# Infinitely Many High Energy Solutions for the Generalized Chern-Simons-Schrödinger System

Authors: Hua Su; Yongqing Wang; Jiafa Xu
Journal: Journal of Function Spaces
 (2020)

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## Abstract

In this paper, by virtue of the critical point theory, we are pleased to investigate the existence of infinitely many high energy solutions for the generalized Chern-Simons-Schrödinger system with a perturbation.


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## Body

## 1. Introduction

In this work, we are concerned with the following generalized Chern-Simons-Schrödinger system with a perturbation:(1)−Δu+λVxu+A0uxu+∑j=12Aj2uxu=fu+μgxuq−2u,∂1A2ux−∂2A1ux=−12u2,∂1A1ux+∂2A2ux=0,ΔA0ux=∂1A2uxu2−∂2A1uxu2,where λ,μ are positive parameters and V,f,g satisfy the following conditions:
(V1)
Vx∈Cℝ2, Vx=V∣x∣, and Vx≥0 on ℝ2;(V2)
There existsb>0 such that Vb=x∈ℝ2:Vx<b is nonempty and has finite measure;(V3)
There existsR>0 such that B0,R=intV−10 and B0,R¯=V−10, where B0,R denotes the ball of radius R centered at 0;(H1)
f∈Cℝ,ℝ, and fu=o∣u∣ as ∣u∣⟶0;(H2)
There existsR0≥0 such that Fu=∫0uftdt≥0 and Fu=1/6fuu−Fu≥0 for ∣u∣≥R0;(H3)
fuu/u6⟶+∞ as ∣u∣⟶∞;(H4)
There existα0,R1>0 and τ∈1,+∞ such that fuτ≤α0Fuuτ,for∣u∣≥R1;(H5)
f−u=−fu for u∈ℝ;(g)
g∈Lq′ℝ2, and x≥0≡0, for x∈ℝ2, where q′∈1,2/2−q,q∈1,2.Recently, the Chern-Simons-Schrödinger system(2)iD0u+D1D1+D2D2u=−fu,∂0A1−∂1A0=−Imu¯D2u,∂0A2−∂2A0=−Imu¯D1u,∂1A2−∂2A1=−12u2has been paid more attention by many researchers (for example, see [1–10]), where i denotes the imaginary unit, ∂0=∂/∂t,∂1=∂/∂x1,and∂2=∂/∂x2 for t,x1,x2∈ℝ1+2,u:ℝ1+2⟶ℂ is the complex scalar field, Aμ:ℝ1+2⟶ℝ is the gauge field, and Dμ=∂μ+iAμ is the covariant derivative for μ=0,1,2.In [1], the authors studied the nonlocal semilinear Schrödinger equation with the gauge field
(3)−Δu+λVxu+h2xx2+∫x∞hssu2sdsu=fx,u,inℝ2,where the potential V satisfies (V1)-(V3) (this potential can also be found in [11, 12]). When f satisfies more general 6 superlinear conditions at infinity, they obtained some existence theorems of nontrivial solutions for (3). Some similar results can also be found in [2, 5] with a constant external potential. In [4], the authors improved the results in [1, 2, 5] and used the concentration-compactness principle to obtain two bound state solutions for the generalized Chern-Simons-Schrödinger system
(4)−Δu+A0u+∑j=12Aj2u=up−2u+λu,∂1A2−∂2A1=−12u2,∂1A1+∂2A2=0,ΔA0=∂1A2u2−∂2A1u2,where p>4. In [3], the authors used some new techniques joined with the manifold of Pohožaev-Nehari type to study the existence of a semiclassical ground state solution for the generalized Chern-Simons-Schrödinger system
(5)ε2Δu+Vxu+A0uxu+∑j=12Aj2uxu=fu,ε∂1A2ux−∂2A1ux=−12u2,∂1A1ux+∂2A2ux=0,εΔA0u=∂1A2uxu2−∂2A1uxu2,where their results are available to the nonlinearity fu~us−2u for s∈4,6.There also are some papers in the literature which consider perturbation terms (see [12–21]) and the references therein (also refer to [22–27]). For example, in [13, 14], the authors used the famous Ambrosetti-Rabinowitz condition (see also [12, 15]) to study the existence of solutions for the following Schrödinger equations:
(6)−Δu+Vxu+ϕu=fx,u+gx,x∈ℝ3,−Δϕ=u2,x∈ℝ3,−Δpαu+Vxup−2u−λup−2u=fx,u+gxuq−2u,x∈ℝN,where −Δpα is the fractional p-Laplace operator. It is generally known that our conditions (H2) and (H4) are weaker than the corresponding (AR) condition: there exists μ>6 such that
(7)0<μFu≤fuuforu∈ℝ\0.So, our results here can be viewed as an extension to the ones in [12–15].In [16], the authors studied the following nonhomogeneous Schrödinger-Kirchhoff-type fourth-order Elliptic equations in ℝN:
(8)Δ2u−a+b∫ℝN∇u2dxΔu+Vxu=fx,u+hxinℝN,u∈H2ℝN.They obtained the existence of infinitely many solutions for this system by means of the symmetry mountain pass theorem and the fountain theorem.Now, we state the main result:Theorem 1.
Suppose that (V1)-(V3), (H1)-(H5), and (g) hold. Then, for arbitrarily smallμ>0, there exists λ0>0 such that system (1) possesses infinitely many high energy solutions when λ≥λ0.Remark 2.
From (H1), (H2), and (H4), we can get a growth condition forf. Using (H2) and (H4), for ∣u∣≥R2≔maxR0,R1, we have fuτ≤α0Fuuτ=α01/6fuu−Fuuτ≤α0/6fuuτ+1 and ∣fu∣≤α0/6τ−1uτ+1/τ−1. Let p=τ+1/τ−1+1=2τ/τ−1. Then, from (H4), we have p∈2,+∞ and ∣fu∣≤α0/6τ−1up−1,for∣u∣≥R2. On the other hand, using (H1) for all ε>0, we have ∣fu∣≤εu,for∣u∣≤R2. Therefore, we obtain
(9)∣fu∣≤ε∣u∣+cεup−1,u∈ℝ,cε≔α06τ−1,and thus,
(10)∣Fu∣≤ε2u2+cεpup,u∈ℝ.


## 2. Preliminaries

Let∥·∥s be the usual Ls-norm for s∈1,+∞, and li,l~ii∈ℝ+ stand for different positive constants. We use Hr1ℝ2 to denote a Sobolev space with the norm
(11)uHr1=∫ℝ2∇u2+u2dx1/2.Define the space(12)E≔u∈Hr1ℝ2:∫ℝ2∇u2+Vxu2dx<∞with the inner product and norm
(13)u,v=∫ℝ2∇u·∇v+Vxuvdx,u=u,u.Note the large parameterλ in Theorem 1, so we need the following inner product and norm:
(14)u,vλ=∫ℝ2∇u·∇v+λVxuvdx,uλ=u,uλ,u,v∈E.DefineEλ=E,∥·∥λ; then, we have Eλ which is a Hilbert space. Using (V1)-(V3), there exist positive constants λ0,l0 (independent of λ) such that
(15)uHr1≤l0uλ,forallu∈Eλ,λ≥λ0.Moreover, by [28], the embedding Eλ°Lpℝ2 is continuous for p∈2,+∞, and Eλ↪Lpℝ2 is compact for p∈2,+∞, i.e., there exists lp>0 such that
(16)up≤lpuHr1≤lpl0uλ,forallu∈Eλ,λ≥λ0,2≤p<+∞.For convenience, letlpl0=l~p.Now, onEλ, we define the following energy functional:
(17)Iu=12∫ℝ2∇u2+λVxu2+A12uu2+A22uu2dx−∫ℝ2Fudx−μq∫ℝ2gxuqdx.By (V1)-(V3), (9), (10), and [8], I is of class C1Eλ,ℝ, and
(18)I′u,ϕ=∫ℝ2∇u·∇ϕ+λVxuϕ+A12u+A22uuϕ+A0uuϕdx−∫ℝ2fuϕdx−μ∫ℝ2gxuq−2uϕdx,forallϕ∈Eλ.Note that (16) in [3], we have
(19)∫ℝ2A0uu∣2dx=2∫ℝ2A12u+A22uu2dx,forallu∈Hr1ℝ2.Consequently, we have(20)I′u,u=∫ℝ2∇u2+λVxu2+3A12u+A22uu2dx−∫ℝ2fuudx−μ∫ℝ2gxuqdx.Lemma 3 (see [4, 7, 8]).
Suppose thatun converges to u a.e. in ℝ2 and un converges weakly to u in Hr1ℝ2. Let Aα,n≔Aαunx,α=0,1,2. Then, ∫ℝ2Ai,n2unudx,∫ℝ2Ai,n2u2dx, and ∫ℝ2Ai,n2un2dx converge to ∫ℝ2Ai2u2dx, for i=1,2;∫ℝ2A0,nunudx and ∫ℝ2A0,nun2dx converge to ∫ℝ2A0u2dx.We say thatI∈C1X,ℝ satisfies Cc-condition if any sequence un such that
(21)Iun⟶c,1+unλI′un⟶0,asn⟶∞,has a convergent subsequence.Lemma 4 (see [29]).
Suppose thatX is an infinite dimensional Banach space, and Y,Z are two subspaces of X with X=Y⊕Z, where Y is finite dimensional. If I∈C1X,ℝ satisfies the Cc-condition for all c>0 and
(C1).I0=0andI−u=Iu for all u∈X;
(C2). there exist constantsρ,α>0 such that I∂Bρ∩Z≥α;
(C3). for any finite dimensional subspacesX~⊂X, there exists R=RX~>0 such that Iu≤0 on X~\BR,thenI possesses an unbounded sequence of critical values.


## 3. Main Results

In order to prove Theorem1, we provide some lemmas.Lemma 5.
Under assumptions (V1)-(V3), (H1)-(H5), and (g), any sequenceun⊂Eλ satisfying
(22)Iun⟶c>0,(23)I′un,un⟶0asn⟶∞is bounded in Eλ.Proof.
To prove the boundedness ofun, argument by contrary, assume that ∥un∥λ⟶∞. Let vn=un/unλ. Then, vnλ=1, and vnp≤l~pvnλ=l~p,p≥2. Note that qq′/q′−1>2 by (g); for large n, from (16), we have
(24)c+1≥Iun−16I′un,un=12∥un∥λ2+12∫ℝ2A12unun2+A22unun2dx−∫ℝ2Fundx−μq∫ℝ2gxunqdx−16∥un∥λ2−12∫ℝ2A12unun2+A22unun2dx+∫ℝ216funundx+μ6∫ℝ2gxunqdx≥13∥un∥λ2+∫ℝ2Fundx−μ1q−16∥g∥q′∥un∥qq′/q′−1q≥13∥un∥λ2+∫ℝ2Fundx−μ1q−16∥g∥q′l~qq′/q′−1q∥un∥λq≥∫ℝ2Fundx,for the fact that q∈1,2 and μ>0 is an arbitrarily small parameter.
In view of (20), we have
(25)1=∥un∥λ2∥un∥λ2=I′un,un∥un∥λ2−3∥un∥λ2∫ℝ2A12un+A22unun2dx+1∥un∥λ2∫ℝ2funundx+μ∥un∥λ2∫ℝ2gxunqdx≤limsupn→∞I′un,un∥un∥λ2+∫ℝ2funun∥un∥λ2dx+μ∥g∥q′l~qq′/q′−1q∥un∥λq∥un∥λ2≤limsupn→∞∫ℝ2funun∥un∥λ2dx.
Recall thatvnλ=1, and there exists a function v∈Eλ such that vn⇀v weakly in Eλ, vn→v strongly in Lrℝ2 with r∈2,+∞ and vnx→vx for a.e. x∈ℝ2. Define a set Ωna,b=x∈ℝ2:a≤∣unx∣<b with 0≤a<b, and we consider the following two possible cases.Case 1.
v=0, and vn⇀0 weakly in Eλ, vnx⟶0 for a.e. x∈ℝ2. From (9), we have
(26)∫Ωn0,R1funun∥un∥λ2dx=∫Ωn0,R1fununun2vn2dx≤ε+cεR1p−2∫Ωn0,R1vn2dx≤ε+cεR1p−2∫ℝ2vn2dx⟶0.On the other hand, by the Hölder inequality, (24), and (H4), we obtain
(27)∫ΩnR1,∞funun∥un∥λ2dx=∫ΩnR1,∞fununun2vn2dx≤∫ΩnR1,∞fununun2τdx1/τ∫ΩnR1,∞vn2τ/τ−1dxτ−1/τ≤∫ΩnR1,∞fununτdx1/τ∫ΩnR1,∞vnpdx2/p≤∫ΩnR1,∞α0Fundx1/τ∫ΩnR1,∞vnpdx2/p≤α0c+11/τ∥vn∥p2⟶0.Combining (26) and (27), we have
(28)∫ℝ2funun∥un∥λ2dx=∫Ωn0,R1funun∥un∥λ2dx+∫ΩnR1,∞funun∥un∥λ2dx⟶0,which contradicts (25).Case 2.
vx≢0,x∈ℝ2. Hence, let A=x∈ℝ2:vx≠0 and then, measA>0. For x∈A, we have limn→∞∣unx∣=∞, and hence, A⊂ΩnR1,∞ for large n. Note that from Proposition 2.1 in [8] and (2.15) in [3], there exists a constant C0>0 such that
(29)∫ℝ2A12u+A22uu2dx≤C0∥u∥λ6,forallu∈Hr1ℝ2.
Therefore, by (H3), (29), and (9), note the nonnegativity of fuu and q∈1,2, Fatou’s lemma enables us to obtain
(30)0=limn→∞I′un,un∥un∥λ6=limn→∞∥un∥λ2∥un∥λ6+3∫ℝ2A12un+A22unun2dx∥un∥λ6−∫ℝ2funundx∥un∥λ6−μ∫ℝ2gxunqdx∥un∥λ6≤limn→∞∥un∥λq∥un∥λ6μ∥g∥q′l~qq′/q′−1q+3C0∥un∥λ6∥un∥λ6−∫Ωn0,R1funun∥un∥λ6dx−∫ΩnR1,∞fununun6vn6dx≤3C0+limsupn→∞∫Ωn0,R1funun∥un∥λ6dx−liminfn→∞∫ΩnR1,∞fununun6vn6dx≤3C0+limsupn→∞εR12+cεR1p∥un∥λ6·measΩn0,R1−liminfn→∞∫ΩnR1,∞fununun6χΩnR1,∞xvn6dx≤3C0−∫ΩnR1,∞liminfn→∞fununun6χΩnR1,∞xvn6dx⟶−∞.This is also a contradiction.From Cases1 and 2, we have that un is bounded in Eλ.Lemma 6.
Under assumptions (V1)-(V3), (H1)-(H5), and (g), any sequenceun⊂Eλ satisfying (22) has a convergent subsequence in Eλ.Proof.
From Lemma5 and the compactness of Eλ°Lrℝ2 for r∈2,+∞, we have
(31)un⇀uweaklyinEλ,un⟶ustronglyinLrℝ2forr∈2,+∞,un⟶ufora.e.x∈ℝ2.
By a standard argument, we can check that(32)∫ℝ2fun−fuun−udx⟶0,asn⟶∞.
From Lemma3, we easily have
(33)∫ℝ2A12un+A22unun−A12u+A22uuun−udx⟶0,∫ℝ2A0unun−A0uuun−udx⟶0,asn⟶∞.
Note that from Lemma 1 in [30], there exists Cq>0 such that unq−2un−uq−2u≤Cqun−uq−1. Therefore, from (g) and the Hölder inequality, we have
(34)∫ℝ2gx∣unq−2un−uq−2uun−u∣dx≤Cq∫ℝ2gxun−uqdx≤Cq∥g∥q′∥un−u∥qq′/q′−1q⟶0,asn⟶∞.
By a simple calculation, we have(35)I′un−I′u,un−u=un−uλ2+∫ℝ2A12un+A22unun−A12u+A22uuun−udx+∫ℝ2A0unun−A0uuun−udx−∫ℝ2fun−fuun−udx−∫ℝ2gxunq−2un−uq−2uun−udx.
It is clear thatI′un−I′u,un−u⟶0. As a result, from (32)-(34), we have
(36)un−uλ⟶0asn⟶∞.
Letwn=un−u. Then, we have
(37)wn⟶0weaklyinEλ,wn⟶0stronglyinLrℝ2forr∈2,+∞,wn⟶0fora.e.x∈ℝ2.
Noteunu in Eλ and from Lemma A.1 of [28], there exists σx∈Lrℝ2 with r∈2,+∞ such that
(38)unx≤σx,ux≤σx,forx∈ℝ2,n∈ℕ.
From this and (2.12), forσ1∈L2ℝ3 and σ2∈Lpℝ3 with p∈4,2α∗, we have
(39)F~x,un−F~x,u≤c12un2+u2+c2punp+up≤c1σ12x+2c2pσ2px∈L1ℝ3.
Recallwn=un−u. From (2.11), (2.12), and (29), we have
(40)∫ℝ3F~x,wndx=∫ℝ314f~x,wnwn−F~x,wndx≤∫ℝ334c1wn2+p+44pc2wnpdx≤∫ℝ33c1σ12x+p+4p2p−2c2σ2pxd≤M~,where M~>0.Lemma 7.
Under assumptions (V1)-(V3), (H1)-(H5), and (g), for any finite dimensional subspaceE~λ⊂Eλ, there hold
(41)Iu⟶−∞,(42)∥u∥⟶∞,u∈E~λ.Proof.
Arguing indirectly, assume that for some sequenceun⊂E~λ with ∥un∥→∞, ∃M>0 such that Iun≥−M, ∀n∈ℕ. Let vn=un/∥un∥. Then, vn=1, and there is a function v∈E~λ such that vn⇀v in E~λ. Since dimE~λ<∞, we have vn⟶v in E~λ, vnx⟶vx for a.e., x∈ℝ2, and v=1. Let Ω=x∈ℝ2:vx≠0. Then, measΩ>0, and limn→∞∣unx∣⟶∞ for a.e., x∈Ω. From q∈1,2, (29), and (g), we have
(43)limn→∞∫ℝ2Fununλ6dx=limn→∞1/2∥un∥λ2+1/2∫ℝ2A12un+A22unun2dx−Iun−μ/q∫ℝ2gxunqdxunλ6≤limn→∞1/2∥un∥λ2+C0/2∥un∥λ6−Iun+μ/q∥g∥q′l~qq′/q′−1q∥un∥λqunλ6=C02.
Note that from the L’Hospital rule and (H3), we have(44)lim∣u∣→∞Fuu6=+∞uniformlyinx∈ℝ2.
Fatou’s lemma implies that(45)limn→∞∫ℝ2Fununλ6dx≥limn→∞∫ΩFununλ6dx≥liminfn→∞∫ΩFunun6vn6dx≥∫Ωliminfn→∞Funun6vn6dx≥∫Ωliminfn→∞Funun6χΩxvn6dx=+∞.
This contradicts ((43)), and thus, ((41)) holds.Proof of Theorem 1.
Note thatEλ is a Hilbert space, and let ej be a total orthonormal basis of Eλ, and define Xj=ℝej,Yk=⊕j=1kXj,Zk=⊕j=k+1∞Xj,k∈ℤ. From the compact embedding Eλ°Lrℝ2 with r∈2,+∞ and Lemma 3.8 in [28], we have
(46)βkr=supu∈Zk,∥u∥λ=1∥u∥r⟶0,k⟶∞.
This, together with (10), implies that
(47)Iu=12∫ℝ2∇u2+λVxu2+A12uu2+A22uu2dx−∫ℝ2Fudx−μq∫ℝ2gxuqdx≥12∥u∥λ2−∫ℝ2Fudx−μq∫ℝ2gxuqdx≥12∥u∥λ2−ε2∥u∥22−cεp∥u∥pp−μq∫ℝ2gxuqdx≥12∥u∥λ2−ε2l~22∥u∥λ2−cεp∥u∥pp−μq∥g∥q′l~qq′q′−1q∥u∥λq.
Note thatp=2τ/τ−1, q∈1,2, and ε,μ can be arbitrarily small. So, there exists k∈ℤ; we can choose ε≤1/2l~222τ−1, μ≤cεβkp2/qq/42τ−1∥g∥q′l~qq′/q′−1q′−1q, and let ∥u∥λ=cεβkp1/2−p with u∈Zk; by ((47)), we find
(48)Iu≥τ−12τ−1cεβkp2/2−p−cεpβkpcεβkpp/2−p≥τ−12τ−1−τ−12τcεβkp2/2−p>0,withτ>1,p>2.In Lemma4, let X=Eλ,Y=Yk, and Z=Zk,k∈ℤ. From Lemma 5-7 and (48), we have that (C2)-(C3) of Lemma 4 hold, and I satisfies the Cc-condition for all c>0. Moreover, (H5) implies that (C1) of Lemma 4 is also satisfied. Thus, I possesses an unbounded sequence of critical values un and then (1) possesses infinitely many high energy solutions, i.e.,
(49)Iun=12∫ℝ2∇un2+λVxun2+A12unun2+A22unun2dx−∫ℝ2Fundx−μq∫ℝ2gxunqdx⟶∞.


## 4. Conclusions

In this paper, we use a variational method and critical point theory to study the existence of infinitely many high energy solutions for the generalized Chern-Simons-Schrödinger system (1) with a perturbation. The conditions used in this paper are weaker than the famous Ambrosetti-Rabinowitz condition. Moreover, we consider the effect of the parameters λ,μ on the existence of solutions.



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