In this paper, we consider the Choquard equations with a local perturbation $-\mathrm{\Delta }u+u=\left({\int }_{{\mathbb{R}}^N}\frac{|u\left(y\right){|}^2}{|x-y{|}^{N-\alpha }}\mathrm{d}y\right)u+|u{|}^{q-2}u\text{in}{\mathbb{R}}^N,$ where $N\ge 3,\alpha \in \left(\right(N-4{)}^{+},N),q\in (2,2N/(N-2))$. By using variational method and approximating approach, we prove that for any given positive integer k, the above equation has a least energy radial solution changing sign exactly k times. This solution is constructed as the limit of such solutions for the following Choquard equations $-\mathrm{\Delta }u+u=\left({\int }_{{\mathbb{R}}^N}\frac{|u\left(y\right){|}^p}{|x-y{|}^{N-\alpha }}\mathrm{d}y\right)|u{|}^{p-2}u+|u{|}^{q-2}u\text{in}{\mathbb{R}}^N,$ as $p\searrow 2_{+}$. Our result improves and extends the previous results in the literature.

中文翻译：

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有微扰的二次Choquard方程的多节点解

在本文中，我们考虑具有局部扰动的 Choquard 方程 $-\mathrm{\Delta }你+你=\left({\int }_{{\mathbb{电阻}}^N}\frac{|你\left(是\right){|}^2}{|X-是{|}^{N-\alpha }}\mathrm{d}是\right)你+|你{|}^{q-2}你\text{在}{\mathbb{电阻}}^N,$ 在哪里 $N\ge 3,\alpha \in \left(\right(N-4{)}^{+},N),q\in (2,2N/(N-2))$. 通过使用变分法和逼近法，我们证明了对于任意给定的正整数k，上述方程有一个最小能量径向解的变化符号恰好为k次。此解被构造为以下 Choquard 方程的此类解的极限$-\mathrm{\Delta }你+你=\left({\int }_{{\mathbb{电阻}}^N}\frac{|你\left(是\right){|}^{磷}}{|X-是{|}^{N-\alpha }}\mathrm{d}是\right)|你{|}^{磷-2}你+|你{|}^{q-2}你\text{在}{\mathbb{电阻}}^N,$ 作为 $磷\searrow 2_{+}$. 我们的结果改进并扩展了之前文献中的结果。