We are concerned with discussing the ground state solutions of the Choquard equation with the Hardy potentials and critical Sobolev exponent: $\left\{\begin{array}{l}-\mathrm{\Delta }u+\left(a-\frac{\mu }{|x{|}^2}\right)u=(I_{\alpha }\ast F(u\left)\right)f\left(u\right)+|u{|}^{2^{\ast }-2}u,x\in {\mathbb{R}}^N\setminus \left\{0\right\},\\ u\in H^1\left({\mathbb{R}}^N\right),\end{array}\right.$ where $N\ge 3$, $\alpha \in (0,N)$, $0\le \mu <\overline{\mu }:=\frac{(N-2{)}^2}{4}$, $I_{\alpha }$ is the Riesz potential, $2^{\ast }:=2N/(N-2)$ is the critical Sobolev exponent, and $f\in \mathcal{C}(\mathbb{R},\mathbb{R})$ satisfies neither the usual Ambrosetti–Rabinowitz type condition nor any monotonicity condition. Using some new variational and analytic techniques, we obtain a ground state solution of Poho$\breve{z}$aev type for the given problem.

中文翻译：

---

具有 Hardy 势和临界非线性的 Choquard 方程的基态解

我们关心的是讨论具有 Hardy 势和临界 Sobolev 指数的 Choquard 方程的基态解：$\left\{\begin{array}{l}-\mathrm{\Delta }你+\left(\mathrm{一个}-\frac{\mu }{|X{|}^2}\right)你=({我}_{\alpha }*F(你\left)\right)F\left(你\right)+|你{|}^{2^{*}-2}你,X\in {\mathbb{R}}^{ñ}\setminus \left\{0\right\},\\ 你\in H^1\left({\mathbb{R}}^{ñ}\right),\end{array}\right.$在哪里$ñ\ge 3$,$\alpha \in (0,ñ)$,$0\le \mu <\overline{\mu }:=\frac{(ñ-2{)}^2}{4}$,${我}_{\alpha }$是 Riesz 势，$2^{*}:=2ñ/(ñ-2)$是临界 Sobolev 指数，并且$F\in \mathcal{C}(\mathbb{R},\mathbb{R})$既不满足通常的 Ambrosetti-Rabinowitz 类型条件也不满足任何单调性条件。使用一些新的变分和解析技术，我们得到了 Poho 的基态解$\breve{z}$给定问题的 aev 类型。