带有负指数 Kirchhoff 方程稳定解的Liouville型定理

Liouville-Type Theorem for Stable Solutions of the Kirchhoff Equation with Negative Exponent

中文关键词:  Kirchhoff 方程  负指数  稳定解  不存在性.

英文关键词:Kirchhoff equation  negative exponent  stable solution  nonexistence

中文摘要:

      该文考虑如下形式的Kirchhoff方程稳定解的 Liouville型定理$$M(\int_{\mathbb{R}^N}|\nabla u|^2dx)\Delta u=&g(x)u^{-q}, x\in \mathbb{R}^N,$$其中$M(t)=a+bt^{\theta}, a>0, b, \theta\ge0,$ 当且仅当 $b=0$ 时, $\theta=0$. $N\geq2, q>0,$ $g(x)\in L^{1}_{loc}(\mathbb{R}^N)$ 是非负函数. 在 $g(x), \theta$ 以及 $q$ 适当假设条件下, 研究了该问题稳定正解的不存在性.

英文摘要:

      In this paper, we consider the Liouville-type theorem for stable solutions of the following Kirchhoff equation $$M\Big(\int_{\mathbb{R}^N}|\nabla u|^2\d x\Big)\Delta u=g(x)u^{-q},\ \ x\in \mathbb{R}^N, $$ where $M(t)=a+bt^{\theta}, a>0, b, \theta\ge0, \theta=0$ if and only if $b=0$. $N\geq2, q>0$ and the nonnegative function $g(x)\in L^{1}_{{\rm loc}}(\mathbb{R}^N)$. Under suitable conditions on $g(x), \theta$ and $q$, we investigate the nonexistence of positive stable solution for this problem.