Abstract
This article focuses on the study of the following Schrödinger-Poisson system with zero mass:
where f is a continuous function satisfying some general growth conditions, and it requires only to be super-quadratic growth at infinity and includes, in particular, the pure power function ∣u∣p−2u with p∈(3,6) . The nonlinear term ∣u∣u is the so-called Coulomb critical nonlinearity because it presents a certain scaling invariance and the mountain-pass geometry cannot be established when f=0 . Few results are known to such case. By developing some delicate analyses and using detailed estimates, we obtain the existence of ground-states and least energy solution for the aforementioned system under some natural assumptions on f .
1 Introduction and main results
This article is dedicated to studying the existence of ground-state solution for the following Schrödinger-Poisson system with zero mass and the Coulomb critical exponent:
where the nonlinearity f satisfies the following basic assumptions:
f∈C(R,R) and
limt→0f(t)∣t∣2=0,lim∣t∣→+∞∣f(t)∣∣t∣5=0;there holds
lim∣t∣→+∞F(t)∣t∣3=+∞,where F(t)≔∫t0f(s)ds ;
for every τ≠0 , f(τ)τ≥0 , and the function t↦2f(tτ)(tτ)−3F(tτ)t3 is increasing on t∈(0,+∞) ;
f(t)t>3F(t)>0,∀t∈R\{0} , and there exist R0>0,κ∈(32,3] , and C0>0 such that
f(t)t≥R0⇒∣f(t)t∣κ≤C0[f(t)t−3F(t)].
Define
As we know, −Δϕu=u2 in a weak sense, and problem (1.1) can be deduced from the following equation:
This is the well-known Schrödinger-Poisson-Slater problem, and it stems from the Slater approximation of the exchange term in the Hartree-Fock model, which appears in quantum mechanics in the study of a system of N particles [35]. ω≥0 represents the phase of the standing wave for the time-dependent equation, the convolution term denotes the Coulombic repulsion between the electrons, and the function ∣u∣2 presents the density of electrons. In such framework, f(u)=∣u∣23u was used by Slater to approximate the exchange term, and other exponents have been used in different approximations (see [3,27,30,32] for more information on these models). Such problems present a combination of repulsive forces (induced by the nonlocal term) and attractive forces (induced by the local term); the interaction between them leads to unexpected situations and also brings some interesting challenges in mathematics.
When ω>0 , the working space is the usual Sobolev space H1(R3) and fruitful results are obtained by researchers under various assumptions imposed on the nonlinearity f (see, e.g., [1,2,5,6,9–13,15,28,32–34, 36–38,42,43] and references therein); particularly, the existence or nonexistence of solutions, radial or nonradial solutions, positive or sign-changing solutions, bound-states and ground-states, semi-classical states, and normalized solutions are established there.
As pointed out by Ruiz [32], the behavior of radial minimizers motivates the study of the static case, i.e., ω=0 . Such case is called “zero mass” problem by Berestycki and Lions [4], because the linearized operator at zero involves only the Laplacian operator. Note that the absence of a phase term makes the space H1(R3) not to be a good framework for studying (1.3). The following space was first introduced by Ruiz [32]:
and it was shown by Ruiz [32, Proposition 2.2] that such space is a uniformly convex Banach space with the following norm:
The double integral expression is the so-called Coulomb energy of wave and has been well studied by Lieb and Loss in their classic book [26]. From this point of view, E is the subspace of D1,2(R3) such that the Coulomb energy of the charge is finite. Denote by Er the subspace of radial functions. Ruiz [32] obtained a general lower bound for the Coulomb energy, taking advantage of which useful embedding result was established there, i.e., Er↪Ls(R3) is continuous for p∈(187,6] , and the inclusion is compact for p∈(187,6) . This explains the significance of the exponent 187 in the radial case. Moreover, the existence of positive solution was shown there [32] for the following equation with p∈(187,3) :
Later, Ianni and Ruiz [16] considered the case p∈(3,6) and obtained the existence of ground-states and infinitely many radial bound-state solutions using the monotonicity trick and Krasnoselskii genus (see also [20] for similar results). Using the Pohozaev identity, we know that (1.5) has no solution in E∩H2loc(R3) for p≥6 (cf. [16]). Liu et al. [29] first considered the following problem with Sobolev critical exponent and lower-order perturbation:
where μ>0 and 187<p<6 . Moreover, they showed that (1.1) has a positive solution for p∈(4,6) and p∈(3,4] with μ large enough via a perturbation procedure. It is worth pointing out that the effect of the additive perturbation is to lower the energy such that the ground-state level of the functional can be controlled by a fine threshold. For similar results, we refer readers to [21,22]. These results were improved recently in previous studies [14,24] where some existence results were obtained by developing a much simpler method. For critical Schrödinger-Poisson-Slater problem (1.6) with non-autonomous nonlinearity, we refer readers to [41,44,45] where existence and multiplicity of positive solutions were established there. For other related results of nonlocal problems, we draw reader’s attention to previous studies [7,23,39].
The case p=3 is called Coulomb critical since it exhibits a certain invariance, i.e., for a solution u of (1.5) and parameter λ∈R , the function λ2u(λx) is also a solution of (1.5). It was shown by Ianni and Ruiz [16, Proposition 5.1] that (1.5) has only trivial solution for μ<2 and p=3 . Moreover, they obtained the following result.
Theorem 1.1
[16, Theorem 1.3] There exists an increasing sequence μk>0,μk→+∞ such that the problem
has a radial solution uk∈Er , where μk is the Lagrange multiplier which is not priori.
By Theorem 1.1, it is reasonable that the solutions in such case (i.e., p=3 in (1.5)) appear only for certain values of μ , and all possible solutions of (1.5) have energy equal to zero using Pohozaev-type equality (cf. [16, Remark 5.2]). We emphasize that we cannot get rid of the Lagrange multiplier μk ; this is not a problem of the method of the proof, but it is something intrinsic of the problem. In such case, the mountain-pass geometry cannot be verified by the energy functional corresponding to (1.5), and (1.5) should be interpreted as an eigenvalue problem rather than an equation (cf. [16,31]). In order to obtain more information about solutions for Schrödinger-Poisson-Slater problem with Coulomb critical exponent, it is reasonable to consider the nonlinearity as the combination of critical term ∣u∣u and perturbation term f(u) , i.e., problem (1.1). Lei et al. [19] considered (1.1) with f(u)=μ∣u∣p−2u , p∈(3,6) and showed that the Nehari-Pohozaev manifold is a C1 manifold, which is a natural constraint set for the corresponding energy functional. Existence of ground-state solution for (1.1) was obtained there by establishing a splitting lemma for a bounded Palais-Smale sequence on the Nehari-Pohozaev manifold. We point out that it is very crucial that the nonlinearity f considered there [19] is a pure power function, which guarantees the Nehari-Pohozaev set is a C1 manifold, and the Nehari-Pohozaev method used there becomes invalid without the C1 smooth restriction on the nonlinearity. For the following nonlinearity f , even though it possesses smoothness, we do not know whether problem (1.1) still has a nontrivial solution:
It is nature to ask whether the result obtained in [19] still holds for general nonlinearity; precisely, the following question will be addressed in this article.
Question 1. Study problem (1.1) with Coulomb critical exponent and zero mass. In this case, how to obtain the ground-state solution and least energy solution without C1 smooth restriction on the nonlinearity.
Since we only require that the nonlinearity f is super-quadratic growth at infinity (i.e., (F2) holds), the main difficulty to obtain a ground-state solution of (1.1) lies in verifying the boundedness of a proper Palais-Smale sequence or Cerami sequence. Different from Theorem 1.1 (cf. [16, Theorem 1.3]) and [29, Theorem 1.4], we work in the space E instead of the radial space Er , and the compact embedding E↪L3(R3) does not hold in such case; thus, solutions obtained here is not limited to be radial. Our purpose in this article is twofold. First, we try to establish a ground-state solution of (1.1) under general assumptions (F1)–(F3); to this end, we employ the strategy used in [37] and apply the deformation lemma together with some delicate estimates. Second, we pay our attention on the existence of least energy solution for (1.1). We introduce a new condition (F4), which is much weaker than the Ambrosetti-Rabinowitz type condition and use a perturbation method employed by Jeanjean [17,18] to achieve our goal. Thus, Question 1 is well solved (see Examples 1 and 2 for functions satisfying (F1)–(F4)). Such existence results, to the best of our knowledge, are up to date, which partially extend and, in fact, complement the aforementioned results of previous studies [14,16,19,29,31,32].
As well known, the energy functional corresponding to (1.1) is
Define the “Nehari-Pohozaev” manifold introduced by Ruiz [33] by
where
Our main results in this article read as follows:
Theorem 1.2
Assume that (F1)–(F3) hold. Then, problem (1.1) has a ground-state solution of Nehari-Pohozaev type, i.e., a nontrivial solution ˉu∈E such that Φ(ˉu)=infℳΦ>0 .
Theorem 1.3
Assume that (F1), (F2), and (F4) hold. Then, problem (1.1) has a least energy solution u0∈E\{0} .
Before ending this section, we give two nonlinear examples: both of them satisfy assumptions (F1)–(F4).
Example 1.4
where 3<t<6 and μ>0 .
Example 1.5
where 3≤p<5 and μ>0 .
Throughout this article, set ut(x)≔u(tx) for t>0 , and denote the norm of Ls(R3) by ‖u‖s=(∫R3∣u∣sdx)1⁄s for s≥1 . Br(x)={y∈R3:∣y−x∣<r} and C1,C2,… stand for positive constants, which may be different in different places.
2 Variational framework and preliminaries
In this section, we collect some basic notations and preliminaries, which will be used in the proofs of our main results.
Consider the space E defined by (1.4), and for any u∈E , set
and
In the following, we list some useful properties about the space E with the norm ‖⋅‖E (see, e.g., [16,29,32,38]).
Lemma 2.1
[32] ‖⋅‖E is a norm, and (E,‖⋅‖E) is a uniformly convex Banach space. Moreover, C∞0(R3) is dense in E, and E↪Ls(R3) is continuous for p∈[3,6] .
For function ϕu defined by (1.2), we have ϕu(x)>0 when u≠0 , and u∈E if and only if both u,ϕu∈D1,2(R3) . Moreover, by −Δϕ=u2 , we have
and
Using Hardy-Littlewood-Sobolev inequality (see [25] or [26, page 98]), one has
Lemma 2.3
For any s∈[3,6] , there exists Cs>0 such that
Proof
This lemma has been proved in [16, Lemma 3.1]; here, we give a direct proof with some information of the embedding constant Cs . Observe that
In view of the Sobolev inequality [40], there exists a best embedding constant S>0 such that
from which we deduce that
It follows from the Hölder inequality that
By setting Cs≔23−2s3S3−s , we obtain (2.8).□
Lemma 2.4
[32] Suppose that {un}⊂E . Then,
un→ˉu in E if and only if un→ˉu and ϕun→ϕˉu in D1,2(R3) ;
un⇀ˉu in E if and only if un⇀ˉu in D1,2(R3) and supN[un]<+∞ . In such case, ϕun⇀ϕˉu in D1,2(R3) .
As in [16,32], we define functionals T and D as follows:
and
Lemma 2.5
[16] Suppose that {un},{vn},{wn}⊂E , z∈E . If un⇀ˉu,vn⇀ˉv,wn⇀ˉw in E, then
We have the following vanishing lemma due to Lions [27, Lemma I.1] (see [16,29]).
Lemma 2.6
If un⇀ˉu in E, and
then
It follows from (F1) and Lemma 2.1 that the functional Φ defined by (1.7) is of class C1 , and
It is not difficult to verify that solutions of (1.1) are critical points of the functional (1.7).
3 Existence of ground-state solution
Lemma 3.1
Assume that (F1) and (F3) hold. Then,
for every τ≠0 , there holds
(3.1) h(t,τ)≔t−3F(t2τ)+2(1−t3)3f(τ)τ−(2−t3)F(τ)>0,∀t∈(0,1)∪(1,+∞);there holds
(3.2) f(τ)τ>3F(τ)>0,∀τ≠0.
Proof
(i). For any τ≠0 , by (F3) and a simple calculation, we have
This shows that h(t,τ)>h(1,τ)=0,∀t∈(0,1)∪(1,+∞) ; thus, (3.1) holds. By (F1) and (3.3), one has limt→0+h(t,τ)>0 .
(ii). Letting t→0+ in (3.1) and using the fact limt→0+h(t,τ)>0 , we have
It follows that (ii) holds.□
Lemma 3.2
Assume that (F1) and (F3) hold. Then,
Proof
Note that
By (1.7), (1.9), and (3.6), one has
This shows that (3.5) holds.□
From Lemmas 3.1 and 3.2, we have the following corollary immediately.
Corollary 3.3
Assume that (F1) and (F3) hold. Then, for u∈ℳ ,
Lemma 3.4
Assume that (F1), (F2), and (F3) hold. Then, for any u∈E\{0} , there exists a tu>0 such that t2uutu∈ℳ .
Proof
Let u∈E\{0} be fixed and define a function ζ(t)≔Φ(t2ut) on [0,∞) . By (1.9) and (3.6), we have for any t>0 ,
By (F1), (2.4), and (3.6), we have
From (3.8) and (F2), it is easy to verify that ζ(0)=0 , ζ(t)>0 for t>0 small and ζ(t)<0 for t large. Therefore, maxt∈[0,∞)ζ(t) is achieved at a tu=t(u)>0 so that ζ′(tu)=0 and t2uutu∈ℳ .□
Both Corollary 3.3 and Lemma 3.4 imply the following lemma.
Lemma 3.5
Assume that (F1), (F2), and (F3) hold. Then,
Lemma 3.6
Assume that (F1), (F2), and (F3) hold. Then,
there exists ρ0>0 such that ‖∇u‖22≥ρ0,∀u∈ℳ ;
m0=infu∈ℳΦ(u)>0 .
Proof
Since J(u)=0,∀u∈ℳ , by (F1), (1.9), (2.4), and (2.9), it follows that
where C1 is a positive constant. This implies
From (F1), (1.7), (2.4), (2.9), (3.6), and (3.7), we have
For any u∈ℳ , by (3.9), we can set t=(17S3150C2‖∇u‖42)16>0 . Then, it follows from (3.10) that
This shows that m0=infu∈ℳΦ(u)>0 .□
In view of [37, Lemma 2.8] and [14, Lemma 3.9], we have the following lemma.
Lemma 3.7
Assume that (F1), (F2), and (F3) hold. If un⇀ˉu in E, then
and
Lemma 3.8
Assume that (F1), (F2), and (F3) hold. Then, m0 is achieved.
Proof
We prove this lemma by using the strategy used in [37]. Let {un}⊂ℳ be such that Φ(un)→m0 . We first prove that supn∈NQ[un]<+∞ . Otherwise, we may assume that limn→∞Q[un]=+∞ . Let tn≔3√Q[un] and vn(x)≔t−2nun(x⁄tn) . Then, Q[vn]=1 . Choose R=23√2m0 . If δ≔limsupn→∞supy∈R3∫B1(y)∣vn∣3dx>0 , then in light of Lemma 2.6, ‖vn‖s→0 for s∈(3,6) . It follows from (F1), (2.4), (2.8), (3.6), and (3.7) that
This contradiction shows that δ>0 . Without loss of generality, we may assume the existence of yn∈R3 such that ∫B1(yn)∣vn∣3dx>δ2 . Let ˆvn(x)=vn(x+yn) . Passing to a subsequence, we have
By (F2), (2.8), (3.6), and (3.15), we obtain
This contradiction shows that {Q[un]} is bounded, and so {un} is bounded in E .
Since J(un)=0 , it follows from (1.7) and (1.9) that
We claim that there exist a δ′>0 and a sequence yn∈R3 such that
Indeed, suppose that (3.18) does not hold. Then, we have
By (F1) and Lemma 2.6, we have
This contradicts with (3.17) and Lemma 3.6 (ii). Therefore, (3.18) holds.
Let ˆun(x)=un(x+yn) . Then, we have ‖ˆun‖E=‖un‖E and
Therefore, there exists ˆu∈E\{0} such that, passing to a subsequence,
Let wn=ˆun−ˆu and
Then, (3.22) and Lemma 3.7 yield
and
From (1.7), (1.9), (3.21), (3.24), and (3.25), one has
and
If there exists a subsequence {wni} of {wn} such that wni=0 , then going to this subsequence, we have
which implies the conclusion of Lemma 3.8 holds. Next, we assume that wn≠0 . In view of Lemma 3.4, there exists tn>0 such that t2n(wn)tn∈ℳ . We claim that J(ˆu)≤0 . Otherwise, if J(ˆu)>0 , then (3.27) implies J(wn)<0 for large n . From (1.7), (1.9), (3.1), (3.5), and (3.26), we obtain
which is a contradiction due to Λ(ˆu)>0 ; thus J(ˆu)≤0 . Since ˆu∈E\{0} , in view of Lemma 3.4, there exists ˉt>0 such that ˉt2ˆuˉt∈ℳ . From (1.7), (1.9), (3.1), (3.5), (3.21), and Fatou’s lemma, one has
which implies (3.28) also holds.□
Lemma 3.9
Assume that (F1), (F2), and (F3) hold. If ˆu∈ℳ and Φ(ˆu)=m0 , then ˆu is a critical point of Φ .
Proof
We prove this lemma by employing the method introduced in [8]. If the conclusion does not hold, we assume that Φ′(ˆu)≠0 . Then, there exist δ>0 and ϱ>0 such that
Let {tn}⊂R such that tn→1 . Since t2nˆutn⇀ˆu in E , it follows from (2.11) and Lemma 2.5 that
and
Combining (3.30) with (3.31), one has
Thus, there exists δ1>0 such that
In view of Lemma 3.2 and J(ˆu)=0 , one has
It follows from (F1) and (1.9) that there exist T1∈(0,1) and T2∈(1,∞) such that
Set Θ≔min{∫R3h(T1,ˆu)dx,∫R3h(T2,ˆu)dx} . Let S≔B(ˆu,δ) and ε≔min{Θ⁄24,1,ϱδ⁄8} . Then, [40, Lemma 2.3] yields a deformation η∈C([0,1]×E,E) such that
η(1,u)=u if Φ(u)<m0−2ε or Φ(u)>m0+2ε ;
η(1,Φm0+ε∩B(ˆu,δ))⊂Φm0−ε ;
Φ(η(1,u))≤Φ(u),∀u∈E ;
η(1,u) is a homeomorphism of E .
By Corollary 3.3, Φ(t2ˆut)≤Φ(ˆu)=m0 for t>0 , then it follows from (3.33) and (ii) that
On the other hand, by (iii), (3.3), and (3.34), one has
where
Combining (3.36) with (3.37), we have
Define Ψ0(t)≔J(η(1,t2ˆut)) for t>0 . It follows from ε≤Θ⁄24 , (3.34) and (i) that η(1,t2ˆut)=t2ˆut for t=T1 and t=T2 , which, together with (3.35), implies
Since Ψ0(t) is continuous on (0,∞) , we have that η(1,t20ˆut0)∩ℳ≠∅ for some t0∈[T1,T2] , which contradicts with (3.38). Thus, Φ′(ˆu)=0 , and the proof is complete.□
Proof of Theorem 1.2
Applying Lemmas 3.6,3.8, and 3.9, we show Theorem 1.2.□
4 Existence of least energy solution
In this section, we give the proof of Theorem 1.3.
Proposition 4.1
[17] Let X be a Banach space, and let J⊂R+ be an interval. We consider a family {Iλ}λ∈D of C1 -functional on X of the form
where B(u)≥0,∀u∈X , and such that either A(u)→+∞ or B(u)→+∞ , as ‖u‖→∞ . We assume that there are two points v1,v2 in X such that
where
Then, for almost every λ∈D , there is a bounded (PS) cλ sequence for Iλ , i.e., there exists a sequence such that
{un(λ)} is bounded in X ;
Iλ(un(λ))→cλ ;
I′λ(un(λ))→0 in X* , where X* is the dual of X.
To apply Proposition 4.1, we let X=E and introduce a family of functional defined by
for λ∈[1⁄2,1] .
Lemma 4.2
[16,29] Assume that (F1) and (F2) hold. Let u be a critical point of Φλ in H1(R3) , and then, we have the following Pohozaev-type identity:
We set Jλ(u)≔2⟨Φ′λ(u),u⟩−Pλ(u) , then
for λ∈[1⁄2,1] .
Choose w∈C∞0(R3) such that ∣supp{w}∣>1 , and then, we have the following lemma.
Lemma 4.3
Assume that (F1) and (F2) hold. Then,
there exists a T>0 independent of λ such that Φλ(T2wT)<0 for all λ∈[1⁄2,1] ;
there exists a positive constant κ0 independent of λ such that for all λ∈[1⁄2,1] ,
cλ≔infγ∈Γmaxt∈[0,1]Φλ(γ(t))≥κ0>max{Φλ(0),Φλ(v2)},where
Γ={γ∈C([0,1],E):γ(0)=0,γ(1)=v2},andΦ12(v2)≤Φ12(T2wT)<0.
Proof
(i). It follows from (4.2) that
By (F2), one has that limt→+∞[t−6∫R3F(t2w)dx]=+∞ . Then, there exists T>0 such that
Hence,
(ii). By (F1), (2.2), (2.4), (2.8), and (4.2), we have
which yields that there exists δ>0 and κ0>0 such that for all λ∈[1⁄2,1] ,
Lemma 4.4
Assume that (F1), (F2), and (F4) hold. Then, for almost every λ∈[1⁄2,1] , there exists a uλ∈E\{0} such that
Proof
In view of (F1), (F3), and Lemma 4.3, we obtain that Iλ(u) satisfies the assumptions of Proposition 4.1 with X=E and Iλ=Φλ . So for almost every λ∈[1⁄2,1] , there exists a bounded sequence {un(λ)}⊂H1(R3) (for simplicity, we denote {un} instead of {un(λ)} such that
Rest of the proof is standard (cf. [16, Proposition 3.6, Corollary 3.7]), so we omit it.□
Proof of Theorem 1.3
In view of Lemma 4.4, there exist two sequences of {λn}⊂[1⁄2,1] and {uλn}⊂E , denoted by {un} , such that
From (4.2), (4.4), and (4.12), one has
We claim that {un} is bounded in E . Otherwise, we may suppose that Q[un]→∞ . Set tn≔3√Q[un] and vn(x)≔t−2nun(x⁄tn) , then Q[vn]=1 . By (F1) and (F2), we can assume that
where C3 is the embedding constant, and R0 is given in (F4). Set κ′=κ⁄(κ−1) and
Then, (F4), (4.14), and (4.15) imply that κ′∈[32,3) and Ω1n⊂Ω2n . By (F1) and (F2), there exist 0<t1<t2<+∞ such that
Hence, it follows from (F1) and (F4) that there exists C1>0 such that
By (2.8), (4.15), and (4.16), one has
From (2.8), (4.12), (4.13), (4.15), (4.16), (4.17) and the Hölder inequality, we have
By virtue of (F4), (2.8), (4.12), (4.13), (4.15), and the Hölder inequality, one can obtain
Hence, it follows from (2.2), (2.4), (4.2), (4.12), (4.18) (4.19), and (4.20) that
This contradiction shows that supn∈NQ[un]<∞ , i.e., {un} is bounded in E .
By (F1), (4.12), (4.13), and Lemma 2.6, it is easy to prove that δ1≔limsupn→∞supy∈R3∫B1(y)∣un∣3dx>0 . Going if necessary to a subsequence, we may assume the existence of yn∈R3 such that ∫B1(yn)∣un∣3dx>δ12 . Let ˜un(x)=un(x+yn) . Then, ‖˜un‖E=‖un‖E=1 , and
Passing to a subsequence, we have ˜un⇀˜u in E , ˜un→˜u in Lsloc(R3) , 3≤s<6 , ˜un→˜u a.e. on R3 . Obviously, (4.21) implies that ˜u≠0 . By a standard argument, we can prove that Φ′(˜u)=0 and Φ(˜u)=c1=infγ∈Γmaxt∈[0,1]Φ(γ(t)) ; these yield that ˜u is a least energy solution of problem (1.1) by noting that for any nontrivial solution u∈E\{0} , there holds Φ(u)≥c1 . (cf. [16, Proposition 3.4]).□
Acknowledgements
The authors would like to thank the anonymous referees for their invaluable suggestions.
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Funding information: This work was supported by the National Natural Science Foundation of China (No. 12171486), the Project of Key Laboratory of Language Engineering and Computing of Guangdong Province (No. LEC2020ZBKT002), the Scientific Research Fund of Hunan Provincial Education Department, China (22B1103), the Science and Technology Innovation Program of Hunan Province (No. 2024RC3021), the Young Backbone Teachers Project of Hunan Province, and Natural Science Foundation for Excellent Young Scholars of Hunan Province (No. 2023JJ20057).
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Author contributions: All authors contributed equally to the writing of this article. All authors read and approved the final manuscript.
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Conflict of interest: The authors declare that they have no competing interests.
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Data availability statement: Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.
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