The aim of this paper is establishing the existence of a nontrivial solution for the following quasilinear Schrödinger–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.

中文翻译：

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具有亚临界增长的拟线性渐近周期Schrödinger-Poisson系统

本文的目的是建立以下拟线性Schrödinger-Poisson系统非平凡解的存在：$$ \ left \ {\ textstyle \ begin {array} {l}-\ Delta u + V（x）uu \ Delta （u ^ {2}）+ K（x）\ phi（x）u = g（x，u），\ quad x \ in \ mathbb {R} ^ {3}，\\-\ Delta \ phi = K （x）u ^ {2}，\ quad x \ in \ mathbb {R} ^ {3}，\\ u \ in H ^ {1}（\ mathbb {R} ^ {3}），\ qquad u> 0，\ end {array} \ displaystyle \ right。$$，其中V，K，g是连续函数。为了克服由准线性项引起的技术难题，我们更改变量以确保应用山口定理解决上述问题的可行性。我们使用山口定理和集中紧凑性原理作为基本工具来获得系统在无穷大的渐近周期性条件下拥有的非平凡解。