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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

www.frontiersin.orgGuofa Li1 www.frontiersin.orgChong Qiu2 www.frontiersin.orgBitao Cheng1 www.frontiersin.orgWenbo 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+bR3g2u|u|2dxdivg2uu+gugu|u|2+Vxu=hu,

where xR3,b0,V:R3R and h:RR are continuous functions, gC1(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:

divg2uu+gugu|u|2+Vxu=hu,xR3.

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

uφ|u|2φ|u|2u+Vxu=hu,xR3.

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 tR, it is reduced to the following classical Kirchhoff equation:

1+bR3|u|2dxΔu+Vxu=hu,xR3.

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

utt+1+bR3|u|2dxΔu+Vxu=hu,xR3,

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(x)=V(|x|),0<V0V(x)Vlim|x|+V(x)<;

(V2) VC1(R3,R) and V(x),x0,xR3;

(h1) hC(R,R), h(t) = 0, ∀ t ≤ 0, and limt0h(t)t=0;

(h2) lim|t|+|h(t)||t|3=γ,γ>bl4λ1, where

λ1infR3|w|2dx2:wH,R3|w|4dx=1

and H is defined in Section 2;

(h3) 14h(t)tH(t) for all t > 0, where H(t)=0th(s)ds.

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

Ht=γt44γt22+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 13l1 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 nN. Then, ubnu0 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 Lp(R3);

• → 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

HuH1R3:ux=u|x|,

equipped with the norm

uH2=R3|u|2+Vxu2dx.

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

Ibu=12R3g2u|u|2dx+12R3Vx|u|2dx+b4R3g2u|u|2dx2R3Hudx,

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

v=Gu=0ugtdt,

and I(u) can be reduced to

Jbv=12R3|v|2dx+12R3Vx|G1v|2dx+b4R3|v|2dx2R3HG1vdx,

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 vH 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 vH is a weak solution of Eq. 1.1 with h(t) = |t|p−2t, p ≥ 6, then v satisfies

12R3|v|2dx+32R3Vx|G1v|2dx+12R3Vx,x|G1v|2dx+b2R3|v|2dx2=3pR3|G1v|pdx.

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

R3|v|2dx+R3VxG1vgG1vvdx+bR3|v|2dx2=R3|G1v|p2G1vgG1vvdx.

Since 13l1, 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,μv12R3|v|2dx+12R3Vx|G1v|2dx+b4R3|v|2dx2μR3HG1vdx,

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 vH\{0} such that Jb,μ(v) < 0.

(ii) there exists ρ, α > 0 such that Jb,μ(v) ≥ α and vH=ρ.

Proof. (i) It is well-known that λ1 > 0 is attained [ ([31]; Section 1.7)]. In other words, ϕH 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<limt+1l2t2ϕH2+b4R3|ϕ|2dx2μR3HG1tϕ|G1tϕ|4|G1tϕ|4|tϕ|4|ϕ|4dxb4R3|ϕ|2dx2bλ14R3|ϕ|4dx=0.

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

(ii) Let ε0,l2V02μ, then we obtain

Jb,μv12R3|v|2dx+12R3V0μεl2|v|2dxCεμqlqR3|v|qdx.

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

Define

Av12R3|v|2+Vx|G1v|2dx+b4R3|v|2dx2,BvR3HG1vdx.

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

Av>12vH2+,asvH,vH.

Moreover, from (h1), it can be observed that B(v)=R3H(G1(v))dx0,vH.

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 vμH such that vnvμ, in H, vnvμ, in Lp(R3) for 2 < p < 6, and vn(x) → vμ(x) a.e. xR3. Obviously, Jb,μ(vμ)=0. Then,

on1=Jb,μvnJb,μvμ,vnvμ=R3|vnvμ|2dx+R3VxG1vngG1vnG1vμgG1vμvnvμdx+b[R3|vn|2dxR3vnvnvμdxR3|vμ|2dxR3vμvnvμdx]μR3hG1vngG1vnhG1vμgG1vμvnvμdx.

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

φt=1g2G1t1G1tgG1tgG1t1g2G1t1.

According to the mean value theorem, for any xR3, there exists a function ξ(x) between vμ(x) and vn(x) such that

R3VxG1vngG1vnG1vμgG1vμvnvμdx=R3Vxφξ|vnvμ|2dxR3Vx|vnvμ|2dx.

It is easy to check that

R3|vn|2dxR3vnvnvμdxR3|vμ|2dxR3vμvnvμdx=R3|vn|2|vμ|2dxR3vnvnvμdx+R3|vμ|2dxR3|vnvμ|2dx0,n.

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

R3hG1vngG1vnhG1vμgG1vμvnvμdxCR3|vn|+|vn|q1+|vμ|+|vμ|q1|vnvμ|dx.

Therefore, vnvμ in H.

It is easy to check the following lemma.

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

12R3|v|2dx+32R3Vx|G1v|2dx+12R3Vx,x|G1v|2dx+b2R3|v|2dx2=3μR3HG1vdx.

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 limnμn=1,vμnH 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

MJb,μnvμn=13R3|vμn|2dx16R3Vx,x|G1vμn|2dx+b12R3|vμn|2dx213R3|vμn|2dx.

It is easy to check that

R3|vμn|2dx+R3VxG1vμngG1vμnvμndx+bR3|vμn|2dx2=μnR3hG1vμngG1vμnvμndxεμnl2R3|vμn|2dx+Cεμnl6R3|vμn|6dxεμnl2R3|vμn|2dx+CεμnSl6R3|vμn|2dx3εμnl2R3|vμn|2dx+CεμnSl63M3.

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

R3V0|vμn|2dxR3|vμn|2dx+R3VxG1vμngG1vμnvμndx+bR3|vμn|2dx2εμnl2R3|vμn|2dx+CεμnSl63M3.

Let ε=l2V02μn, then we obtain

R3|vμn|2dx2SCεμnl6V03M3.

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,

limnJbvμn=limnJb,μnvμn+μn1R3HG1vμndx=limncμn=c̃.

In addition,

limnJbvμn,ψ=limnJb,μnvμn,ψ+μn1R3hG1vμngG1vμnψdx=0,

for any ψC0(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=Jbv,v=R3|v|2+VxG1vgG1vvdxR3|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)tH(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 vH\{0} such that J0(v) < 0.

(ii) there exist ρ, α > 0 such that J0(v) ≥ α, vH=ρ.

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

Δv+VG1vgG1v=hG1vgG1v,xR3.

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

c=infξΓmaxt0,1J0,ξt,

where

Γ=ξC0,1,H:ξ0=0ξ1,J0,ξ1<0.

By the standard arguments, it shows that wH1(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=12R3|w|2dx+32t2R3V|G1w|2dx3t2R3HG1wdx.

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

12R3|w|2dx+32R3V|G1w|2dx=3R3HG1wdx.

Therefore,

ddtJ0,αt=121t2R3|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)wytL, noting that (V1), we obtain

J0ζy1=J0,ζy1+12R3Vx+yV|G1ζy1|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

J0v12R3|v|2+Vx|v|2dxR3ε2|G1v|2+Cεq|G1v|qdxC4vH2C1CεqlqvHq.

Hence, choosing vH=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 G1(vn)g(G1(vn))H, jointly with (h3) and [29], (2) of Lemma 2.1], we obtain

c+on1=J0vn14J0vn,G1vngG1vn14vnH2.

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,xR3.

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 nN. 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 vbnv0 in H. It is easy to obtain

vbnv0H2JbnvbnJ0v0,vbnv0bnR3|vbn|2dxR3vbnvbnv0dx+R3hG1vbngG1vbnhG1v0gG1v0vbnv0dx=on1.

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

limnR3VxG1vbnϕgG1vbndx=R3VxG1v0ϕgG1v0dx,
limnR3hG1vbnϕgG1vbndx=R3hG1v0ϕgG1v0dx.

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

limnR3vbnϕdx=R3v0ϕdx,
limnbnR3|vbn|2dxR3vbnϕdx=0.

Thus,

R3v0ϕdx+R3VxG1v0gG1v0ϕdx=R3hG1v0gG1v0ϕdx.

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

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