Planar Schrödinger–Poisson system with exponential critical growth: Local well-posedness and standing waves with prescribed mass

Abstract

This paper investigates a class of planar Schrödinger–Poisson systems with critical exponential growth. The conditions for the local well-posedness of the Cauchy problem in the energy space are defined. By introducing innovative ideas and relaxing some of the classical growth assumptions on nonlinearity, this study shows that such a system has at least two standing waves with a prescribed mass. One wave is a ground-state standing wave with positive energy, and another one is a high-energy standing wave with positive energy. In addition, with the help of the local well-posedness, it is shown that the set of ground-state standing waves is orbitally stable.