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Existence and concentration of solution for Schrödinger-Poisson system with local potential (2009.07272v2)
Published 15 Sep 2020 in math.AP
Abstract: In this paper, we study the following nonlinear Schr\"odinger-Poisson type equation \begin{equation*} \begin{cases} -\varepsilon2\Delta u+V(x)u+K(x)\phi u=f(u)&\text{in}\ \mathbb{R}3,\ -\varepsilon2\Delta \phi=K(x)u2&\text{in}\ \mathbb{R}3, \end{cases} \end{equation*} where $\varepsilon>0$ is a small parameter, $V: \mathbb{R}3\rightarrow \mathbb{R}$ is a continuous potential and $K: \mathbb{R}3\rightarrow \mathbb{R}$ is used to describe the electron charge. Under suitable assumptions on $V(x), K(x)$ and $f$, we prove existence and concentration properties of ground state solutions for $\varepsilon>0$ small. Moreover, we summarize some open problems for the Schr\"odinger-Poisson system.