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

Abstract

In this paper we study the following zero mass Schrödinger-Bopp-Podolsky system with critical growth \[ \begin{cases} -Δu +q^2ϕu=μ|u|^{p-2}u+|u|^4u\\ -Δϕ+a^2Δ^2ϕ=4πu^2, \end{cases} \] where $a>0$, $q\ne 0$, $\mu >0$ 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}.