Abstract We consider the following Choquard equation: { - Δ ⁢ u = ( ∫ Ω | u ⁢ ( y ) | 2 μ * | x - y | μ ⁢ 𝑑 y ) ⁢ | u | 2 μ * - 2 ⁢ u in ⁢ Ω , u = 0 on ⁢ ∂ ⁡ Ω , \left\{\begin{aligned} \displaystyle-\Delta u&\displaystyle=\Bigg{(}\int_{% \Omega}\frac{|u(y)|^{2^{*}_{\mu}}}{|x-y|^{\mu}}\,dy\Bigg{)}|u|^{2^{*}_{\mu}-2}% u&&\displaystyle\phantom{}\text{in }\Omega,\\ \displaystyle u&\displaystyle=0&&\displaystyle\phantom{}\text{on }\partial% \Omega,\end{aligned}\right. where Ω is a smooth bounded domain in ℝ N {\mathbb{R}^{N}} ( N ≥ 3 {N\geq 3} ), 2 μ * = 2 ⁢ N - μ N - 2 {2^{*}_{\mu}=\frac{2N-\mu}{N-2}} . This paper is concerned with the existence of a positive high-energy solution of the above problem in an annular-type domain when the inner hole is sufficiently small.

中文翻译：

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涉及 Choquard 非线性的非局部方程的 Coron 问题

摘要 我们考虑以下 Choquard 方程：{ - Δ ⁢ u = ( ∫ Ω | u ⁢ ( y ) | 2 μ * | x - y | μ ⁢ 𝑑 y ) ⁢ | 你| 2 μ * - 2 ⁢ u in ⁢ Ω , u = 0 on ⁢ ∂ ⁡ Ω , \left\{\begin{aligned} \displaystyle-\Delta u&\displaystyle=\Bigg{(}\int_{% \Omega} \frac{|u(y)|^{2^{*}_{\mu}}}{|xy|^{\mu}}\,dy\Bigg{)}|u|^{2^{* }_{\mu}-2}% u&&\displaystyle\phantom{}\text{in }\Omega,\\ \displaystyle u&\displaystyle=0&&\displaystyle\phantom{}\text{on }\partial% \Omega ,\end{对齐}\right。其中 Ω 是 ℝ N {\mathbb{R}^{N}} ( N ≥ 3 {N\geq 3} ) 中的光滑有界域，2 μ * = 2 ⁢ N - μ N - 2 {2^{* }_{\mu}=\frac{2N-\mu}{N-2}} 。本文关注的是当内孔足够小时，在环形域中上述问题的正高能解的存在性。