Abstract
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 aswhere and are continuous functions, satisfies (g1), g is even, g′(t) ≤ 0, , and ∀ t ≥ 0.
When b = 0, Eq. 1.1 is reduced to the following quasilinear Schrödinger equation:According to [4], let , then, Eq. 1.2 is transformed intoIt is well-known that the classical case is φ(s) = s or [5–12].
For Eq. 1.1, another interesting question is b > 0. When g(t) = 1 for all , it is reduced to the following classical Kirchhoff equation:
It is well-known that Eq. 1.4 is related to the stationary analog of the following Kirchhoff-type equation: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 [
16–
20]. However, to the best of our knowledge, there are no works involving Eq.
1.1when 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) ;
(V2) and ;
(h1) , h(t) = 0, ∀ t ≤ 0, and ;
(h2) , where
and
is defined in
Section 2;
(h3) for all t > 0, where .
Remark 1.1: For example, . By direct calculations, we haveIt 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 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 , 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 are the positive radial solutions obtained in Theorem 1.2 for each . Then, in 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 2provides some preliminaries, and
Section 3is 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:
• ‖u‖p (1 < p ≤ ∞) is the norm in ;
• → 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 spaceequipped with the normAccording to [16], the energy functional associated with Eq. 1.1 iswhere . We require the change of variable [24–27]and I(u) can be reduced towhere 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 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 is a weak solution of Eq. 1.1 with h(t) = |t|p−2t, p ≥ 6, then v satisfiesBased on the identity, we can provide the proof of Theorem 1.1. Indeed, v satisfiesSince , 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, whereμ ∈ [1, 2]. We have the following lemma.
Lemma 3.2: Assume that (
h1)–(
h2) are satisfied, then
(i) for μ ∈ [1, 2], there exists such that Jb,μ(v) < 0.
(ii) there exists ρ, α > 0 such that Jb,μ(v) ≥ α and .
Proof(i) It is well-known that λ1 > 0 is attained [ ([31]; Section 1.7)]. In other words, satisfied and ϕ > 0 such thatIn view of (h2), , and 1 ≤ μ ≤ 2, jointly with (3) and (4) of Lemma 2.1 in [29], we haveHence, when t is large, let v ≔ tϕ, and we obtain the results.(ii) Let , then we obtainHence, we can choose small enough such that Jb,μ(v) > 0.DefineIt is deduced from (V1) and (3) of Lemma 2.1 in [29] thatMoreover, from (h1), it can be observed that .Using [30], Theorem 1.1 or [16], Theorem 4.1, it shows that for a.e. μ ∈ [1, 2], there is a bounded sequence , where cμ is the mountain pass level.
Lemma 3.3: Up to a subsequence, vn → vμ in .
Proof: Since is bounded, up to a subsequence, there exists such that vn ⇀ vμ, in , vn → vμ, in for 2 < p < 6, and vn(x) → vμ(x) a.e. . Obviously, . Then,We define by φ(t) = G−1(t)/g(G−1(t)). Noting that l < g(t) ≤ 1 for , jointly with [29], (2) of Lemma 2.1, we haveAccording to the mean value theorem, for any , there exists a function ξ(x) between vμ(x) and vn(x) such thatIt is easy to check thatNoting that [29], (3) of Lemma 2.1, we obtainTherefore, vn → vμ in .It is easy to check the following lemma.
Lemma 3.4: If is a critical point of Jb,μ(v), then v satisfiesUp 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 satisfies . Next, we prove is bounded in . Since the map μ → cμ is non-increasing, combining with Lemma 3.4 and condition (V2), we obtainIt is easy to check thatMoreover, using (3) and (5) of Lemma 2.1 in [29], it is deduced from condition (V1) thatLet , then we obtainFrom Eqs 3.6, 3.7, we know that is bounded in .A subsequence of is selected and also denoted by {vn}, such that vn ⇀ v in . Similar to the proof of Lemma 3.3, we obtain vn → v in . It is well-known that μ↦cμ is continuous from the left [ ([16], Theorem 4.1)]. So,In addition,for any , which means that satisfies . Let v− = min{v, 0}. Using (3) and (5) of Lemma 2.1 in [29], we haveIt 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 for all t > 0 and without the Kirchhoff term . 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 such that J0(v) < 0.
(ii) there exist ρ, α > 0 such that J0(v) ≥ α,.
Proof (i) Motivated by Lemma 2.2 of [23], we need to study the following equation:The corresponding functional is J0,∞(v). We also define the mountain pass min–max levelwhereBy the standard arguments, it shows that is a solution of Eq. 3.9, which satisfies J0,∞(w) = c∞. A continuous path is defined by α(t) (x) = w(x/t), if t > 0 and α(0) = 0. Taking the derivative, we know thatSince w is a solution of Eq. 3.9, it satisfies the Pohožaev identity,Therefore,The map t ↦ J0,∞(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 , noting that (V1), we obtainChoosing e = ζy(1), we can obtain the result.(ii) Similar to (ii) of Lemma 3.2, we obtainHence, choosing small enough, we can obtain the desired conclusion.Therefore, there is a (PS) sequence where c0 is the mountain pass level of the J0.
Lemma 3.6: {vn} is bounded.
Proof: Since , jointly with (h3) and [29], (2) of Lemma 2.1], we obtainHence, {vn} is bounded in .Similar to Lemma 3.3, we obtain the following result.
4 Asymptotic properties of the positive radial solution
Proof of Theorem 1.4: If is a critical point of , which is obtained in Theorem 1.2 for each . Similar to the proof of Lemma 3.2, for bn → 0, n → ∞, is a (PS)c sequence, which is bounded in . There exists a subsequence of {bn}, still denoted by {bn}, such that in . It is easy to obtain
On one hand, in view of (3) of Lemma 2.1 in [29], we can use the Lebesgue dominated convergence theorem to obtain
On the other hand, we have and . Moreover,Thus,It shows that v0 is a positive solution of Eq. 1.2.
Statements
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
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors, and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.
References
1.
FeitMFleckJSteigerA. Solution of the Schrödinger equation by a spectral method. J Comput Phys (1982) 47:412–33. 10.1016/0021-9991(82)90091-2
2.
HeJHeCSaeedT. A fractal modification of chen-lee-liu equation and its fractal variational principle. Internat J Mod Phys. B (2021) 35:2150214. 10.1142/S0217979221502143
3.
AlmutairiA. Stochastic solutions to the non-linear Schrödinger equation in optical fiber. Therm Sci (2022) 26:185–90. 10.2298/tsci22s1185a
4.
DengYPengSYanS. Positive soliton solutions for generalized quasilinear Schrödinger equations with critical growth. J Differential Equations (2015) 258:115–47. 10.1016/j.jde.2014.09.006
5.
AlvesCWangYShenY. Soliton solutions for a class of quasilinear Schrödinger equations with a parameter. J Differential Equations (2015) 259:318–43. 10.1016/j.jde.2015.02.030
6.
BouardAHayashiNSautJ. Global existence of small solutions to a relativistic non-linear Schrödinger equation. Comm Math Phys (1997) 189:73–105. 10.1007/s002200050191
7.
BrizhikLEremkoAPietteBZakrzewskiW. Static solutions of aD-dimensional modified non-linear Schr dinger equation. Non-linearity (2003) 16:1481–97. 10.1088/0951-7715/16/4/317
8.
LiGHuangYLiuZ. Positive solutions for quasilinear Schrödinger equations with superlinear term. Complex Var. Elliptic Equ (2020) 65:936–55. 10.1080/17476933.2019.1636791
9.
LiQWuX. Existence, multiplicity, and concentration of solutions for generalized quasilinear Schrödinger equations with critical growth. J Math Phys (2017) 58:041501. 10.1063/1.4982035
10.
ShenYWangY. Soliton solutions for generalized quasilinear Schrödinger equations. Nonlinear Anal (2013) 80:194–201. 10.1016/j.na.2012.10.005
11.
ShiHChenH. Positive solutions for generalized quasilinear Schrödinger equations with potential vanishing at infinity. Appl Math Lett (2016) 61:137–42. 10.1016/j.aml.2016.06.004
12.
ShenYWangY. Standing waves for a class of quasilinear Schrödinger equations. Complex Var. Elliptic Equ (2016) 61:817–42. 10.1080/17476933.2015.1119818
13.
KirchhoffG. Mechanik (leipzig: Teubner) (1883).
14.
HeJEl-dibY. The enhanced homotopy perturbation method for axial vibration of strings. Facta Universitatis Ser Mech Eng (2021) 19:735–50. 10.22190/FUME210125033H
15.
WangK. Construction of fractal soliton solutions for the fractional evolution equations with conformable derivative. Fractals (2023) 31:2350014. 10.1142/S0218348X23500147
16.
ChenJTangXChengB. Existence and nonexistence of positive solutions for a class of generalized quasilinear Schrödinger equations involving a Kirchhoff-type perturbation with critical sobolev exponent. J Math Phys (2018) 59:021505. 10.1063/1.5024898
17.
ChengBTangX. Ground state sign-changing solutions for asymptotically 3-linear Kirchhoff-type problems. Complex Var. Elliptic Equ (2017) 62:1093–116. 10.1080/17476933.2016.1270272
18.
FengRTangC. Ground state sign-changing solutions for a Kirchhoff equation with asymptotically 3-linear nonlinearity. Qual Theor Dyn. Syst. (2021) 20:91–19. 10.1007/s12346-021-00529-y
19.
LiFZhuXLiangZ. Multiple solutions to a class of generalized quasilinear Schrödinger equations with a Kirchhoff-type perturbation. J Math Anal Appl (2016) 443:11–38. 10.1016/j.jmaa.2016.05.005
20.
ShenL. Existence and nonexistence results for generalized quasilinear Schrödinger equations of Kirchhoff type in. Appl Anal (2020) 99:2465–88. 10.1080/00036811.2019.1569225
21.
LiGChengBHuangY. Positive solutions for asymptotically 3-linear quasilinear Schrödinger equations. Electron J Differential Equations (2020) 2020:1–17. Available at: http://ejde.math.txstate.edu.
22.
WangWYuYLiY. On the asymptotically cubic fractional Schrödinger-Poisson system. Appl Anal (2021) 100:695–713. 10.1080/00036811.2019.1616695
23.
LehrerRMaiaLSquassinaM. Asymptotically linear fractional Schrödinger equations. Complex Var. Elliptic Equ (2015) 60:529–58. 10.1080/17476933.2014.948434
24.
ChuCLiuH. Existence of positive solutions for a quasilinear Schrödinger equation. Nonlinear Anal RWA (2018) 44:118–27. 10.1016/j.nonrwa.2018.04.007
25.
ColinMJeanjeanL. Solutions for a quasilinear schrödinger equation: A dual approach. Nonlinear Anal (2004) 56:213–26. 10.1016/j.na.2003.09.008
26.
LiangZGaoJLiA. Infinitely many solutions to a quasilinear Schrödinger equation with a local sublinear term. Appl Math Lett (2019) 89:22–7. 10.1016/j.aml.2018.09.015
27.
ShenYWangY. Standing waves for a relativistic quasilinear asymptotically Schrödinger equation. Appl Anal (2016) 95:2553–64. 10.1080/00036811.2015.1100296
28.
SeveroUGlossESilvaE. On a class of quasilinear Schrödinger equations with superlinear or asymptotically linear terms. J Differential Equations (2017) 263:3550–80. 10.1016/j.jde.2017.04.040
29.
LiGHuangY. Positive solutions for generalized quasilinear Schrödinger equations with asymptotically linear nonlinearities. Appl Anal (2021) 100:1051–66. 10.1080/00036811.2019.1634256
30.
JeanjeanL. On the existence of bounded Palais–Smale sequences and application to a Landesman–Lazer-type problem set on ℝN. Proc Roy Soc Edin (1999) 129A:787–809. 10.1017/S0308210500013147
31.
WillemM. Minimax theorems, progress in nonlinear differential equations and their applications. Boston, MA: Birkhäuser Boston, Inc. (1996). p. 24.
Summary
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
Volume
11 - 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
Updates
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
Disclaimer
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article or claim that may be made by its manufacturer is not guaranteed or endorsed by the publisher.