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

Consider the following Schrödinger–Bopp–Podolsky system in R 3 under an L 2 -norm constraint, - Δ u + ω u + ϕ u = u | u | p - 2 , - Δ ϕ + a 2 Δ 2 ϕ = 4 π u 2 , ‖ u ‖ L 2 = ρ , where a , ρ > 0 are fixed, with our unknowns being u , ϕ : R 3 → 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 L 2 -norm constraint.