Existence and multiplicity results for the zero mass Schrödinger-Bopp-Podolsky system with critical growth

In this paper we study the following zero mass Schrödinger-Bopp-Podolsky system with critical growth $$\{\begin{array}{l}-\mathrm{\Delta }u+q^2\phi u=\mu \mathrm{\mid }u{\mathrm{\mid }}^{p-2}u+\mathrm{\mid }u{\mathrm{\mid }}^{4}u\\ -\mathrm{\Delta }\phi +a^2{\mathrm{\Delta }}^2\phi =4\pi u^2,\end{array}$$ where $a>0$, $q\mathrm{\ne }0$, $\mu >0$ is a parameter and $p\in (3,6)$. By introducing a new functional framework developed by Caponio et al. \cite{Cd}, we first establish the existence of positive ground state solutions for the case of $p\in (3,6)$. Moreover, for the case of $p\in (4,6)$, multiplicity results are obtained by applying an abstract critical point theorem due to Perera \cite{Pe}.