This paper is dedicated to investigating the existence of sign-changing solutions to the following Schrödinger–Poisson system $\left\{\begin{array}{cc}-\Delta u+V\left(x\right)u+\lambda \varphi u=f\left(u\right)\hfill & \text{in}{\mathbb{R}}^3,\hfill \\ -\Delta \varphi =u^2\hfill & \text{in}{\mathbb{R}}^3,\hfill \end{array}\right.$where $\lambda >0$ and $f$ is subquadratic or quadratic at infinity. By using the method of invariant sets of descending flow, we prove that the problem above admits multiple radial sign-changing solutions in the subquadratic case as $\lambda$ small. As for the quadratic case, by virtue of a perturbation argument, we show that this problem admits at least one sign-changing solution as $\lambda$ small.

中文翻译：

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具有无限二次或二次增长的非线性Schrödinger-Poisson系统的符号转换解决方案

本文致力于调查以下Schrödinger-Poisson系统的符号转换解决方案的存在 $\left\{\begin{array}{cc}-\Delta ü+V（X）ü+\lambda \varphi ü=F（ü）\hfill & \text{在}{\mathbb{[R}}^3，\hfill \\ -\Delta \varphi ={ü}^2\hfill & \text{在}{\mathbb{[R}}^3，\hfill \end{array}\right.$哪里 $\lambda >0$ 和 $F$是次二次或二次于无限远。通过使用下降流不变集的方法，我们证明了上述问题在次二次情况下允许多个径向符号改变解为$\lambda$小。对于二次情况，通过摄动论证，我们证明该问题至少允许一个符号转换解为$\lambda$ 小。