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

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 Λ_T > 0 such that for λ > Λ_T, equation (1.5) has a sign-changing multi-bump solution u_λ, which possesses the following property: for any sequence {λ_n} with λ_n → +∞ as n → ∞, there is a subsequences {u_λni} converges strongly to u in Hr1 (ℝ²), where u ∈ Hr1 (ℝ²) is a nontrivial solution of the equation

Moreover, u|_Ωi is positive for i ∈ T₁, u|_Ωi is negative for i ∈ T₂, and u|_Ωi changes sign exactly once for i ∈ T₃.

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 u_λ for (1.10), and they also proved that, up to a subsequence, u_λ converges strongly in H¹(ℝ^N) to a function u, which satisfies u = 0 outside Ω_T 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 Ω_i. 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 Ω_i, i ∈ T.

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 i ∈ T₁, u|_Ωi is negative for i ∈ T₂, and u|_Ωi changes sign exactly once for i ∈ T₃. 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 𝓜_T (see (2.1)) is nonempty and then we seek minimizers of the energy functional on 𝓜_T. Observe that 𝓜_T 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 Ω_T. 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 ∥u∥p=∫R2|u|pdx1p by L^p(ℝ²), where 1 ≤ p < ∞, and C denotes different positive constant in different place.

Lemma 2.1

Assume that (f₁)-(f₄) hold. For u ∈ H with u1+≠0,u2−≠0 and u3±≠0, then there exists a unique 4-tuple (s₁, s₂, s₃, s₄) ∈ (ℝ₊)⁴ such that

Proof

For u ∈ H with u1+≠0,u2−≠0 and u3±≠0, we define

where

Obviously, t₁u₁ + t₂u₂ + t₃ u3+ + t₄ u3− ∈ 𝓜 if and only if

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

Claim 1

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

Firstly, there exists a unique t̄₁ > 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 t̄₁ > 0 such that k(t̄₁) = 0. Moreover, by (f₄) we can deduce that t̄₁ is unique. Thus, we can get a function η₁ : (ℝ₊)³ → (0, +∞) defined by

such that F₁(t̄₁u₁ + t₂u₂ + t₃ u3+ + t₄ u3− ) = 0.

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

satisfying F₂(t₁u₁ + t̄₂u₂ + t₃ u3+ + t₄ u3− ) = 0, F₃(t₁u₁ + t₂u₂ + t̄₃ u3+ + t₄ u3− ) = 0, and F₄(t₁u₁ + t₂u₂ + t₃ u3+ + t̄₄ u3− ) = 0.

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

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

2. η_i are continuous on [0, +∞)³;

3. If amaxi large enough, then

   a¯i=ηi(ai|a1,a2,a3,a4)<amaxi:=max{a1,...,ai−1,ai+1,...,a4}

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

By the property (iii), there exists M₁ > 0 such that η_i(a_i|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 η_i(a_i|a₁, a₂, a₃, a₄) ≤ M₀.

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

where ā_i = η_i(a_i| a₁, a₂, a₃, a₄) satisfying F₁(ā₁u₁ + a₂u₂ + a₃ u3+ + a₄ u3− ) = 0, F₂(a₁u₁ + ā₂u₂ + a₃ u3+ + a₄ u3− ) = 0, F₃(a₁u₁ + a₂u₂ + ā₃ u3+ + a₄ u3− ) = 0, F₄(a₁u₁ + a₂u₂ + a₃ u3+ + ā₄ u3− ) = 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.

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

   Fi(a1u1+a2u2+a3u3++a4u3−)=0, i=1,2,3,4.

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

   a12∫Ω|∇u1|2dx+∫Ω(∫0|x|s2(a12u12(s)+a22u22(s)+a32u3+2(s)+a42u4+2(s))ds)2a12u12|x|2dx  +∫Ωa12u12+a22u22+a32u3+2+a42u4+2|x|2(∫0|x|s2(a12u12(s)+a22u22(s)+a32u3+2(s)+a42u4+2(s))ds)    (∫0|x|sa12u12(s)ds)dx=∫Ωf~(x,a1u1)a1u1dx.

   Since u ∈ 𝓜, we have

   ∫Ω|∇u1|2dx+∫Ω∫0|x|s2(u12(s)+u22(s)+u3+2(s)+u4+2(s))ds2u12|x|2dx  +∫Ωu12+u22+u3+2+u4+2|x|2(∫0|x|s2(u12(s)+u22(s)+u3+2(s)+u4+2(s))ds)    (∫0|x|su12(s)ds)dx=∫Ωf~(x,u1)u1dx.

   So we obtain

   1a14−1∫Ω|∇u1|2dx≥∫Ω(f~(x,a1u1)(a1u1)5−f~(x,u1)(u1)5)u16dx.

   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).

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

   s1′s1s1u1+s2′s2s2u2+s3′s3s3u3++s4′s4s4u3−∈M.

   Since s₁u₁ + s₂u₂ + s₃ u3+ + s₄ u3− ∈ 𝓜, by Case 1 we get

   s1′s1=s2′s2=s3′s3=s4′s4=1.

   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

Proof

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 u ∈ H with u1+ ≠ 0, u2− ≠ 0 and u3± ≠ 0 such that, F_i(u) ≤ 0 for i = 1, 2, 3, 4, where F_i 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.

Proof

Without loss of generality, we suppose that t₁ = max{t₁, t₂, t₃, t₄}. Since t₁u₁ + t₂u₂ + t₃ u3+ + t₄ u3− ∈ 𝓜, we get

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

Therefore, (2.7) and (2.8) imply that

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

Lemma 2.4

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

Proof

Let u ∈ 𝓜, then

and

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

Therefore,

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

we can easily to get that

Passing to a subsequences, one has

Assumptions (f₁) and (f₂) give

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 {v_n} ⊂ 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

Thus, t₁ = t₂ = t₃ = t₄ = 1 and m is attained by v. Since the restriction on f̃(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