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Critical fractional elliptic equations with exponential growth

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Abstract

In this paper we establish, using variational methods combined with the Moser–Trudinger inequality, existence and multiplicity of weak solutions for a class of critical fractional elliptic equations with exponential growth without a Ambrosetti–Rabinowitz-type condition. The interaction of the nonlinearities with the spectrum of the fractional operator will be used to study the existence and multiplicity of solutions. The main technical result proves that a local minimum in \(C_{s}^0 (\overline{\Omega })\) is also a local minimum in \(W^{s,p}_0\) for exponentially growing nonlinearities.

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Acknowledgements

The authors would like to thank A. Iannizzotto for fruitful discussions with respect to Theorem 1. H. P. Bueno takes part in the project 422806/2018-8 by CNPq/Brazil. E. H. Caqui was supported by CAPES/Brazil. O. H. Miyagaki was supported by Grant 2019/24901-3 by São Paulo Research Foundation (FAPESP) and Grant 307061/2018-3 by CNPq/Brazil.

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Authors and Affiliations

  1. Department of Matematics, Universidade Federal de Minas Gerais, 31270-901, Belo Horizonte, MG, Brazil

    Hamilton P. Bueno & Eduardo Huerto Caqui

  2. Department of Mathematics, Universidade Federal de São Carlos, 13565-905, São Carlos, SP, Brazil

    Olimpio H. Miyagaki

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  1. Hamilton P. Bueno
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Correspondence to Hamilton P. Bueno.

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Bueno, H.P., Caqui, E.H. & Miyagaki, O.H. Critical fractional elliptic equations with exponential growth. J Elliptic Parabol Equ 7, 75–99 (2021). https://doi.org/10.1007/s41808-021-00095-z

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