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.

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