Existence of solutions for nonlinear Dirac equations in the Bopp–Podolsky electrodynamics

In this paper, we study the following nonlinear Dirac–Bopp–Podolsky system −i∑k=13αk∂ku+[V(x)+q]βu+wu−ϕu=f(x,u),in R3,−△ϕ+a2△2ϕ=4π|u|2,in R3,where a,q>0,w∈R, V(x) is a potential function, and f(x,u) interaction term (nonlinearity). First, give physical motivation for new kind of system. Second, under suitable assumptions on f V, by means minimax techniques involving Cerami sequences, prove existence at least one pair solutions (u,ϕu).