Abstract
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.
