Author(s)

Qi Xue

School of Mathematics, Liaoning Normal University, Dalian, China.

This paper is devoted to the normalized solutions of a planar $L^2$ -critical Schrödinger-Poisson system with an external potential $V\left(x\right)={\left|x\right|}^2$ and inhomogeneous attractive interactions $K\left(x\right)\in \left(0,1\right)$ . Applying the constraint variational method, we prove that the normalized solutions exist if and only if the interaction strength $a$ satisfies $a\in \left(0,a^{\text{*}}\right):={\Vert Q\Vert }_{L^2\left({ℝ}^2\right)}^2$ , where $Q$ is the unique positive solution of $\text{Δ}u-u+u^3=0$ in ${ℝ}^2$ . Particularly, the refined limiting behavior of positive minimizers is also analyzed as $a\nearrow a^{\text{*}}$ .

Schrödinger-Poisson System, Logarithmic Convolution, Inhomogeneous Attractive Interaction, Normalized Solution

Xue, Q. (2025) Normalized Solutions for a Planer Schrödinger-Poisson System with Inhomogeneous Attractive Interactions. Open Access Library Journal, 12, 1-1. doi: 10.4236/oalib.1113315.