NON-SIMPLE CONCENTRATIONS FOR NONLINEAR MAGNETIC SCHRÖDINGER EQUATIONS WITH CONSTANT ELECTRIC POTENTIALS

The paper investigates the influence of magnetic fields on the non-simple concentration phenomenon in the complex-valued nonlinear Schrödinger equations with a constant electric potential (iε∇ + A)²u + u − |u|^p−1u = 0 in R^N. We demonstrate that a multi-peak solution always exists at a non-degenerate local maximum or minimum point of the Frobenius norm ∥B∥²_F, where B is the magnetic field generated from the magnetic potential A. Interestingly, the locations of peaks form a regular simplex near such a maximum point. It is also 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 [24].