Least energy sign-changing solutions of fractional Kirchhoff–Schrödinger–Poisson system with critical growth

Applied Mathematics Letters 106 (2020) 106372

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Applied Mathematics Letters www.elsevier.com/locate/aml

Least energy sign-changing solutions of fractional Kirchhoff–Schrödinger–Poisson system with critical growth✩ Da-Bin Wang ∗, Jin-Long Zhang Department of Applied Mathematics, Lanzhou University of Technology Lanzhou, Gansu, 730050, People’s Republic of China

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Article history: Received 15 February 2020 Received in revised form 2 April 2020 Accepted 2 April 2020 Available online 8 April 2020

abstract This paper deals with the following nonlinear fractional Kirchhoff–Schrödinger– Poisson system

⎧( ∫ ∫ ⎪ ⎨ a + b R3 R3

Keywords: Sign-changing solutions Nonlocal term Constraint variation methods

|u(x)−u(y)|2 dxdy |x−y|3+2s

)

(−∆)s u + V (x)u + ϕu

2∗ s −2

= |u| ⎪ ⎩ t

u + λf (u), (−∆) ϕ = u2 ,

x ∈ R3 , x ∈ R3 ,

where s ∈ ( 43 , 1), t ∈ (0, 1), λ, a, b > 0. Under some conditions on f , we use constraint variational method to prove the existence and energy characteristics of least energy sign-changing solution to this problem. © 2020 Elsevier Ltd. All rights reserved.

1. Introduction and main results In this article, we are interested in the existence of the least energy sign-changing solution for the following fractional Schr¨ odinger–Poisson system with critical growth ) { ( ∫ ∫ 2 2∗ −2 a + b R3 R3 |u(x)−u(y)| dxdy (−∆)s u + V (x)u + ϕu = |u| s u + λf (u), x ∈ R3 , 3+2s |x−y| (1.1) (−∆)t ϕ = u2 , x ∈ R3 , where s ∈ ( 34 , 1), t ∈ (0, 1), λ, a, b > 0. As in [1], we assume that V ∈ C(R3 , R+ ) and satisfies: ∫ (V ) H ⊂ H s (R3 ) and H ↪→ Lp (R3 ), 2 < p < 2∗s is compact, where H = {u ∈ H s (R3 ) : R3 V (x)u2 dx < ∞} ∫ ∫ ∫ 2 with the norm ∥u∥2 = (u, u) = a R3 R3 |u(x)−u(y)| dxdy + R3 V (x)u2 dx. |x−y|3+2s ✩ This research was supported by the National Natural Science Foundation of China (11561043, 11961043). ∗ Corresponding author. E-mail address: [email protected] (D.-B. Wang).

https://doi.org/10.1016/j.aml.2020.106372 0893-9659/© 2020 Elsevier Ltd. All rights reserved.

D.-B. Wang and J.-L. Zhang / Applied Mathematics Letters 106 (2020) 106372

2

We suppose that f ∈ C 1 (R, R) and satisfies: (f1 ) limt→0

f (t) t3

= 0;

(f2 ) There exist q ∈ (4, 2∗s ) and C > 0 such that |f (t)| ≤ C(1 + |t| (f3 )

f (t) |t|3

q−1

), for all t ∈ R;

is increasing on t ∈ R\{0}.

The system (1.1) is related to fractional Kirchhoff equation ∫ ∫ 2 ( ) |u(x) − u(y)| a+b dxdy (−∆)s u + V (x)u = f (u), x ∈ R3 . 3+2s R3 R3 |x − y|

(1.2)

Many interesting results for (1.2) were obtained in last decades. Especially, some authors pay attention to find sign-changing solutions to problem like (1.2), see for examples, [2–4] and the references therein. When a = 1, b = 0, the system (1.1) stems from the following fractional Schr¨odinger–Poisson system { (−∆)s u + V (x)u + ϕu = f (u), x ∈ R3 , (1.3) (−∆)t ϕ = u2 , x ∈ R3 , where s, t ∈ (0, 1). Recently, there are many results about problems like system (1.3). Moreover, some people considered sign-changing solutions to problems like system (1.3) (see [5–9] and the references therein). However, as far as we know, there are few results involving sign-changing solutions to problems like (1.2) or (1.3) with critical growth. In [9], Yu, Zhao ang Zhao considered the following system with critical growth { 2∗ −2 p−2 (−∆)s u + u + k(x)ϕu = |u| s u + h(x)|u| u, x ∈ R3 , (1.4) (−∆)t ϕ = k(x)u2 , x ∈ R3 , where s ∈ ( 34 , 1), t ∈ (0, 1). They investigated the existence and energy characteristics of least energy signchanging solutions to system (1.4). Since their results depend on the case k ∈ Lp (R3 ) ∩ L∞ (R3 )\{0} for some p ∈ [6/(4s + 2t − 3), ∞), the methods used in [9] seem not valid if k(x) ≡ 1. It is worth pointing out that there are some interesting results, for example [10–25], considered signchanging solutions for other nonlocal problems. ∫ 1 p Let Lp (R3 ) be a Lebesgue space with the norm |u|p := ( R3 |u| dx) p , 1 ≤ p < ∞. For any γ ∈ (0, 1), we define the fractional space Dγ,2 (R3 ) as the closure of C0∞ (R3 ) with respect to the Gagliardo seminorm ∫ ∫ 2 dxdy. It is well known that, by the Lax–Milgram Theorem, for given ∥u∥21 = (u, u)1 = R3 R3 |u(x)−u(y)| |x−y|3+2γ t u ∈ H, there is a unique ϕu ∈ Dt,2 (R3 ) that satisfies (−∆)t ϕtu = u2 , x ∈ R3 . Moreover, ϕtu (x) = ∫ Γ ( 3−2t u2 (y) −3 3 2 ) 2 2−2t C(t) R3 |x−y| 3−2t dy, x ∈ R , where C(t) = π Γ (t) . Therefore, system (1.1) has equivalent form (

∫

∫

a+b R3

R3

2

|u(x) − u(y)| |x − y|

3+2s

) 2∗ −2 dxdy (−∆)s u + V (x)u + ϕtu u = |u| s u + λf (u), x ∈ R3 .

So, the energy functional associated with system (1.1) is defined by ∫ ∫ ∫ 1 b 1 1 2∗ Φλ (u) = ∥u∥2 + ∥u∥41 + ϕtu u2 dx − λ F (u)dx − ∗ |u| s dx, for any u ∈ H. 2 4 4 R3 2s R3 R3 From (f1 ) and (f2 ), for any ε > 0, there is Cε > 0 such that q−1

|f (t)| ≤ ε|t| + Cε |t|

, for all t ∈ R.

(1.5)

Moreover, by (f1 ) and (f3 ), we have that f (t)t > 0, t ̸= 0; F (t) ≥ 0, t ∈ R.

(1.6)

D.-B. Wang and J.-L. Zhang / Applied Mathematics Letters 106 (2020) 106372

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Under conditions of this paper, Φλ (u) belongs to C 1 , and the Fr´echet derivative of Φλ is ∫ ∫ ∫ 2∗ −2 ′ 2 t ⟨Φλ (u), v⟩ = (u, v) + b∥u∥1 (u, v)1 + ϕu uvdx − λ f (u)vdx − |u| s uvdx, R3

R3

R3

for any u, v ∈ H. The solution of system (1.1) is the critical point of the functional Φλ (u). Furthermore, if u ∈ H is a solution of system (1.1) and u± ̸= 0, then u is a sign-changing solution of system (1.1), where u+ = max{u(x), 0} and u− = min{u(x), 0}. It is worth noting that, thanks to (1.14) in [13], we get u± ∈ H if u ∈ H ⊂ H s (R3 ), which is important for finding sign-changing solutions. The main purpose of present paper is to study the existence and energy characteristics of least energy signchanging to system (1.1). As is usually done, for example [1,14,17,19,20,22,26], we first try to seek a minimizer of the energy functional Φλ over the constraint M = {u ∈ H, u± ̸= 0 and ⟨Φλ′ (u), u+ ⟩ = ⟨Φλ′ (u), u− ⟩ = 0}, and then will prove that the minimizer is a sign-changing solution of system (1.1). However, compared with (1.2) and (1.3), it is more difficult to deal with (1.1): (i) Since there are multiple nonlocal terms, the research of system (1.1) becomes more complicated; (ii) System (1.1) involves critical term, so we must deal with difficulty coming from the competing effect between nonlocal terms and critical term. The main results can be stated as follows. Theorem 1.1. Assume that (f1 )–(f3 ) hold. Then, there exists λ⋆ > 0 such that for all λ ≥ λ⋆ , the system (1.1) has a least energy sign-changing solution. Theorem 1.2. Assume that (f1 )–(f3 ) hold. Then, there exists λ⋆⋆ > 0 such that for all λ ≥ λ⋆⋆ , the c∗ := inf u∈N Φλ (u) > 0 is achieved and Φλ (u) > 2c∗ , where N = {u ∈ H\ {0}|⟨Φλ′ (u), u⟩ = 0}, and u is the least energy sign-changing solution obtained in Theorem 1.1. In particular, c∗ > 0 is achieved either by a positive or a negative function. 2. Technical lemmas Inspired by [17,26] and similar to that of in [17,26], we have Lemmas 2.1–2.3. Lemma 2.1. Suppose (f1 )–(f3 ) hold, if u ∈ H with u± ̸= 0, then there is a unique pair (αu , βu ) of positive numbers such that αu u+ + βu u− ∈ M. Furthermore, if ⟨Φλ′ (u), u± ⟩ ≤ 0, then 0 < αu , βu ≤ 1. Lemma 2.2. For u ∈ H with u± ̸= 0, (αu , βu ) is the unique maximum point of the function ψu (α, β) := Φλ (αu+ + βu− ), (α, β) ∈ [0, ∞) × [0, ∞). Lemma 2.3. There exists ρ > 0 such that ∥u± ∥ ≥ ρ for all u ∈ M. Lemma 2.4. Let cλ = inf u∈M Φλ (u), then limλ→∞ cλ = 0. Proof . For any u ∈ M, it is obvious that ⟨Φλ′ (u), u⟩ = 0. Thanks to (f3 ), Θ(t) := f (t)t − 4F (t) ≥ 0, and is increasing when t > 0 and decreasing when t < 0. Then, for any u ∈ M, one gets ∫ ∫ 1 1 1 1 λ 1 2∗ Φλ (u) = Φλ (u) − ⟨Φλ′ (u), u⟩ = ∥u∥2 + ( − ∗ ) |u| s dx + (f (u)u − 4F (u))dx ≥ ∥u∥2 . 4 4 4 2s R3 4 R3 4 So, we have that Φλ (u) > 0, for all u ∈ M. That is, cλ = inf u∈M Φλ (u) is well-defined. Let u ∈ H with u± ̸= 0 be fixed. According to Lemma 2.1, for each λ > 0, there exist αλ , βλ > 0 such that αλ u+ + βλ u− ∈ M. Hence, by (1.6) and properties of ϕu [27], we get 0 ≤ cλ = inf Φλ (u) ≤ Φλ (αλ u+ + βλ u− ) ≤

u∈M αλ2 ∥u+ ∥2

+ βλ2 ∥u− ∥2 + 2bαλ4 ∥u+ ∥41 + 2bβλ4 ∥u− ∥41 + 2λCαλ4 ∥u+ ∥4 + 2λCβλ4 ∥u− ∥4 .

D.-B. Wang and J.-L. Zhang / Applied Mathematics Letters 106 (2020) 106372

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To our end, we just prove that αλ → 0 and βλ → 0, as λ → ∞. Let Γu = {(αλ , βλ ) ∈ [0, ∞) × [0, ∞) : Wu (αλ , βλ ) = (0, 0), λ > 0}, where Wu = (⟨Φλ′ (αλ u+ + βλ u− ), αλ u+ ⟩, ⟨Φλ′ (αλ u+ + βλ u− ), βλ u− ⟩). Then, we have that ∫ ∫ 2∗ 2∗ 2∗ 2∗ αλs |u+ | s dx + βλs |u− | s dx ≤ 2αλ2 ∥u+ ∥2 + 2βλ2 ∥u− ∥2 + 4bαλ4 ∥u+ ∥41 + 4bβλ4 ∥u− ∥41 R3

R3

+ 8λCαλ4 ∥u+ ∥4 + 8λCβλ4 ∥u− ∥4 .

Therefore, Γu is bounded. Let {λn } ⊂ (0, ∞) be such that λn → ∞ as n → ∞. Then, in subsequence sense, there exist α0 and β0 such that (αλn , βλn ) → (α0 , β0 ), as n → ∞. We claim α0 = β0 = 0. Suppose, by contradiction, that α0 > 0 or β0 > 0. Thanks to αλn u+ + βλn u− ∈ M, for any n ∈ N, we have ∫ ∥αλn u+ + βλn u− ∥2 + b∥αλn u+ + βλn u− ∥41 + ϕtα u+ +β u− (αλn u+ + βλn u− )2 dx λn λn 3 R ∫ ∫ (2.1) ∗ 2 = |αλn u+ + βλn u− | s dx + λn f (αλn u+ + βλn u− )(αλn u+ + βλn u− )dx. R3

R3

Then, according to (1.5) and (1.6), we have that ∫ ∫ + − + − f (αλn u + βλn u )(αλn u + βλn u )dx → R3

f (α0 u+ + β0 u− )(α0 u+ + β0 u− )dx > 0,

R3

as n → ∞. Then, we conclude a contradiction with the equality (2.1). Hence, α0 = β0 = 0. □ Lemma 2.5. There exists λ⋆ > 0 such that for all λ ≥ λ⋆ , cλ is achieved. Proof . According to definition of cλ , there is a sequence {un } ⊂ M such that limn→∞ Φλ (un ) = cλ . Obviously, {un } is bounded in H. Then, in subsequence sense, there is u ∈ H such that un ⇀ u. Since the embedding H ↪→ Lp (R3 ) is compact, for all p ∈ (2, 2∗s ), we have un → u in Lp (R3 ), un (x) → ± ± ± 3 u(x) a.e. x ∈ R3 . So, u± in H, u± in Lp (R3 ), u± n ⇀u n →u n (x) → u (x) a.e. x ∈ R . 2 3 ∥u∥ ⋆ Denote δ := 3s S 2s , where S := inf u∈H\{0} ∫ 1 . According to Lemma 2.4, there is λ > 0 such ∗ (

|u|2s dx) 3 R3

− that cλ < δ for all λ ≥ λ . Fix λ ≥ λ , it follows from Lemma 2.2 that Φλ (αu+ n + βun ) ≤ Φλ (un ) for all α, β ≥ 0. So, by using Brezis–Lieb Lemma, Fatou’s Lemma and Hardy–Littlewood–Sobolev inequality, we get ⋆

α2 β2 + 2 + 2 lim (∥u+ lim (∥u− − u− ∥2 + ∥u− ∥2 ) n − u ∥ + ∥u ∥ ) + 2 n→∞ 2 n→∞ n bα4 bβ 4 + 2 + 2 2 − 2 − 2 2 + ( lim (∥u+ − u ∥ + ∥u ∥ )) + ( lim (∥u− 1 1 n − u ∥1 + ∥u ∥1 )) n→∞ 4 n→∞ n 4 ∫ ∫ α4 β4 2 2 + αβ lim inf H(un ) + lim ϕtu+ |u+ | dx + lim ϕt − |u− | dx n→∞ 4 n→∞ R3 n n 4 n→∞ R3 un n ∗ ∗ ∗ ∗ ∗ α 2s β 2s 2∗ + 2s + 2s − 2s − ∗ lim (|u+ − u | + |u | ) − lim (|u− + |u− |2s∗s ) ∗ ∗ n n − u | 2∗ 2 2 ∗ s s s 2s n→∞ 2s n→∞ ∫ ∫ bα2 β 2 2 − 2 −λ F (αu+ )dx − λ F (βu− )dx + lim inf (∥u+ n ∥1 ∥un ∥1 ) n→∞ 2 R3 R3 bα2 β 2 bα3 β bαβ 3 2 2 + lim inf H 2 (un ) + lim inf (H(un )∥u+ lim inf (H(un )∥u− n ∥1 ) + n ∥1 ) n→∞ n→∞ n→∞ 4 2 2 ∫ ∫ α2 β 2 α2 β 2 2 2 + lim inf ϕtu+ |u− | dx + lim inf ϕtu− |u+ n n | dx n→∞ n→∞ n n 4 4 R3 R3 ∗ 2 4 4 bα bα α 2s α A1 + A23 ∥u+ ∥21 + A43 − ∗ B1 ≥ Φλ (αu+ + βu− ) + 2 2 4 2s ∗ β2 bβ 4 2 − 2 bβ 4 4 β 2s + A2 + A ∥u ∥1 + A − ∗ B2 , 2 2 4 4 4 2s

− lim inf Φλ (αu+ n + βun ) ≥ n→∞

⋆

D.-B. Wang and J.-L. Zhang / Applied Mathematics Letters 106 (2020) 106372

where H(u) =

∫

R3

∫

R3

−(u+ (x)u− (y)+u− (x)u+ (y)) dxdy |x−y|3+2s

5

and

+ 2 − − 2 + + 2 A1 = lim ∥u+ n − u ∥ , A2 = lim ∥un − u ∥ , A3 = lim ∥un − u ∥1 , n→∞

n→∞

n→∞

2∗

2∗

− 2 + + s − s A4 = lim ∥u− , B2 = lim |u− . n − u ∥1 , B1 = lim |un − u |2∗ n − u |2∗ s s n→∞

n→∞

n→∞

So, one has that ∗

β2 bα4 2 + 2 bα4 4 α2s α2 A1 + A3 ∥u ∥1 + A3 − ∗ B 1 + A2 cλ ≥ Φλ (αu + βu ) + 2 2 4 2s 2 ∗ bβ 4 2 − 2 bβ 4 4 β 2s + A ∥u ∥1 + A − ∗ B2 , for all α ≥ 0 and all β ≥ 0. 2 4 4 4 2s −

+

(2.2)

Firstly, we prove that u± ̸= 0. By contradiction, we suppose u+ = 0. + + Case 1: B1 = 0. If A1 = 0, that is, u+ n → u in H. According to Lemma 2.3, we obtain ∥u ∥ > 0, which aα2 contradicts supposition. If A1 > 0, by (2.2) we get 2 A1 ≤ cλ for all α ≥ 0, which is a contradiction. 3 3 Case 2: B1 > 0. According to definition of S, we have that δ = 3s S 2s ≤ 3s ( A1 2 ) 2s . By direct ∗

(B1 ) 2s

calculation, we have that s ( 3

∗

A1

3

2 ∗

) 2s = max{

(B1 ) 2s

α≥0

∗

aα2 α 2s aα2 bα4 2 + 2 bα4 4 α2s A1 − ∗ B1 } ≤ max{ A1 + A ∥u ∥1 + A − ∗ B1 } α≥0 2 2s 2 2 3 4 3 2s

Thanks to cλ < δ, by (2.2) we have that ∗

δ ≤ max{ α≥0

∗

aα2 α 2s aα2 bα4 2 + 2 bα4 4 α2s A1 − ∗ B1 } ≤ max{ A1 + A ∥u ∥1 + A − ∗ B1 } < δ. α≥0 2 2s 2 2 3 4 3 2s

From above discussion, we have that u+ ̸= 0. Similarly, we obtain u− ̸= 0. Secondly, we prove that B1 = B2 = 0. We just prove B1 = 0 (the proof of B2 = 0 is analogous). By contradiction, we suppose that B1 > 0. Case 1: B2 > 0. Let α ˜ and β˜ satisfy ∗

∗

b˜ α4 2 + 2 b˜ α4 4 α ˜ 2s α2 bα4 2 + 2 bα4 4 α2s α ˜2 A1 + A3 ∥u ∥1 + A3 − ∗ B1 = max{ A1 + A ∥u ∥1 + A − ∗ B1 }, α≥0 2 2 4 2s 2 2 3 4 3 2s ∗ ∗ β˜2 bβ˜4 2 − 2 bβ˜4 4 β˜2s β2 bβ 4 2 − 2 bβ 4 4 β 2s A2 + A ∥u ∥1 + A − ∗ B2 = max{ A2 + A ∥u ∥1 + A − ∗ B2 }. β≥0 2 2 4 4 4 2s 2 2 4 4 4 2s

˜ is compact, there exist (αu , βu ) ∈ [0, α ˜ such that ψu (αu , βu ) = According to [0, α ˜] × [0, β] ˜] × [0, β] max(α,β)∈[0,α ˜]×[0,β˜] ψu (α, β), where ψu was defined as in Lemma 2.2. ˜ Obviously, if β small enough, we have that In the following, we prove that (αu , βu ) ∈ (0, α ˜) × (0, β). ψu (α, 0) < Φλ (αu+ ) + Φλ (βu− ) ≤ Φλ (αu+ + βu− ) = ψu (α, β), for all α ∈ [0, α ˜]. ˜ such that ψu (α, 0) ≤ ψu (α, β0 ), for all α ∈ [0, α So, there exists β0 ∈ [0, β] ˜]. That is, (αu , βu ) ̸∈ [0, α ˜]×{0}. ˜ By similar discussion, we conclude that (αu , βu ) ̸∈ {0} × [0, β]. Obviously, we get α2 A1 + 2 β2 A2 + 2

bα4 2 + 2 A ∥u ∥1 + 2 3 bβ 4 2 − 2 A ∥u ∥1 + 2 4

∗

bα4 4 α2s A − ∗ B1 > 0, α ∈ (0, α ˜] 4 3 2s ∗ bβ 4 4 β 2s ˜ A4 − ∗ B2 > 0, β ∈ (0, β]. 4 2s

(2.3) (2.4)

D.-B. Wang and J.-L. Zhang / Applied Mathematics Letters 106 (2020) 106372

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Then, ∗

∗

β2 b˜ α4 2 + 2 b˜ bβ 4 2 − 2 bβ 4 4 β 2s α ˜2 α4 4 α ˜ 2s A1 + A3 ∥u ∥1 + A3 − ∗ B 1 + A2 + A ∥u ∥1 + A − ∗ B2 , δ≤ 2 2 4 2s 2 2 4 4 4 2s ∗ ∗ β˜2 α2 bβ˜4 2 − 2 bβ˜4 4 β˜2s bα4 2 + 2 bα4 4 α2s δ≤ A2 + A ∥u ∥1 + A − ∗ B2 + A1 + A ∥u ∥1 + A − ∗ B1 , 2 2 4 4 4 2s 2 2 3 4 3 2s

˜ for all α ∈ [0, α ˜] and all β ∈ [0, β]. ˜ ≤ 0, ψu (˜ ˜ That is, Together with (2.2), we obtain ψu (α, β) α, β) ≤ 0, for all α ∈ [0, α ˜] and all β ∈ [0, β]. ˜ ˜ (αu , βu ) ̸∈ {˜ α} × [0, β] and (αu , βu ) ̸∈ ×[0, α ˜] × {β}. ˜ Hence, αu u+ + βu u− ∈ M. So, combining (2.2), (2.3) In conclusion, we get that (αu , βu ) ∈ (0, α ˜) × (0, β). with (2.4), we have that 2∗

cλ ≥ Φλ (αu u+ + βu u− ) +

αu2 bα4 bα4 αus β2 A1 + u A23 ∥u+ ∥21 + u A43 − ∗ B1 + u A2 2 2 4 2s 2 2∗

bβu4 2 − 2 bβu4 4 βus + A ∥u ∥1 + A − ∗ B2 > Φλ (αu u+ + βu u− ) ≥ cλ . 2 4 4 4 2s Therefore, we have a contradiction. Case 2: B2 = 0. In this case, we can maximize in [0, α ˜] × [0, ∞). Indeed, it is possible to show that there exists β0 ∈ [0, ∞) such that Φλ (αu+ + βu− ) ≤ 0, for all (α, β) ∈ [0, α ˜] × [β0 , ∞). Hence, there is (αu , βu ) ∈ [0, α ˜] × [0, ∞) that satisfies ψu (αu , βu ) = maxα∈[0,α ˜]×[0,∞) ψu (α, β). Following, we prove that (αu , βu ) ∈ (0, α ˜) × (0, ∞). Since ψu (α, 0) < ψu (α, β) for α ∈ [0, α ˜] and β small enough, we have (αu , βu ) ̸∈ [0, α ˜] × {0}. Meantime, ψu (0, β) < ψu (α, β) for β ∈ [0, ∞) and α small enough, then we have (αu , βu ) ̸∈ {0} × [0, ∞). On the other hand, for all β ∈ [0, ∞), it is obvious that ∗

b˜ α4 2 + 2 b˜ α 4 4 α 2s β2 bβ 4 2 − 2 bβ 4 4 α ˜2 A1 + A3 ∥u ∥1 + A3 − ∗ B1 + A2 + A ∥u ∥1 + A . δ≤ 2 2 4 2s 2 2 4 4 4 Hence, we have that ψu (˜ α, β) ≤ 0 for all β ∈ [0, ∞). Thus, (αu , βu ) ̸∈ {˜ α} × [0, ∞). And so (αu , βu ) ∈ + − (0, α ˜) × (0, ∞). That is, αu u + βu u ∈ M. Therefore, according to (2.3), we have that 2∗

α2 bα4 bα4 αus cλ ≥ Φλ (αu u + βu u ) + u A1 + u A23 ∥u+ ∥21 + u A43 − ∗ B1 2 2 4 2s 4 4 2 bβ bβ β + u A2 + u A24 ∥u− ∥21 + u A44 > Φλ (αu u+ + βu u− ) ≥ cλ , 2 2 4 +

−

which is a contradiction. Therefore, from above discussion, we have that B1 = B2 = 0. Lastly, we prove that cλ is achieved. Since B1 = B2 = 0, by discussions similar to subcritical problem, we easily conclude that cλ is achieved by u := u+ + u− ∈ M. □ 3. The proof of main results Proof of Theorem 1.1. Thanks to Lemma 2.5, we just prove that the minimizer u for cλ is indeed a signchanging solution of system (1.1). By an argument similar to [17,26], using the quantitative deformation lemma, we can prove that u is a sign-changing solution for system (1.1). □

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Proof of Theorem 1.2. Similar as the proof of Lemma 2.5, there exists λ⋆1 > 0 such that for all λ ≥ λ⋆1 , there exists v ∈ N such that Φλ (v) = c∗ > 0. By standard arguments, the critical points of the functional Φλ on N are critical points of Φλ in H and we obtain Φλ′ (v) = 0. That is, v is a ground state solution of (1.1). According to Theorem 1.1, we know that the system (1.1) has a least energy sign-changing solution u. Let λ⋆⋆ = max{λ⋆ , λ⋆1 }. As the proof of Lemma 2.1, there exist αu+ , βu− ∈ (0, 1) such that αu+ u+ ∈ N , βu− u− ∈ N . Therefore, in view of Lemma 2.2, we have that 2c∗ ≤ Φλ (αu+ u+ ) + Φλ (βu− u− ) ≤ Φλ (αu+ u+ + βu− u− ) < Φλ (u+ + u− ) = cλ . □ CRediT authorship contribution statement Da-Bin Wang: Conceptualization, Methodology, Formal analysis, Writing - review & editing. Jin-Long Zhang: Formal analysis, Writing - original draft. References [1] Z. Wang, H. Zhou, Sign-changing solutions for the nonlinear Schr¨ odinger–Poisson system in R3 , Calc. Var. Partial Differential Equations 52 (2015) 927–943. [2] S. Chen, X.H. Tang, F. Liao, Existence and asymptotic behavior of sign-changing solutions for fractional Kirchhoff-type problems in low dimensions, Nonlinear Differential Equations Appl. 25 (2018) 40. [3] K. Cheng, Q. Gao, Sign-changing solutions for the stationary Kirchhoff problems involving the fractional Laplacian in RN , Acta Math. Sci. 38B (2018) 1712–1730. [4] H. Luo, X.H. Tang, Z. Gao, Ground state sign-changing solutions for fractional Kirchhoff equations in bounded domains, J. Math. Phys. 59 (2018) 031504. [5] L. Guo, Sign-changing solutions for fractional Schr¨ odinger–Poisson system in R3 , Appl. Anal. 98 (2019) 2085–2104. [6] W. Long, S.J. Peng, J. Yang, Infinitely many positive solutions and sign-changing solutions for nonlinear fractional scalar field equations, Discrete Contin. Dyn. Syst. 36 (2016) 917–939. [7] W. Long, J.F. Yang, W. Yu, Nodal solutions for fractional Schr¨ odinger–Poisson problems, Sci. China Math. (2020) http://dx.doi.org/10.1007/s11425-018-9452-y. [8] D.B. Wang, Y. Ma, W. Guan, Least energy sign-changing solutions for the fractional Schr¨ odinger–Poisson systems in R3 , Bound. Value Probl. 2019 (2019) 25, http://dx.doi.org/10.1186/s13661-019-1128-x. [9] Y. Yu, F. Zhao, L. Zhao, Positive and sign-changing least energy solutions for a fractional Schr¨ odinger–Poisson system with critical exponent, Appl. Anal. (2020) http://dx.doi.org/10.1080/00036811.2018.1557325. [10] A.M. Batista, M.F. Furtado, Positive and nodal solutions for a nonlinear Schr¨ odinger–Poisson system with sign-changing potentials, Nonlinear Anal. RWA 39 (2018) 142–156. [11] D. Cassani, Z. Liu, C. Tarsi, J. Zhang, Multiplicity of sign-changing solutions for Kirchhoff-type equations, Nonlinear Anal. 186 (2019) 145–161. [12] Y.B. Deng, S.J. Peng, W. Shuai, Existence and asymptotic behavior of nodal solutions for the Kirchhoff-type problems in R3 , J. Funct. Anal. 269 (2015) 3500–3527. [13] Y.B. Deng, W. Shuai, Sign-changing solutions for non-local elliptic equations involving the fractional Laplacian, Adv. Differential Equations 23 (2018) 109–134. [14] R.F. Gabert, R.S. Rodrigues, Existence of sign-changing solution for a problem involving the fractional Laplacian with critical growth nonlinearities, Complex Var. Elliptic Equations 65 (2020) 272–292. [15] S. Kim, J. Seok, On nodal solutions of the nonlinear Schr¨ odinger–Poisson equations, Commun. Contemp. Math. 14 (2012) 16, 1250041. [16] Y. Li, D.B. Wang, J. Zhang, Sign-changing solutions for a class of p-laplacian kirchhoff-type problem with logarithmic nonlinearity, AIMS Math. 5 (2020) 2100–2112. [17] W. Shuai, Sign-changing solutions for a class of Kirchhoff-type problem in bounded domains, J. Differential Equations 259 (2015) 1256–1274. [18] W. Shuai, Q. Wang, Existence and asymptotic behavior of sign-changing solutions for the nonlinear Schr¨ odinger–Poisson system in R3 , Z. Angew. Math. Phys. 66 (2015) 3267–3282. [19] X.H. Tang, B. Cheng, Ground state sign-changing solutions for Kirchhoff type problems in bounded domains, J. Differential Equations 261 (2016) 2384–2402. [20] D.B. Wang, Least energy sign-changing solutions of kirchhoff-type equation with critical growth, J. Math. Phys. 61 (2020) 011501. [21] D.B. Wang, T. Li, X. Hao, Least-energy sign-changing solutions for Kirchhoff-Schr¨ odinger-Poisson systems in R3 , Bound. Value Probl. 2019 (2019) 75, http://dx.doi.org/10.1186/s13661-019-1183-3. [22] D.B. Wang, H. Zhang, W. Guan, Existence of least-energy sign-changing solutions for Schr¨ odinger–Poisson system with critical growth, J. Math. Anal. Appl. 479 (2019) 2284–2301.

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