutions forf(u)=|u|p-2u,24.The existence and concentration of semiclassical ground state solutions to system(1.5)with a general nonlinearityfwas studied in[6,29].We also note that there are two results about the normalized solutions of the Chern-Simons-Schr?dinger system inH1(R2);see [7,20].Liang and Zhai [20]obtained the existence of normalized solutions for system (1.4)withf(u)=|u|p-2uandp>4.In [7],Gou and Zhang researched the normalized solutions of system (1.4) withf(u)=|u|p-2uandp>2.

Inspired by the above works,we will investigate the existence of ground state normalized solutions to the system

whereV ∈C1(R2,R) andf ∈C(R,R) satisfy the following conditions:

(V1)is finite,for anyb>0;

(V2) there existsK ∈R+such that-2V(x)≤?V(x)·x ≤KV(x) a.e.in R2;

(V3) there existsa ∈C(R+,R+) such thatV(tx)≤a(t)V(x) for anyx ∈R2andt>0;

(f1)

(f2) there existp ≥μ>4 such that 0<μF(t)≤f(t)t ≤pF(t),where

We will work in the space

which is endowed with the inner product and norm

Lemma 1.1([1,Theorem 2.1]) IfV ∈C(R2,R) satisfies (V1),thenEis compactly embedded inLq(R2)for anyq ∈[2,+∞).In particular,for anyq ∈[2,+∞),there existsνq>0 such that

ProofAssertion (i) is from [9,Propositions 4.2 and 4.3](see also [32,Proposition 2.1]).By (i) and H?lder’s inequality,we deduce that,for anyu ∈H1(R2),

Thus (ii) holds.The proof is finished.

Foru ∈E,we define the energy functional

By (f1),(f2),and Lemmas 1.1 and 1.2,it is easy to check thatI ∈C1(E,R) and,for anyu,? ∈E,one has that

As is well known,a normalized solution to system (1.6) with a prescribedL2-normcis obtained as a critical point ofIconstrained on

It is worth pointing out that the frequencyλis determined as a Lagrange multiplier.For any fixedc>0,uc ∈Scis said to be a ground state normalized solution to system (1.6) if

Our main result reads as follows:

Theorem 1.3Suppose that (V1)–(V3),(f1) and (f2) hold.Then there existsc0>0 such that system (1.6) has at least a ground state normalized solution inH1(R2) for anyc ∈(0,c0].

Remark 1.4We point out that there exist many functions satisfying (V1)–(V3);these includeV(x)=|x|2α,α>0.Moreover,the special caseV(x)=|x|2is said to be a harmonic potential that is related to an external uniform magnetic field.As in [1],our condition (V1) is weaker than=+∞.Theorem 1.3 seems to be the first attempt to study the existence of ground state normalized solutions to the nonautonomous Chern-Simons-Schr?dinger system inH1(R2).Compared with [24],in which the author considered equation (1.3) withf(u)=|u|p-2u,p>4 andV(x)=|x|2,here the more general potential and nonlinearity are considered.

Remark 1.5Though the condition (V1) ensures that the embedding(R2) is compact for anyq ∈[2,+∞),it is difficult to obtain the boundedness of the Palais-Smale sequence for the energy functional of system (1.6) restricted onScunder the assumptions (f1)and(f2).Inspired by[12],we construct a Palais-Smale sequence which satisfies,asymptotically,a Nehari-Poho?aev type identity.We would like to point out that the approach used in [12]is only valid for autonomous equations.Therefore,to study system (1.6) with a non-constant potentialV(x),we will impose condition (V3).

Throughout this paper,we will use the following notations:

?is endowed with the same inner product and norm as inH1(R2).

?(E?,‖·‖?) denotes the dual space of (E,‖·‖E).

?R+=(0,+∞).

? Cdenotes positive constant that possibly varies in different places.

2 Proof of Theorem 1.3

Before proving Theorem 1.3,we give some preliminaries.

Lemma 2.1([33],Gagliardo-Nirenberg inequality) For anyq ∈[2,+∞),there existsC(q)>0 such that

which implies that

Lemma 2.2([7,Lemma 2.3]) Suppose thatun ?uinH1(R2) andun(x)→u(x)a.e.in R2.Then,forj=1,2 and any? ∈H1(R2),asn →∞,

Lemma 2.3([6,Lemma 3.1]) Letu ∈Ebe a weak solution of system (1.6).Thenusatisfies the following Poho?aev identity:

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In the following lemma,we will prove thatIsatisfies the mountain pass geometry:

Lemma 2.4If (V1)–(V3),(f1) and (f2) hold,then there existsc0>0 such thatIhas a mountain pass geometry onScfor anyc ∈(0,c0].That is,there existu1,u2∈Scsuch that

ProofFor anyk>0,we define that

It follows from (1.7) and (2.1) that,for anyu ∈Bk,

which implies that

Byf ∈C(R,R),(f1) and (f2),for anyε>0,there existsCε>0 such that

Then,by (1.7) and (2.4),we have,for anyu ∈E,that

Sincep>4,onceε>0 is small enough,there existsk1>0 small enough such that

Consequently,there existsu1∈Scsuch that‖u1‖≤k2andI(u1)>0.By (f1) and (f2),there existC1,C2>0 such that

Then,by (V3) and (2.9),one obtains,for anyu ∈E{0} andt>0,that

Sinceμ>4 anda ∈C(R+,R+),one checks thatI(tu(t·))→-∞ast →∞.Note thattu(t·)∈Scfor anyt>0 andu ∈Sc.Thus,there existst1>0 large enough such thatu2(·)=t1u1(t1·)∈Scsatisfies‖u2‖>k1andI(u2)<0.Define the following minimax class:

Sinceg(t)(·)=(1+tt1-t)u1(·+t(t1-1)·)∈Γc,we get that Γc≠?.Then we define that

which,combined the arbitrariness ofg ∈Γc,implies that

Thus we have completed the proof.

It is easy to check thatI ?Ψ∈C1(E1,R).Based on Lemma 2.4,we define that

Repeating the arguments in [12,Proposition 2.2],we can get the following proposition:

Proposition 2.5Suppose that (V1)–(V3),(f1) and (f2) hold.Let∈satisfy

Recall that{vn}?Eis a Palais-Smale sequence forIonScifI(vn)→γ(c)andI|(vn)→0.In the next lemma,applying Proposition 2.5,we construct a Palais-Smale sequence forIwhich satisfies,asymptotically,the following Nehari-Poho?aev identity:

Lemma 2.6Suppose that (V1)–(V3),(f1) and (f2) hold.Then,for anyc ∈(0,c0],there exists a Palais-Smale sequence{vn}?Scsatisfying,forn →∞,that

ProofBy the definition ofγ(c),for eachn ∈N,there exists somegn ∈Γcsuch that

Since (0,1)∈(un,θn),by taking (w,s)=(0,1) in (2.11),we derive from (c) that,asn →∞,

It follows from (b) that,for alln,

From (V3) and (2.13) we deduce that｛a(eθn)｝is bounded.Therefore,for allnandx ∈R2,

Hence,we can infer,for alln,that

Now,by (2.12) and (2.14),one has that

Consequently,asn →∞,

Proposition 2.7Suppose that (V1)–(V3),(f1) and (f2) hold.Then,for anyc ∈(0,c0],if{vn} ?Scsatisfies (2.10),there existvc ∈Sc,a sequence{λn} ?R andλc ∈R such that,up to a subsequence,asn →∞,

(i)vn →vcinE;(ii)λn →λcin R;

(iii)I′(vn)+λnvn →0 inE?;

(iv)I′(vc)+λcvc=0 inE?.

ProofSince{vn}?Scsatisfies (2.10),by (V2) and (f2),we deduce that

which shows that{vn} is bounded inE.Then,up to a subsequence,there exists avc ∈Esuch that,asn →∞,

It is clear that|vc|=c.Noting that(vn)=on(1) and applying [2,Lemma 3],we have that

which means,for any? ∈E,that

Thus (iii) holds.Since{vn}?Scis bounded inE,it is easy to get that each term on the right hand side of (2.17) is bounded.Therefore,{λn} is bounded.Then,up to a subsequence,there existsλc ∈R such thatλn →λcasn →∞.Thus (ii) holds.Sincevn ?vcinE,by using Lemma 2.2,we get,for any? ∈E,that

From (ii) and (2.16),one deduces,for any? ∈E,that

Therefore,one infers from(iii),(2.18)and(2.19)that(iv)holds.Byf ∈C(R,R),(f1)and(f2),for anyε>0,there exists>0 such that

By (2.20),H?lder’s inequality and Young’s inequality,we obtain that

Byvn →vcinLp(R2) and the arbitrariness ofε,we deduce that

By Lemma 1.2 and H?lder’s inequality,we get,for∈(1,2) and,that

Since{vn} is bounded inLq(R2) for anyq ∈[2,+∞),one infers thatis bounded inL2(R2).In addition,by H?lder’s inequality,we conclude that

Similarly,one has that

Therefore,we have that

By (ii)–(iv),one obtains that

Thus,combining (2.21)–(2.23) indicates that

Sincevn →vcinL2(R2),(2.24) implies thatvn →vcinE.Thus Proposition 2.7 is proven.

Proof of Theorem 1.3Letc ∈(0,c0].Define that

By Proposition 2.7,there existsvc ∈Scsatisfying that(vc)=0.ThusMcis unempty.Take{un} ?Mcas a minimizing sequence ofmcsatisfying thatI(un)→mcasn →∞.By{un} ?Mc,one has that(un)=0.According to Proposition 2.7,there exists{λn} ?R such thatI′(un)+λnun=0.MultiplyingI′(un)+λnun=0 byun,we have that

FromI′(un)+λnun=0 and Lemma 2.3,we know thatunsatisfies the Poho?aev identity

Combining (2.25) and (2.26),we get that

By Proposition 2.7,there existsuc ∈Scsuch thatun →ucinEasn →∞.ThusI(uc)=mcand(uc)=0;that is,uc ∈Scis a ground state normalized solution of system (1.6).Theorem 1.3 is proven.

Conflict of InterestThe authors declare no conflict of interest.