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January  2019, 18(1): 181-193. doi: 10.3934/cpaa.2019010

## Kirchhoff type equations with strong singularities

 Department of Mathematics, University of Chinese Academy of Sciences, Beijing 100049, China

* Corresponding author

Received  October 2017 Revised  January 2018 Published  August 2018

Fund Project: The authors are supported by NSFC grants 11571339 and 11771468.

An optimal condition is given for the existence of positive solutions of nonlinear Kirchhoff PDE with strong singularities. A byproduct is that $-2$ is no longer the critical position for the existence of positive solutions of PDE's with singular potentials and negative powers of the form: $- |x{|^\alpha }\Delta u = {u^{{\rm{ - }}\gamma }}$ in $Ω$, $u = 0$ on $\partial \Omega$, where $\Omega$ is a bounded domain of ${\mathbb{R}}^{N}$ containing 0, with $N \ge 3$, $\alpha \in \left( {0, N} \right)$ and $- \gamma \in \left( { - 3, - 1} \right)$.

Citation: Yijing Sun, Yuxin Tan. Kirchhoff type equations with strong singularities. Communications on Pure & Applied Analysis, 2019, 18 (1) : 181-193. doi: 10.3934/cpaa.2019010
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##### References:
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