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

Consider the following Schr¨ odinger–Bopp–Podolsky system in R3 under an L2-norm constraint, −Δφ + a2Δ2φ = 4πu2 ⎧ ⎪ ⎨ ⎪ ⎩−Δu + ωu + φu= u|u|p−2 , , ∥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 Schr¨ odinger–Poisson–Slater system under the same L2-norm constraint.