Existence and Asymptotic Behavior of Solutions of a Nonhomogeneous Quasilinear Schrödinger–Poisson System

In this paper, we study the existence and asymptotic behaviour of solutions of the nonhomogeneous quasilinear Schrödinger–Poisson system $$\begin{cases} -\Delta u +V(x)u+\lambda \phi u=f(x, u)+g(x),&x\in \mathbb{R}^3 , \\ -\Delta \phi -\varepsilon^4 \Delta_4 \phi=\lambda u^2 ,&x \in \mathbb{R}^3, \end{cases}$$ where $$\lambda$$ and $$\varepsilon$$ are positive parameters, $$\Delta _4\phi =\operatorname{div}(|\nabla \phi|^2 \nabla \phi),$$ $$V$$ is a continuous and coercive potential function with positive infimum, and $$f$$ is a Carathéodory function defined on $$\mathbb{R}^3 \times \mathbb{R}$$ and satisfying the classic Ambrosetti–Rabinowitz condition. Under some suitable assumptions on $$V(x)$$ , $$f(x,u)$$ , and $$g(x)$$ , we obtain the existence of two different energy nontrivial solutions by use of variational methods and truncation technique for sufficiently small $$\lambda$$ and fixed $$\varepsilon$$ . Moreover, the asymptotic behaviour of these solutions is studied whenever $$\varepsilon$$ and $$\lambda$$ , respectively, tend to zero.