We consider the following Schrödinger-Bopp-Podolsky system with critical and sublinear terms \(\begin{aligned} {\left\{ \begin{array}{ll} - \Delta u+ u+Q(x)\phi u= \vert u\vert ^4 u+ \lambda K(x)\vert u \vert ^{p-1}u& \text{ in } \ \mathbb {R}^3 \\ - \Delta \phi + a^{2}\Delta ^{2} \phi = 4\pi Q(x) u^{2}& \text{ in } \ \mathbb {R}^3. \end{array}\right. } \end{aligned}\) Here \(u,\phi :\mathbb {R}^{3}\rightarrow \mathbb {R}\) are the unknowns, Q and K are given functions satisfying mild assumptions, \(a\ge 0, \lambda >0\) are parameters and \(p\in (0,1)\) . We first show existence of infinitely many solutions at negative energy level, including the ground state, when the parameter \(\lambda \) is small. Then we give general results concerning the structure of the set of solutions. We show also the behaviour of the solutions as the parameters \(a,\lambda \) tend to zero. In particular the ground states solutions tends to a ground state solution of the Schrödinger-Poisson system as a tends to zero.
