Global existence of positive solution for nonlocal singular fourth order Kirchhoff equation with Hardy potential

This paper is concerned with the existence of positive solution for a class of singular fourth order elliptic equation of Kirchhoff type \triangle^2 u-\lambda M(\Vert \nabla u\Vert^2)\triangle u-\frac{\mu}{\vert x\vert^4}u=\frac{h(x)}{u^\gamma}+k(x)u^\alpha under Navier boundary condition, $u=\triangle u=0$. Here $\Omega\subset \mathbb{R}^N$, $N\ge 1$ is a bounded smooth domain, $0\in \Omega$, $h(x)$ and $k(x)$ are positive continuous functions, $\gamma\in [0,1]$, $\alpha\in (0,1)$ and $M:\mathbb{R}^+\rightarrow \mathbb{R}^+$ is continuous function. By using Galerkin method, we will show that this problem has a positive solution for $\lambda \frac{\mu}{\mu^*m_0}$ and $0 \mu \mu^*$. Here $\mu^*=\Big(\frac{N(N-4)}{4}\Big)^2$ is the best constant in the Hardy inequality .For a case of discontinuity of $M$, we also show the existence of positive solution for $\lambda \frac{\mu}{\mu^*m_0}$ and $0 \mu \mu^*$ with the same metho .