Consider the general Choquard equation$\{\begin{array}{cc}\hfill & {\mathrm{\Delta }}_{p}u+\left(\underset{{\mathbb{R}}^N}{\int }\frac{|u(y){|}^{p_{\alpha }^{⁎}}}{|x-y{|}^{\alpha }}\text{d}y\right)|u{|}^{p_{\alpha }^{⁎}-2}u=0,\text{in}{\mathbb{R}}^N,\hfill \\ \hfill & u\in {\mathcal{D}}^{1,p}({\mathbb{R}}^N),\hfill \end{array}$ where $1<p<N,0<\alpha <N,p<p_{\alpha }^{⁎}=\frac{(N-\frac{\alpha }{2})p}{N-p}$ and ${\mathrm{\Delta }}_{p}u=\text{div}(|\mathrm{\nabla }u{|}^{p-2}\mathrm{\nabla }u)$ is the p-Laplacian operator. By using the Wolff potential theory, we prove that for any solution u of the equation, there exists a constant c such that$|u(x)|\le c{(1+|x|)}^{-\frac{N-p}{p-1}},\text{ for }x\in {\mathbb{R}}^N.$ The decay estimate is sharp in the sense that the positive solution u satisfies$u(x)\ge c{(1+|x|)}^{-\frac{N-p}{p-1}}.$

中文翻译：

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一般 Choquard 方程解的急剧衰减估计

考虑一般 Choquard 方程$\{\begin{array}{cc}\hfill & {\mathrm{\Delta }}_p你+（\underset{{\mathbb{右}}^{氮}}{\int }\frac{|你（y）{|}^{p_{\alpha }^{⁎}}}{|X-y{|}^{\alpha }}\text{d}y）|你{|}^{p_{\alpha }^{⁎}-2}你=0,\text{在}{\mathbb{右}}^{氮},\hfill \\ \hfill & 你\epsilon {\mathcal{D}}^{1,p}（{\mathbb{右}}^{氮}）,\hfill \end{array}$在哪里$1<p<氮,0<\alpha <氮,p<p_{\alpha }^{⁎}=\frac{（氮-\frac{\alpha }{2}）p}{氮-p}$和${\mathrm{\Delta }}_p你=\text{分区}（|\mathrm{\nabla }你{|}^{p-2}\mathrm{\nabla }你）$是p -拉普拉斯算子。通过使用 Wolff 势理论，我们证明对于方程的任何解u ，都存在一个常数c使得$|你（X）|\le C{（1+|X|）}^{-\frac{氮-p}{p-1}},\text{ 为了 }X\epsilon {\mathbb{右}}^{氮}。$衰减估计是尖锐的，因为正解u满足$你（X）\ge C{（1+|X|）}^{-\frac{氮-p}{p-1}}。$