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Advances in Nonlinear Analysis

Editor-in-Chief: Radulescu, Vicentiu / Squassina, Marco

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Sign-changing multi-bump solutions for the Chern-Simons-Schrödinger equations in ℝ2

Zhi Chen / Xianhua Tang / Jian Zhang
  • Corresponding author
  • School of Mathematics and Statistics, Hunan University of Technology and Business, Changsha, 410205 Hunan, P. R. China
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Published Online: 2019-09-20 | DOI: https://doi.org/10.1515/anona-2020-0041


In this paper we consider the nonlinear Chern-Simons-Schrödinger equations with general nonlinearity


where λ > 0, V is an external potential and


is the so-called Chern-Simons term. Assuming that the external potential V is nonnegative continuous function with a potential well Ω := int V–1(0) consisting of k + 1 disjoint components Ω0, Ω1, Ω2 ⋯, Ωk, and the nonlinearity f has a general subcritical growth condition, we are able to establish the existence of sign-changing multi-bump solutions by using variational methods. Moreover, the concentration behavior of solutions as λ → +∞ are also considered.

Keywords: Chern-Simons-Schrödinger equations; sign-changing solution; potential well; concentration behavior

MSC 2010: 35J20; 58E50

1 Introduction and main results

In this paper we are interested in the following nonlinear Schrödinger system with the gauge field


where i denotes the imaginary unit, 0=t,1=x1,2=x2 for (t, x1, x2) ∈ ℝ1+2, ϕ : ℝ1+2 → ℂ is the complex scalar field, Aκ : ℝ1+2 → ℝ is the gauge field and Dκ = κ + iAκ is the covariant derivative for κ = 0, 1, 2. This model (1.1) was first proposed and studied in [22, 23, 24], and is sometimes called the Chern-Simons-Schrödinger equations. The two-dimensional Chern-Simons-Schrödinger equations is a nonrelativistic quantum model describing the dynamics of a large number of particles in the plane, which interact both directly and via a self-generated electromagnetic field. Moreover, it describes an external uniform magnetic field which is of great phenomenological interest for applications of Chern-Simons theory to the quantum Hall effect.

As usual in Chern-Simons theory, system (1.1) is invariant under gauge transformation


for any arbitrary C function χ. The existence of standing wave solutions for system (1.1) with power type nonlinearity, that is, g(u) = λ|u|p–1u (p > 1 and λ > 0), has been investigated recently by a number of authors. For example, see [6, 7, 20, 25, 32] and the references therein. The standing wave solutions of system (1.1) have the following form


where ω > 0 is a given frequency, u, k, h are real valued functions depending only on |x|. Note that the ansatz (1.2) satisfies the Coulomb gauge condition 1A1 + 2A2 = 0. Inserting the ansatz (1.2) into the system (1.1), Byeon et al.[6] got the following nonlocal semilinear elliptic equation




Mathematically, equation (1.3) is not a pointwise identity as the appearance of the Chern-Simons term


Hence problem (1.3) is called a nonlocal problem and is quite different from the usual semi-linear Schrödinger equation. From the variational point of view, the nonlocal term causes some mathematical difficulties that make the study of problem (1.3) more interesting.

Following [6], equation (1.3) possesses a variational structure, that is, the standing wave solutions are obtained as critical points of the energy functional associated to (1.3) defined by


uHr1(ℝ2), where Hr1(ℝ2) := {uH1(ℝ2)| u(x) = u(|x|)}. In [6, 10, 20, 29, 30, 33, 39, 40, 45], the critical points of 𝓛 are found by using variational methods. It is shown that the value p = 4 is critical for this problem. Indeed, for p > 4, it is known that the energy functional is unbounded from below and satisfies a mountain-pass geometry. In a certain sense, in this case the local nonlinearity dominates the nonlocal term. However the existence of a solution is not so direct, since for p ∈ (4, 6) the (PS)-condition is not known to hold. In the spirit of [34], this problem is bypassed in [6] by using a constrained minimization taking into account the Nehari-Pohozaev manifold.

A special case is p = 4: in this case, solutions have been explicitly found in [6, 7] as optimizers of a certain inequality. An alternative approach would be to pass to a self-dual equation, which leads to a Liouville equation that can be solved explicitly. For more information on the self-dual equations, see [14, 24]. For the case p ≥ 4, [29] and [30] proved the existence, multiplicity, quantitative property and asymptotic behavior of normalized solutions with prescribed L2-norm.

The situation is different if p ∈ (2, 4), solutions are found in [6] as minimizers on a L2-sphere. Later, the results has been extended by Pomponio and Ruiz [32] by investigating the geometry of the energy functional under the different range of frequency ω. Moreover, Pomponio and Ruiz in [33] also studied the bounded domain case for p ∈ (2, 4). By using singular perturbation arguments based on a Lyapunov-Schmidt reduction, they obtained some results on boundary concentration of solutions. Wan and Tan [40] studied the existence and multiplicity of standing waves for asymptotically linear nonlinearity case, and see [44] for the sublinear case. Cunha et al.[10] obtained a multiplicity result when the nonlinearity satisfies the general hypotheses introduced by Berestycki and Lions [8]. For more results about the initial value problem, well-posedness, existence and blow-up, scattering and uniqueness results for some nonlocal problems, we refer readers to [5, 11, 12, 19, 21, 26, 27, 42, 43] and references therein.

When p > 6, by using the symmetric mountain pass theorem, Huh [20] obtained the existence of infinitely many radially symmetric solutions for equation (1.3). Recently, this result has been extended to more general nonlinearity model by Zhang et al.[45]. Besides, Deng et al.[16] and Li et al.[31] investigated the existence and asymptotic behavior of radial sign-changing solutions by using constraint minimization method and quantitative deformation lemma for equation (1.3). Liu et al.[28] obtained a multiplicity result of sign-changing solutions via a novel perturbation approach and the method of invariant sets of descending flow.

Very recently, for the general 6-superlinear nonlinearity case, Tang et al.[39] considered the following nonlocal Schrödinger equation with the gauge field and deepening potential well


where the potential V is a continuous function satisfies

  • (V1′$\begin{array}{} V'_{1} \end{array}$)

    V(|x|) ∈ C(ℝ2) and V(|x|) ≥ 0 on ℝ2;

  • (V2)

    there exists a constant b > 0 such that the set Vb := {x ∈ ℝ2|V(|x|) < b} is nonempty and has finite measure;

  • (V3)

    there is bounded symmetric domain Ω such that Ω = int V–1(0) with smooth boundary ∂Ω and Ω̄ = V–1(0).

Under some suitable conditions on the nonlinearity f, the second and third authors proved the existence and multiplicity of solutions (possibly positive, negative or sign-changing) by using mountain pass theorem. Moreover, the concentration behavior of these solutions on the set Ω as λ → +∞ are also studied.

Involving the Chern-Simons-Schrödinger equations with potential wells, there is only the work [39] so far. As described above, the shape of solutions obtained in [39] may be single-bump. However, nothing is known for the existence of multi-bump type solutions. Motivated by the above facts, we intend in the present paper to study the existence of sign-changing multi-bump solutions for equation (1.5) with deepening potential well. To the best of our knowledge, it seems that such a problem was not considered in literature before. In order to state our statements, for the potential V we need to assume that the following conditions besides (V2) and (V3),

  • (V1)

    VC1(ℝ2) and V(|x|) ≥ 0 on ℝ2;

  • (V4)

    there are k + 1 disjoint open bounded components Ω0, Ω1, Ω2, ⋯, Ωk (k ≥ 2) such that Ω = int V–1(0) = i=0kΩi and dist(Ωi, Ωj) > 0 for ij, i, j = 0, 1, 2, ⋯, k, where Ω0 = {x ∈ ℝ2, r0 = 0 ≤ |x| ≤ r0}, Ωi = {x ∈ ℝ2, ri ≤ |x| ≤ ri}.

Moreover, we suppose that the nonlinearity f satisfies

  • (f1)

    fC(ℝ, ℝ), and there exist constants C > 0 and q0 ∈ (4, +∞) such that


  • (f2)

    f(t) = o(t) as t → 0;

  • (f3)

    limtF(t)t6=+ where F(t) = 0tf(s)ds;

  • (f4)

    the function f(t)t5 is increasing on (0, +∞) and decreasing on (–∞, 0).

Before stating our results we first need to introduce some notations. Throughout this paper, we define


with the norm


Clearly, the embedding HλHr1(ℝ2) is continuous due to (V1) and (V2). We will give the proof later. Define the energy functional 𝓘λ : Hλ → ℝ by


Then, our hypotheses imply that the functional 𝓘λC1(Hλ, ℝ), and for any u, φHλ, we have


Obviously, critical points of 𝓘λ are the weak solutions for equation (1.5). Furthermore, if uHλ is a solution of (1.5) and u± ≠ 0, then u is a sign-changing solution of (1.5), where


Our main result can be stated as follows.

Theorem 1.1

Suppose that (V1)-(V4) and (f1)-(f4) hold. Then, for any non-empty subset T ⊂ {0, 1, 2, 3, …, k} with


There exists a constant ΛT > 0 such that for λ > ΛT, equation (1.5) has a sign-changing multi-bump solution uλ, which possesses the following property: for any sequence {λn} with λn → +∞ as n → ∞, there is a subsequences {uλni} converges strongly to u in Hr1(ℝ2), where uHr1(ℝ2) is a nontrivial solution of the equation


Moreover, u|Ωi is positive for iT1, u|Ωi is negative for iT2, and u|Ωi changes sign exactly once for iT3.

The motivation of the present paper arises from the study of the local Schrödinger equations with deepening potential well


We remark that conditions (V1)-(V3) have been first introduced by Bartsch, Pankov and Wang [9] in studying the Schrödinger equation (1.10). They obtained some results on the existence of multiple solutions, and also showed that the solutions concentrated at the bottom of the potential well as λ → +∞. The existence and characterization of the solutions for equation (1.10) were considered in [1, 2, 15, 35] under conditions (V1)-(V4). For example, Ding and Tanaka [15] first constructed the existence of multi-bump positive solutions uλ for (1.10), and they also proved that, up to a subsequence, uλ converges strongly in H1(ℝN) to a function u, which satisfies u = 0 outside ΩT and u|Ωi is the positive least energy solution to the equation


Inspired by [15], Alves [1] and Sato and Tanaka [35] investigated the sign-changing multi-bump solutions to equation (1.10) independently. Later, Alves and Pereira [2] obtained a similar result for the critical growth case. We must point out that the equation (1.11) (called limit equation of (1.10)) plays an important role in the study of multi-bump solutions for equation (1.10). Because the positive, negative and sign-changing solutions of (1.11) are used as building bricks to construct the multi-bump solutions of (1.10) by using of gluing techniques. Recently, there are some works focused on study of multi-bump solutions and ground states solutions for other nonlocal problems. For instance, see [18, 36, 37] for Schrödinger-Kirchhoff equation, [3, 13, 38] for Schrödinger-Poisson system, and [4] for Choquard equation and so on.

From the commentaries above, it is quite natural to ask if the results in [1, [35] still hold for the Chern-Simons-Schrödinger equations. Unfortunately, we can not draw a similar conclusion in a straight way. Since problem (1.5) is a nonlocal one as the appearance of the Chern-Simons term, then the solutions of problem (1.5) need their global information. Thus, we cannot use the same arguments explored in [1, 35] to solve the corresponding limit equation (1.9) separately on each Ωi. This result in the effective methods used [1, 35] for local Schrödinger equations cannot be applied to nonlocal problem (1.5) directly, there arises a technical problem one should overcome. As we all know, in order to obtain the existence of sign-changing multi-bump solutions for problem (1.5), we need to use the existence and some properties of the least energy sign-changing solutions of limit equation (1.9). However, it is not easy to prove the existence and sign properties of the least energy solution for limit equation (1.9). Hence, the first step is to consider the limit equation (1.9) and to look for the existence of least energy sign-changing solution that is nonzero on each component Ωi, iT.

Our result on the limit equation (1.9) can be stated as follows.

Theorem 1.2

Suppose that (f1)-(f4) hold, then, for any non-empty subset T, equation (1.9) has a nontrivial solution u with u|Ωi is positive for iT1, u|Ωi is negative for iT2, and u|Ωi changes sign exactly once for iT3. Moreover, u is the least energy solution among all solutions with those sign properties.

To prove our results, some arguments are in order. First, to obtain the least energy sign-changing solution solutions with prescribed sign properties for limit equation (1.9), we first prove the set 𝓜T (see (2.1)) is nonempty and then we seek minimizers of the energy functional on 𝓜T. Observe that 𝓜T is not a C1-manifold, we will take advantage of constraint minimization method and quantitative deformation lemma to obtain the existence of minimizers of the energy functional. Second, we will use the penalization technique explored by del Pino and Felmer in [17] to cut off the nonlinearity f, then, to control the order of growth of nonlinearity f outside the potential well ΩT. In such a way, we build a modification of the energy functional associated to (1.5) and give some energy relations of problems (1.5) and (1.9) which play a key role in getting the critical point of (1.5) (see Lemma 4.3). Moreover, in order to show a critical point associated to the modified functional is indeed a solution to the original problem, we also need give a delicate L-estimation for the solutions of the modified problem. Finally, we study a special minimax value of the modified functional, which is crucial for proving Theorem 1.1. Furthermore, via a rather precise analysis of deformation flow to the modified functional, we prove the existence of sign-changing multi-bump solutions for (1.5).

The paper is organized as follows. In Section 2, we give some preliminary lemmas and the proof of Theorem 1.2. In Section 3, we define a penalization problem and modified functional, and give a L-estimation for the solutions of the modified problem. In Section 4, we study a special minimax value of the modified functional. At last, we give the proof of Theorem 1.1 in Section 5.

Throughout the sequel, we denote the usual Lebesgue space with norms up=R2|u|pdx1p by Lp(ℝ2), where 1 ≤ p < ∞, and C denotes different positive constant in different place.

2 Mixed type sign-changing solutions to limit problem

In this section, we study the existence of solutions for the limited equation (1.9) with prescribed sign properties. Firstly, we restrict the nonlinearity (x, t) = 0 if




with the norm


Now we define the energy functional JT corresponding to limit problem (1.9) on HT


and the set


where ui := u|Ωi and T = T1T2T3 ⊂ {0, 1, 2, 3, …, k} satisfies (1.8), and (x, u) = 0u(x, t)dt. Let


If m is attained by u0 ∈ 𝓜T and JT(u0) = 0, then u0 be called a the least energy sign-changing solution of limit problem (1.9).

Without loss of generality, we consider the case T1 = {1}, T2 = {2} and T3 = {3} for simplify. In this case, ΩT = i=13Ωi with dist(Ωi, Ωj) > 0 for ij, i, j = 1, 2, 3. To simplify the notations, we use Ω, 𝓜, H to denote the sets ΩT, 𝓜T, HT respectively. Moreover, we define the functional on H as follows


The functional J is well-defined and belongs to C1(H, ℝ). Moreover, for any u, φH, we have


Clearly, critical points of J are the weak solutions for limit problem (1.9).

We use constraint minimizer on 𝓜 to seek a critical point of J with nonzero component. We first check that the set 𝓜 is nonempty in H.

Lemma 2.1

Assume that (f1)-(f4) hold. For uH with u1+0,u20 and u3±0, then there exists a unique 4-tuple (s1, s2, s3, s4) ∈ (ℝ+)4 such that



For uH with u1+0,u20 and u3±0, we define




Obviously, t1u1 + t2u2 + t3u3+ + t4u3 ∈ 𝓜 if and only if


Next, we obtain the desired results by proving two claims.

Claim 1

For uH with u1 ≠ 0, u2 ≠ 0 and u3± ≠ 0, there exists at least one solution for (2.5).

Firstly, there exists a unique 1 > 0 such that


for fixed (t2, t3, t4) ∈ (ℝ+)3. In fact, we define


By (f2), (f3) and (f4), it is easy to see that k(t) > 0 for t > 0 small enough and k(t) < 0 for t > 0 large enough. Hence, there exists 1 > 0 such that k(1) = 0. Moreover, by (f4) we can deduce that 1 is unique. Thus, we can get a function η1 : (ℝ+)3 → (0, +∞) defined by


such that F1(1u1 + t2u2 + t3u3+ + t4u3) = 0.

By the same arguments as above, we can define function ηi : (ℝ+)3 → (0, +∞), i = 2, 3, 4, given by


satisfying F2(t1u1 + 2u2 + t3u3+ + t4u3) = 0, F3(t1u1 + t2u2 + 3u3+ + t4u3) = 0, and F4(t1u1 + t2u2 + t3u3+ + 4u3) = 0.

Moreover, the functions ηi : (ℝ+)3 → (0, +∞), i = 1, 2, 3, 4, have the following three properties:

  1. For any (a1, a2, a3, a4) ∈ (ℝ+)4, we have ηi(ai|a1, a2, a3, a4) > 0;

  2. ηi are continuous on [0, +∞)3;

  3. If amaxi large enough, then


    where (ai|a1, a2, a3, a4) := (a1, …, ai–1, ai+1, …, a4).

By the property (iii), there exists M1 > 0 such that ηi(ai|a1, a2, a3, a4) ≤ amaxi for amaxi > M1. From the property (i), we get


Thus, M0 := max{M1, M2} > 0. For any (a1, a2, a3, a4) ∈ [0, M0]4, it follows from (iii) that ηi(ai|a1, a2, a3, a4) ≤ M0.

Hence, we can define L : [0, M0]4 → [0, M0]4 by


where āi = ηi(ai| a1, a2, a3, a4) satisfying F1(ā1u1 + a2u2 + a3u3+ + a4u3) = 0, F2(a1u1 + ā2u2 + a3u3+ + a4u3) = 0, F3(a1u1 + a2u2 + ā3u3+ + a4u3) = 0, F4(a1u1 + a2u2 + a3u3+ + ā4u3) = 0.

Obviously, L(a1, a2, a3, a4) is continuous on [0, M0]4. Now, by applying the Brouwer Fixed Point Theorem, there exists (s1, s2, s3, s4) ∈ [0, M0]4 such that


Thus, (s1, s2, s3, s4) is a solution of (2.5).

Claim 2

(s1, s2, s3, s4) obtained by Claim 1 is the unique solution of (2.5). To show Claim 2, the proof will be carried out in two cases.

  • Case 1

    u ∈ 𝓜. In this case, the 4-tuple (1, 1, 1, 1) is a solution of (2.5). We prove that (1, 1, 1, 1) is the unique solution of (2.5) in (ℝ+)4. Indeed, suppose (a1, a2, a3, a4) be any other solution, then


    Without loss of generality, we suppose a1 = max{a1, a2, a3, a4}. That is


    Since u ∈ 𝓜, we have


    So we obtain


    By (f1)-(f4), it imply that a1 = max{a1, a2, a3, a4} ≤ 1. By the same arguments, we can easily conclude that min{a1, a2, a3, a4} ≥ 1. Consequently, the 4-tuple (1, 1, 1, 1) is the unique solution of (2.5).

  • Case 2

    u ∉ 𝓜. If (u1, u2, u3, u4) ∉ 𝓜, then by Claim 1, we know that (2.5) has a solution (s1, s2, s3, s4). Assume that (s1,s2,s3,s4) also be a solution. Then we have


    Since s1u1 + s2u2 + s3u3+ + s4u3 ∈ 𝓜, by Case 1 we get


    Thus, (s1, s2, s3, s4) is the unique solution of (2.5) in (ℝ+)4.

Lemma 2.2

Assume that (f1)-(f4) hold, then the unique 4-tuple (t1, t2, t3, t4) ∈ (ℝ+)4 obtained by Lemma 2.1 is the unique maximum point of the function φ : (ℝ+)4 → ℝ defined by



From the proof of Lemma 2.1, we know that (t1, t2, t3, t4) is the unique critical point of φ in (ℝ+)4. By the assumption (f3), we deduce that φ(t1, t2, t3, t4) → –∞ as |(t1, t2, t3, t4)| → ∞, so it is sufficient to check that a maximum point cannot be achieved on the boundary of (ℝ+)4. Choose (s1, s2, s3, s4) ∈ (ℝ+)4, without loss of generality, we may assume that s1 = 0. But, it is obviously that


is an increasing function with respect to s if s > 0 is small enough, (0, s2, s3, s4) is impossible to be a maximum point of φ.

Lemma 2.3

Assume that (f1)-(f4) hold, let uH with u1+ ≠ 0, u2 ≠ 0 and u3± ≠ 0 such that, Fi(u) ≤ 0 for i = 1, 2, 3, 4, where Fi are given by (2.4). Then the unique 4-tuples (t1, t2, t3, t4) ∈ (ℝ+)4 obtained by Lemma 2.1 satisfied (t1, t2, t3, t4) ∈ (0, 1]4.


Without loss of generality, we suppose that t1 = max{t1, t2, t3, t4}. Since t1u1 + t2u2 + t3u3+ + t4u3 ∈ 𝓜, we get


At the same time, using Fi(u) ≤ 0, one has


Therefore, (2.7) and (2.8) imply that


By (f4), we obtain t1 ≤ 1. Thus we complete the proof.

Next, we consider the following constrained minimization problem m := infu∈𝓜 J(u).

Lemma 2.4

Assume that (f1)-(f4) hold, then m > 0 and m can be achieved by a function v ∈ 𝓜.


Let u ∈ 𝓜, then




Thus, by (f1), (f2) and Sobolev embedding theorem, there exists a positive constant C such that




this implies that m > 0. Suppose that {vn} ⊂ 𝓜 satisfying


we can easily to get that


Passing to a subsequences, one has

vnv weakly inH,vnvstrongl inLp(Ω)for2p<,vnva.e. inΩ.(2.9)

Assumptions (f1) and (f2) give

Ωf~(x,v3±)v3±dx=limn+Ωf~(x,vn,3±)vn,3±dxlim infn+vn,3±2C1.

Thus, v3± ≠ 0. By the same arguments, we conclude that v1+ ≠ 0 and v2 ≠ 0. By Lemma 2.1, there exists a unique 4-tuple (t1, t2, t3, t4) ∈ (ℝ+)4 such that


On the other hand, due to {vn} ⊂ 𝓜, we have that


Thus, Lemma 2.3 deduces that


Thanks to the function sf(s) – 6F(s) is a non-negative function, increasing on (0, +∞), decreasing on (–∞, 0), it follow from (2.10) and Lemma 2.2, we have

6m6J(v¯)=6J(v¯)i=12J(v¯),tiviJ(v¯),t3v3+J(v¯),t4v4=i=122ti2vi2+2t32v3+2+2t42v32+i=12Ω(f~(x,tivi)tivi6F~(x,tivi))dx+Ω(f~(x,t3v3+)t3v3+6F~(x,t3v3+))dx+Ω(f~(x,t4v3)t4v36F~(x,t4v3))dxlim infn+{i=122vn,i2+2vn,3+2+2vn,32+i=12Ω(f~(x,vn,i)vn,i6F~(x,vn,i))dx+Ω(f~(x,vn,3+)vn,3+6F~(x,vn,3+))dx+Ω(f~(x,vn,3)vn,36F~(x,vn,3))dx}lim infn+6J(vn)=6m.

Thus, t1 = t2 = t3 = t4 = 1 and m is attained by v. Since the restriction on (x, u), we can get that v1 ≥ 0, v2 ≤ 0 and (v3±) ≠ 0. So we prove the conclusion.

Proof of Theorem 1.2

We prove indirectly and assume that Φ′(v) ≠ 0. Then there exist δ > 0 and ϱ > 0 such that


For convenient, we define the function


By Lemma 2.2, we have


For ε := min{(m)/2, ϱδ/8} and S := B(v, δ), then [41] yields a deformation ηC([0, 1] × H, H) such that

  1. η(1, u) = u if J(u) < m – 2ε or J(u) > m + 2ε;

  2. η(1, Jm0+εB(v, δ)) ⊂ Jmε;

  3. J(η(1, u)) ≤ J(u), ∀ uH;

  4. η(1, u) is a homeomorphism of H.

It is clear that


We claim that η(1, g(D)) ∩ 𝓜 ∉ ∅. In fact, define




Then the degree theory and Lemma 2.1 yield


It follows from (i), one has g = h on ∂D. Thus, we obtain


Thus, Ψ1(1, 2, 3, 4) = 0 for some (1, 2, 3, 4) ∈ D, this implies that η(1, g(D)) ∩ 𝓜 ≠ 0.

By (2.13) and the definition of m we reach a contradiction. Thus, J′(v) = 0. We can get vC2(Ω) by standard elliptic regularity theory. So v1 > 0, v2 < 0 by the Maximum Principle. At last, we show that v3 indeed has two nodal domains. Arguing by contradiction, we may assume that


with wi ≠ 0, w1 ≥ 0, w2 ≤ 0 and suppt(wi) ∩ suppt(wj) = ∅, for ij, i, j = 1, 2, 3 and


Let u := w1 + w2, we see that u+ = w1 and u = w2. Setting


By Lemma 2.1, there exists a unique 4-tuple (1, 2, 3, 4) ∈ (ℝ+)4 such that




Moreover, J′(v)wi = 0 implies that Fi() < 0 for i = 1, 2, 3, 4. Using Lemma 2.3, we deduce that (1, 2, 3, 4) ∈ (0, 1]4. At the same time,




By using (2.14)-(2.16) and the fact that u+ = w1, u = w2, we have


which is a contradiction. So w3 = 0, and v3 has indeed two nodal domains.

3 Penalization of the nonlinearity and L-estimation

In this section, we will modify the functional Iλ by penalizing the nonlinearity f(u). It plays a key role in establishing the relation between mλ and m (will be defined later). On the other hand, we also give a delicate L-estimation for the critical points of the modified functional.



there exist open sets Ωiρ = {x ∈ ℝ2 : dist(x, Ωi) < ρ} for i = 1, 2, 3 with smooth boundary such that dist(Ωiρ,Ωjρ)>0 for ij, i, j = 1, 2, 3. Denote Ωρ:=i=13Ωiρ. For open set Θ ⊂ ℝ2, we define


with norm


By (V1) and (V2), there exists a positive constant ν0 such that

ν0R2Ωρu2dx12uλ,R2Ωρ2for alluHλ(R2Ωρ).(3.1)

Let a0 > 0 satisfy


and , : ℝ → ℝ are the functions given by




Using the above notations, we denote




where χΩρ denotes the characteristic function of the set Ωρ. We define the functional Φλ : Hλ → ℝ


and the critical points of Φλ are weak solutions of


The next Proposition is about the asymptotic behavior of the critical points of Φλ as λ → +∞.

Proposition 3.1

Suppose λn → +∞ as n → ∞ and {uλn} ⊂ Hλn satisfying


Then, up to a subsequence, there exists uHr1(ℝ2) such that

  1. unuλn → 0, consequently uλnu in Hr1(ℝ2);

  2. u = 0 in2Ω and u is a solution to equation (1.9);

  3. Φλn(uλn)J(u)=12Ω|u|2dx+12Ωu2|x|20|x|s2u2(s)ds2dxΩF(u)dx.


It is easy to know that {uλn} is bounded in Hλn(ℝ2) and hence {uλn} is bounded in Hr1(ℝ2). Passing to a subsequences, one has

unu weakly inHr1(R2);unustrongly inLp(RN)for2<p<;unua.e. inR2.(3.4)

We prove (ii) firstly. Let m ∈ ℕ+, set Sm = {x ∈ ℝ2 : V(|x|) ≥ 1m}, one has


and hence

Smu2dxlim infnSmuλn2dx=0.

It implies that u ≡ 0 in Sm. So we prove the u ≡ 0 in ℝ2Ω̄. Now, for any φC0(Ω), since 〈Φλn(un), φ〉 = 0, we can deduce that u is a solution to equation (1.9).

Next, we prove uλnu in Hr1(ℝ2). For convenience, let


and un = uλn. By virtue of 〈Φλn(un), unu〉 = 〈Φλn(u), unu〉 = 0 when n → +∞, we have



There, by Lemma 3.2 in [6], we have


Using the standard argument, we can deduce that


Therefore, by (3.6), (3.7) and (3.8), we get


On the other hand, the embedding HλHr1(ℝ2) is continuous. Indeed, by (V1) and (V2), we can deduce that


so it is easy to get


Thus we only need to show that


We choose a cut-off function ΨC(ℝ2) such that 0 ≤ Ψ ≤ 1 in ℝ2, Ψ(x) = 1 for each xΩρ and Ψ(x) = 0 for x ∈ ℝ2Ω2ρ and |∇Ψ| < C. By Sobolev’s embedding inequality,


So we get ∫2|∇u|2 + u2dxC2|∇u|2 + λV(|x|)u2dx. Thus we can get that


Combining with (i), it is easy to prove the (iii).

The next Lemma is important which indicates that the critical points uλ of Φλ with bounded energy are the solutions of the original problem (1.9) if λ large enough.

Lemma 3.2

Fix M > 0, for any critical points uλ of Φλ(uλ) ≤ M. Then there exists Λ0 > 0 such that λΛ0, one has



We prove Lemma 3.2 by Moser’s iteration. By Proposition 3.1, it is easy to get that


So, for any small η0 > 0 and λ large enough,


Let ψ be a smooth cut-off function and β > 1, both of them will be specified later. For R > 0, we define


and multiply (3.3) by ψ2|uλR|β1uλ, then


That is


On the other hand, by Hölder’s inequality and Young’s inequality, one has


Note |g(|x|, u)u| ≤ |u|2 + C0|u|q0, so the inequality (3.13) leads to


By sobolev imbedding theorem, we have


where S(p) is imbedding constant. Using (3.14) and (3.15), one has


Now, for y ∈ ℝ2Ωρ and fix a r which 0<r<ρ8. Then take the cut-off function ψ by


and 0ψ1,|ψ|Cr. Using Hölder’s inequality and (3.10), we have


Combing (3.16) and (3.17), we have


Taking limit R → +∞ and β = 5 in (3.18), which implies that for any y ∈ ℝ2Ωρ,


Because p ∈ (2, +∞), we choose p=32q03>2 since q0 > 4, one has


Now, we use the above estimation combining with Moser’s iteration argument to complete the proof. Let Zλ=|uλR|β12uλ, where β > 1 will choose later, then (3.16) becomes


where ψ is a cut-off function supported in B2r(y) with y ∈ ℝ3Ωρ and rρ4. By the Hölder’s inequality, we get


Since 2<187<6, thus, for any ϵ > 0,


By (3.21) and above estimate, it deduces that




Setting ϵ = S(6)[2C0C1(r)(β + 1)2]–1, we obtain from (3.22) that


Now, for rr2 < r1 ≤ 2r, we choose ψ such that ψ ≡ 1 in Br2(y), ψ ≡ 0 in ℝ2Br1(y) and 0 ≤ ψ ≤ 1, |ψ|Cr1r2. Then we obtain that


where h = (1 + β). Set


When R → +∞ in (3.24), then we have


Let h = hm = 6 ⋅ 3m, rm = r(1 + 2m) for m = 0, 1, 2, ⋯, by (3.25), we get


Let m → ∞, we have


We can choose Λ0, when λΛ0, we have C4(r)(2η0)16a0. So we can get


for λΛ0. Therefore we complete the proof.

4 A special minimax value for the modified functional

We investigate a special minimax value for the modified functional Φλ, which is used to get a key Lemma. We define a new functional


which is well defined and belongs to C1(Hλ(Ωρ), ℝ). We define set




Similar to the proof of Section 2, we deduce that there exists λHλ(Ωρ) such that


Lemma 4.1

There holds that

  1. 0 < λm, for all λ > 0;

  2. λm, as λ → +∞.


The proof of (i) is trivial since u ∈ 𝓜 which also belongs to 𝓜̄ by zero extension.

Now we are going to prove (ii). Let {λn} be a sequence with λn → +∞. For each λn, there exists uλnHλn(Ωρ) with


Since λnm, we can suppose {(uλn)} convergence (up to a subsequence) and J¯λn(uλn) = 0. It is easy to know that there exists uH01(Ω) ∩ Hr1(Ω) ⊂ Hλ(Ωρ) such that


and (u|Ω1)+, (u|Ω2), (u|Ω3)± ≠ 0. Moreover,




By the definition of m, one has that


Using conclusion (i), we obtain that λnm as n → +∞.

In Section 2, we have known that there exists vH, that is


and v1 = v|Ω1 is positive, v2 = v|Ω2 is negative and v3 = v|Ω3 changes sign exactly once. At the same time, we can find two positive constants τ2 > τ1 such that


We define y0 : [12,32]Hλ by






It is easy to check y0Σλ, so Σλ ≠ ∅ and mλ is well defined.

The next Lemma is trivial by degree theory, so we omit the detail.

Lemma 4.2

For any yΣλ, there exists an 4-tuple t=(t1,t2,t3,t4)D=(12,32)4 such that J¯λ(y(t)|Ωρ),y1+(t)=J¯λ(y(t)|Ωρ),y2(t)=0 and J¯λ(y(t)|Ωρ),y3±(t)=0 where yi(t)=y(t)Ωiρ for i = 1, 2, 3.

Lemma 4.3

There holds that

  1. λmλm for all λ ≥ 1;

  2. mλm as λ → +∞;

  3. There exists ϵ0 > 0 such that Φλ(y(t)) < mϵ0 for all λ ≥ 0, yΣλ and t = (t1, t2, t3, t4) ∈ [12,32]4.


  1. Since y0Σλ, we have


    Now, fixing t(12,32)4 given by Lemma 4.2, it implies


    By the definition of g(|x|, u), we deduce that |G(|x|,u)|ν02u2 for x ∈ ℝ2Ωρ. By (3.1) we can get



    maxt[12,32]4Φλ(y(t))J¯λ(y(t)|Ωρ)m¯λ,for eachyΣλ.

    So mλλ.

  2. it is obtained by Lemma 4.1 (ii) and Lemma 4.3 (i).

  3. For t = (t1, t2, t3, t4) ∈ [12,32]4, we have


By Lemma 2.1, it is to get


where ϵ0 is a small positive constant.

5 Proof of Theorem 1.1

In this section, we prove our main results. Define


Then we need further to study the properties of the set S.

Lemma 5.1

S is compact in H.


The proof is standard, we omit it immediately.

Lemma 5.2

Let d > 0 be a fixed number and let {un} ⊂ Sd be a sequence. Then, up to a subsequence, unu0 in Hλ as n → ∞, and u0S2d where


and distλ denotes the distance in Hλ.


Since S is compact in H, we can choose {ūn} ⊂ S satisfy


On the other hand, there exists ūS such that, up to a subsequence, ūnū in H. Hence, dist(un, u) ≤ d for n large enough. Thus {un} is bounded in Hλ. Up to a subsequence, unu0 weakly in Hλ. Since B2d(u) is weakly closed in Hλ, so u0B2d(u) ⊂ S2d.

Lemma 5.3

Let d ∈ (0, τ1), where τ1 is given by (4.4). Suppose that there exist a sequence λn > 0 with λn → ∞, and {un} ⊂ Sd satisfying


Then, up to a subsequence, {un} converges strongly in Hr1(ℝ2) to an element uS.


Observe that, by limn→∞ Φλn(un) ≤ m and limn→∞ Φλn(un) = 0 we deduce that {∥uuλn} and {Φλn(un)} are bounded. Up to a subsequence, we may assume that Φλn(un) → cm. By Proposition 3.1, there exists uHr1(ℝ2) such that unu in Hr1(ℝ2), u = 0 in ℝ2Ω and Φλn(un) → J(u). Moreover, u is a solution of equation (1.9). Next we prove that uS. Since {un} ⊂ Sd and d ∈ (0, τ1), we can deduce that (u|Ω1)+ ≠ 0, (u|Ω2) ≠ 0 and (u|Ω3)± ≠ 0. Indeed, if the conclusion is not correct, we can assume (u|Ω1)+ = 0, we can choose {ūn} ⊂ S satisfy




which implies that a contradiction. Hence, by Proposition 3.1 again, we get J′(u) = 0, u = 0 in ℝ2Ω. Then we get that J(u) ≥ m. At the same time, Φλn(un) → J(u) ≤ m, therefore uS.

Lemma 5.4

Let δ ∈ (0, τ1), where τ1 is given by (4.4). Then there exist constants 0 < σ < 1 and Λ1 > 0 such that Φλ(u)Hλσ for any uΦλm(SδSδ2) and λΛ1.


We prove it by contradiction. Suppose that there exist a number δ0 ∈ (0, τ1), a positive sequence {λj} with λj → 0, and a sequence of function {uj}Φλjm(Sδ0Sδ02) such that


Up to a subsequence, we get {uj} ⊂ Sδ0 and limj→∞ Φλj(uj) ≤ m. By Lemma 5.3, we can deduce that there exists uS such that uju in Hλj(ℝ2). Therefore, distλj(uj, S) → 0 as j → +∞. This contradict the assumption that ujSδ02.

Lemma 5.5

There exist Λ2Λ1 and α > 0 such that for any λΛ2,


implies that y0(t1, t2, t3, t4) ∈ Sδ2 for some δ ∈ (0, τ1).


We argue by contradiction. There exist λn → ∞, αn → 0 and (t1(n),t2(n),t3(n),t4(n))[12,32]4 such that


We can choose a subsequence (t1(n),t2(n),t3(n),t4(n))(t¯1,t¯2,t¯3,t¯4)[12,32]4. Then we have


By the unique of 4-tuple, it is easy to have (1, 2, 3, 4) = (1, 1, 1, 1). It implies that


Since y0(1, 1, 1, 1) = vS, which contradicts the assumption.

Now, we define


where δ, σ, α, ϵ0 are from Lemma 5.4, Lemma 5.5 and Lemma 4.3-(iii) respectively. Using Lemma 4.2, one has that there exists Λ3Λ2 such that

|mλm|<α0for allλΛ3.(5.2)

Lemma 5.6

There exists a critical point uλ of Φλ with uλSδΦλm for λΛ3.


We argue by contradiction. Fix a λΛ3, by the Lemma 5.3, we can assume that there exists 0 < ρλ < 1 such that ∥Φλ(u)∥ ≥ ρλ on SδΦλm. There exists a pseudo-gradient vector field Kλ in Hλ which is defined on a neighborhood Zλ of SδΦλm such that for any uZλ there holds


Define ψλ be a Lipschitz function on Hλ such that 0 ≤ ψλ ≤ 1, ψλ ≡ 1 on SδΦλm and ψλ ≡ 0 on HλZλ. Define ξλ be a Lipschitz function on ℝ such that 0 ≤ ξλ ≤ 1, ξλ(t) ≡ 1 if |tmλ| ≤ α2 and ξλ(t) ≡ 0 if |tmλ| ≥ α. Let


Then there exists a global solution ηλ : Hλ × [0, +∞) → Hλ for the initial value problem


We can deduce that ηλ has the following properties:

  1. ηλ(u, θ) = u if θ = 0 or uHλZλ or |Φλ(u) – mλ| ≥ α;

  2. ddθηλ(u, θ)∥ ≤ 2;

  3. ddθΦλ(ηλ(u, θ)) = 〈Φλ(ηλ(u, θ)), eλ(θλ(u, θ))〉 ≤ 0.

Assertion 1

For any (t1, t2, t3, t4) ∈ [12,32]4, there exists θ̄ = θ(t1, t2, t3, t4) ∈ [0, +∞) such that ηλ(y0(t1, t2, t3, t4), θ̄) ∈ Φλmλα0.

Assuming by contradiction that there exists (t1, t2, t3, t4) ∈ [12,32]4 such that


for any θ ≥ 0. By Lemma 5.5, we get y0(t1, t2, t3, t4) ∈ Sδ2. Note Φλ(y0(t1, t2, t3, t4)) ≤ mλ + α0, due the property (3) of ηλ,


So we can deduce that ξλ(Φλ(ηλ(y0(t1, t2, t3, t4), θ))) ≡ 1. If ηλ(y0(t1, t2, t3, t4), θ) ∈ Sδ for all θ ≥ 0, so it imply that

ψ(ηλ(y0(t1,t2,t3.t4),θ))1andΦλ(ηλ(y0(t1,t2,t3,t4),θ))ρλfor allθ>0.

It follows that


which is a contradiction. Thus, there exists θ3 > 0 such that ηλ(y0(t1, t2, t3, t4), θ3) ∉ Sδ. Note that y0(t1, t2, t3, t4) ∈ Sδ2, there exist 0 < θ1 < θ2 < θ3 such that


and ηλ(y0(t1, t2, t3, t4), θ) ∈ SδSδ2 for all θ ∈ (θ1, θ2). By Lemma 5.4, one has

Φλ(ηλ(y0(t1,t2,t3,t4),θ))σfor allθ(θ1,θ2).

Using the property (2) of ηλ we get that


This deduce that


which is a contradiction. Therefore, we prove the assertion 1.

Now, we define




Then Φλ((t1, t2, t3, t4)) ≤ mλα0 for all (t1, t2, t3, t4) ∈ [12,32]4.

Assertion 2

(t1, t2, t3, t4) = ηλ(y0(t1, t2, t3, t4), Γ(t1, t2, t3, t4)) ∈ Σλ.

For any (t1, t2, t3, t4) ∈ [12,32]4, we have


which implies that Γ(t1, t2, t3, t4) = 0. So (t1, t2, t3, t4) = y0(t1, t2, t3, t4) for (t1, t2, t3, t4) ∈ [12,32]4. We also need to prove ∥(t1, t2, t3, t4)∥ ≤ 6τ2 + τ1 for all [12,32]4 and Γ(t1, t2, t3, t4) is continuous with respect to (t1, t2, t3, t4).

For any (t1, t2, t3, t4) ∈ [12,32]4, we have Γ(t1, t2, t3, t4) = 0 if Φλ(y0(t1, t2, t3, t4)) ≤ mλα0, so (t1, t2, t3, t4) = y0(t1, t2, t3, t4). By the definition of y0(t), we have ∥(t1, t2, t3, t4)∥ ≤ 6τ2 < 6τ2 + τ1.

On the other hand, if Φλ(y0(t1, t2, t3, t4)) > mλα0, it implies that y0(t1, t2, t3, t4) ∈ Sδ2 and

mλα0<Φλ(ηλ(y0(t1,t2,t3,t4),θ))<mλ+α0,for allθ[0,Γ(t1,t2,t3,t4)).

So one has

ξλ(Φλ(ηλ(y0(t1,t2,t3,t4),θ)))1for allθ[0,Γ(t1,t2,t3,t4)).

Now, we will prove (t1, t2, t3, t4) ∈ Sδ. Otherwise, (t1, t2, t3, t4) ∉ Sδ, similar to the proof of assertion 1, we can find two constants 0 < θ1 < θ2 < Γ(t1, t2, t3, t4) such that


It contradicts to the definition of Γ(t1, t2, t3, t4). Therefore


Thus, there exists uS, such that


To prove the continuity of Γ(t1, t2, t3, t4), we fix arbitrarily (t1, t2, t3, t4) ∈ [12,32]4. First, we assume that Φλ((t1, t2, t3, t4)) < mλα0. In this case, it is to see that Γ(t1, t2, t3, t4) = 0, which gives that Φλ(y0(t1, t2, t3, t4)) < mλα0. By the continuity of y0, there exists r > 0 such that for any (s1, s2, s3, s4) ∈ Br(t1, t2, t3, t4) ∩ [12,32]4, we have Φλ(y0(s1, s2, s3, s4)) < mλα0, so Γ(s1, s2, s3, s4) = 0, and hence Γ is continuous at (t1, t2, t3, t4). On the other hand, we assume that Φλ((t1, t2, t3, t4)) = mλα0. Similar to the proof of assertion 1, we can deduce that (t1, t2, t3, t4) = ηλ(y0(t1, t2, t3, t4), Γ(t1, t2, t3, t4)) ∈ Sδ, thus


Therefore, for any w > 0, we have


Using the continuity of ηλ, there exists r > 0 such that


for any (s1, s2, s3, s4) ∈ Br(t1, t2, t3, t4) ∩ [12,32]4. Thus, Γ(s1, s2, s3, s4) ≤ Γ(t1, t2, t3, t4) + w. It follows that

0lim sup(s1,s2,s3,s4)(t1,t2,t3,t4)Γ(s1,s2,s3,s4)Γ(t1,t2,t3,t4).(5.3)

If Γ(t1, t2, t3, t4) = 0, we immediately obtain that


If Γ(t1, t2, t3, t4) > 0, we can similarly deduce that


for any 0 < w < Γ(t1, t2, t3, t4).

By the continuity of ηλ again, we see that

lim inf(s1,s2,s3,s4)(t1,t2,t3,t4)Γ(s1,s2,s3,s4)Γ(t1,t2,t3,t4).(5.4)

Combining (5.3) and (5.4), it is easy to see Γ is continuous at (t1, t2, t3, t4). This completes the proof of Assertion 2.

Thus, we have proved that (t1, t2, t3, t4) ∈ Σλ and


which contradicts the definition of mλ. This completes the proof.

Proof of Theorem 1.1

We still prove it with T1 = {1}, T2 = {2} and T3 = {3}. By Lemma 5.6, when λ > Λ3, we can get that there exists a solution uλSδΦλm for equation (3.3). By Lemma 3.2, we can know that uλ is a solution of equation (1.5) when λ > Λ := max{Λ0, Λ3}. Moreover, combining with Lemma 5.3, uλuS (up to subsequence) strongly in Hr1(ℝ2). So, we complete the proof of Theorem 1.1.


This work was supported by the NNSF (Nos. 11571370, 11601145, 11701173), by the Natural Science Foundation of Hunan Province (Nos. 2017JJ3130, 2017JJ3131), by the Excellent youth project of Education Department of Hunan Province (17B143, 17A113, 18B342), by the Hunan University of Commerce Innovation Driven Project for Young Teacher (16QD008), and by the Project of China Postdoctoral Science Foundation (2019M652790).


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About the article

Received: 2019-03-20

Accepted: 2019-07-11

Published Online: 2019-09-20

Published in Print: 2019-03-01

Citation Information: Advances in Nonlinear Analysis, Volume 9, Issue 1, Pages 1066–1091, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496, DOI: https://doi.org/10.1515/anona-2020-0041.

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© 2020 Z. Chen et al., published by De Gruyter. This work is licensed under the Creative Commons Attribution 4.0 Public License. BY 4.0

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