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arXiv 1607.07580math.AP
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关于不定权分数次Hardy-Sobolev算子特征值的注记

A note on the eigenvalues of fractional Hardy-Sobolev operator with indefinite weight

Sarika Goyal

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30 pages
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摘要：

本文研究了非线性$p-$分数Hardy算子\begin{align*} (-Δ)_p^αu - μ\frac{|u|^{p-2}u}{|x|^{pα}} = λV(x) |u|^{p-2}u \; \text{in}\; Ω, \quad u = 0 \; \mbox{in}\; \mathbb{R}^n \setminusΩ, \end{align*}的特征值，其中$n>pα$，$p\geq2$，$α\in(0,1)$，$0\leq μ<C_{n,α,p}$和$Ω$是$\mathbb{R}^n$中的一个区域，Lipschitz边界包含$0$ .特别地，允许$Ω=\mathbb{R}^n$。权重函数$V$可以改变符号并且可以具有奇异点。我们还表明，最小的正特征值是简单的，它是唯一的非负本征函数相关联。证明了存在一个特征值序列$λ_k \to \infty$为$k\to\infty$ .

英文摘要：

In this article, we study the eigenvalue of nonlinear $p-$fractional Hardy operator \begin{align*} (-Δ)_p^αu - μ\frac{|u|^{p-2}u}{|x|^{pα}} = λV(x) |u|^{p-2}u \; \text{in}\; Ω, \quad u = 0 \; \mbox{in}\; \mathbb{R}^n \setminusΩ, \end{align*} where $n>pα$, $p\geq2$, $α\in(0,1)$, $0\leq μ<C_{n,α,p}$ and $Ω$ is a domain in $\mathbb{R}^n$ with Lipschitz boundary containing $0$. In particular, $Ω=\mathbb{R}^n$ is admitted. The weight function $V$ may change sign and may have singular points. We also show that the least positive eigenvalue is simple and it is unique associated to a non-negative eigenfunction. Moreover, we proved that there exists a sequence of eigenvalues $λ_k \to \infty$ as $k\to\infty$.

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