In this paper we are interested in the existence of ground state sign-changing solutions to the following Schrödinger–Poisson system $\left\{\begin{array}{cc}-\Delta u+\varphi u=\lambda u+\mu {\left|u\right|}^{2}u\hfill & \text{in}\Omega \hfill \\ -\Delta \varphi =u^2\hfill & \text{in}\Omega \hfill \\ u=\varphi =0\hfill & \text{on}\partial \Omega ,\hfill \\ \hfill \end{array}\right.$where $\Omega$ is a bounded smooth domain of ${\mathbb{R}}^3$, $\mu >0$, $\lambda <{\lambda }_1$ and ${\lambda }_1$ is the first eigenvalue of $\left(-\Delta ,H_0^1\left(\Omega \right)\right)$. Using construction technique, we show that the above system possesses one ground state sign-changing solution if and only if $\mu >0$, which improves the recent results by Khoutir (2021).

中文翻译：

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有界域上具有三次非线性的薛定谔-泊松系统的基态符号变化解的充分必要条件

在本文中，我们对以下薛定谔-泊松系统的基态符号变化解的存在感兴趣 $\left\{\begin{array}{cc}-\Delta 你+\phi 你=\lambda 你+\mu {\left|你\right|}^2你\hfill & \text{在}\Omega \hfill \\ -\Delta \phi ={你}^2\hfill & \text{在}\Omega \hfill \\ 你=\phi =0\hfill & \text{在}\partial \Omega ,\hfill \\ \hfill \end{array}\right.$在哪里 $\Omega$ 是一个有界光滑域 ${\mathbb{电阻}}^3$, $\mu >0$, $\lambda <{\lambda }_1$ 和 ${\lambda }_1$ 是的第一个特征值 $\left(-\Delta ,H_0^1\left(\Omega \right)\right)$. 使用构造技术，我们证明上述系统具有一个基态符号变化解当且仅当$\mu >0$，这改进了 Khoutir (2021) 最近的结果。