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

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

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