Liouville-Type Theorem for Stable Solutions of the Kirchhoff Equation with Negative Exponent
Received:July 04, 2019  Revised:December 08, 2019
Key Word: Kirchhoff equation   negative exponent   stable solution   nonexistence  
Fund ProjectL:Supported by the National Natural Science Foundation of China (Grant No.11571092), the Natural Science Foundation of the Higher Education Institutions of Jiangsu Province (Grant No.19KJD100002), the Natural Science Foundation of Shandong Province (Grant No.ZR2018MA017) and the China Postdoctoral Science Foundation (Grant No.2017M610436).
Author NameAffiliation
Yunfeng WEI School of Statistics and Mathematics, Nanjing Audit University, Jiangsu 211815, P. R. China
College of Science, Hohai University, Jiangsu 210098, P. R. China 
Caisheng CHEN College of Science, Hohai University, Jiangsu 210098, P. R. China 
Hongwei YANG College of Mathematics and Systems Science, Shandong University of Science and Technology, Shandong 266590, P. R. China 
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      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.
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