Qualitative analysis of solutions of obstacle elliptic inclusion problem with fractional Laplacian

Abstract

In this paper, we study an elliptic obstacle problem with a generalized fractional Laplacian and a multivalued operator which is described by a generalized gradient. Under quite general assumptions on the data, we employ a surjectivity theorem for multivalued mappings generated by the sum of a maximal monotone multivalued operator and a bounded multivalued pseudomonotone mapping to prove that the set of weak solutions to the problem is nonempty, bounded and closed. Then, we introduce a sequence of penalized problems without obstacle constraints. Finally, we prove that the Kuratowski upper limit of the sets of solutions to penalized problems is nonempty and is contained in the set of solutions to original elliptic obstacle problem, i.e., \(\emptyset \ne w\text{- }\limsup _{n\rightarrow \infty }{\mathcal {S}}_n=s\text{- }\limsup _{n\rightarrow \infty }{\mathcal {S}}_n\subset \mathcal S\).

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Acknowledgements

The authors wish to thank the two knowledgeable referees for their useful remarks in order to improve the paper. Project is supported by the NNSF of China Grants Nos. 12001478 and 12026256, the H2020-MSCA-RISE-2018 Research and Innovation Staff Exchange Scheme Fellowship within the Project No. 823731 CONMECH and National Science Center of Poland under Preludium Project No. 2017/25/N/ST1/00611. It is also supported by the Startup Project of Doctor Scientific Research of Yulin Normal University No. G2020ZK07, NSF of Guangxi Grants No. 2020JJB110001, and the International Project co-financed by the Ministry of Science and Higher Education of Republic of Poland under Grant No. 3792/GGPJ/H2020/2017/0.

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Correspondence to Shengda Zeng.

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Zeng, S., Cen, J., Atangana, A. et al. Qualitative analysis of solutions of obstacle elliptic inclusion problem with fractional Laplacian. Z. Angew. Math. Phys. 72, 30 (2021). https://doi.org/10.1007/s00033-020-01460-z

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Keywords

  • Elliptic obstacle problem
  • Fractional Laplace operator
  • Clarke generalized gradient
  • Surjectivity theorem
  • Kuratowski upper limit

Mathematics Subject Classification

  • 35J20
  • 35J25
  • 35J60
  • 35R11
  • 49Jxx