Critical Schrödinger–Bopp–Podolsky systems: solutions in the semiclassical limit

In this paper we consider the following critical Schrödinger–Bopp–Podolsky system (Formula presented.) in the unknowns u,ϕ:R3→R and where ε,a>0 are parameters. The functions V, K, Q satisfy suitable assumptions as well as the nonlinearity h which is subcritical. For any fixed a>0, we show existence of “small” solutions in the semiclassical limit, namely whenever ε→0. We give also estimates of the norm of this solutions in terms of ε. Moreover, we show also that fixed ε suitably small, when a→0 the solutions found strongly converge to solutions of the Schrödinger-Poisson system.