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

Abstract

This paper is concerned with the existence of positive solution to a class of singular fourth order elliptic equation of Kirchhoff type

$$\begin{aligned} \triangle ^2 u-\lambda M(\Vert \nabla u\Vert ^2)\triangle u-\frac{\mu }{\vert x\vert ^4}u=\frac{h(x)}{u^\gamma }+k(x)u^\alpha , \end{aligned}$$

under Navier boundary conditions, $u=\triangle u=0$. Here $\varOmega \subset {\mathbf {R}}^N$, $N\ge 1$ is a bounded $C^4$-domain, $0\in \varOmega$, h(x) and k(x) are positive continuous functions, $\gamma \in (0,1)$, $\alpha \in (0,1)$ and $M:{\mathbf {R}}^+\rightarrow {\mathbf {R}}^+$ is a continuous function. By using Galerkin method and sharp angle lemma, 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. Besides, if $\mu =0$, $\lambda >0$ and h, k are Lipschitz functions, we show that this problem has a positive smooth solution. If $h,k\in C^{2,\,\theta _0}(\overline{\varOmega })$ for some $\theta _0\in (0,1)$, then this problem has a positive classical solution.