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.

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