Quasilinear asymptotically periodic Schrödinger–Poisson system with subcritical growth

The aim of this paper is establishing the existence of a nontrivial solution for the following quasilinear Schrodinger–Poisson system: $$ \left \{ \textstyle\begin{array}{l} -\Delta u+V(x)u-u\Delta(u^{2})+K(x)\phi(x)u=g(x, u),\quad x\in\mathbb {R}^{3}, \\ -\Delta\phi=K(x)u^{2}, \quad x\in\mathbb{R}^{3},\\ u\in H^{1}(\mathbb{R}^{3}),\qquad u>0, \end{array}\displaystyle \right . $$ where V, K, g are continuous functions. To overcome the technical difficulties caused by the quasilinear term, we change the variable to guarantee the feasibility of applying the mountain pass theorem to solve the above problems. We use the mountain pass theorem and the concentration–compactness principle as basic tools to gain a nontrivial solution the system possesses under an asymptotic periodicity condition at infinity.