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| In this paper, we consider the following Schr\"{o}dinger-Poisson system \begin{equation} \left\{\begin{matrix} -\Delta u+V(x)u+\phi u=\left(\int_{\mathbb{R}^3}\frac{1}{p}|u|^pdx\right)^{\frac{2}{p}}|u|^{p-2}u+g(x)|u|^{q-2}u,& \mbox{in}\ \mathbb{R}^3,\\ -\Delta\phi=u^2,& \mbox{in}\ \mathbb{R}^3,\hfill\label{0.1} \end{matrix}\right. \end{equation} where $1<q<2<p<6$ and the functions $V(x), g(x)$ satisfy the certain conditions. Using variational methods and invariant sets of descending flow, we prove that system (\ref{0.1}) possesses three nontrivial solutions of mountain pass type (one positive, one negative and one sign-changing) and infinitely many high-energy sign-changing solutions. |
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| Keywords:Basic mathematics; Schr\"{o}dinger-Poisson system; Sign-changing solution; Invariant sets; Descending flow \par |
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