Existence and limit behavior of least energy solutions to constrained Schrödinger–Bopp–Podolsky systems in R3

Consider the following Schrödinger–Bopp–Podolsky system in R3 under an L2-norm constraint, {-Δu+ωu+ϕu=u|u|p-2,-Δϕ+a2Δ2ϕ=4πu2,‖u‖L2=ρ,where a, ρ> 0 are fixed, with our unknowns being u, ϕ: R3→ R and ω∈ R. We prove that if 2 < p< 3 (resp., 3 < p< 10 / 3) and ρ> 0 is sufficiently small (resp., sufficiently large), then this system admits a least energy solution. Moreover, we prove that if 2 < p< 14 / 5 and ρ> 0 is sufficiently small, then least energy solutions are radially symmetric up to translation, and as a→ 0 , they converge to a least energy solution of the Schrödinger–Poisson–Slater system under the same L2-norm constraint.