Positive solutions of Kirchhoff type problem with singular and critical nonlinearities in dimension four

Rui-Qi  Liu; Chun-Lei Tang; Jia-Feng  Liao; Xing-Ping Wu

doi:10.3934/cpaa.2016006

Article Contents

2016, Volume 15, Issue 5: 1841-1856. Doi: 10.3934/cpaa.2016006

This issue Previous Article Liouville type theorems for singular integral equations and integral systems Next Article Smooth quasi-periodic solutions for the perturbed mKdV equation

Positive solutions of Kirchhoff type problem with singular and critical nonlinearities in dimension four

1.
School of Mathematics and Statistics, Southwest University, Chongqing 400715

Received: August 2015

Revised: April 2016

Published: September 2016

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Abstract

In this article, we study the existence and multiplicity of positive solutions for the Kirchhoff type problem with singular and critical nonlinearities \begin{eqnarray} \begin{cases} -\left(a+b\displaystyle\int_\Omega|\nabla u|^2dx\right)\Delta u=\mu u^{3}+\frac{\lambda}{|x|^{\beta}u^{\gamma}}, &\rm \mathrm{in}\ \ \Omega, \\ u>0, &\rm \mathrm{in}\ \ \Omega, \\ u=0, &\rm \mathrm{on}\ \ \partial\Omega, \end{cases} \end{eqnarray} where $\Omega\subset \mathbb{R}^{4}$ is a bounded smooth domain and $a, b>0$, $\lambda>0$. For all $\mu>0$ and $\gamma\in(0,1)$, $0\leq\beta < 3$, we obtain one positive solution. Particularly, we prove that this problem has at least two positive solutions for all $\mu> bS^{2}$ and $\gamma\in(0,\frac{1}{2})$, $2+2\gamma < \beta < 3$.

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Mathematics Subject Classification: Primary: 35J75; Secondary: 35J20.

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