一类带临界指标的非自治Kirchhoff型方程非平凡解的存在性

摘要/Abstract

摘要：

该文研究下列非自治Kirchhoff型方程 $ \begin{equation} M\left(\int_{{{\Bbb R}} ^{3}}|\nabla u(x)|^{2}+\int_{{{\Bbb R}} ^{3}}V(x)|u(x)|^{2}\right) \left(-\Delta u+V(x)u \right)=\lambda K(x)f(u)+u^{5} x\in {{\Bbb R}} ^{3} \end{equation} $ 非平凡解的存在性. 其中, 位势V(x) 和K(x) 在无穷远处消失, λ是一个大于零的参数. 该文证明: 存在λ^*>0, 当$ \lambda\geq\lambda^* $时, 上述方程至少有一个非平凡解u_λ.

关键词: Kirchhoff型方程, 临界指标, 变分法

Abstract:

We are concerned with the following non-autonomous Kirchhoff type equation in $ {{\Bbb R}} ^{3} $ $ \begin{eqnarray*} M\left(\int_{{{\Bbb R}} ^{N}}|\nabla u(x)|^{2}+\int_{{{\Bbb R}} ^{N}}V(x)|u(x)|^{2}\right) \left(-\Delta u+V(x)u \right)=\lambda K(x)f(u)+u^{5}, \end{eqnarray*} $ with vanishing potentials $ V(x), K(x) $ at infinity and a Sobolev critical term $ u^{5} $, where $ \lambda\geq 0 $ is a parameter. We prove that there exists $ \lambda^*>0 $ such that the equation has a nontrivial solution $ u_{\lambda} $ for all $ \lambda\geq\lambda^* $.

Key words: Kirchhoff type equation, Critical growth, Variational methods

中图分类号:

O175.2

张鹏辉,韩志清. 一类带临界指标的非自治Kirchhoff型方程非平凡解的存在性[J]. 数学物理学报, 2022, 42(5): 1424-1432.

Penghui Zhang,Zhiqing Han. Existence of Nontrivial Solutions for Non-autonomous Kirchhoff-type Equations with Critical Growth in $ {{\Bbb R}} ^{3} $[J]. Acta mathematica scientia,Series A, 2022, 42(5): 1424-1432.

参考文献 20

1	Alves C O , Corrêa F J S A , Figueiredo G . On a class of nonlocal elliptic problems with critical growth. Differ Equ Appl, 2010, 2 (3): 409- 417
2	Alves C O , Figueiredo G . Nonlinear perturbations of periodic Krichhoff equation in $ {\rm {{\Bbb R}} .{N}} $. Nonlinear Anal, 2012, 75 (5): 2750- 2759 doi: 10.1016/j.na.2011.11.017
3	Alves C O , Souto M A S . Existence of solutions for a class of nonlinear Schrodinger equations with potentials vanishing at infinitly. J Differential Equations, 2013, 254 (4): 1977- 1991 doi: 10.1016/j.jde.2012.11.013
4	Ambrosetti A , Felli V , Malchiodi A . Ground states of nonlinear Schrodinger equations with potentials vanishing at infinity. J Eur Math Soc, 2005, 7 (1): 117- 144
5	Arosio A , Panizzi S . On the well-posedness of the Kirchhoff string. Trans Amer Math Soc, 1996, 348 (1): 305- 330 doi: 10.1090/S0002-9947-96-01532-2
6	Bonheure D , Van Schaftingen J . Ground states for the nonlinear Schrodinger equation with potential vanishing at infinity. Ann Mat Pura Appl, 2010, 189 (2): 273- 301 doi: 10.1007/s10231-009-0109-6
7	Chen P , Liu X . Multiplicity of solutions to Kirchhoff type equations with critical Sobolev exponent. Commun Pure Appl Anal, 2018, 17 (1): 113- 125 doi: 10.3934/cpaa.2018007
8	成艺群, 滕凯民. 非线性临界Kirchhoff型问题的正基态解. 数学物理学报, 2021, 41 (3): 666- 685 doi: 10.3969/j.issn.1003-3998.2021.03.008
	Cheng Y , Teng K . Positive ground state solutions for nonlinear critical kirchhoff type problem. Acta Mathematica Scientia, 2021, 41 (3): 666- 685 doi: 10.3969/j.issn.1003-3998.2021.03.008
9	D'Ancona A , Spagnolo S . Global solvability for the degenerate Kirchhoff equation with real analytic data. Invent Math, 1992, 108 (2): 247- 262
10	Faraci F , Farkas C . On a critical Kirchhoff-type problem. Nonlinear Anal, 2020, 192, 1- 14
11	Figueiredo G M , Santos Junior J R . Multiplicity of solutions for a Kirchhoff equation with subcritical or critical growth. Differential Integral Equations, 2012, 25 (9/10): 853- 868
12	Figueiredo G M . Existence of a positive solution for a Kirchhoff problem type with critical growth via truncation argument. J Math Anal Appl, 2013, 401 (2): 706- 713 doi: 10.1016/j.jmaa.2012.12.053
13	Fiscella A , Valdinoci E . A critical Kirchhoff type problem involving a nonlocal operator. Nonlinear Anal, 2014, 94, 156- 170 doi: 10.1016/j.na.2013.08.011
14	Kirchhoff G . Vorlesungen über Mechanik. Leipzig: Teubner, 1883
15	Lions J L . On some questions in boundary value problems of mathematical physics. North-Holland Math Stud, 1978, 30, 284- 346
16	Liu Z , Guo S . On ground states for the Kirchhoh-type problem with a general critical nonlinearity. J Math Anal Appl, 2005, 426 (1): 267- 287
17	Naimen D . Positive solutions of Kirchhoff type elliptic equations involving a critical Sobolev exponent. NoDEA Nonlinear Differential Equations Appl, 2014, 21 (6): 885- 914
18	Naimen D , Shibata M . Two positive solutions for the Kirchhoff type elliptic problem with critical nonlinearity in high dimension. Nonlinear Anal, 2019, 186, 187- 208
19	Willem W . Minimax Theorems. Boston: Birkhäuser, 1996
20	Zhong X , Tang C . Multiple positive solutions to a Kirchhoff type problem involving a critical nonlinearity in ${{\Bbb R}} .{3} $. Adv Nonlinear Stud, 2017, 17 (4): 661- 676

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