Concentration phenomena for the Schrödinger-Poisson system in ${\mathbb{R}}^2$

Abstract

We perform a semiclassical analysis for the planar Schrödinger-Poisson system \[ \cases{ -\varepsilon^{2} Δψ+V(x)ψ= E(x) ψ\quad \text{in ${\mathbb{R}}^2$},\cr -ΔE= |ψ|^{2} \quad \text{in ${\mathbb{R}}^2$}, \cr } \tag{$S{P}_{\epsilon }$} \] where $\epsilon$ is a positive parameter corresponding to the Planck constant and $V$ is a bounded external potential. We detect solution pairs $(u_{\epsilon },E_{\epsilon })$ of the system $(S{P}_{\epsilon })$ as~$\ge \to 0$.