This paper is concerned with the following fractional Hénon equation$\{\begin{array}{cc}{(-\mathrm{\Delta })}^{s}u=|x{|}^{\alpha }{u}^{p-1},\hfill & x\in \mathrm{\Omega },\hfill \\ u>0,\hfill & x\in \mathrm{\Omega },\hfill \\ u=0,\hfill & x\in {\mathbb{R}}^N\setminus \mathrm{\Omega },\hfill \end{array}$ where $s\in (0,1)$, $N>2s$, $\alpha >0$ and $\mathrm{\Omega }\subset {\mathbb{R}}^N$ is a bounded domain with the origin and smooth boundary, we consider the limiting behaviour of the ground state solution $u_p$ as $p\to 2_s^{⁎}=2N/(N-2s)$. Moreover, we prove that $u_p$ concentrates at the point farthest from the origin on the boundary of Ω. This work can be seen as a nonlocal analogue of the results of Cao and Peng [8].

中文翻译：

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分数阶 Hénon 方程基态解的渐近行为

本文涉及以下分数阶Hénon方程$\{\begin{array}{cc}{(-\mathrm{\Delta })}^{秒}你=|X{|}^{\alpha }{你}^{磷-1},\hfill & X\in \mathrm{\Omega },\hfill \\ 你>0,\hfill & X\in \mathrm{\Omega },\hfill \\ 你=0,\hfill & X\in {\mathbb{电阻}}^N\setminus \mathrm{\Omega },\hfill \end{array}$ 在哪里 $秒\in (0,1)$, $N>2秒$, $\alpha >0$ 和 $\mathrm{\Omega }\subset {\mathbb{电阻}}^N$ 是具有原点和平滑边界的有界域，我们考虑基态解的极限行为 ${你}_{磷}$ 作为 $磷\to 2_{秒}^{⁎}=2N/(N-2秒)$. 此外，我们证明${你}_{磷}$集中在Ω的边界上离原点最远的点。这项工作可以看作是曹和彭 [8] 结果的非局部模拟。