Key points are not available for this paper at this time.
In the present paper, we give some remarks on the magnetic field operators iA. As its applications, we study the Schr\"odinger equation with a magnetic field equation* - u+|A (x) |^2u+iA (x) u= u+|u|^pu, ~x R^N, equation* where u is a complex-valued function and R. When N>2, for 2 p+2 2NN-2 or N=2, for 2 p+2 +, the existence and nonexistence of minimizers of the corresponding minimization problem are given via constrained variational methods. As a by-product, the above equation admits a normalized solution. We point out that the condition divA (x) =0 plays a crucial role in our study.
Wenbo Wang (2024) studied this question.