Non-autonomous Schrödinger-Poisson system in   $\mathbb{R}^{3}$

Juntao Sun; Tsung-Fang Wu; Zhaosheng Feng

doi:10.3934/dcds.2018077

Article Contents

2018, Volume 38, Issue 4: 1889-1933. Doi: 10.3934/dcds.2018077

This issue Previous Article Spectral asymptotics of one-dimensional fractal Laplacians in the absence of second-order identities Next Article Mean-square almost automorphic solutions for stochastic differential equations with hyperbolicity

Non-autonomous Schrödinger-Poisson system in $\mathbb{R}^{3}$

1.
School of Mathematics and Statistics, Shandong University of Technology Zibo 255049, China
2.
School of Mathematical Sciences, Qufu Normal University, Qufu 273165, China
3.
Department of Applied Mathematics, National University of Kaohsiung, Kaohsiung 811, Taiwan
4.
School of Mathematical and Statistical Sciences, University of Texas-Rio Grande Valley, Edinburg, Texas 78539, USA

* Corresponding author
* Corresponding author

Received: November 2016

Revised: October 2017

Published: July 2018

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Abstract

We study the existence of positive solutions for the non-autonomous Schrödinger-Poisson system:

$\left\{ {\begin{array}{*{20}{l}} { - \Delta u + u + \lambda K\left( x \right)\phi u = a\left( x \right){{\left| u \right|}^{p - 2}}u}&{{\text{in }}{\mathbb{R}^3},} \\ { - \Delta \phi = K\left( x \right){u^2}}&{{\text{in }}{\mathbb{R}^3},} \end{array}} \right.$

where $\lambda >0$, $2 < p \le 4$ and both $K\left( x\right) $ and $a\left( x\right) $ are nonnegative functions in $\mathbb{R}^{3}$, which satisfy the given conditions, but not require any symmetry property. Assuming that $% \lim_{\left\vert x\right\vert \rightarrow \infty }K\left( x\right) = K_{\infty }\geq 0$ and $\lim_{\left\vert x\right\vert \rightarrow \infty }a\left( x\right) = a_{\infty }>0$, we explore the existence of positive solutions, depending on the parameters $\lambda$ and $p$. More importantly, we establish the existence of ground state solutions in the case of $3.18 \approx \frac{{1 + \sqrt {73} }}{3} < P \le 4$.

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Mathematics Subject Classification: Primary: 35J20, 35J61, 35A01; Secondary: 35B40.

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