We prove the existence of infinitely many solutions of the nonlinear Chern-Simons-Schrödinger
equations under a wide class of nonlinearities. This class includes the standard power-type nonlinearity with exponent p>4. This extends the previous result which covers the exponent p>6.

1. Introduction

In [1, 2], Jackiw and Pi introduce a nonrelativistic model that the nonlinear Schrödinger dynamics is coupled with the Chern-Simons gauge terms as follows:(1)iD0ϕ+D1D1+D2D2ϕ=-λϕp-2ϕ,∂0A1-∂1A0=-Imϕ¯D2ϕ,∂0A2-∂2A0=Imϕ¯D1ϕ,∂1A2-∂2A1=-12ϕ2.Here, i denotes the imaginary unit, ∂0=∂/∂t, ∂1=∂/∂x1, ∂2=∂/∂x2 for (t,x1,x2)∈R1+2, ϕ:R1+2→C is a complex scalar field, Aμ:R1+2→R is a component of gauge potential and Dμ=∂μ+iAμ is a covariant derivative for μ running over 0,1,2, and λ>0 is a parameter. The Chern-Simons gauge theory appears in the 1980s to explain electromagnetic phenomena of anyon physics such as the high temperature superconductivity or the fractional quantum Hall effect. In this paper, we are interested in standing wave solutions of (1). In [3], the authors introduce a standing wave ansatz of the following form:(2)ϕt,x=uxeiωt,A0t,x=kx,A1t,x=x2x2hx,A2t,x=-x1x2hx,where ω>0 is a phase frequency and u,k,h are real valued functions on [0,∞) such that h(0)=0. Inserting (2) into (1), one may check from direct computation that (1) is reduced to the following nonlinear nonlocal elliptic equation:(3)Δu-ωu-∫x∞hussu2sdsu-hu2xx2u+λup-2u=0inR2,where hu(s)=∫0s1/2u2(σ)dσ. See [3] for its derivation. It is shown in [3] that (3) is an Euler-Lagrange equation of a C1 functional,(4)Ju≔12∫R2∇u2+ωu2+hu2xx2u2dx-1p∫R2updxonHr1R2,where Hr1(R2) denotes the set of radially symmetric functions in standard Sobolev space H1(R2). Investigating the structure of J, the authors of [3] obtain several existence and nonexistence results for (3), depending on the range of p>2 and λ>0. Recently, Pomponio and Ruiz [4] improve the results in [3] for the case p∈(2,4). They find a threshold for the behavior of J, depending on ω>0. They also study (3) on bounded domain in [5].

In this paper, we are concerned with the existence of infinitely many solutions of (3). It is proved in [6] that if p>6, J enjoys the symmetric mountain pass geometry and satisfies the (PS) condition so that the well-known symmetric mountain pass lemma (see [7]) applies to show there exist infinitely many critical points of J. For p∈(4,6], it turns out that J still enjoys the symmetric mountain pass geometry although checking the (PS) condition is not easy job. One aim of this paper is to show nevertheless J still admits infinitely many critical points for p>4. Moreover, we will replace the power-type nonlinearity λ|u|p-2u of (3) with more general one as follows:(5)Δu-ωu-∫x∞hussu2sdsu-hu2xx2u+Vu2u=0.The structure conditions for V are given by the following:

Consider V∈C1(R+,R) such that V(0)=V′(0)=0.

Consider limsups→∞V′s/sp-1<∞, limsups→∞Vs/sp<∞ for some p>1.

There exists some α>1 such that αV′(s)/s2-1/α-V(s)/s3-1/α is monotonically increasing to ∞ as s→∞.

Observe that assumptions (V1)–(V3) include the power-type nonlinearity λ/p|u|p, p>4.
Theorem 1.

Assume (V1)–(V3). Then, (5) admits infinitely many solutions.

We refer to the work of Cunha et al. [8] that if we insert a sufficiently small parameter q>0 into (5) as in(6)Δu-ωu-q∫x∞hussu2sdsu-qhu2xx2u+Vu2u=0,then much more general assumptions for V, the so-called Berestycki-Lions conditions [9], are sufficient for guaranteeing the existence and multiplicity of solutions of (6). In our work, we assume further than the Berestycki-Lions conditions but we do not need small parameter q>0. See also [10] in which Tan and Wan consider asymptotically linear nonlinearities.

To prove Theorem 1, we will apply the method employed in author’s former paper [11] in which the Schrödinger-Poisson equation, another nonlocal field equation similar to (5), is dealt with. Instead of generating (PS) sequences, we will show the existence and compactness of the so-called approximate solution sequences of J which may be considered as more refined version of (PS) sequences. In Section 2, we give a definition of the approximate solution sequences of J. Some auxiliary lemmas are also prepared in Section 2. In Section 3, we prove the compactness of approximate solution sequences. In Section 4, we construct infinitely many approximate solution sequences whose energy levels go to infinity and complete the proof of Theorem 1.

2. Mathematical Settings and Preliminaries

Let Hr1(R2) be the completion of D≔u∈Cc∞R2∣ux=ux with respect to the norm(7)u=∫R2∇u2+ωu2dx1/2.The dual space of Hr1(R2) is denoted by Hr1(R2)∗. Arguing similarly to [3], it is easy to show (5) is an Euler-Lagrange equation of the C1 functional(8)Ju≔12∫R2∇u2+ωu2+hu2xx2u2-Vu2dxonHr1R2.In this paper, we search for infinitely many critical points of J to prove Theorem 1. To do this, we insert parameter λ into J as follows:(9)Jλu≔12∫R2∇u2+ωu2+hu2xx2u2-λVu2dx.Here λ ranges over [1/2,1]. For a sequence {λj}∈[1/2,1] which converges to 1 as j→∞, we say {λj,uj} is an approximate solution sequence of J if Jλj′(uj)=0 for all j. In the following subsection, we state a variant of the famous Struwe’s monotonicity trick [12], which plays a crucial role in constructing approximate solution sequences.

2.1. A Variant of Struwe’s Monotonicity Trick

Let X be Banach space. We say a subset A⊂X is symmetric if -v∈A for every v∈A. Let A be a compact subset of X and B a closed subset of A. We denote by Γ the set of every continuous odd function γ:A→X such that γ(v)=v on B. Let I be a closed interval in R and Jλ one parameter family of even C1 functional on X. We define a minimax level by(10)cλ≔minγ∈Γmaxv∈AJλγv.The following theorem is a variant of so-called Struwe’s monotonicity trick [12]. A more general version of it is given in [11]. The property (H) below is first proposed by Jeanjean and Toland in [13].

Theorem 2 (see [<xref ref-type="bibr" rid="B11">11</xref>]).

Suppose that, for all λ∈I,(11)cλ>maxv∈BJλv.Then, for almost every λ∈I, there exists a norm-bounded (PS) sequence of Jλ at level c(λ), provided the following property (H) for Jλ holds:

For given λ0∈I, let {λj}∈I be a sequence strictly increasing to λ0 and {vj} a sequence in X such that(12)-Jλ0vj,Jλjvj,Jλjvj-Jλ0vjλ0-λj

are all uniformly bounded above for j. Then the following holds:

{vj} is norm-bounded in X.

For given ɛ>0, there exists N>0 such that(13)Jλ0vj≤Jλjvj+ɛ∀j≥N.

2.2. Some Auxiliary Lemmas

Here, we prepare some lemmas which will be necessarily used for proving the main result. Define(14)J^λu≔12∫R2hu2xx2u2-λVu2dx.

Lemma 3 (Lemma 3.2 in [<xref ref-type="bibr" rid="B2">3</xref>]).

Let {uj}∈Hr1(R2) be a sequence weakly converging to some u0 in Hr1(R2) as j→∞. Then, for each φ∈Hr1(R2), it holds that J^λ(un)→J^λ(u0), J^λ′(un)φ→J^λ′(u0)φ, and J^λ′(un)un→J^λ′(u0)u0 as j→∞, up to a subsequence.

Lemma 4 (Pohozaev identity).

Let u∈Hr1(R2) be a critical point of Jλ. Then one has(15)∫R2ωu2dx+2∫R2hu2xx2u2dx-∫R2λVu2dx=0.

For each u∈Hr1(R2) and t>0, we define one parameter family of functions ut∈Hr1(R2) by(16)utx≔tαutx.For fixed t>0, we define a map Lt:H1(R2)→H1(R2) by Lt(u)=ut. It is easy to see that Lt is a continuous and linear map with the inverse L1/t. Thus Lt is a linear isomorphism.

For each u∈H1(R2) and λ∈[1/2,1], let cu,λ:[0,∞)→R be a function defined by(17)cu,λt≔Jλut.

Lemma 5.

For any u∈Hr1(R2) and λ∈[1/2,1], cu,λ admits a unique critical point t0>0 on (0,∞); that is, cu,λ′(t0)=0, such that cu,λ(t) is increasing on (0,t0), attains its maximum at t0, and is decreasing to -∞ on (t0,∞).

Proof.

By the change of variable, one can compute(18)cu,Î»t=JÎ»tÎ±utÂ·=12t2Î±âˆ«R2âˆ‡u2dx+t2Î±-2âˆ«R2Ï‰u2dx+t6Î±-4âˆ«R2hu2xx2u2dx-t-2âˆ«R2Î»Vt2Î±u2dx.We differentiate it with respect to t to get(19)cu,Î»â€²t=Î±t2Î±-1âˆ«R2âˆ‡u2dx+Î±-1t2Î±-3âˆ«R2Ï‰u2dx+3Î±-2t6Î±-5âˆ«R2hu2xx2u2dx+t-3âˆ«R2Î»Vt2Î±u2dx-t-2âˆ«R2Î»Vâ€²t2Î±u2Î±t2Î±-1u2dx=t6Î±-5Î±t4Î±-4âˆ«R2âˆ‡u2dx+Î±-1t4Î±-2âˆ«R2Ï‰u2dx+3Î±-2âˆ«R2hu2xx2u2dxï¸¸At-âˆ«R2Î»Î±Vâ€²tÎ±u2tÎ±u22-1/Î±-VtÎ±u2tÎ±u23-1/Î±u23-1/Î±dxï¸¸Bt.Observe from assumption (V3) that A(t) is strictly monotonically decreasing from infinity to the positive number (3α-2)∫R2hu2(x)/x2u2dx on (0,∞) and B(t) is monotonically increasing to infinity on (0,∞). Therefore there is t0>0 such that A(t)>B(t) on (0,t0), A(t0)=B(t0), and A(t)<B(t) on (0,t0). Also from assumption (V3), we deduce B(t)→∞ as t→∞. This proves the proposition.

We define a function T:H1(R2)→R by assigning a positive number t satisfying t-αu(t-1·)=1 for any nonzero u∈Hr1(R2). The value T(0) is defined by 0.

Lemma 6.

The function T is well-defined and continuous even map on [0,∞).

Proof.

To show the well-definedness of T, we have to show that there exists unique t>0 satisfying t-αu(t-1·)=1 for given nonzero u∈Hr1(R2). We note that this is equivalent to prove there is a unique solution t>0 of the equation(20)gt≔t2α-∫R2ωu2dxt2-∫R2∇u2dx=0.Arguing similarly to the proof of Lemma 5, we are able to see that g(t) is monotonically decreasing on (0,t0) for some t0>0, attains its unique local minimum at t0, and is monotonically increasing to infinity on (t0,∞). Therefore there is a unique positive zero of g(t) since g(0)=0. Also, the implicit function theorem says that T(u) is continuous on [0,∞) because g′(T(u))≠0. The evenness of T follows from the fact that each coefficient of (20) is even. This completes the proof.

3. Compactness of Approximate Solution Sequences

In this section, we prove the compactness of an approximate solution sequence {uj} of Jλj when its energy {Jλj(uj)} is bounded above.

Proposition 7.

Let λj∈[1/2,1] be such that λj→1 as j→∞. Let {uj}∈Hr1(R2) be a sequence of critical points of Jλj; that is, Jλj′(uj)=0. Suppose that Jλj(uj)<C for some C>0, independent of j. Then uj→u0 in H1(R2) for some critical point u0∈Hr1(R2) of J(=J1) up to a subsequence.

Proof.

We divide the proof into two steps.

Step 1 (boundedness of uj). We first prove that {uj} is bounded in Hr1(R2). Arguing indirectly, suppose that {uj} is unbounded. Let Tj≔T(uj), where the function T is defined in Section 2. Equation (20) says Tj is unbounded. Let vj(x)≔Tj-αuj(Tj-1x) so that vj=1. Then, up to a subsequence, {vj} converges weakly in Hr1(R2) and strongly in Lq(R2) for all q>2 to some v0∈Hr1(R2). Since uj is a critical point of Jλj, we see that(21)∫R2∇uj2dx+∫R2ωuj2dx+3∫R2huj2xx2uj2dx-∫R2λjV′uj2uj2dx=Jλj′ujuj=0.Combining this with the Pohozaev identity (15), we obtain(22)α∫R2∇uj2dx+α-1∫R2ωuj2dx+3α-2∫R2huj2xx2uj2dx+∫R2λjVuj2dx-α∫R2λjV′uj2uj2dx=0.Then, from the change of variable and dividing by Tj6α-4, (22) transforms to(23)αTj4α-4∫R2∇vj2dx+α-1Tj4α-2∫R2ωvj2dx+3α-2∫R2hvj2xx2vj2dx︸Aj=λj∫R2αV′Tjαvj2Tjαvj22-1/α-VTjαvj2Tjαvj23-1/αvj23-1/αdx︸Bj.Since vj=1 and Tj is unbounded, Aj is bounded for j but the structure assumption (V3) implies that Bj tends to infinity as j→∞ provided v0 is not identically zero. We claim that v0 is nonzero. Suppose v0 is identically zero. From (19) and (22), we see that(24)cuj,λj′1=α∫R2∇uj2dx+α-1∫R2ωuj2dx+3α-2∫R2huj2xx2uj2dx+∫R2λjVuj2dx-α∫R2λjV′uj2uj2dx=0.Then, Lemma 5 implies that cuj,λj(1) is the global maximum of cuj,λj(t) on (0,∞). Thus we see that, for each R>1,(25)C>JÎ»juj=cuj,Î»j1â‰¥cuj,Î»jRTj-1=12R2Î±âˆ«R2âˆ‡vj2dx+R2Î±-2âˆ«R2Ï‰vjdx+R6Î±-4âˆ«R2hvj2xx2vj2dx-âˆ«R2Î»jVRÎ±vjRx2dxâ‰¥12R2Î±-2vj2-12âˆ«R2Î»jVRÎ±vjRx2dx=12R2Î±-2+o1.The last equality follows from vj=1, the convergence of {vj} to 0 in Lq(R2) for all q>2, and the structure conditions (V1)-(V2). However, taking large R>1, this makes a contradiction and shows v0 is not identically zero. This proves the boundedness of {uj} in Hr1(R2).

Step 2 (compactness of uj). Compactness of {uj} follows from a standard procedure. Since {uj} is bounded, there exists u0∈Hr1(R2) such that {uj} converges, up to a subsequence, to u0 weakly in Hr1(R2) and strongly in Lq(R2) for all q>2. Then it follows from Lemma 3 that u0 is a critical point of J. Also, it is easy to see from the boundedness of {uj} that J′uj→0 in Hr1(R2)∗. Recall that(26)Ju=12u2+J^u.Using Lemma 3 once again, one can observe that(27)o1uj=J′ujuj=uj2+J^′ujuj=uj2+J^′u0u0+o1=uj2-u02+o1,which shows uj→u0 as j→∞. Therefore we have uj→u0 in Hr1(R2) as j→∞.

4. Construction of Approximate Solution Sequences

In this section, we construct infinitely many approximate solution sequences. Choose an orthonormal basis {ui}i=1∞ of Hr1(R2). For given k∈N, let Yk and Zk be linear subspaces of Hr1(R2) spanned by {u1,u2,…,uk} and {uk,uk+1,…}, respectively. We will show that Jλ enjoys a variant of symmetric mountain pass geometry (see [14]).

Lemma 8.

There exist a sequence {kn}∈N such that kn→∞ as n→∞ and sequences {ρn},{rn}∈R satisfying ρn>rn>0 for each n and

infJλu∣u∈Zkn,Tu=rn≥n for all λ∈[1/2,1] and all n;

maxJλu∣u∈Ykn,Tu=ρn≤0 for all λ∈[1/2,1] and all n.

Proof.

We first show (i). The structure assumptions (V1)-(V2) imply that(28)Vs≤ω2s+Cspfor someC>0 and alls≥0.Then,(29)Jλu≥12∫R2∇u2dx+12-λ4∫R2ωu2dx+12∫R2hu2xx2u2dx-Cλ2∫R2u2pdx≥14∫R2∇u2dx+14∫R2ωu2dx-C2∫R2u2pdx.We recall the function T and the linear isomorphism Lt in Section 2. Let v(x)≔T(u)-αu(T(u)-1x) so that v=1. By a change of variable, we get from (29) that, for each u∈Zk,(30)Jλu≥14Tu2α∫R2∇v2dx+14Tu2α-2∫R2ωv2dx-C2Tu2pα-2∫R2v2pdx≥14Tu2α-2-C2Tu2pα-2βk,Tu,where(31)βk,t≔supv∈Wk,t∫R3v2pα-2dx,Wk,t=v∈Hr1R2∣v=1,vt=tαvt·∈Zk.We claim that, for each t>0, βk,t→0 as k→∞. To see this, suppose that βk,t→βt>0 as k→∞. Choose vk,t∈Wk,t satisfying(32)∫R2vk,tpdx>βk,t2.Since Wk,t is the unit sphere of a linear subspace Lt-1(Zk) of Hr1(R2) with codimension k-1, we deduce {vk,t} converges to 0 weakly in Hr1(R2) and strongly in Lp(R2), up to a subsequence. This however contradicts the fact that(33)limk→∞∫R2vk,tpdx>βt2>0and the claim is true. We take rn=(8n)1/2α-2. Then, for any u∈Zk satisfying T(u)=rn,(34)Jλu≥2n-C2rn2pα-2βk,rn.For each n, by taking sufficiently large {kn} satisfying(35)βkn,rn<2nCrn2pα-2,we can see that the proof of (i) is complete.

Next we show (ii). Lemma 5 says that, for each u∈Hr1(R2), Jλ(ut)→-∞ as t→∞. Also, we see from Lemma 6 and (20) that the set u∈Yk∣Tu=ρ is closed and bounded in finite dimensional space so it is compact. Combining these two facts with the compactness of [1/2,1], we can deduce easily (ii) holds.

Define(36)Bn≔u∈Ykn∣Tu≤ρn,Sn≔u∈Zkn∣Tu=rn,with kn,ρn, and rn given in Lemma 8. Let Γn be the set of continuous functions γ:Bn→Hr1(R2) satisfying γ(u)=u on ∂Bn=u∈Bn∣Tu=ρn. By γ(Bn), we denote the set γu∈Hr1R2∣u∈Bn.

Lemma 9 (intersection property).

For any n∈N, the intersection γ(Bn)∩Sn≠∅ for every γ∈Γn.

Proof.

Choose and fix arbitrary n and γ∈Γn. Define(37)U≔u∈intBn∣Tγu<rn,where int(Bn) denotes the interior of Bn in Ykn. Then U is a symmetric open neighborhood of 0 since T is a continuous even map by Lemma 6 and γ is a continuous odd map. Equation (20) in Lemma 6 says Bn is bounded so that U is also bounded. We claim that T(γ(∂U))=rn. From the continuity of T and γ, it holds that T(γ(∂U))≤rn. Suppose that there is some u0∈∂U such that T(γ(u0))<rn. Then there is a neighborhood of V of u0 in Ykn such that T(γ(V))<rn. Choose some v0∈V∖U. From the definition of U, v0∉int(Bn). Since v0∈∂U⊂Bn, we see that v0∈∂Bn. Then, from the definition of γ, we have T(γ(v0))=T(v0)=ρn>rn, which is a contradiction. This shows the claim is true.

Now, consider a map Pn∘γ:∂U→Ykn-1, where Pn is the projection map from Hr1(R2)→Ykn-1. Then the well-known Borsuk-Ulam theorem applies to see the continuous odd map Pn∘γ has a vanishing point u∈∂U; that is, Pn(γ(u))=0. This means that γ(u)∈Zkn. Therefore γ(u)∈γ(Bn)∩Sn. The proof is complete.

Now, we are ready to prove the existence of infinitely many approximate solution sequences of J. For each λ∈[1/2,1], we define infinitely many minimax levels as follows:(38)Cnλ≔minγ∈Γnmaxu∈BnJλγu.It follows from Lemmas 8 and 9 that Cn(λ)≥n for all λ∈[1/2,1].

Proposition 10.

For every fixed n∈N, there exists an approximate solution sequence {(λj,uj)} of J such that Jλj(uj)=Cn(λj).

Proof.

We invoke Theorem 2. From Lemmas 8 and 9, it holds that(39)Cnλ≥n>0≥maxv∈∂BnJλv.Let us check Jλ enjoys property (H). Let {λj}∈[1/2,1] be a sequence strictly increasing to some λ0∈[1/2,1] and {vj}∈Hr1(R2) a sequence such that(40)-Jλ0vj,Jλjvj,Jλjvj-Jλ0vjλ0-λj<C.We need to show (i) {vj} is bounded in Hr1(R2) and (ii), for given ɛ>0, there exists N>0 satisfying(41)Jλ0,vj≤Jλj,vj+ɛ∀n≥N.We first show (i). We see from (40) that(42)12vj2+12∫R2hvj2xx2vj2dx<2C,-C<12∫R2Vuj2dx<C,which shows {vj} is bounded in Hr1(R2). Also, for given ɛ>0,(43)Jλ0,vj-Jλj,vj=λj-λ012∫R2Vuj2dx≤λ0-λjC≤ɛif j is sufficiently large. This shows (ii). Therefore, there exists a subset M⊂[1/2,1] with full measure in [1/2,1] that, for every λ∈M, there exists a bounded (PS) sequence {uλ,k}k=1∞ of Jλ at level Cn(λ). Arguing similarly to Step 2 of Proposition 7, we also deduce {uλ,k}k=1∞ converges, up to a subsequence, to some critical point uλ of Jλ with Jλ(uλ)=Cn(λ). Since M has full measure in [1/2,1], this completes the proof.

Completion of the Proof of Theorem 1. Now we complete the proof of Theorem 1. We first choose and fix arbitrary n∈N. Let {uj} be an approximate solution sequence of Jλj, given by Proposition 10. Take γ∈Γn satisfying(44)maxu∈BnJγu≤Cn1+12.It follows from the compactness of Bn that(45)Cnλj≤maxu∈BnJλjγu≤maxu∈BnJγu+1-λjmaxu∈Bn∫R212Vγu2dx≤Cn1+12+12=Cn1+1for sufficiently large j. Then Proposition 7 applies to see uj converges, up to a subsequence, to some u0 which is a critical point of J. Recall that Jλj(uj)=Cn(λj)≥n. By taking a limit j→∞, we deduce J(u0)≥n. Since n is arbitrary, this shows the existence of infinitely many critical points of J. This completes the proof.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The author was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2014R1A1A2054805) and also was supported by the POSCO TJ Park Science Fellowship.

JackiwR.PiS.-Y.Soliton solutions to the gauged nonlinear Schrödinger equation on the planeJackiwR.PiS.-Y.Classical and quantal nonrelativistic Chern-Simons theoryByeonJ.HuhH.SeokJ.Standing waves of nonlinear Schrödinger equations with the gauge fieldPomponioA.RuizD.A variational analysis of a gauged nonlinear Schrödinger equationPomponioA.RuizD.Boundary concentration of a Gauged nonlinear Schrödinger equation on large ballsHuhH.Standing waves of the Schrödinger equation coupled with the Chern-Simons gauge fieldAmbrosettiA.RabinowitzP. H.Dual variational methods in critical point theory and applicationsCunhaP.d'AveniaP.PomponioA.SicilianoG.A multiplicity result for Chern-Simons-Schrödinger equation with a general nonlinearityBerestyckiH.LionsP.-L.Nonlinear scalar field equations. I. Existence of a ground stateTanJ.WanY.Standing waves for the Chern-Simons-Schrödinger systems without (AR) conditionSeokJ.On nonlinear Schrödinger-Poisson equations with general potentialsStruweM.The existence of surfaces of constant mean curvature with free boundariesJeanjeanL.TolandJ. F.Bounded Palais-Smale mountain-pass sequencesWillemM.