Neumann problem driven by discontinuous nonlinearity

Jinxia Cen; Debajyoti Choudhuri; Van Thien Nguyen

doi:10.3934/dcdss.2025019

Article Contents

2026, Volume 23: 99-109. Doi: 10.3934/dcdss.2025019

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Neumann problem driven by discontinuous nonlinearity

1.
Center for Applied Mathematics of Guangxi, Guangxi Colleges and Universities, Key Laboratory of Complex System Optimization and Big Data Processing, Yulin Normal University, Yulin, 537000, Guangxi, China
2.
Department of Mathematics, School of Basic Sciences, Indian Institute of Technology Bhubaneswar, Khordha-752050, Odisha, India
3.
Departement of Mathematics, FPT University, Education zone, Hoa Lac high tech park, Km29 Thang Long highway, Thach That ward, Hanoi, Vietnam

^*Corresponding author: Debajyoti Choudhuri
^*Corresponding author: Debajyoti Choudhuri

Received: November 2024

Revised: January 2025

Early access: February 17, 2025

Published online: February 17, 2025

The first author is supported by [Natural Science Foundation of Guangxi Grant Nos. 2021GXNSFFA196004 and 2024GXNSFBA010337, the NNSF of China Grant No. 12371312, the Natural Science Foundation of Chongqing Grant No. CSTB2024NSCQ-JQX0033. It is also supported by the Systematic Project of Center for Applied Mathematics of Guangxi (Yulin Normal University) Nos. 2023CAM002 and 2023CAM003.]
The second author is supported by [National Baord for Higher Mathematics, Department of Atomic Energy (DAE) India, [02011/47/2021/NBHM(R.P.)/R & D II/2615].

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Abstract

This paper is devoted to the study of a class of elliptic $ p $-Laplacian Neumann problems (pLNP) involving a discontinuous power nonlinearity on a metric measure space. The main difficulties of this paper are twofold. On one hand, the framework of the studied problem is assumed in a metric measure space, rather than a Euclidean space. On the other hand, the presence of discontinuous nonlinearity leads to the invalidity of classical smooth variational methods. Under mild assumptions, we apply techniques from the theory of metric measure space and critical point theory of nonsmooth optimization to establish the nonemptyiness, convexity, and closedness of the solution set to the pLNP under consideration.

Keywords:

p-Laplacian Neumann problem,
metric measure space.

Mathematics Subject Classification: Primary: 35J35, 35J60, 35R11, 46E35, 35J75.

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