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A class of elliptic mixed boundary value problems with (p, q)-Laplacian: existence, comparison and optimal control

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Abstract

The paper deals with two nonlinear elliptic equations with (p, q)-Laplacian and the Dirichlet–Neumann–Dirichlet (DND) boundary conditions, and Dirichlet–Neumann–Neumann (DNN) boundary conditions, respectively. Under mild hypotheses, we prove the unique weak solvability of the elliptic mixed boundary value problems. Then, a comparison and a monotonicity results for the solutions of elliptic mixed boundary value problems are established. We examine a convergence result which shows that the solution of (DND) can be approached by the solution of (DNN). Moreover, two optimal control problems governed by (DND) and (DNN), respectively, are considered, and an existence result for optimal control problems is obtained. Finally, we provide a result on asymptotic behavior of optimal controls and system states, when a parameter tends to infinity.

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Acknowledgements

The authors wish to thank the two knowledgeable referees for their useful remarks in order to improve the paper. This project has received funding from the NNSF of China Grant Nos. 12001478, 12026255, 12026256 and 12001463, the European Union’s Horizon 2020 Research and Innovation Programme under the Marie Skłodowska-Curie grant agreement No. 823731 CONMECH, National Science Center of Poland under Project No. 2021/41/B/ST1/01636, and the Startup Project of Doctor Scientific Research of Yulin Normal University No. G2020ZK07. It is also supported by Natural Science Foundation of Guangxi Grant Nos. 2021GXNSFFA196004, 2020GXNSFBA297137 and 2018GXNSFAA281353, and the Ministry of Science and Higher Education of Republic of Poland under Grants Nos. 4004/GGPJII/H2020/2018/0 and 440328/PnH2/2019.

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Authors and Affiliations

  1. Guangxi Colleges and Universities Key Laboratory of Complex System Optimization and Big Data Processing, Yulin Normal University, Yulin, 537000, People’s Republic of China

    Shengda Zeng

  2. Department of Mathematics, Nanjing University, Nanjing, 210093, Jiangsu, People’s Republic of China

    Shengda Zeng

  3. Faculty of Mathematics and Computer Science, Jagiellonian University in Krakow, ul. Lojasiewicza 6, 30348, Kraków, Poland

    Shengda Zeng

  4. College of Applied Mathematics, Chengdu University of Information Technology, Chengdu, 610225, Sichuan Province, People’s Republic of China

    Stanisław Migórski

  5. Chair of Optimization and Control, Jagiellonian University in Krakow, ul. Lojasiewicza 6, 30348, Kraków, Poland

    Stanisław Migórski

  6. Depto. Matemática, FCE, Universidad Austral, Paraguay 1950, S2000FZF, Rosario, Argentina

    Domingo A. Tarzia

  7. CONICET, Buenos Aires, Argentina

    Domingo A. Tarzia

  8. School of Mathematics and Computational Science, Xiangtan University, Xiangtan City, 411105, Hunan Province, People’s Republic of China

    Lang Zou

  9. Department of Mathematics, FPT University, Education Zone, Hoa Lac High Tech Park, Km29 Thang Long Highway, Thach That ward, Hanoi, Vietnam

    Van Thien Nguyen

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  1. Shengda Zeng
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  2. Stanisław Migórski
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  3. Domingo A. Tarzia
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  4. Lang Zou
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  5. Van Thien Nguyen
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Correspondence to Lang Zou.

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Zeng, S., Migórski, S., Tarzia, D.A. et al. A class of elliptic mixed boundary value problems with (p, q)-Laplacian: existence, comparison and optimal control. Z. Angew. Math. Phys. 73, 151 (2022). https://doi.org/10.1007/s00033-022-01789-7

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  • DOI: https://doi.org/10.1007/s00033-022-01789-7

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