Solutions of a quasilinear Schrödinger–Poisson system with linearly bounded nonlinearities

In this paper, we are concerned with the following quasilinear Schrödinger–Poisson system \(\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u+V(x)u+ K(x)\phi u=f(x,u),\quad &{}x\in {\mathbb {R}}^3,\\ -\Delta \phi -\varepsilon ^4\Delta _4\phi = K(x) u^2, &{}x\in {\mathbb {R}}^3, \end{array}\right. } \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{cc}-\mathrm{\Delta }u+V(x)u+K(x)\varphi u=f(x,u),\hfill & x\in {\mathbb{R}}^3,\hfill \\ -\mathrm{\Delta }\varphi -{\epsilon }^4{\mathrm{\Delta }}_4\varphi =K(x)u^2,\hfill & x\in {\mathbb{R}}^3,\hfill \end{array}\right)\end{array}$$ where \(\varepsilon \) $\epsilon$ is a positive parameter and f is linearly bounded in u at infinity. Under suitable assumptions on V, K and f, we establish the existence and asymptotic behavior of ground state solutions to the system. We prove that they converge to the solutions of the classic Schrödinger–Poisson system associated as \(\varepsilon \) $\epsilon$ tends to zero.