Multiple solutions for nonhomogeneous generalized quasilinear Kirchhoff-Schrödinger-Poisson system

In this paper, we consider a class of modified Schrödinger-Poisson system with Kirchhoff-type perturbation by use of variational methods: 0.1$$ \left \{ \textstyle\begin{array}{l@{\quad}l} \bigg(1+b\int _{\mathbb{R}^{3}}g^{2}(u)|\nabla u|^{2}dx\bigg)[- \textrm{div}(g^{2}(u)\nabla u)+g(u)g'(u)|\nabla u|^{2}] \ + V(x)u + \phi u=h(u)+{\vartheta (x)},&x\in \mathbb{R}^{3}, \ -\Delta \phi =u^{2},&x\in \mathbb{R}^{3}, \end{array}\displaystyle \right . $$ where $b\geq 0$, $g\in C^{1}(\mathbb{R},\,\mathbb{R}^{+})$, V and h are continuous functions, $\vartheta \not \equiv 0$. There are three major ingredients in two cases. Firstly, for $\vartheta =0$, we demonstrated the system (0.1) has a nontrivial ground state solution for $h(u)=\lambda f(u)+g(u)G^{5}(u)$, where $G(u)=\int _{0}^{u}g(t)dt$. Moreover, when nonlinear term $h(u)$ satisfies appropriate assumptions, a sequence of weak solutions were obtained by Clark’s theorem. Finally, for $\vartheta \neq 0$, with the help of Jeanjean theorem and cut-off function, at least two positive solutions of (0.1) were gained. To the best of our knowledge, this paper is one of the first contribution to study the nonhomogeneous generalized Kirchhoff-Schrödinger-Poisson system.