Concentrations for nonlinear Schrödinger equations with magnetic potentials and constant electric potentials

Abstract

This article studies point concentration phenomena of nonlinear Schrödinger equations with magnetic potentials and constant electric potentials. The existing results show that a common magnetic field has no effect on the locations of point concentrations, as long as the electric potential is not a constant. This article finds out the role of the magnetic fields in the locations of point concentrations when the electric potential is a constant.

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

“…Kang and Wei [25] proved the nonexistence of multi-peak solutions tending to a non-degenerate minimum point of the electric potential V (x) for real-valued Schrödinger equation. Our paper [38] demonstrates for our problem that ∥B∥ F plays the similar role as the nonconstant V (x) in real-valued nonlinear Schrödinger problem. From this point, Theorem 1.2 presents a total different phenomenon compared to the real-valued case.…”

“…To find the mechanism of the magnetic field, an improved approximation solution is needed. According our result in [38], we introduce the following functions. Define…”

“…The magnetic Laplacian introduces a phase parameter for each peak. These phase parameters have a significant impact on the energy in non-simple peaks case, unlike the simple peak phenomena in [38]. To determine them, we require some geometric characteristic of the location, such as an approximate regular simplex.…”

“…Our recent work [38] has proven that if the Forbenius norm ∥B∥ F exibits local maximum or minimum points, or non-degenerate critical points, problem (3) will have a solution that concentrates near these critical points. Roughly speaking, in our case of constant electric potentials, ∥B∥ F serves a similar function as the potential V (x) does in the case of non-constant electric potentials.…”

“…Roughly speaking, in our case of constant electric potentials, ∥B∥ F serves a similar function as the potential V (x) does in the case of non-constant electric potentials. Moreover, it is worth noting that the peaks of the solution in [38] are simple. This paper aims to further explore the existence of a non-simple peak solution and examine the role of A. Interestingly, we discover several noteworthy differences in our analysis.…”