Electron. J. Differential Equations, Vol. 2016 (2016), No. 277, pp. 1-21.

Nodal solutions for Schrödinger-Poisson type equations in R3

Jin Deng, Jianfu Yang

Abstract:
In this article, we consider the existence of nodal solutions for the Schrodinger-Poisson type problem
$$\displaylines{ -\Big(a+b\int_{\mathbb{R}^3}|\nabla u|^2\,dx\Big)\Delta u+V(|x|)u+\varphi u =|u|^{p-2}u,\quad\text{in }\mathbb{R}^3, \cr -\Delta\varphi=u^2, \quad \lim_{|x|\to \infty}\varphi (x) =0, }$$
where a and b are positive constants, $p\in(4,6)$ and V(x) is a radial smooth function. For each $k\in \mathbb{N}_+$, we show the existence of to nodal solution changing sign exactly k times.

Submitted August 30, 2016. Published October 18, 2016.
Math Subject Classifications: 35J20, 35J25, 35J61.
Key Words: Schrödinger-Poisson equation; Kirchhoff type problem; nodal solution.

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  Jin Deng
Department of Mathematics
Jiangxi Normal University
Nanchang, Jiangxi 330022, China
email: jindeng_2016@126.com
Jianfu Yang
Department of Mathematics
Jiangxi Normal University
Nanchang, Jiangxi 330022, China
email: jfyang_2000@yahoo.com

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