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arXiv 2104.12940math.AP
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带洞半空间中一类非局部椭圆型问题正解的存在性

Existence of positive solution for a class of nonlocal elliptic problems in the half space with a hole

Xing Yi

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arXiv admin note: text overlap with arXiv:1812.04878 by other authors
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摘要：

本文研究了一类非局部椭圆型问题\begin{eqnarray}\label{eq:0.1} &&\left\{\begin{array}{l} (-Δ)^{s} u+u=|u|^{p-2} u \text { in } Ω_{r} \\ u \geq 0 \quad \text { in }Ω_{r} \text { and } u \neq 0 \\ u=0 \quad \mathbb{R}^{N} \backslash Ω_{r} \end{array}\right., \end{eqnarray}的解的存在性，其中\begin{eqnarray}\label{eq:0.1} &&\left\{\begin{array}{l} (-Δ)^{s} u+u=|u|^{p-2} u \text { in } Ω_{r} \\ u \geq 0 \quad \text { in }Ω_{r} \text { and } u \neq 0 \\ u=0 \quad \mathbb{R}^{N} \backslash Ω_{r} \end{array}\right., \end{eqnarray}，$Ω_{r}$是$\mathbb{R}^N$和$p \in\left(2,2_{s}^{*}\right) .$中带洞的半空间。主要的技术方法是基于变分和拓扑方法。

英文摘要：

This work concerns with the existence of solutions for the following class of nonlocal elliptic problems \begin{eqnarray}\label{eq:0.1} &&\left\{\begin{array}{l} (-Δ)^{s} u+u=|u|^{p-2} u \text { in } Ω_{r} \\ u \geq 0 \quad \text { in }Ω_{r} \text { and } u \neq 0 \\ u=0 \quad \mathbb{R}^{N} \backslash Ω_{r} \end{array}\right., \end{eqnarray} involving the fractional Laplacian operator $(-Δ)^{s},$ where $s \in(0,1), N>2 s$, $Ω_{r}$ is the half space with a hole in $\mathbb{R}^N$ and $p \in\left(2,2_{s}^{*}\right) .$ The main technical approach is based on variational and topological methods.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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