The existence of solutions for Schrödinger-Poisson equation with zero mass and a convolution nonlinearity

In this article, we study the Schrödinger-Poisson system with zero mass and a convolution nonlinearity:$-\Delta u+(\frac{1}{4\pi \left|x\right|}⁎u^2)u=\mu (I_{\alpha }⁎|u{|}^p\left)\right|u{|}^{p-2}u,x\in R^3,$ where $p\in (\frac{17(3+\alpha )}{48},3+\alpha )$ and $\mu >0$. We prove that the aforementioned system admits a ground state solution when $p\in (\frac{17(3+\alpha )}{48},\frac{6+\alpha }{4})$ or $p\in (\frac{3+\alpha }{2},3+\alpha )$; and a radially symmetric ground state solution when $p\in (\frac{6+\alpha }{4},\frac{3+\alpha }{2}]$. Specifically, when $p=\frac{6+\alpha }{4}$, we demonstrate that the system only admits trivial solution for $0<\mu <{\mu }_0$, but admits a positive solution if μ takes some particular value, where ${\mu }_0$ is a specific value dependent on α.