POSITIVE SOLUTIONS FOR A KIRCHHOFF EQUATION WITH PERTURBED SOURCE TERMS

doi:10.1007/s10473-022-0507-z

数学物理学报(英文版) ›› 2022, Vol. 42 ›› Issue (5): 1817-1830.doi: 10.1007/s10473-022-0507-z

• 论文 • 上一篇

POSITIVE SOLUTIONS FOR A KIRCHHOFF EQUATION WITH PERTURBED SOURCE TERMS

Narimane AISSAOUI¹, Wei LONG²

1. School of Mathematics and Statistics, Hubei Key Laboratory of Mathematical Sciences, Central China Normal University, Wuhan, 430079, China;
2. College of Mathematics and Information Science, Jiangxi Normal University, Nanchang, 330022, China

收稿日期:2021-03-12 修回日期:2022-05-19 发布日期:2022-11-02
通讯作者: Wei Long,E-mail:lwhope@jxnu.edu.cn E-mail:lwhope@jxnu.edu.cn
基金资助:
The second author was supported by NSF of China (11871253), supported by Jiangxi Provincial Natural Science Foundation (20212ACB201003), Jiangxi Two Thousand Talents Program (jxsq2019101001), Double-high talents in Jiangxi Province and Jiangxi Provincial Department of Education Fund (GJJ191687).

POSITIVE SOLUTIONS FOR A KIRCHHOFF EQUATION WITH PERTURBED SOURCE TERMS

Narimane AISSAOUI¹, Wei LONG²

1. School of Mathematics and Statistics, Hubei Key Laboratory of Mathematical Sciences, Central China Normal University, Wuhan, 430079, China;
2. College of Mathematics and Information Science, Jiangxi Normal University, Nanchang, 330022, China

Received:2021-03-12 Revised:2022-05-19 Published:2022-11-02
Contact: Wei Long,E-mail:lwhope@jxnu.edu.cn E-mail:lwhope@jxnu.edu.cn
Supported by:
The second author was supported by NSF of China (11871253), supported by Jiangxi Provincial Natural Science Foundation (20212ACB201003), Jiangxi Two Thousand Talents Program (jxsq2019101001), Double-high talents in Jiangxi Province and Jiangxi Provincial Department of Education Fund (GJJ191687).

摘要/Abstract

摘要： This paper deals with the existence of positive solutions to the following nonlinear Kirchhoff equation with perturbed external source terms: $$ \left\{ \begin{array}{ll} -\Big(a+b \int_{\mathbb{R}^3} | \nabla u |^2 {\rm d}x\Big) \Delta u+ V(x)u=Q(x)u^p+\varepsilon f(x),\quad &x\in \mathbb{R}^3, \\ u>0,\quad &u\in H^1(\mathbb{R}^3). \end{array} \right.$$ Here $a,b$ are positive constants, $V(x),Q(x)$ are positive radial potentials, 1<p<5, $\varepsilon$ >0 is a small parameter, $f(x)$ is an external source term in $L^2(\mathbb{R}^3)\cap L^{\infty}(\mathbb{R}^3)$.

关键词: nonlinear Kirchhoff problem, Lyapunov-Schmidt reduction, positive solutions, level set

Abstract: This paper deals with the existence of positive solutions to the following nonlinear Kirchhoff equation with perturbed external source terms: $$ \left\{ \begin{array}{ll} -\Big(a+b \int_{\mathbb{R}^3} | \nabla u |^2 {\rm d}x\Big) \Delta u+ V(x)u=Q(x)u^p+\varepsilon f(x),\quad &x\in \mathbb{R}^3, \\ u>0,\quad &u\in H^1(\mathbb{R}^3). \end{array} \right.$$ Here $a,b$ are positive constants, $V(x),Q(x)$ are positive radial potentials, 1<p<5, $\varepsilon$ >0 is a small parameter, $f(x)$ is an external source term in $L^2(\mathbb{R}^3)\cap L^{\infty}(\mathbb{R}^3)$.

Key words: nonlinear Kirchhoff problem, Lyapunov-Schmidt reduction, positive solutions, level set

中图分类号:

35J20

Narimane AISSAOUI, Wei LONG. POSITIVE SOLUTIONS FOR A KIRCHHOFF EQUATION WITH PERTURBED SOURCE TERMS[J]. 数学物理学报(英文版), 2022, 42(5): 1817-1830.

Narimane AISSAOUI, Wei LONG. POSITIVE SOLUTIONS FOR A KIRCHHOFF EQUATION WITH PERTURBED SOURCE TERMS[J]. Acta mathematica scientia,Series B, 2022, 42(5): 1817-1830.

参考文献

[1] Adachi S, Tanaka K. Four positive solutions for the semilinear elliptic equation: -Δu + u = a(x)u^p + f(x) in $\mathbb{R}^N$. Calc Var Partial Differential Equations, 2000, 11: 63–95
[2] Aissaoui N, Li Q, Zheng B. A perturbed Kirchhoff problem with critical exponent. Applicable Analysis, acctepted
[3] Alves C O, Corrê F J S A, Ma T F. Positive solutions for a quasilinear elliptic equation of Kirchhoff type. Comput Math Appl, 2005, 49: 85–93
[4] Arosio A, Panizzi S. On the well-posedness of the Kirchhoff string. Transl Amer Math Soc, 1996, 348: 305–330
[5] Berestycki H, Lions P L. Nonlinear scalarfield equations I, Existence of a ground state. Arch Rational Mech Anal, 1983, 82: 313–375
[6] Bernstein S. Surune classe d’equations fonctionelles aux dérivées partielles. Bull Acad Sci USSR Sér, 1940, 4: 17–26
[7] Zhang B L, Rădulescu V D, Wang L. Existence results for Kirchhoff-type superlinear problems involving the fractional Laplacian. Proc Roy Soc Edinburgh Sect A, 2019, 149: 1061–1081
[8] Carrier G F. On the non-linear vibration problem of the elastic string. Quart Appl Math, 1945, 3: 157–165
[9] Cao D, Zhou H S. Multiple positive solution of nonhomogeneous semilinear elliptic equations in $\mathbb{R}^N$. Proc Roy Soc Edinburgh Sect A, 1996, 126: 443–463
[10] Chen C Y, Kuo Y C, Wu T F. The Nehari manifold for a Kirchhoff type problem involving sign-changing weight functions. J Differential Equations, 2011, 250: 1876–1908
[11] Deng Y, Peng S, Shuai W. Existence and asymptotic behavior of nodal solutions for the Kirchhoff-type problems in $\mathbb{R}^3$. J Funct Anal, 2015, 269: 3500–3527
[12] Fiscella A, Pucci P. Kirchhoff-Hardy fractional problems with lack of compactness. Adv Nonlinear Stud, 2017, 17: 429–456
[13] Fiscella A, Pucci P, Zhang B L. p-Fractional Hardy-Schrödinger-Kirchhoff systems with critical nonlinearities. Adv Nonlinear Anal, 2019, 8: 1111–1131
[14] He X, Zou W. Existence and concentration behavior of positive solutions for a Kirchhoff equation in $\mathbb{R}^3$. J Differential Equations, 2012, 252: 1813–1834
[15] Hu T, Lu L. Infinitely many positive solutions for Kirchhoff equations with competing coefficients. Z Angew Math Phys, 2019, 70: 53
[16] Hu T, Shuai W. Multi-peak solutions to Kirchhoff equations in $mathbb{R}^3$ with general nonlinearity. J Differential Equations, 2018, 265: 3587–3617
[17] Jeanjean L. Two positive solutions for a class of nonhomogeneous elliptic equations. Differential Integral Equations, 1997, 10: 609–624
[18] Kang X, Wei J. On interacting bumps of semi-classical states of nonlinear Schrödinger equations. Adv Diff Equ, 2000, 5: 899–928
[19] Kirchhoff G. Vorlesungen über Mechanik. Teubner, Leipzig, 1883
[20] Li G, Luo P, Peng S, Wang C, Xiang C L. A singularly perturbed Kirchhoff problem revisited. J Differential Equations, 2020, 268: 541–589
[21] Lions J L. On some questions in boundary value problems of mathematical physics//Contemporary Development in Continuum Mechanics and Partial Differential Equations. North-Holland Math Stud, Vol 30. Amsterdam, New York: North-Holland, 1978: 284–346
[22] Li G B, Niu Y H. The existence and local uniqueness of multi-peak positve solutions to a class of Kirchhoff equation. Acta Mathematica Scientia, 2020, 40B(1): 90–112
[23] Long W, Peng S. Positive vector solutions for a Schrödinger system with external source terms. Nonlinear Differential Equations Appl, 2020, 27(1): 5–36
[24] Luo P, Peng S, Wang C. Multi-peak positive solutions to a class of Kirchhoff equations. arXiv:1708.01770 [math.AP]
[25] Xiang M Q, Rădulescu V D, Zhang B. Combined effects for fractional Schrödinger-Kirchhoff systems with critical nonlinearities. ESAIM Control Optim Calc Var, 2018, 24: 1249–1273
[26] Xiang M Q, Rădulescu V D, Zhang B. A critical fractional Choquard-Kirchhoff problem with magnetic field. Commun Contemp Math, 2019, 21: 36pp
[27] Xiang M Q, Rădulescu V D, Zhang B. Fractional Kirchhoff problems with critical Trudinger-Moser nonlinearity. Calc Var Partial Differential Equations, 2019, 58: 27pp
[28] Ni W M, Takagi I. Locating the peaks of least energy solutions to a semilinear Neumann problem. Duke Math J, 1993, 70: 247–281
[29] Oplinger D. Frequency response of a nonlinear stretched string. J Acoust Soc Amer, 1960, 32: 1529–1538
[30] Pohozaev S I. A certain class of quasilinear hyperbolic equations. Mat Sb (NS), 1975, 96(138): 152–166
[31] Shuai W. Sign-changing solutions for a class of Kirchhoff-type problem in bounded domains. J Differential Equations, 2015, 259: 1256–1274
[32] Tang X, Chen S. Ground state solutions of Nehari-Pohožaev type for Kirchhoff-type problems with general potentials. Calc Var Partial Differential Equations, 2017, 56: 25pp
[33] Tang X, Cheng B. Ground state sign-changing solutions for Kirchhoff type problems in bounded domains. J Differential Equations, 2016, 261: 2384–2402
[34] Xu S M, Wang C H. Local uniqueness of a single peak solution of a subcritical Kirchhoff problem in $mathbb{R}^3$. Acta Mathematica Scientia, 2020, 40A(2): 432–440
[35] Zhu X P. A perturbation result on positive entire solutions of a semilinear elliptic equation. J Differential Equations, 1991, 92: 163–178

POSITIVE SOLUTIONS FOR A KIRCHHOFF EQUATION WITH PERTURBED SOURCE TERMS

POSITIVE SOLUTIONS FOR A KIRCHHOFF EQUATION WITH PERTURBED SOURCE TERMS

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