Existence and concentration of solution for Schrödinger-Poisson system with local potential

Zhipeng Yang () and Yuanyang Yu ()
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Zhipeng Yang: Georg-August-University of Göttingen
Yuanyang Yu: University of Chinese Academy of Sciences

Partial Differential Equations and Applications, 2021, vol. 2, issue 4, 1-22

Abstract: Abstract In this paper, we study the following nonlinear Schrödinger-Poisson type equation $$\begin{aligned} {\left\{ \begin{array}{ll} -\varepsilon ^2\Delta u+V(x)u+K(x)\phi u=f(u)&{}\text {in}\ {\mathbb {R}}^3,\\ -\varepsilon ^2\Delta \phi =K(x)u^2&{}\text {in}\ {\mathbb {R}}^3, \end{array}\right. } \end{aligned}$$ - ε 2 Δ u + V ( x ) u + K ( x ) ϕ u = f ( u ) in R 3 , - ε 2 Δ ϕ = K ( x ) u 2 in R 3 , where $$\varepsilon >0$$ ε > 0 is a small parameter, $$V: {\mathbb {R}}^3\rightarrow {\mathbb {R}}$$ V : R 3 → R is a continuous potential and $$K: {\mathbb {R}}^3\rightarrow {\mathbb {R}}$$ K : R 3 → 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 $$\varepsilon >0$$ ε > 0 small. Moreover, we summarize some open problems for the Schrödinger-Poisson system.

Keywords: Ground state solution; Concentration; Schrödinger-Poisson system; 35A15; 35B40; 35J20 (search for similar items in EconPapers)
Date: 2021
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DOI: 10.1007/s42985-021-00105-8

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