Liouville-type theorem for stable solutions of the Kirchhoff equation with negative exponent
Received:July 04, 2019  Revised:November 28, 2019
Key Word: Kirchhoff equation   Negative exponent   Stable solution   Nonexistence.  
Fund ProjectL:Supported by the National Natural Science Foundation of China(Grant No. 11571092); Jiangsu Province Natural Science Research Projects in Colleges and Universities(19KJD100002); the Natural Science Foundation of Shandong Province(Grant No. ZR2018MA017) and the China Postdoctoral Science Foundation (Grant No. 2017M610436).
Author NameAffiliationE-mail
Yunfeng Wei School of Statistics and Mathematics, Nanjing Audit University weiyunfeng@nau.edu.cn 
Caisheng Chen College of Science, Hohai University  
Hongwei Yang College of Mathematics and Systems Science, Shandong University of Science and Technology  
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Abstract:
      In this paper, we are concerned with Liouville-type theorem for stable solutions of the Kirchhoff equation $$ \align M(\int_{\mathbb{R}^N}|\nabla u|^2dx)\Delta u=&g(x)u^{-q}, x\in \mathbb{R}^N,\tag 0.1 \endalign $$ 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 $g(x)\in L^{1}_{loc}(\mathbb{R}^N)$. Under suitable conditions on $g(x), \theta$ and $q$, the nonexistence of positive stable solution to equation (0.1) is investigated.
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