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Maximum principles and qualitative properties of solutions for nonlocal double phase operator

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Abstract

In this paper, we are concerned with the following nonlocal double phase problems with a gradient term:

$$\begin{aligned} {\mathcal {L}}u(x)=f(x,u,\nabla u), \end{aligned}$$

where \({\mathcal {L}}\) is a nonlocal double phase operator. We first establish various maximum principles for nonlocal double phase operators in bounded or unbounded domains. Together these maximum principles with the direct method of moving planes and direct sliding methods, we further derive qualitative properties of solutions such as Liouville type theorem, monotonicity, symmetry and uniqueness results for solutions to the nonlocal double phase problems in bounded domains, unbounded domains, epigraph, and \({\mathbb {R}}^{n}\) respectively. We believe that the new ideas and methods employed here can be conveniently applied to study a variety of nonlinear elliptic problems involving other nonlocal operators.

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Acknowledgements

The authors are grateful to the anonymous referees for their careful reading and valuable comments and suggestions that improved the presentation of the paper.

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Authors and Affiliations

  1. Department of Mathematical Science, Tsinghua University, Beijing, People’s Republic of China

    Yichen Hu

  2. Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, People’s Republic of China

    Shaolong Peng

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  1. Yichen Hu
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Shaolong Peng is supported by the National Natural Science Foundation of China (No. 11971049).

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Hu, Y., Peng, S. Maximum principles and qualitative properties of solutions for nonlocal double phase operator. Math. Z. 306, 9 (2024). https://doi.org/10.1007/s00209-023-03405-4

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