This paper deals with the following Kirchhoff–Schrödinger–Poisson system: $$ \textstyle\begin{cases} -(a+b\int _{\mathbb{R}^{3}} \vert \nabla u \vert ^{2}\,dx)\Delta u+V(x)u+\phi u=K(x)f(u)&\text{in } \mathbb{R}^{3}, \\ -\Delta \phi =u^{2}&\text{in } \mathbb{R}^{3}, \end{cases} $$ where a, b are positive constants, $K(x)$, $V(x)$ are positive continuous functions vanishing at infinity, and $f(u)$ is a continuous function. Using the Nehari manifold and variational methods, we prove that this problem has a least energy nodal solution. Furthermore, if f is an odd function, then we obtain that the equation has infinitely many nontrivial solutions.

中文翻译：

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潜在消失的Kirchhoff–Schrödinger–Poisson系统的最小能量节点解的存在

本文处理以下Kirchhoff–Schrödinger–Poisson系统：$$ \ textstyle \ begin {cases}-（a + b \ int _ {\ mathbb {R} ^ {3}} \ vert \ nabla u \ vert ^ { 2} \，dx）\ Delta u + V（x）u + \ phi u = K（x）f（u）＆\ text {in} \ mathbb {R} ^ {3}，\\-\ Delta \ phi = u ^ {2}＆\ text {in} \ mathbb {R} ^ {3}，\ end {cases} $$，其中a，b为正常数，$ K（x）$，$ V（x）$是无穷大的正连续函数，而$ f（u）$是连续函数。使用Nehari流形和变分方法，我们证明该问题具有最小能量节点解。此外，如果f是一个奇数函数，则我们得出该方程具有无限多个非平凡解。