Normalized Solutions for a Planer Schrödinger-Poisson System with Inhomogeneous Attractive Interactions

This paper is devoted to the normalized solutions of a planer 2 L -critical Schrdinger-Poisson system with an external potential ( )and inhomogeneous attractive interactions ( ) ( ). Applying the constraint variational method, we prove that the normalized solutions exist if and only if the interaction strength a satisfies ( ) ( ) 2 2 2 * 0, : L a a Q = , where Q is the unique positive solution of 3 0 u u u -+ = in 2 .Particularly, the refined limiting behavior of positive minimizers is also analyzed as * a a .