A Fractional Elliptic System With Strongly Coupled Critical Terms and Concave-Convex Nonlinearities

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DOI: https://doi.org/10.28924/2291-8639-22-2024-107
Published: Jun 24, 2024
Cite as:
Rachid Echarghaoui, Abdelouhab Hatimi, Mohamed Hatimi, A Fractional Elliptic System With Strongly Coupled Critical Terms and Concave-Convex Nonlinearities, Int. J. Anal. Appl., 22 (2024), 107.

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Rachid Echarghaoui, Abdelouhab Hatimi, Mohamed Hatimi

Abstract

By the Nehari method and variational method, two positive solutions are obtained for a fractional elliptic system with strongly coupled critical terms and concave-convex nonlinearities. Recent results from the literature are extended to the fractional case.

Article Details

References

  1. C.O. Alves, D.C. de Morais Filho, M.A.S. Souto, On systems of elliptic equations involving subcritical or critical Sobolev exponents, Nonlinear Anal.: Theory Meth. Appl. 42 (2000), 771–787. https://doi.org/10.1016/s0362-546x(99)00121-2.
  2. C.O. Alves, Y.H. Ding, Multiplicity of positive solutions to a p-Laplacian equation involving critical nonlinearity, J. Math. Anal. Appl. 279 (2003), 508–521. https://doi.org/10.1016/s0022-247x(03)00026-x.
  3. B. Barrios, E. Colorado, A. de Pablo, U. Sánchez, On some critical problems for the fractional Laplacian operator, J. Diff. Equ. 252 (2012), 6133–6162. https://doi.org/10.1016/j.jde.2012.02.023.
  4. S. Benmouloud, R. Echarghaoui, Si.M. Sbaï, Multiplicity of positive solutions for a critical quasilinear elliptic system with concave and convex nonlinearities, J. Math. Anal. Appl. 396 (2012), 375–385. https://doi.org/10.1016/j.jmaa.2012.05.078.
  5. C. Brändle, E. Colorado, A. de Pablo, U. Sánchez, A concave–convex elliptic problem involving the fractional Laplacian, Proc. R. Soc. Edinburgh: Sect. A Math. 143 (2013), 39–71. https://doi.org/10.1017/s0308210511000175.
  6. H. Brézis, E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc. 88 (1983), 486–490.
  7. H. Brézis, L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math. 36 (1983), 437–477.
  8. L. Caffarelli, L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Part. Diff. Equ. 32 (2007), 1245–1260. https://doi.org/10.1080/03605300600987306.
  9. A. Capella, J. Dávila, L. Dupaigne, Y. Sire, Regularity of radial extremal solutions for some non-local semilinear equations, Comm. Part. Diff. Equ. 36 (2011), 1353–1384. https://doi.org/10.1080/03605302.2011.562954.
  10. I. Ekeland, On the variational principle, J. Math. Anal. Appl. 47 (1974), 324–353. https://doi.org/10.1016/0022-247x(74)90025-0.
  11. P.G. Han, The effect of the domain topology on the number of positive solutions of elliptic systems involving critical Sobolev exponents, Houston J. Math. 32 (2006), 1241–1257.
  12. T.S. Hsu, Multiple positive solutions for a critical quasilinear elliptic system with concave–convex nonlinearities, Nonlinear Anal.: Theory Meth. Appl. 71 (2009), 2688–2698. https://doi.org/10.1016/j.na.2009.01.110.
  13. T.S. Hsu, H.L. Lin, Multiple positive solutions for a critical elliptic system with concave—convex nonlinearities, Proc. R. Soc. Edinburgh: Sect. A Math. 139 (2009), 1163–1177. https://doi.org/10.1017/s0308210508000875.
  14. R. Servadei, E. Valdinoci, On the spectrum of two different fractional operators, Proc. R. Soc. Edinburgh: Sect. A Math. 144 (2014), 831–855. https://doi.org/10.1017/s0308210512001783.
  15. J.L. Vázquez, Recent progress in the theory of nonlinear diffusion with fractional Laplacian operators, preprint, (2014). http://arxiv.org/abs/1401.3640.
  16. Y. Wei, X. Su, Multiplicity of solutions for non-local elliptic equations driven by the fractional Laplacian, Calc. Var. Part. Differ. Equ. 52 (2014), 95–124. https://doi.org/10.1007/s00526-013-0706-5.
  17. T.F. Wu, On semilinear elliptic equations involving concave-convex nonlinearities and sign-changing weight function, J. Math. Anal. Appl. 318 (2006), 253–270. https://doi.org/10.1016/j.jmaa.2005.05.057.
  18. X. Zhou, H.Y. Li, J.F. Liao, Multiplicity of positive solutions for a semilinear elliptic system with strongly coupled critical terms and concave nonlinearities, Qual. Theory Dyn. Syst. 22 (2023), 126. https://doi.org/10.1007/s12346-023-00825-9.