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

Abstract

<p style='text-indent:20px;'>In this paper, we consider the following Schrödinger-Poisson system with double quasi-linear terms</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \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*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ \lambda,\varepsilon $\end{document}</tex-math></inline-formula> are positive parameters. Under suitable assumptions on <inline-formula><tex-math id="M2">\begin{document}$ V $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M3">\begin{document}$ f $\end{document}</tex-math></inline-formula>, we prove that the above system admits at least one pair of positive solutions for <inline-formula><tex-math id="M4">\begin{document}$ \lambda $\end{document}</tex-math></inline-formula> large by using perturbation method and truncation technique. Furthermore, we research the asymptotical behavior of solutions with respect to the parameters <inline-formula><tex-math id="M5">\begin{document}$ \lambda $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M6">\begin{document}$ \varepsilon $\end{document}</tex-math></inline-formula> respectively. These results extend and improve some existing results in the literature.</p>