Existence and asymptotical behavior of positive solutions for the Schrödinger-Poisson system with double quasi-linear terms

Xueqin Peng; Gao Jia

doi:10.3934/dcdsb.2021134

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2022, Volume 27, Issue 4: 2325-2344. Doi: 10.3934/dcdsb.2021134

This issue Previous Article Existence of unstable stationary solutions for nonlinear stochastic differential equations with additive white noise Next Article On a generalized diffusion problem: A complex network approach

Existence and asymptotical behavior of positive solutions for the Schrödinger-Poisson system with double quasi-linear terms

Xueqin Peng^, and
Gao Jia

College of Science, University of Shanghai for Science and Technology, Shanghai, 200093, China

^* Corresponding author: Xueqin Peng
^* Corresponding author: Xueqin Peng

Received: January 2021

Revised: March 2021

Early access: April 2021

Published: April 2022

This research is supported by the National Natural Science Foundation of China, Grant No. 11171220

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Abstract

In this paper, we consider the following Schrödinger-Poisson system with double quasi-linear terms

$ \begin{equation*} \label{1.1} \begin{cases} -\Delta u+V(x)u+\phi u-\frac{1}{2}u\Delta u^2 = \lambda f(x,u),\; &\; {\rm{in}}\; \mathbb{R}^{3},\\ -\triangle\phi-\varepsilon^4\Delta_4\phi = u^{2},\; &\; {\rm{in}}\; \mathbb{R}^{3},\\ \end{cases} \end{equation*} $

where $ \lambda,\varepsilon $ are positive parameters. Under suitable assumptions on $ V $ and $ f $, we prove that the above system admits at least one pair of positive solutions for $ \lambda $ large by using perturbation method and truncation technique. Furthermore, we research the asymptotical behavior of solutions with respect to the parameters $ \lambda $ and $ \varepsilon $ respectively. These results extend and improve some existing results in the literature.

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Mathematics Subject Classification: Primary: 35J10, 35J60; Secondary: 35J20.

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