Multiple positive solutions for nonhomogeneous Schr\"{o}dinger-Poisson system with Berestycki-Lions type conditions
首发时间:2020-01-23
Abstract:In this paper, we study the following Schr\"{o}dinger-Poisson system \begin{align*} \begin{cases} -\Delta u+\lambda\phi u=g(u)+h(x), &\mathrm{in}\ \mathbb{R}^{3},\\ -\Delta \phi=u^2, & \mathrm{in}\ \mathbb{R}^{3}, \end{cases} \end{align*}where $\lambda >0$ is a parameter, $h(x) \not \equiv0$. Under the Berestycki-Lions type conditions, we prove that there exists $\lambda_{0}>0$ such that the system has at least two positive radial solutions for $\lambda\in(0,\lambda_{0})$ by using variational methods.
keywords: Partial differential equation Nonhomogeneous Schr\"{o}dinger-Poisson system Variational methods Multiple positive solutions Berestycki-Lions type conditions
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带~Berestycki--Lions~型条件的非齐次~Schr\"{o}dinger--Poisson~系统的多个正解
摘要:在这篇文章中,我们研究了这样的~Schr\"{o}dinger--Poisson~系统\begin{align*} \begin{cases} -\Delta u+\lambda\phi u=g(u)+h(x), &\mathrm{in}\ \mathbb{R}^{3},\\ -\Delta \phi=u^2, & \mathrm{in}\ \mathbb{R}^{3}, \end{cases} \end{align*}$\lambda >0$~是一个参数,$h(x)\not\equiv0$。在~Berestycki-Lions~型条件下,通过变分法我们证明到存在~$\lambda_{0}>0$~使得当~$\lambda\in(0,\lambda_{0})$~时,系统至少有两个正的径向解。
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带~Berestycki--Lions~型条件的非齐次~Schr\"{o}dinger--Poisson~系统的多个正解
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