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Fractional elliptic problems with nonlinear gradient sources and measures

Abstract

In this manuscript we deal with existence/uniqueness and regularity issues of suitable weak solutions to nonlocal problems driven by fractional Laplace type operators. Different from previous researches, in our approach we consider gradient non-linearity sources with subcritical growth, as well as appropriated measures as sources and boundary datum. We provide an in-depth discussion on the notions of solutions involved together with existence/uniqueness results in different regimes and for different boundary value problems. Finally, this work extends previous ones by dealing with more general nonlocal operators, source terms and boundary data.

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Acknowledgements

We would like to thanks the referees for their useful corrections and suggestions. J.V. da Silva was partially supported by Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (PNPD/CAPES-UnB-Brazil) Grant No. 88887.357992/2019-00, CNPq-Brazil under Grant No. 310303/2019-2 and FONCyT - PICT-2018-03183. A.S. is supported by PICT 2017-0704, by Universidad Nacional de San Luis under grants PROIPRO 03-2418 and PROICO 03-1916. P. O. is supported by Proyecto Bienal B080 Tipo 1 (Res. 4142/2019-R).

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Correspondence to Pablo Ochoa.

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This article is dedicated to the memory of Prof. Ireneo Peral.

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da Silva, J.V., Ochoa, P. & Silva, A. Fractional elliptic problems with nonlinear gradient sources and measures. Rev Mat Complut 35, 485–514 (2022). https://doi.org/10.1007/s13163-021-00391-1

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  • DOI: https://doi.org/10.1007/s13163-021-00391-1

Keywords

  • Existence/regularity of solutions
  • Weak solutions
  • Nonlocal operators
  • Fractional Laplace

Mathematics Subject Classification

  • 35J61
  • 35R06
  • 35R11