Abstract

In this paper, we are concerned with the following quasilinear Schrödinger–Poisson system $$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u+V(x)u+ K(x)\phi u=f(x,u),\quad &{}x\in {\mathbb {R}}^3,\\ -\Delta \phi -\varepsilon ^4\Delta _4\phi = K(x) u^2, &{}x\in {\mathbb {R}}^3, \end{array}\right. } \end{aligned}$$ where \(\varepsilon \) 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 \(\varepsilon \) tends to zero.

中文翻译：

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具有线性有界非线性的拟线性薛定谔-泊松系统的解

摘要

在本文中，我们关注以下拟线性薛定谔-泊松系统$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u+V(x)u+ K(x)\ phi u=f(x,u),\quad &{}x\in {\mathbb {R}}^3,\\ -\Delta \phi -\varepsilon ^4\Delta _4\phi = K(x) u^2, &{}x\in {\mathbb {R}}^3, \end{array}\right。 } \end{aligned}$$其中\(\varepsilon \)是一个正参数，并且f在u 的无穷大处线性有界。在V、K和f的适当假设下，我们建立了系统基态解的存在性和渐近行为。我们证明它们收敛于经典薛定谔-泊松系统的解，因为\(\varepsilon \)趋于零。