Nonlinear Schr\"odinger equation in the Bopp-Podolsky electrodynamics: solutions in the electrostatic case

We study the following nonlinear Schr\"odinger-Bopp-Podolsky system \[ \begin{cases} -\Delta u + \omega u + q^{2}\phi u = |u|^{p-2}u -\Delta \phi + a^2 \Delta^2 \phi = 4\pi u^2 \end{cases} \hbox{ in }\mathbb{R}^3 \] with $a,\omega>0$. We prove existence and nonexistence results depending on the parameters $q,p$. Moreover we also show that, in the radial case, the solutions we find tend to solutions of the classical Schr\"odinger-Poisson system as $a\to0$.