Multiple solutions for a class of Kirchhoff equation with singular nonlinearity

Abstract: In this article, we investigate the existence and multiplicity of solutions of Kirchhoff equation \begin{equation*} \left{ \begin{aligned} -(1+b \int_{\mathbb{R}^3}|\nabla u|^2)\Delta u= k(x)\frac{|u|^2 u}{|x|} +\lambda h(x)u,x\in\mathbb{R}^3\ u(x)\rightarrow 0 ~~~~~~~~~~~~~~~~~~~as ~~|x|\rightarrow\infty \end{aligned} \right. \end{equation*} where the potential $k(x)$ allows sign changing. Making use of Nehari manifold method and Concentration-compactness principle, we obtain the existence and multiplicity of solutions for this equation.