Multiple semiclassical states for coupled Schrödinger-Poisson equations with critical exponential growth

In this paper, we study the multiplicity of positive solutions for a class of Schrödinger-Poisson systems. Working in a variational setting, we prove the existence and multiplicity of positive solutions for the system when the Plank's constant is small and the potential satisfies some suitable conditions. We show that the number of positive solutions depends on the profile of the potential and that each solution concentrates around its corresponding global minimum point of the potential in the semi-classical limit. We also study the exponential decay.