非线性临界Kirchhoff型问题的正基态解

数学物理学报 ›› 2021, Vol. 41 ›› Issue (3): 666-685.

非线性临界Kirchhoff型问题的正基态解

成艺群(),滕凯民*()

^{太原理工大学数学学院太原 030024}

收稿日期:2020-04-17 出版日期:2021-06-01 发布日期:2021-06-09
通讯作者: 滕凯民 E-mail:1906157258@qq.com;tengkaimin2013@163.com
作者简介:成艺群, E-mail: 1906157258@qq.com
基金资助:
国家自然科学基金(11501403);山西省留学回国择优项目(2018);山西省自然科学基金面上项目(201901D111085)

Positive Ground State Solutions for Nonlinear Critical Kirchhoff Type Problem

Yiqun Cheng(),Kaimin Teng*()

^{School of Mathematical Sciences, Taiyuan University of Technology, Taiyuan 030024}

Received:2020-04-17 Online:2021-06-01 Published:2021-06-09
Contact: Kaimin Teng E-mail:1906157258@qq.com;tengkaimin2013@163.com
Supported by:
the NSFC(11501403);the Scientific Activities of Selected Returned Overseas Professionals in Shanxi Province(2018);the NSF of Shanxi Province(201901D111085)

摘要/Abstract

摘要：

该文研究如下Kirchhoff型方程 $\left\{ \begin{array}{l}-\big(a+b\int_{\mathbb{R}^3}|\nabla u|^2{\rm d}x\big)\triangle u+V(x)u=|u|^{p-2}u+\varepsilon|u|^4u，\quad x\in\mathbb{R}^3, \\ u\in H^{1}(\mathbb{R}^3), \end{array}\right.$ 其中 $a>0$ ， $b>0$ ， $4 < p < 6$ ， $V(x)\in L_{\rm loc}^{\frac{3}{2}}(\mathbb{R}^3)$ 是一个给定的非负函数且满足 $\lim\limits_{|x|\rightarrow\infty}V(x): =V_{\infty}$ .对 $V(x)$ 给定适当的假设条件，当 $\varepsilon$ 充分小时，证明了基态解的存在性.

关键词: Kirchhoff型方程, 临界非线性, 基态解

Abstract:

In this paper, we consider the following Kirchhoff type problem (3.36) $\begin{equation}\left\{ \begin{array}{l}-\Big(a+b\int_{\mathbb{R}^3}|\nabla u|^2{\rm d}x\Big)\triangle u+V(x)u=|u|^{p-2}u+\varepsilon|u|^4u, \, \, \, x\in\mathbb{R}^3, \\ u\in H^{1}(\mathbb{R}^3), \end{array} \right. \end{equation}$ where $a>0$ , $b>0$ , $4< p < 6$ and $V (x)\in L_{\rm loc}^{\frac{3}{2}}(\mathbb{R}^3)$ is a given nonnegative function such that $\lim\limits_{|x|\rightarrow\infty}V(x): =V_{\infty}$ . Under suitable conditions on $V(x)$ , we prove that the existence of ground state solutions for small $\varepsilon$ .

Key words: Kirchhoff type problem, Critical nonlinearity, Ground state

中图分类号:

O175.2

成艺群,滕凯民. 非线性临界Kirchhoff型问题的正基态解[J]. 数学物理学报, 2021, 41(3): 666-685.

Yiqun Cheng,Kaimin Teng. Positive Ground State Solutions for Nonlinear Critical Kirchhoff Type Problem[J]. Acta mathematica scientia,Series A, 2021, 41(3): 666-685.

参考文献 29

1	Kirchhoff G . Vorlesungen über Mechanik. Teubner: Leipzig, 1883
2	Xu L P , Chen H B . Gound state solutions for Kirchhoff-type equations with a general nonlinearity in the critical growth. Adv Nonlinear Anal, 2018, 7, 535- 546 doi: 10.1515/anona-2016-0073
3	Liu Z S , Guo S J . On ground states for the Kirchhoff-type problem with a general critical nonlinearity. J Math Anal Appl, 2015, 426, 267- 287 doi: 10.1016/j.jmaa.2015.01.044
4	Li H Y . Existence of positive ground state solutions for a critical Kirchhoff type problem with sign-changing potential. Comput Math Appl, 2018, 75, 2858- 2873 doi: 10.1016/j.camwa.2018.01.015
5	Chen B T , Tang X H . Ground state sign-changing solutions for asymptotically 3-linear Kirchhoff-type problems. Complex Var Elliptic Equ, 2017, 62, 1093- 1116 doi: 10.1080/17476933.2016.1270272
6	Qin D D , He Y B , Tang X H . Ground state solutions for Kirchhoff type equations with asymptotically 4-linear nonlinearity. Comput Math Appl, 2016, 71, 1524- 1536 doi: 10.1016/j.camwa.2016.02.037
7	Qin D D , He Y B , Tang X H . Ground state and multiple solutions for Kirchhoff type equations with critical exponent. Canad Math Bull Vol, 2018, 61 (2): 353- 369 doi: 10.4153/CMB-2017-041-x
8	Li G B , Ye H Y . Existence of positive ground state solutions for the nonlinear Kirchhoff type equations in $\mathbb{R}^3$ . J Differential Equations, 2014, 257, 566- 600 doi: 10.1016/j.jde.2014.04.011
9	Wu K . Existence of ground states for a Kirchhoff type problem without 4-superlinear condition. Comput Math Appl, 2018, 75, 755- 763 doi: 10.1016/j.camwa.2017.10.005
10	Guo Z J . Ground states for Kirchhoff equations without compact condition. J Differential Equations, 2015, 259, 2884- 2902 doi: 10.1016/j.jde.2015.04.005
11	Zhang H , Zhang F B . Ground states for the nonlinear Kirchhoff type problems. J Math Anal Appl, 2015, 423, 1671- 1692 doi: 10.1016/j.jmaa.2014.10.062
12	Liu Z S , Guo S J . Existence of positive ground state solutions for Kirchhoff type problems. Nonlinear Anal, 2015, 120, 1- 13 doi: 10.1016/j.na.2014.12.008
13	Tang X H , Chen S T . Ground state solutions of Nehari-Pohozaev type for Kirchhoff-type problems with general potentials. Calc Var Partial Differential Equations, 2017, 56, 110- 134 doi: 10.1007/s00526-017-1214-9
14	Chen S T , Tang X H . Berestycki-Lions conditions on ground state solutions for Kirchhoff-type problems with variable potentials. J Math Phys, 2019, 60, 121509 doi: 10.1063/1.5128177
15	Ye Y W . Ground state solutions for Kirchhoff-type problems with critical nonlinearity. Taiwanese J Math, 2020, 24, 63- 79
16	Sun J T , Wu T F . Ground state solutions for an indefine Kirchhoff type problem with steep potential well. J Differential Equations, 2014, 256, 1771- 1792 doi: 10.1016/j.jde.2013.12.006
17	Chen B T , Li G F , Tang X H . Nehari-type ground state solutions for Kirchhoff type problems in $\mathbb{R}^N$ . Appl Anal, 2019, 98, 1255- 1266 doi: 10.1080/00036811.2017.1419202
18	Hu T X , Lu L . On some nonlocal equations with competing coefficients. J Math Anal Appl, 2018, 460, 863- 884 doi: 10.1016/j.jmaa.2017.12.027
19	He X M , Zou W M . Ground states for nonlinear Kirchhoff equations with critical growth. Annali di Mate, 2014, 193, 473- 500 doi: 10.1007/s10231-012-0286-6
20	Liu Z S , Guo S J . Existence and concentration of positive ground states for a Kirchhoff equation involving critical Sobolev exponent. Z Angew Math Phys, 2015, 66, 747- 769 doi: 10.1007/s00033-014-0431-8
21	Liu Z S , Luo C L . Existence of positive ground state solutions for Kirchhoff type equation with general critical growth. Top Methods Nonlinear Anal, 2016, 49, 165- 182
22	Liu J , Liao J F , Pan H L . Ground state solution on a non-autonomous Kirchhoff type equation. Comput Math Appl, 2019, 78, 878- 888 doi: 10.1016/j.camwa.2019.03.009
23	Lin X Y , Wei J Y . Existence and concentration of ground state solutions for a class of Kirchhoff-type problems. Nonlinear Anal, 2020, 195, 111715 doi: 10.1016/j.na.2019.111715
24	Li G B , Ye H Y . Existence of positive solutions for nonlinear Kirchhoff type problems in $\mathbb{R}^3$ with critical Sobolev exponent. Math Meth Appl Sci, 2014, 37, 2570- 2584 doi: 10.1002/mma.3000
25	Sun D D , Zhang Z T . Existence and asymptotic behaviour of ground state solutions for Kirchhoff-type equations with vanishing potentials. Z Angew Math Phys, 2019, 70 (37): 1- 18 doi: 10.1007/s00033-019-1082-6
26	Tang X H , Chen B T . Ground state sign-changing solutions for Kirchhoff type problems in bounded domains. J Differential Equations, 2016, 261, 2384- 2402 doi: 10.1016/j.jde.2016.04.032
27	Hu T , Lu L . Multiplicity of positive solutions for Kirchhoff type problems in $\mathbb{R}^3$ . Topol Methods Nonlinear Anal, 2017, 50, 231- 252
28	Willem M . Minimax Theorems. Boston: Birkhäuser, 1996
29	Lions P L . The concentration-compactness principle in the calculus of variation. The locally compact case, Part Ⅱ. Ann Inst H Poincaré Anal Non Linéaire, 1984, 1, 223- 283 doi: 10.1016/S0294-1449(16)30422-X

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