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A note on Hopf’s lemma and strong minimum principle for nonlocal equations with non-standard growth

  • Abhrojyoti Sen EMAIL logo
Published/Copyright: May 6, 2023

Abstract

Let Ω R n be any open set and 𝑢 a weak supersolution of L u = c ( x ) g ( | u | ) u | u | , where

L u ( x ) = p.v. R n g ( | u ( x ) u ( y ) | | x y | s ) u ( x ) u ( y ) | u ( x ) u ( y ) | K ( x , y ) d y | x y | s

and g = G for some Young function 𝐺. This note imparts a Hopf type lemma and strong minimum principle for 𝑢 when c ( x ) is continuous in Ω ¯ that extend the results of Del Pezzo and Quaas [A Hopf’s lemma and a strong minimum principle for the fractional 𝑝-Laplacian, J. Differential Equations 263 (2017), 1, 765–778] in fractional Orlicz–Sobolev setting.

MSC 2010: 35R11; 47G20; 35D30; 35B50

Acknowledgements

The author is indebted to Anup Biswas and Saibal Khan for helpful discussions and suggestions. The author also thanks the referee for his/her careful reading of the manuscript and valuable suggestions.

  1. Communicated by: Christopher D. Sogge

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Received: 2022-11-05
Revised: 2023-02-22
Published Online: 2023-05-06
Published in Print: 2023-11-01

© 2023 Walter de Gruyter GmbH, Berlin/Boston

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