On a zero mass Schrödinger-Bopp-Podolsky system: ground states, nonexistence results and asymptotic behaviour

In this paper, we consider the following zero mass Schrödinger-Bopp-Podolsky system $$\{\begin{array}{l}-\mathrm{\Delta }u+q^2\phi u=\mathrm{\mid }u{\mathrm{\mid }}^{p-2}u,-\mathrm{\Delta }\phi +a^2{\mathrm{\Delta }}^2\phi =4\pi u^2,\end{array}\text{ in }{\mathbb{R}}^3,$$ where $a>0$ and $q\mathrm{\ne }0$. We complete the study initiated in [2], which relied on a perturbation argument to establish the existence of weak solutions. Here, in contrast, our approach, based on the Mountain Pass Theorem and the splitting lemma, directly yields a ground state solution for $p\in (4,6)$. Moreover, by deriving a Pohozaev identity, we further obtain some nonexistence results for suitable $p$. Finally, based on the minimax characterization, we also analyse, in the radial case, the asymptotic behaviour of the solutions obtained as $a\to 0$, thereby establishing a link with the zero mass Schrödinger-Poisson system.