Two solutions for fractional Schrödinger-Poisson system involving a degenerate Kirchhoff term

In this paper, we investigate the multiplicity of solutions for the following nonlinear fractional Schrödinger-Poisson system of Kirchhoff type:

$$\begin{aligned} \left\{ \begin{array}{ll} [u]_{s}^{2(\theta -1)}(-\Delta )^{s}u+ \phi (x)u = f(x)|u|^{r-2}u + \lambda \frac{|u|^{q - 2} u}{|x|^{\alpha }}, & \text {in} \,\,\Omega , \\ (-\Delta )^{t} \phi = u^2, & \text {in} \,\,\Omega ,\\ u=\phi =0, & \text {in} ~\mathbb {R}^{N} \backslash \Omega , \end{array} \right. \end{aligned}$$

where \(s, t\in (0,1)\), \(\Omega \subset \mathbb {R}^N\) is a smooth bounded domain containing 0 with Lipschitz boundary, \(\left( -\Delta \right) ^{\gamma }\) \((\gamma =s,t)\) is the fractional Laplace operator, \(\lambda \) is a positive parameter, \(0\le \alpha<2s<N\), \(2<r<2\theta<4<q<2_{\alpha }^{*}\) and \(f(x)\in L^{\frac{2_\alpha ^*}{2_\alpha ^*-r}}(\Omega )\) is positive almost everywhere in \({\Omega }\). By using variational methods, we get over some tricky difficulties stemming from degenerate feature of Kirchhoff term. As a result, by employing the Nehari manifold method, under some certain conditions, we prove that the above system has at least two distinct positive solutions for \(\lambda \) small.