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

Abstract

In this article, we study the existence and multiplicity of positivesolutions 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 $\gamma\in(0,1)$, $0\leq\beta bS^{2}$ and $\gamma\in(0,\frac{1}{2})$, $2+2\gamma < \beta < 3$.