Non-Simple Concentrations for Nonlinear Magnetic Schrodinger Equations with Constant Electric Potentials

Abstract The paper investigates the influence of magnetic fields on the non-simple concentration phenomenon in the complex-valued nonlinear Schr¨odinger equations with a constant electric potential (iε∇ + A)2u + u − |u|p−1u = 0. We demonstrate that a multi-peak solution always exists at a non-degenerate local maximum or minimum point of the Frobenius norm ∥B∥F, where B is the magnetic field corresponding to the magnetic potential A. It is surprising that at such a minimum point, we can find a two-peak solution, which is distinct from the real-valued case. This is unexpected given that the non-existence of a multi-peak solution at a non-degenerate local minimum point of the electric potential has been proven in [25]. 2020 MSC 35J10, 35A01, 35B25