Normalized solution to Schrödinger system with Bopp–Podolsky-type self-attraction

This paper is concerned with the existence and nonexistence of solutions to the following Schrödinger system with Bopp–Podolsky-type self-attraction under the constraint of prescribed mass −Δu+(V+λ)u−1−e−|x|a|x|∗u2u=f(u) in R3,∫R3u2=c, where a > 0, c > 0 and λ∈R severs as a Lagrange multiplier. Under the suitable assumptions on the nonpositive potential V and the L2 subcritical nonlinear term f, we prove that there exists c0 ⩾ 0 such that: when c > c0, the above system has a positive ground state normalized solution; when c0 > 0 and c ∈ (0, c0), there is no ground state normalized solution. We further study the distinction between the cases of V ≡ 0 and V ⩽ 0. Subsequently, we obtain a sufficient condition for c0 = 0. Our innovation lies in the use of different transformations to obtain the subadditivity of the minimum value of the restricted functional in both cases. Moreover, we also prove the nonexistence of the ground state normalized solution when V is nonnegative.