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arXiv · 2008.08403

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

Denis Bonheure·Silvia Cingolani·Simone Secchi
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Abstract

We perform a semiclassical analysis for the planar Schr\"odinger-Poisson system \[ \cases{ -\varepsilon^{2} \Delta\psi+V(x)\psi= E(x) \psi \quad \text{in $\mathbb{R}^2$},\cr -\Delta E= |\psi|^{2} \quad \text{in $\mathbb{R}^2$}, \cr } \tag{$SP_\varepsilon$} \] 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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Denis Bonheure, Silvia Cingolani, Simone Secchi. 2020-08-19. Concentration phenomena for the Schr\"odinger-Poisson system in $\mathbb{R}^2$. https://arxiv.org/abs/2008.08403

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