数学物理学报 ›› 2019, Vol. 39 ›› Issue (2): 264-276.

• 论文 • 上一篇    下一篇

非线性Kirchhoff型椭圆方程的最低能量解

柳志德,王征平*()   

  1. 武汉理工大学数学科学研究中心, 理学院数学系 武汉 430070
  • 收稿日期:2018-03-13 出版日期:2019-04-26 发布日期:2019-05-05
  • 通讯作者: 王征平 E-mail:zpwang@whut.edu.cn
  • 基金资助:
    国家自然科学基金(11471331);国家自然科学基金(11871386)

Least Energy Solution for Nonlinear Kirchhoff Type Elliptic Equation

Zhide Liu,Zhengping Wang*()   

  1. Center for Mathematical Sciences and Department of Mathematics, School of Science, Wuhan University of Technology, Wuhan 430070
  • Received:2018-03-13 Online:2019-04-26 Published:2019-05-05
  • Contact: Zhengping Wang E-mail:zpwang@whut.edu.cn
  • Supported by:
    the NSFC(11471331);the NSFC(11871386)

摘要:

该文讨论以下非线性Kirchhoff型椭圆方程非平凡解和非负最低能量解的存在性

$\left\{ \begin{align} & -(a+b\int_{{{\mathbb{R}}^{3}}}{|}\nabla u{{|}^{2}}\text{d}x)\Delta u+V(x)u=\mu u+|u{{|}^{p-1}}u,\ \ \ \ \ \ \ \ \ \ x\in {{\mathbb{R}}^{3}}, \\ & u\in {{H}^{1}}({{\mathbb{R}}^{3}}),\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x\in {{\mathbb{R}}^{3}}, \\ \end{align} \right.$

其中$p\in (3, 5)$, $a, b>0$, $V\in C({{\mathbb{R}}^{3}}, {{\mathbb{R}}^{+}})$并且$\lim\limits_{|x|\to +\infty}V(x)=\infty$.通过变分方法,该文首先证明了对于任何$b>0$,存在$\delta(b)>0$,使得当$\mu_1\leq\mu <\mu_1+\delta(b)$时,方程(0.1)有非平凡解.其次,进一步证明了存在$\delta_1(b)\in(0, \delta(b))$,当$\mu_1 <\mu <\mu_1+\delta_1(b)$时,方程(0.1)有非负的最低能量解,这里$\mu_1$是Schrödinger算子$-\triangle+V$的第一特征值.最后利用对称山路引理证明了对任意的$\mu\in\mathbb{R}$,方程(0.1)存在无穷多个非平凡解.

关键词: Kirchhoff方程, 非平凡解, 最低能量解

Abstract:

In this paper, we study the existence of nontrivial solution and nonnegative least energy solution for the following nonlinear Kirchhoff type elliptic equation

where $p\in (3, 5)$, $a, b>0$, $V\in C(\mathbb{R} ^3, \mathbb{R} ^+)$ and $\lim\limits_{|x|\to +\infty}V(x)=\infty$. By using variational methods, firstly we prove that for any $b>0$, there exists $\delta(b)>0$ such that problem (0.1) (0.1) with $\mu_1\leq\mu <\mu_1+\delta(b)$ has a nontrivial solution, where $\mu_1$ denotes the first eigenvalue of the Schrödinger operator $-\triangle+V$. Secondly, we show that there exists $\delta_1(b)\in(0, \delta(b))$ such that problem (0.1) (0.1) with $\mu_1 <\mu <\mu_1+\delta_1(b)$ has a nonnegative least energy solution. Finally, by using the symmetric Mountain Pass lemma we prove that problem (0.1) (0.1) has infinitely many nontrivial solutions for any $\mu\in \mathbb{R} $.

Key words: Kirchhoff equation, Nontrivial solution, Least energy solution

中图分类号: 

  • O175.2