Abstract
In this research, we analyze the existence and multiplicity of nonnegative solutions for a class of non-local elliptic problems with Dirichlet boundary conditions. The nonlinearity of the problem, in general, does not satisfy the Ambrosetti–Rabinowitz condition and is characterized by a concave–convex variable exponent function, exhibiting critical behavior at infinity. Using minimization arguments and Lebourg’s mean value theorem, and applying Ekeland’s variational principle together with the inverse function theorem, we obtain a ground state solution to the non-local elliptic problem in appropriate fractional Musielak spaces. Our main results generalize some recent findings in the literature to non-smooth cases.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Data availability
No data were used to support this study.
References
Azroul, E., Benkirane, A., Shimi, M., Srati, M.: On a class of nonlocal problems in new fractional Musielak–Sobolev spaces. Appl. Anal. 101(6), 1933–1952 (2022)
Azroul, E., Benkirane, A., Shimi, M., Srati, M.: Embedding and extension results in fractional Musielak–Sobolev spaces. Appl. Anal. (2020). https://doi.org/10.1080/00036811.2021.1948019
Bahrouni, S., Ounaies, H., Tavares, L.S.: Basic results of fractional Orlicz–Sobolev space and applications to non-local problems (2019)
Chadli, L.S., El-Houari, H., Moussa, H.: Multiplicity of solutions for nonlocal parametric elliptic systems in fractional Orlicz–Sobolev spaces. J. Ellipt. Parabol. Equ. 9, 1131–1164 (2023)
da Silva, E.D., Carvalho, M.L.M., Goncalves, J.V., Goulart, C.: Critical quasilinear elliptic problems using concave–convex nonlinearities. Ann. Mat. Pura Appl. (1923) 198(3), 693–726 (2019)
Di Nezza, E., Palatucci, G., Valdinoci, E.: Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136(5), 521–573 (2012)
Diening, L., Harjulehto, P., Hästö, P., Ruzicka, M.: Lebesgue and Sobolev Spaces with Variable Exponents. Springer, Berlin (2011)
El-Houari, H., Chadli, L.S., Moussa, H.: Existence of a solution to a nonlocal Schrödinger system problem in fractional modular spaces. Adv. Oper. Theory 7(1), 1–30 (2022)
El-houari, H., Moussa, H., Chadli, L.S.: A class of elliptic inclusion in fractional Orlicz–Sobolev spaces. Complex Var. Ellipt. Equ. (2022). https://doi.org/10.1080/17476933.2022.2159955
El-Houari, H., Chadli, L.S., Moussa, H.: A class of non local elliptic system in non reflexive fractional Orlicz–Sobolev spaces. Asian-Eur. J. Math. 16, 2350114 (2023)
El-Houari, H., Moussa, H., Chadli, L.S.: Ground state solutions for a nonlocal system in fractional Orlicz–Sobolev spaces. Int. J. Differ. Equ. 2022, 3849217 (2022)
El-Houari, H., Sabiki, H., Hicham, M.: Multiplicity and concentration properties of solutions for double-phase problem in fractional modular spaces (in preparation)
Fan, X., Zhao, D.: On the spaces \(L^{p(x)}(Q)\) and \(W^{m, p(x)}(Q)\). J. Math. Anal. Appl. 263(2), 424–446 (2001)
Gossez, J.P.: Orlicz–Sobolev spaces and nonlinear elliptic boundary value problems. Nonlinear analysis, function spaces and applications (Proc. Spring School, Horni Bradlo, 1978). Teubner, Leipzig, pp. 59–94 (1979)
Hamza, E.H., Chadli, L.S., Moussa, H.: Existence of ground state solutions of elliptic system in fractional Orlicz–Sobolev spaces. Results Nonlinear Anal. 5(2), 112–130 (2022)
Krasnosel’skii, M.A., Rutickii, Y.B.: Convex Functions and Orlicz Spaces, vol. 9. Noordhoff, Groningen (1961)
Molica Bisci, G., Servadei, R.: A Brezis–Nirenberg splitting approach for nonlocal fractional equations. Nonlinear Anal. Theory Methods Appl. 119, 341–353 (2015)
Musielak, J.: Orlicz Spaces and Modular Spaces, vol. 1034. Springer, Berlin (2006)
Saoudi, K.: A critical fractional elliptic equation with singular nonlinearities. Fract. Calc. Appl. Anal. 20(6), 1507–1530 (2017)
Acknowledgements
We would like to express our special thanks to the Editor-in-Chief and the anonymous referees for their valuable suggestions and advisory comments improving the quality of this paper.
Funding
Not applicable.
Author information
Authors and Affiliations
Contributions
All authors of this manuscript contributed equally to this work.
Corresponding author
Ethics declarations
Conflicts of Interest
The authors declare that there are no conflicts of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
El-Houari, H., Hicham, M., Kassimi, S. et al. Fractional Musielak spaces: a class of non-local problem involving concave–convex nonlinearity. J Elliptic Parabol Equ (2023). https://doi.org/10.1007/s41808-023-00252-6
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s41808-023-00252-6
Keywords
- Fractional Musielak spaces
- Minimization arguments
- Ekeland’s variational principle
- Inverse function theorem
- Ground state solution