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

  • * Corresponding author: Xueqin Peng

    * Corresponding author: Xueqin Peng 

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

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  • 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.

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


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