Critical Schrodinger-Bopp-Podolsky systems: solutions in the semiclassical limit

In this paper we consider the following critical Schrodinger-Bopp-Podolsky system {-& varepsilon;(2)Delta u+V(x)u+Q(x)phi u=h(x,u)+K(x)|u|(4)u in R-3 (-)Delta phi+a(2)Delta(2)phi=4 pi Q(x)u(2 )in R-3 in the unknowns u,phi:R-3 -> R and where epsilon,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 epsilon -> 0. We give also estimates of the norm of this solutions in terms of epsilon. Moreover, we show also that fixed epsilon suitably small, when a -> 0 the solutions found strongly converge to solutions of the Schr & ouml;dinger-Poisson system.