Supercritical Schrödinger equations with vanishing potential and double weighted nonlocal interaction

In this paper, we discuss the existence of positive solutions for the following class of Schrödinger equations involving double weighted nonlocal interaction$\begin{align} -\Delta u+V(x)u = \left(\int_{\mathbb{R}^N}\frac{F(u)}{|y|^{\alpha}|x-y|^{\mu}}\,\mathrm{d}y\right)\frac{f(u)}{|x|^{\alpha}} + \psi(x,u), \quad \text{in } \mathbb{R}^{N}, \end{align}$where $ N\geq 3 $, $ 0<\mu <N $, $ \alpha\geq 0 $ and $ 0<2\alpha+\mu < \min\{\frac{N+2}{2},4\} $, the potential $ V:\mathbb{R}^N \rightarrow \mathbb{R} $ decays to zero at infinity and $ F $ is the primitive of function $ f. $ Under two types of supercritical function $ \psi $, we obtain the existence of solutions of the equation with subcritical or critical homogeneous $ f $ in the sense of the Stein-Weiss inequality. The approach is based on variational methods combined with a truncation argument, an adapted version of penalization technique and the Moser's iteration method.