Kirchhoff type equations with strong singularities

Yijing Sun; Yuxin Tan

doi:10.3934/cpaa.2019010

Article Contents

2019, Volume 18, Issue 1: 181-193. Doi: 10.3934/cpaa.2019010

This issue Previous Article Existence and a general decay results for a viscoelastic plate equation with a logarithmic nonlinearity Next Article Existence of weak solutions for a diffuse interface model for two-phase flow with surfactants

Kirchhoff type equations with strong singularities

Yijing Sun^, and
Yuxin Tan

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

* Corresponding author
* Corresponding author

Received: October 2017

Revised: January 2018

Published online: August 4, 2018

The authors are supported by NSFC grants 11571339 and 11771468.

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Abstract

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)$ .

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Mathematics Subject Classification: Primary: 35J70, 35J20; Secondary: 35J60.

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