论文标题
本地/非本地$ p $ -laplacians的系统：特征值问题及其渐近限制为$ p \ to \ infty $
A System of Local/Nonlocal $p$-Laplacians: The Eigenvalue Problem and Its Asymptotic Limit as $p\to\infty$
论文作者
论文摘要
在这项工作中，给定（1，\ infty）$中的$ p \，我们证明了第一个eigenvalue $λ_p$及其相应的eigenVector $（u_p，v_p，v_p）$的存在和简单性，用于以下本地/非核心PDE PDE System \ begin {equination} \ begin {array} {rclcl}-Δ_pu +（-ux） \ frac {2β} {α+β}λ| u |^α| v |^{β-2} v＆\ mbox {in}＆ω u＆=＆0＆\ text {on}＆\ mathbb {r}^n \setMinusΩ v＆=＆0＆\ text {on}＆\ mathbb {r}^n \ setMinusω，\ end {array} \ right。 \ end {equation}其中$ω$$ \ subset $ $ $ \ mathbb {r}^n $是一个有限的开放域，$ 0 <r，s <1 $和$α（p）+β（p）= p $。此外，我们将渐近限制定为$ p \ to \ infty $，证明了相应的第一个$ \ infty- $ egenvalue的明确几何表征，即$λ_ {\ infty} $，以及对$ _p，v_p，v_p，v_p，v_p）的均匀融合$（u _ {\ infty}，v _ {\ infty}）$。最后，三重$（u _ {\ infty}，v _ {\ infty}，λ_ {\ infty}）$在粘度意义上验证了一个极限的PDE系统。
In this work, given $p\in (1,\infty)$, we prove the existence and simplicity of the first eigenvalue $λ_p$ and its corresponding eigenvector $(u_p,v_p)$, for the following local/nonlocal PDE system \begin{equation}\label{Eq0} \left\{ \begin{array}{rclcl} -Δ_p u + (-Δ)^r_p u & = & \frac{2α}{α+β}λ|u|^{α-2}|v|^βu & \mbox{in} & Ω\\ -Δ_p v + (-Δ)^s_p v& = & \frac{2β}{α+β}λ|u|^α|v|^{β-2}v & \mbox{in} & Ω u& =& 0&\text{ on } & \mathbb{R}^N \setminus Ω v& =& 0&\text{ on } & \mathbb{R}^N \setminus Ω, \end{array} \right. \end{equation} where $Ω$$\subset$ $\mathbb{R}^N$ is a bounded open domain, $0<r, s<1$ and $α(p)+β(p) = p$. Moreover, we address the asymptotic limit as $p \to \infty$, proving the explicit geometric characterization of the corresponding first $\infty-$eigenvalue, namely $λ_{\infty}$, and the uniformly convergence of the pair $(u_p,v_p)$ to the $\infty-$eigenvector $(u_{\infty},v_{\infty})$. Finally, the triple $(u_{\infty},v_{\infty},λ_{\infty})$ verifies, in the viscosity sense, a limiting PDE system.
