Multiple nodal solutions for a class of Kirchhoff-type equations in high dimensions

被引:0
作者
He Zhang
Haibo Chen
机构
[1] Central South University,School of Mathematics and Statistics
来源
Zeitschrift für angewandte Mathematik und Physik | 2022年 / 73卷
关键词
Kirchhoff equations; Nonlocal term; Nodal solution; Nehari manifold; 35J20; 35J60;
D O I
暂无
中图分类号
学科分类号
摘要
In this paper, we consider the existence of nodal solutions for a class of Kirchhoff-type problem -a∫RN|∇u|2dx+bΔu+u=f(x)|u|p-2u,inRN,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} -\left( a\int \limits _{{\mathbb {R}}^N}|\nabla u|^2\mathrm {d}x+b\right) \Delta {u}+u=f(x)|u|^{p-2}u,\quad \text {in}\, {\mathbb {R}}^N, \end{aligned}$$\end{document}where a,b>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a,\ b >0$$\end{document}, N≥4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N\ge 4$$\end{document}, 2<p<2∗=2NN-2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2<p<2^{*}=\frac{2N}{N-2}$$\end{document} and the positive continuous function f(x)∈L∞(RN)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(x)\in L^{\infty }({\mathbb {R}}^N)$$\end{document}. Combining a novel constraint manifold method with the detailed energy estimates, we prove that the above problem admits a nodal solution when N=4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N=4$$\end{document} while two nodal solutions with opposite energy level for corresponding action functional are achieved when N≥5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N\ge 5$$\end{document} by controlling the parameter a sufficiently small.
引用
收藏
相关论文
共 50 条
[21]  
Li YH(1992)On nonhomogeneous elliptic equations involving critical Sobolev exponent Ann. Inst. Henri Poincar Anal. Non Linaire 9 281-304
[22]  
Li FY(2011)Existence of nontrivial solutions and high energy solutions for Schrödinger–Kirchhoff-type equations in Nonlinear Anal. RWA 12 1278-1287
[23]  
Shi JP(2015)The existence of least energy nodal solutions for some class of Kirchhoff equations and Choquard equations in J. Math. Anal. Appl. 431 935-954
[24]  
Li GB(2016)Infinitely many sign-changing solutions for Kirchhoff-type equations with power nonlinearity Electron. J. Differ. Equ. 59 1-7
[25]  
Ye HY(2006)Sign changing solutions of Kirchhoff type problems via invariant sets of descent flow J. Math. Anal. Appl. 317 456-463
[26]  
Li L(2020)Existence and asymptotic behavior of positive solutions for Kirchhoff type problems with steep potential well J. Differ. Equ. 269 10085-10106
[27]  
Sun JJ(undefined)undefined undefined undefined undefined-undefined
[28]  
Lu SS(undefined)undefined undefined undefined undefined-undefined
[29]  
Ma TF(undefined)undefined undefined undefined undefined-undefined
[30]  
Muoz Rivera JE(undefined)undefined undefined undefined undefined-undefined