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On a critical time-harmonic Maxwell equation in nonlocal media

Part of: Singular integral equations Elliptic equations and systems Nonlinear integral equations

Published online by Cambridge University Press:  29 February 2024

Minbo Yang ,
Weiwei Ye  and
Shuijin Zhang
Minbo Yang
Affiliation:
Department of Mathematics, Zhejiang Normal University, Jinhua, Zhejiang 321004, People's Republic of China (mbyang@zjnu.edu.cn)
Weiwei Ye
Affiliation:
Department of Mathematics, Zhejiang Normal University, Jinhua, Zhejiang 321004, People's Republic of China Department of Mathematics, Fuyang Normal University, Fuyang, Anhui 236037, People's Republic of China (yeweiweime@163.com)
Shuijin Zhang
Affiliation:
Department of Mathematics, Zhejiang Normal University, Jinhua, Zhejiang 321004, People's Republic of China (shuijinzhang@zjnu.edu.cn)
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Abstract

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

\[ \begin{cases} \nabla\times(\nabla\times u)+\lambda u=\left(I_{\alpha}\ast|u|^{2^{{\ast}}_{\alpha}}\right)|u|^{2^{{\ast}}_{\alpha}-2}u & \mathrm{in}\ \Omega,\\ \nu\times u=0 & \mathrm{on}\ \partial\Omega, \end{cases} \]
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 concentration–compactness principle.

Keywords

time–harmonic Maxwell equationBrezis–Nirenberg problemnonlocal nonlinearitycoulomb spacesharp constant

MSC classification

Primary: 35J15: Second-order elliptic equations
Secondary: 45E10: Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type) 45G05: Singular nonlinear integral equations
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , First View , pp. 1 - 45
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Copyright
Copyright © The Author(s), 2024. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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