一类带临界指标的非自治Kirchhoff型方程非平凡解的存在性

数学物理学报 ›› 2022, Vol. 42 ›› Issue (5): 1424-1432.

一类带临界指标的非自治Kirchhoff型方程非平凡解的存在性

张鹏辉,韩志清*()

^{大连理工大学数学科学学院辽宁大连 116023}

收稿日期:2021-09-28 出版日期:2022-10-26 发布日期:2022-09-30
通讯作者: 韩志清 E-mail:hanzhiq@dlut.edu.cn

Existence of Nontrivial Solutions for Non-autonomous Kirchhoff-type Equations with Critical Growth in $ {{\Bbb R}} ^{3} $

Penghui Zhang,Zhiqing Han*()

^{School of Mathematical Sciences, Dalian University of Technology, Liaoning Dalian 116023}

Received:2021-09-28 Online:2022-10-26 Published:2022-09-30
Contact: Zhiqing Han E-mail:hanzhiq@dlut.edu.cn

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

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