## ORIGINAL RESEARCH article

Front. Phys., 17 May 2023
Sec. Interdisciplinary Physics
Volume 11 - 2023 | https://doi.org/10.3389/fphy.2023.1185846

# On the asymptotically cubic generalized quasilinear Schrödinger equations with a Kirchhoff-type perturbation

Guofa Li1 Chong Qiu2 Bitao Cheng1 Wenbo Wang3*
• 1Key Laboratory of Analytical Mathematics and Intelligent Computing for Yunnan Provincial, Department of Education and College of Mathematics and Statistics, Qujing Normal University, Qujing, China
• 2Faculty of Mathematics and Physics, Huaiyin Institute of Technology, Huaian, China
• 3Department of Mathematics and Statistics, Yunnan University, Kunming, China

In this paper, we consider the non-existence and existence of solutions for a generalized quasilinear Schrödinger equation with a Kirchhoff-type perturbation. When the non-linearity h(u) shows critical or supercritical growth at infinity, the non-existence result for a quasilinear Schrödinger equation is proved via the Pohožaev identity. If h(u) shows asymptotically cubic growth at infinity, the existence of positive radial solutions for the quasilinear Schrödinger equation is obtained when b is large or equal to 0 and b is equal to 0 by the variational methods. Moreover, some properties are established as the parameter b tends to be 0.

## 1 Introduction

The Schrödinger equation [1] is of paramount importance in physics, and there are many modifications in literature, for example, the Chen–Lee–Liu equation [2] and stochastic Schrödinger equation [3]. However, the generalized quasilinear Schrödinger equation with a Kirchhoff-type perturbation was rarely studied in literature, which can be written as

$1+b∫R3g2u|∇u|2dx−divg2u∇u+gug′u|∇u|2+Vxu=hu,$

where $x∈R3,b≥0,V:R3→R$ and $h:R→R$ are continuous functions, $g∈C1(R,R+)$ satisfies (g1), g is even, g′(t) ≤ 0, $g(0)=1,limt→+∞g(t)=l,l∈(0,1)$, and ∀ t ≥ 0.

When b = 0, Eq. 1.1 is reduced to the following quasilinear Schrödinger equation:

$−divg2u∇u+gug′u|∇u|2+Vxu=hu,x∈R3.$

According to [4], let $g(u)=1+2(φ′(|u|2))2u2$, then, Eq. 1.2 is transformed into

$−△u−△φ|u|2φ′|u|2u+Vxu=hu,x∈R3.$

It is well-known that the classical case is φ(s) = s or $φ(s)=1+s$ [512].

For Eq. 1.1, another interesting question is b > 0. When g(t) = 1 for all $t∈R$, it is reduced to the following classical Kirchhoff equation:

$−1+b∫R3|∇u|2dxΔu+Vxu=hu,x∈R3.$

It is well-known that Eq. 1.4 is related to the stationary analog of the following Kirchhoff-type equation:

$utt+1+b∫R3|∇u|2dxΔu+Vxu=hu,x∈R3,$

which was proposed by Kirchhoff as an extension of the classical D’Alembert’s wave equation for free vibrations of elastic strings [13,14]. More physical background can be found in [15] and the references therein. Based on the aforementioned analysis, it is necessary to study Eq. 1.1.

### 1.1 Related works and main results

At first, let us briefly review the predecessors’ pioneering works about the problem [1620]. However, to the best of our knowledge, there are no works involving Eq. 1.1 when the non-linearity h(u) is asymptotically cubic at infinity. More information about the asymptotically cubic problems is given in [21,22] and the references therein. The main goal of the present paper is to investigate this problem. Precisely, we suppose that

(V1) $V\left(x\right)=V\left(|x|\right),0<{V}_{0}\le V\left(x\right)\le {V}_{\infty }≔\underset{|x|\to +\infty }{\mathrm{lim}}V\left(x\right)<\infty$;

(V2) $V\in {C}^{1}\left({\mathbb{R}}^{3},\mathbb{R}\right)$ and $⟨\nabla V\left(x\right),x⟩\le 0,\forall \phantom{\rule{0.3333em}{0ex}}x\in {\mathbb{R}}^{3}$;

(h1) $h\in C\left(\mathbb{R},\mathbb{R}\right)$, h(t) = 0, ∀ t ≤ 0, and $\underset{t\to 0}{\mathrm{lim}}\frac{h\left(t\right)}{t}=0$;

(h2) $\underset{|t|\to +\infty }{\mathrm{lim}}\frac{|h\left(t\right)|}{|t{|}^{3}}=\gamma ,\gamma >b{l}^{4}{\lambda }_{1}$, where

$λ1≔inf∫R3|∇w|2dx2:w∈H,∫R3|w|4dx=1$

and $H$ is defined in Section 2;

(h3) $\frac{1}{4}h\left(t\right)t\ge H\left(t\right)$ for all t > 0, where $H\left(t\right)={\int }_{0}^{t}h\left(s\right)ds$.

Remark 1.1: For example, $h(t)=γt51+t2$. By direct calculations, we have

$Ht=γt44−γt22+γ2ln1+t2.$

It is easy to observe that h satisfies the assumption (h1) − (h3).

The first result involves non-existence for the Kirchhoff-type perturbation problem.

Theorem 1.1: Assume that (g1) holds with $13≤l≤1$ and ⟨∇V(x), x⟩≥ 0. For any b > 0, Eq. 1.1 has no non-trivial solutions with h(u) = |u|p−2u, p ≥ 6.

The next result describes the existence for generalized quasilinear Schrödinger equations with the Kirchhoff term.

Theorem 1.2: Assume that (V1), (V2), (g1), (h1), and (h2) are satisfied. Then, Eq. 1.1 has a positive radial solution.

The third result shows the existence for generalized quasilinear Schrödinger equations without the Kirchhoff term.

Theorem 1.3: Assume that (V1), (V2), (g1), and (h1) − (h3) are satisfied. Then, Eq. 1.2 has a positive radial solution.

Compared with Theorem 1.2, without the Kirchhoff term $∫R3g2(u)|∇u|2dx$, we find that we need to add the condition (h3). Until now, we have not been able to remove it. A natural question is that what happens if Kirchhoff-type perturbation occurs, that is, when b → 0, can we build a relationship between Theorem 1.2 and 1.3? In this regard, we state the following.

Theorem 1.4: Assume that (V1), (V2), (g1), and (h1) − (h3) hold and ${ubn}⊂H$ are the positive radial solutions obtained in Theorem 1.2 for each $n∈N$. Then, $ubn→u0$ in $H$ as bn → 0, n∞, where u0 is a positive radial solution for Eq. 1.2.

### 1.2 Our contributions and methods

We should mention that our results are new since we focus on the asymptotically cubic case. Compared with [16,19,20], we know that in Theorem 1.1, our non-linear term in the autonomy problem Eq. 1.1 is supercritical, so we invoke the Pohožaev-type identity. As for Theorem 1.2, the problem is asymptotically 3-linear at infinity (i.e., h(t) ∼ t3), so it is different from [16]. We take full advantage of the condition h2, and this is our paper’s highlight. We borrow the idea from [16], but we require more elaborate estimates (see Lemma 3.2–3.4) to prove Theorem 1.3. It is worth pointing out that in Theorem 1.3, it seems that the condition (h3) is fussy, but our pursuit is not to relax the condition. Our condition (h3) is different from ([16], h5), and we adopt the idea from [23], Lemma 2.2 to obtain mountain pass geometry (see Lemma 3.5). Finally, we study the behavior of the positive radial solutions as b → 0. Since we do not know whether u0 is unique, we cannot draw the conclusion that u0 is obtained in Theorem 1.2.

### 1.3 Organization

This paper is organized as follows. Section 2 provides some preliminaries, and Section 3 is divided into three parts, which will prove Theorems 1.1–1.3, respectively. The proof of Theorem 1.4 is given in Section 3. Throughout this paper, the following notations are used:

• ‖up (1 < p) is the norm in ${L}^{p}\left({\mathbb{R}}^{3}\right)$;

• → and ⇀ denote strong and weak convergence, respectively;

• ⟨⋅, ⋅⟩ denotes the duality pairing between a Banach space and its dual space;

on(1) denotes on(1) → 0 as n.

## 2 Preliminary results

Since the condition (V1), we use the work space

$H≔u∈H1R3:ux=u|x|,$

equipped with the norm

$‖u‖H2=∫R3|∇u|2+Vxu2dx.$

According to [16], the energy functional associated with Eq. 1.1 is

$Ibu=12∫R3g2u|∇u|2dx+12∫R3Vx|u|2dx+b4∫R3g2u|∇u|2dx2−∫R3Hudx,$

where $H(t)=∫0th(s)ds$. We require the change of variable [2427]

$v=Gu=∫0ugtdt,$

and I(u) can be reduced to

$Jbv=12∫R3|∇v|2dx+12∫R3Vx|G−1v|2dx+b4∫R3|∇v|2dx2−∫R3HG−1vdx,$

where G−1(v) is the inverse of G(u).

Clearly, we have the following lemma (see [16]).

Lemma 2.1: Assume that (V1) holds. If $v∈H$ is a critical point of Jb, then u = G−1(v) is a weak solution of Eq. 1.1.

## 3 Proof of the main results

### 3.1 Proof of Theorem 1.1

By a standard argument in [28], we can obtain the following Pohožaev type.

Lemma 3.1: If $v∈H$ is a weak solution of Eq. 1.1 with h(t) = |t|p−2t, p ≥ 6, then v satisfies

$12∫R3|∇v|2dx+32∫R3Vx|G−1v|2dx+12∫R3⟨∇Vx,x⟩|G−1v|2dx+b2∫R3|∇v|2dx2=3p∫R3|G−1v|pdx.$

Based on the identity, we can provide the proof of Theorem 1.1. Indeed, v satisfies

$∫R3|∇v|2dx+∫R3VxG−1vgG−1vvdx+b∫R3|∇v|2dx2=∫R3|G−1v|p−2G−1vgG−1vvdx.$

Since $13≤l≤1$, using (5) of Lemma 2.1 in [29], jointly with ⟨∇V(x), x⟩≥ 0, we can obtain 0 = u = G−1(v).

### 3.2 Proof of Theorem 1.2

This section provides the proof of Theorem 1.2. Clearly, as mentioned previously, we are devoted to studying the functional Jb [Eq. 2.3]. Since our case is asymptotically cubic, it is hard to prove the boundedness of the PS-sequences of Jb. Hence, we use [30], Theorem 1.1 to find a special bounded PS-sequence of Jb,μ, where

$Jb,μv≔12∫R3|∇v|2dx+12∫R3Vx|G−1v|2dx+b4∫R3|∇v|2dx2−μ∫R3HG−1vdx,$

μ ∈ [1, 2]. We have the following lemma.

Lemma 3.2: Assume that (h1)–(h2) are satisfied, then

(i) for μ ∈ [1, 2], there exists $v\in \mathcal{H}\\left\{0\right\}$ such that Jb,μ(v) < 0.

(ii) there exists ρ, α > 0 such that Jb,μ(v) ≥ α and $‖v{‖}_{\mathcal{H}}=\rho$.

Proof. (i) It is well-known that λ1 > 0 is attained [ ([31]; Section 1.7)]. In other words, $ϕ∈H$ satisfied $∫R3|ϕ|4dx=1$ and ϕ > 0 such that

$λ1=∫R3|∇ϕ|2dx2.$

In view of (h2), $1<1l2$, and 1 ≤ μ ≤ 2, jointly with (3) and (4) of Lemma 2.1 in [29], we have

$limt→+∞Jb,μtϕt4

Hence, when t is large, let v ≔ , and we obtain the results.

(ii) Let $ε∈0,l2V02μ$, then we obtain

$Jb,μv≥12∫R3|∇v|2dx+12∫R3V0−μεl2|v|2dx−Cεμqlq∫R3|v|qdx.$

Hence, we can choose $‖v‖H=ρ>0$ small enough such that Jb,μ(v) > 0.

Define

$Av≔12∫R3|∇v|2+Vx|G−1v|2dx+b4∫R3|∇v|2dx2,Bv≔∫R3HG−1vdx.$

It is deduced from (V1) and (3) of Lemma 2.1 in [29] that

$Av>12‖v‖H2→+∞,as‖v‖H→∞,∀v∈H.$

Moreover, from (h1), it can be observed that $B(v)=∫R3H(G−1(v))dx≥0,∀v∈H$.

Using [30], Theorem 1.1 or [16], Theorem 4.1, it shows that for a.e. μ ∈ [1, 2], there is a bounded $(PS)cμ$ sequence ${vn}⊂H$, where cμ is the mountain pass level.

Lemma 3.3: Up to a subsequence, vnvμ in $H$.

Proof: Since ${vn}⊂H$ is bounded, up to a subsequence, there exists $vμ∈H$ such that vnvμ, in $H$, vnvμ, in $Lp(R3)$ for 2 < p < 6, and vn(x) → vμ(x) a.e. $x∈R3$. Obviously, $Jb,μ′(vμ)=0$. Then,

$on1=⟨Jb,μ′vn−Jb,μ′vμ,vn−vμ⟩=∫R3|∇vn−vμ|2dx+∫R3VxG−1vngG−1vn−G−1vμgG−1vμvn−vμdx+b[∫R3|∇vn|2dx∫R3∇vn∇vn−vμdx−∫R3|∇vμ|2dx∫R3∇vμ∇vn−vμdx]−μ∫R3hG−1vngG−1vn−hG−1vμgG−1vμvn−vμdx.$

We define $φ:R→R$ by φ(t) = G−1(t)/g(G−1(t)). Noting that l < g(t) ≤ 1 for $t∈R$, jointly with [29], (2) of Lemma 2.1, we have

$φ′t=1g2G−1t1−G−1tg′G−1tgG−1t≥1g2G−1t≥1.$

According to the mean value theorem, for any $x∈R3$, there exists a function ξ(x) between vμ(x) and vn(x) such that

$∫R3VxG−1vngG−1vn−G−1vμgG−1vμvn−vμdx=∫R3Vxφ′ξ|vn−vμ|2dx≥∫R3Vx|vn−vμ|2dx.$

It is easy to check that

$∫R3|∇vn|2dx∫R3∇vn∇vn−vμdx−∫R3|∇vμ|2dx∫R3∇vμ∇vn−vμdx=∫R3|∇vn|2−|∇vμ|2dx∫R3∇vn∇vn−vμdx+∫R3|∇vμ|2dx∫R3|∇vn−vμ|2dx→0,n→∞.$

Noting that [29], (3) of Lemma 2.1, we obtain

$∫R3hG−1vngG−1vn−hG−1vμgG−1vμvn−vμdx≤C∫R3|vn|+|vn|q−1+|vμ|+|vμ|q−1|vn−vμ|dx.$

Therefore, vnvμ in $H$.

It is easy to check the following lemma.

Lemma 3.4: If $v∈H$ is a critical point of Jb,μ(v), then v satisfies

$12∫R3|∇v|2dx+32∫R3Vx|G−1v|2dx+12∫R3⟨∇Vx,x⟩|G−1v|2dx+b2∫R3|∇v|2dx2=3μ∫R3HG−1vdx.$

Up to this point, we can prove Theorem 1.3. In fact, it is deduced from Lemma 3.2 and 3.3 that there exists {μn} ⊂ [1, 2] such that $limn→∞μn=1,vμn∈H$ satisfies $Jb,μn(vμn)=cμn>0,Jb,μn′(vμn)=0$. Next, we prove ${vμn}$ is bounded in $H$. Since the map μcμ is non-increasing, combining with Lemma 3.4 and condition (V2), we obtain

$M≥Jb,μnvμn=13∫R3|∇vμn|2dx−16∫R3⟨∇Vx,x⟩|G−1vμn|2dx+b12∫R3|∇vμn|2dx2≥13∫R3|∇vμn|2dx.$

It is easy to check that

$∫R3|∇vμn|2dx+∫R3VxG−1vμngG−1vμnvμndx+b∫R3|∇vμn|2dx2=μn∫R3hG−1vμngG−1vμnvμndx≤εμnl2∫R3|vμn|2dx+Cεμnl6∫R3|vμn|6dx≤εμnl2∫R3|vμn|2dx+CεμnSl6∫R3|∇vμn|2dx3≤εμnl2∫R3|vμn|2dx+CεμnSl63M3.$

Moreover, using (3) and (5) of Lemma 2.1 in [29], it is deduced from condition (V1) that

$∫R3V0|vμn|2dx≤∫R3|∇vμn|2dx+∫R3VxG−1vμngG−1vμnvμndx+b∫R3|∇vμn|2dx2≤εμnl2∫R3|vμn|2dx+CεμnSl63M3.$

Let $ε=l2V02μn$, then we obtain

$∫R3|vμn|2dx≤2SCεμnl6V03M3.$

From Eqs 3.6, 3.7, we know that ${vμn}$ is bounded in $H$.

A subsequence of ${vμn}$ is selected and also denoted by {vn}, such that vnv in $H$. Similar to the proof of Lemma 3.3, we obtain vnv in $H$. It is well-known that μcμ is continuous from the left [ ([16], Theorem 4.1)]. So,

$limn→∞Jbvμn=limn→∞Jb,μnvμn+μn−1∫R3HG−1vμndx=limn→∞cμn=c̃.$

$limn→∞〈Jb′vμn,ψ〉=limn→∞〈Jb,μn′vμn, ψ〉+μn−1∫R3hG−1vμngG−1vμnψdx=0,$

for any $ψ∈C0∞(R3)$, which means that $Jb′(v)=0$ satisfies $Jb(v)=c̃>0$. Let v = min{v, 0}. Using (3) and (5) of Lemma 2.1 in [29], we have

$0=⟨Jb′v,v−⟩=∫R3|∇v−|2+VxG−1v−gG−1v−v−dx≥∫R3|∇v−|2+Vx|v−|2dx.$

It shows that v≡ 0. Applying the strong maximum principle, we obtain v(x) > 0.

### 3.3 Proof of Theorem 1.3

This section studies the case $14h(t)t≥H(t)$ for all t > 0 and without the Kirchhoff term $∫R3g2(u)|∇u|2dx$. At first, let us check the mountain pass geometry of the functional J0.

Lemma 3.5: Assume that (h1)–(h2) are satisfied, then

(i) there exists $v\in \mathcal{H}\\left\{0\right\}$ such that J0(v) < 0.

(ii) there exist ρ, α > 0 such that J0(v) ≥ α, $‖v{‖}_{\mathcal{H}}=\rho$.

Proof (i) Motivated by Lemma 2.2 of [23], we need to study the following equation:

$−Δv+V∞G−1vgG−1v=hG−1vgG−1v,x∈R3.$

The corresponding functional is J0,(v). We also define the mountain pass min–max level

$c∞=infξ∈Γ∞maxt∈0,1J0,∞ξt,$

where

$Γ∞=ξ∈C0,1,H:ξ0=0≠ξ1,J0,∞ξ1<0.$

By the standard arguments, it shows that $w∈H1(RN)$ is a solution of Eq. 3.9, which satisfies J0,(w) = c. A continuous path $α:[0,+∞)→H$ is defined by α(t) (x) = w(x/t), if t > 0 and α(0) = 0. Taking the derivative, we know that

$ddtJ0,∞αt=12∫R3|∇w|2dx+32t2∫R3V∞|G−1w|2dx−3t2∫R3HG−1wdx.$

Since w is a solution of Eq. 3.9, it satisfies the Pohožaev identity,

$12∫R3|∇w|2dx+32∫R3V∞|G−1w|2dx=3∫R3HG−1wdx.$

Therefore,

$ddtJ0,∞αt=121−t2∫R3|∇w|2dx.$

The map tJ0,(α(t)) achieves the maximum value at t = 1. By choosing L > 0 sufficiently large, we have J0,(α(L)) < 0. Taking ζ(t) = α(tL), we have ζ ∈ Γ. If $ζy(t)≔w⋅−ytL$, noting that (V1), we obtain

$J0ζy1=J0,∞ζy1+12∫R3Vx+y−V∞|G−1ζy1|2dx<0,for|y|is large.$

Choosing e = ζy(1), we can obtain the result.

(ii) Similar to (ii) of Lemma 3.2, we obtain

$J0v≥12∫R3|∇v|2+Vx|v|2dx−∫R3ε2|G−1v|2+Cεq|G−1v|qdx≥C4‖v‖H2−C1Cεqlq‖v‖Hq.$

Hence, choosing $‖v‖H=ρ>0$ small enough, we can obtain the desired conclusion.

Therefore, there is a (PS) $c0$ sequence ${vn}⊂H$ where c0 is the mountain pass level of the J0.

Lemma 3.6: {vn} is bounded.

Proof: Since $G−1(vn)g(G−1(vn))∈H$, jointly with (h3) and [29], (2) of Lemma 2.1], we obtain

$c+on1=J0vn−14⟨J0′vn,G−1vngG−1vn⟩≥14‖vn‖H2.$

Hence, {vn} is bounded in $H$.

Similar to Lemma 3.3, we obtain the following result.

Lemma 3.7: Up to a subsequence, vnv in $H$.

Proof of Theorem 1.3: It deduces from lemmas 3.5, 3.6, and 3.7 that Eq. 1.2 has a non-trivial solution v. Similar to Eq. 3.8, we know that $v(x)>0,x∈R3$.

## 4 Asymptotic properties of the positive radial solution

Proof of Theorem 1.4: If $vbn$ is a critical point of $Jbn$, which is obtained in Theorem 1.2 for each $n∈N$. Similar to the proof of Lemma 3.2, for bn → 0, n, ${vbn}$ is a (PS)c sequence, which is bounded in $H$. There exists a subsequence of {bn}, still denoted by {bn}, such that $vbn⇀v0$ in $H$. It is easy to obtain

$‖vbn−v0‖H2≤⟨Jbn′vbn−J0′v0,vbn−v0⟩−bn∫R3|∇vbn|2dx∫R3∇vbn∇vbn−v0dx+∫R3hG−1vbngG−1vbn−hG−1v0gG−1v0vbn−v0dx=on1.$

On one hand, in view of (3) of Lemma 2.1 in [29], we can use the Lebesgue dominated convergence theorem to obtain

$limn→∞∫R3VxG−1vbnϕgG−1vbndx=∫R3VxG−1v0ϕgG−1v0dx,$
$limn→∞∫R3hG−1vbnϕgG−1vbndx=∫R3hG−1v0ϕgG−1v0dx.$

On the other hand, we have $⟨Jbn′(vbn),ϕ⟩=on(1)$ and $⟨J0′(v0),ϕ⟩=on(1)$. Moreover,

$limn→∞∫R3∇vbn∇ϕdx=∫R3∇v0∇ϕdx,$
$limn→∞bn∫R3|∇vbn|2dx∫R3∇vbn∇ϕdx=0.$

Thus,

$∫R3∇v0∇ϕdx+∫R3VxG−1v0gG−1v0ϕdx=∫R3hG−1v0gG−1v0ϕdx.$

It shows that v0 is a positive solution of Eq. 1.2.

## Data availability statement

The original contributions presented in the study are included in the article/Supplementary Material, further inquiries can be directed to the corresponding author.

## Author contributions

All authors listed have made a substantial, direct, and intellectual contribution to the work and approved it for publication.

## Funding

This work was supported by the National Natural Science Foundation of China (Grant Nos 12261075 and 12261076), Yunnan Local Colleges Applied Basic Research Projects (Grant Nos 202001BA070001-032 and 202101BA070001-280), the Technology Innovation Team of University in Yunnan Province (Grant No. 2020CXTD25), and Yunnan Fundamental Research Projects (Grant Nos 202201AT070018 and 202105AC160087). WW was supported in part by the Yunnan Province Basic Research Project for Youths (202301AU070001) and the Xingdian Talents Support Program of Yunnan Province.

## Conflict of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

## Publisher’s note

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Keywords: quasilinear Schrödinger equations, Kirchhoff-type perturbation, asymptotically cubic growth, non-existence, positive solutions

Citation: Li G, Qiu C, Cheng B and Wang W (2023) On the asymptotically cubic generalized quasilinear Schrödinger equations with a Kirchhoff-type perturbation. Front. Phys. 11:1185846. doi: 10.3389/fphy.2023.1185846

Received: 14 March 2023; Accepted: 02 May 2023;
Published: 17 May 2023.

Edited by:

Ji-Huan He, Soochow University, China

Reviewed by:

Ying Wang, Xi’an University of Architecture and Technology, China
Kangle Wang, Henan Polytechnic University, China
Baojian Hong, Nanjing Institute of Technology (NJIT), China

Copyright © 2023 Li, Qiu, Cheng and Wang. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Wenbo Wang, wenbowangmath@ynu.edu.cn