On a zero mass Schr\"odinger-Bopp-Podolsky system: ground states, nonexistence results and asymptotic behaviour

Abstract

In this paper, we consider the following zero mass Schr\"{o}dinger-Bopp-Podolsky system \[ \begin{cases} -\Delta u +q^2\phi u=|u|^{p-2}u, -\Delta \phi+a^2\Delta^2\phi=4\pi u^2, \end{cases} \text{ in } \mathbb{R}^3, \] where $a>0$ and $q\ne 0$. 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 Schr\"odinger-Poisson system.