This paper is concerned with the static Choquard equation $$\begin{eqnarray}-\unicode[STIX]{x1D6E5}u=\bigg(\frac{1}{|x|^{N-\unicode[STIX]{x1D6FC}}}\ast |u|^{p}\bigg)|u|^{p-2}u\quad \text{in }\mathbb{R}^{N},\end{eqnarray}$$ where $N,p>2$ and $\max \{0,N-4\}<\unicode[STIX]{x1D6FC}<N$. We prove that if $u\in C^{1}(\mathbb{R}^{N})$ is a stable weak solution of the equation, then $u\equiv 0$. This phenomenon is quite different from that of the local Lane–Emden equation, where such a result only holds for low exponents in high dimensions. Our result is the first Liouville theorem for Choquard-type equations with supercritical exponents and $\unicode[STIX]{x1D6FC}\neq 2$.

中文翻译：

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静态 CHOQUARD 方程的稳定解

本文关注静态 Choquard 方程$$\begin{eqnarray}-\unicode[STIX]{x1D6E5}u=\bigg(\frac{1}{|x|^{N-\unicode[STIX]{x1D6FC}}}\ast |u|^ {p}\bigg)|u|^{p-2}u\quad \text{in }\mathbb{R}^{N},\end{eqnarray}$$在哪里$N,p>2$和$\max \{0,N-4\}<\unicode[STIX]{x1D6FC}<N$. 我们证明如果$u\in C^{1}(\mathbb{R}^{N})$是方程的稳定弱解，则$u\equiv 0$. 这种现象与局部 Lane-Emden 方程的现象完全不同，后者的结果仅适用于高维中的低指数。我们的结果是具有超临界指数和超临界指数的 Choquard 型方程的第一个刘维尔定理$\unicode[STIX]{x1D6FC}\neq 2$.