Proceedings of the Royal Society of Edinburgh Section A-Mathematics最新文献

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On a critical time-harmonic Maxwell equation in nonlocal media 关于非局部介质中的临界时谐麦克斯韦方程

IF 1.3 3区数学 Q1 Mathematics

Proceedings of the Royal Society of Edinburgh Section A-Mathematics

Pub Date : 2024-02-29 DOI: 10.1017/prm.2024.11

Minbo Yang, Weiwei Ye, Shuijin Zhang

In this paper, we study the existence of solutions for a critical time–harmonic Maxwell equation in nonlocal media

[ begin{cases} nablatimes(nablatimes u)+lambda u=left(I_{alpha}ast|u|^{2^{{ast}}_{alpha}}right)|u|^{2^{{ast}}_{alpha}-2}u & mathrm{in} Omega, nutimes u=0 & mathrm{on} partialOmega, end{cases} ]

where

$Omega subset mathbb {R}^{3}$

is a bounded domain, either convex or with

$mathcal {C}^{1,1}$

boundary,

$nu$

is the exterior normal,

$lambda <0$

is a real parameter,

$2^{ast }_{alpha }=3+alpha$

with

$0<alpha <3$

is the upper critical exponent due to the Hardy–Littlewood–Sobolev inequality. By introducing some suitable Coulomb spaces involving curl operator

$W^{alpha,2^{ast }_{alpha }}_{0}(mathrm {curl};Omega )$

, we are able to obtain the ground state solutions of the curl–curl equation via the method of constraining Nehari–Pankov manifold. Correspondingly, some sharp constants of the Sobolev-like inequalities with curl operator are obtained by a nonlocal version of the con

本文研究了非局部介质中临界时谐麦克斯韦方程的解的存在性。nablatimes(nablatimes u)+lambda u=left(I_{alpha}ast|u|^{2^{{ast}}_{alpha}}right)|u|^{2^{{ast}}_{alpha}-2}u &mathrm{in}Omega,nutimes u=0 & mathrm{on}partialOmega, end{cases}.其中 $Omega subset mathbb {R}^{3}$ 是一个有界域，要么是凸域，要么是有 $mathcal {C}^{1,1}$ 边界的域，$nu$ 是外部法线，$lambda <；0$ 是一个实数参数，$2^{ast }_{alpha }=3+alpha$ 中的 $0<alpha <3$ 是由于哈代-利特尔伍德-索博列夫不等式产生的上临界指数。通过引入涉及卷曲算子 $W^{alpha,2^{ast }_{alpha }}_{0}(mathrm {curl};Omega )$ 的一些合适的库仑空间，我们能够通过约束奈哈里-潘科夫流形的方法得到卷曲-卷曲方程的基态解。相应地，通过非局部版的集中-紧凑性原理，我们得到了带卷曲算子的索波列夫不等式的一些尖锐常数。

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