Existence of ground state solutions of elliptic system in Fractional Orlicz-Sobolev Spaces

Abstract

We employing a minimization arguments on appropriate Nehari manifolds, we obtain ground state solutions
for a non-local elliptic system driven by the fractional a(.)-Laplacian operator, with Dirichlet boundary
conditions type.

Keywords

• Ground state solutions.
• Fractional Orlicz-Sobolev Spaces
• Nehari manifolds and febring map aproach

References

1. [1] R.A. Adams, J.F. Fournier, Sobolev spaces, Second edition, Pure and Applied Mathematics (Amsterdam), Elsevier/Academic Press, Amsterdam, (2003).
2. [2] K. Adriouch, A. El Hamidi, The Nehari manifold for systems of nonlinear elliptic equations. Nonlinear Analysis: Theory, Methods Applications, 64(10), (2006), 2149-2167.
3. [3] G.A. Afrouzi, S. Heidarkhani, Existence of three solutions for a class of Dirichlet quasilinear ellip- tic systems involving the (p1,, pn)-Laplacian. Nonlinear Analysis: Theory, Methods Applications, 70(1) (2009), 135-143.
4. [4] K.B. Ali, M. Hsini, K. Kefi, N.T. Chung, On a nonlocal fractional p (.,.)-Laplacian problem with competing nonlinearities. Complex Analysis and Operator Theory, 13(3) (2019), 1377-1399.
5. [5] E. Azroul, A. Benkirane, & M. Srati, Nonlocal eigenvalue type problem in fractional Orlicz- Sobolev space. Advances in Operator Theory, 5(4) (2020), 1599-1617.
6. [6] E. Azroul, A. Benkirane, M. Srati, & M. Shimi, Existence of solutions for a nonlocal Kirchhoff type problem in Fractional Orlicz-Sobolev spaces. arXiv preprint arXiv:1901.05216, (2019).
7. [7] E. Azroul, A. Benkirane, M. Srati, Existence of solutions for a non-local type problem in Frac- tional Orlicz Sobolev Spaces, Adv. Oper. Theory (2020).
8. [8] E. Azroul, A. Benkirane, M. Srati, Mountain pass type solution for a nonlacal fractional a- Kirchhoff type problem J. Nonlinear Funct. Anal. 2020 (2020), Article ID 22.

9. [9] E. Azroul, A. Boumazourh, On a class of fractional systems with nonstandard growth conditions, J. PseudoDiffer. Oper. Appl. 11 (2020), 805-820.
10. [10] S. Bahrouni, H. Ounaies, L.S. Tavares, Basic results of fractional Orlicz-Sobolev space and applications to non-local problems, (2019).
11. [11] A. Boumazourh and M. Srati, Leray-Schauder’s solution for a nonlocal problem in a fractional Orlicz-Sobolev space. Moroccan J. of Pure and Appl. Anal. (MJPAA) doi : 10.2478/mjpaa-2020-0004 (2020) 42-52.
12. [12] J.F. Bonder, and A.M. Salort, Fractional order Orlicz-Sobolev Spaces, Journal of Functional Analysis, 2019, https://doi.org/10.1016/j.jfa.2019.04.003.
13. [13] L. Boccardo, D. Guedes de Figueiredo, Some remarks on a system of quasilinear elliptic equa- tions, NoDEA Nonlinear Differential Equations Appl., 9 (2002), 309-323.
14. [14] F.J.S.A. Corrˆ ea, M.L.M. Carvalho, Jose VA Goncalves, et al., Positive solutions of strongly nonlinear elliptic problems. Asymptotic Analysis, 2015, vol. 93, no 1-2, p. 1-20.
15. [15] F.J.S.A. Corrˆ ea, M.L.M. Carvalho, Jose VA Goncalves, et al., Sign changing solutions for quasi- linear superlinear elliptic problems. Quarterly Journal of Mathematics, 2017, vol. 68, no 2, p. 391-420.
16. [16] Y. Chen, S. Levine, M. Rao, Variable exponent linear growth functionals in image processing, SIAM J. Appl. Math., 66 (2006), 1383-1406.
17. [17] L. Diening, Theorical and numerical results for electrorheological fluids, Ph.D. thesis, University of Freiburg, Germany (2002)
18. [18] N. Fukagai, M. Ito, K. Narukawa, Positive solutions of quasilinear elliptic equations with critical Orlicz-Sobolev nonlinearity on RN, Funkcial. Ekvac., 49 (2006), 235-267.
19. [19] T.C. Halsey, Electrorheological fluids, Science, 258 (1992), 761-766. https://doi.org/10.1016/j.jfa.2019.04.003.
20. [20] El-houari Hamza, L.S. Chadli, and H. Moussa, Existence of Solution To M-Kirchhoff System Type, 2021 7th International Conference on Optimization and Applications (ICOA). IEEE, 2021.
21. [21] M.A. Krasnosel’skii, Y.B. Rutickii, Convex functions and Orlicz spaces (Vol. 9) (1961), Gronin- gen: Noordhoff.