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Original Articles

Existence results for an anisotropic nonlocal problem involving critical and discontinuous nonlinearities

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Pages 731-755 | Received 31 Oct 2019, Accepted 06 Mar 2020, Published online: 23 Mar 2020

References

  • Bebernes JW, Talaga P. Nonlocal problems modelling shear banding. Nonlinear Anal. 1996;3:79–103.
  • Furter J, Grinfeld M. Local vs. nonlocal interactions in population dynamics. J Math Biol. 1989;27:65–80. doi: 10.1007/BF00276081
  • Enguića R, Sanchez L. Radial solutions for a nonlocal boundary value problem. Bound Value Prob. 2006;2006:32950.
  • Fijalkowski P, Przeradzki B. On a radial positive solution to a nonlocal elliptic equation. Topol Methods Nonlinear Anal. 2003;21:293–300. doi: 10.12775/TMNA.2003.017
  • A. Corrêa FJS, Delgado M, Suárez A. Some nonlinear heterogeneous problems with nonlocal reaction term. Adv Differ Equ. 2011; 16:623–641.
  • Corrêa FJSA, Delgado M, Suárez A. A variational approach to a nonlocal elliptic problem with sign-changing nonlinearity. Adv Nonlinear Stud. 2011;11:361–375. doi: 10.1515/ans-2011-0207
  • Gomes JM, Sanchez L. On a variational approach to some non-local boundary value problems. Appl Anal. 2005;84:909–925. doi: 10.1080/00036810500048202
  • A. Corrêa FJS, Costa ACR. On a bi-nonlocal p(x)-Kirchhoff equation via Krasnoselskii's genus. Math Methods Appl Sci. 2014;38:87–93. doi: 10.1002/mma.3051
  • Arcoya D, Leonori T, Primo A. Existence of solutions for semilinear nonlocal elliptic problems via a Bolzano theorem. Acta Appl Math. 2013;127:87–104. doi: 10.1007/s10440-012-9792-1
  • Corrêa FJSA, Menezes SDB. Positive solutions for a class of nonlocal problems. Progress in nonlinear differential equations and their applications, Volume in honor of Djairo G. de Figueiredo, Vol. 66, 195–206 (2005).
  • Alves CO, Covei D-P. Existence of solution for a class of nonlocal elliptic problem via sub-supersolution method. Nonlinear Anal Real World Appl. 2015;23:1–8. doi: 10.1016/j.nonrwa.2014.11.003
  • Santos CA, Santos LM. How to break the uniqueness of Wloc1,p(Ω)-solutions for very singular elliptic problems by non-local terms. Angew Math Phys. 2018;69:145. doi: 10.1007/s00033-018-1040-8
  • Figueiredo GM, Moussaoui A, dos Santos GCG, Tavares LS. A sub-supersolution approach for some classes of nonlocal problems involving Orlicz spaces. J Differ Equ. 2019;267:4148–4169. doi: 10.1016/j.jde.2019.04.039
  • Papageorgiou NS, Rădulescu VD, Repovš DD. Nonlinear analysis-theory and methods. Cham: Springer; 2019. (Springer Monographs in Mathematics).
  • Dos Santos GCG, Figueiredo GM. Positive solutions for a class of nonlocal problems involving Lebesgue generalized spaces: scalar and system cases. J Elliptic Parabol Equ. 2016;2(1–2):235–266. doi: 10.1007/BF03377404
  • dos Santos GCG, Figueiredo GM, Tavares LS. A sub-supersolution method for a class of nonlocal problems involving the p(x)-Laplacian operator and applications. Acta Appl Math. 2018;153:171–187. doi: 10.1007/s10440-017-0126-1
  • dos Santos GCG, Figueiredo GM, Tavares LS. A sub-supersolution method for a class of nonlocal system involving the p(x)-Laplacian operator and applications; (2017). arXiv:1710.11488.
  • Alves CO, Duarte RC, Souto MAS. A Berestycki-Lions type result and applications. Rev Mat Iberoam. 2019;35(6):1859–1884. doi: 10.4171/rmi/1104
  • Ambrosetti A, Badiale M. The dual variational principle and elliptic problems with discontinuous nonlinearities. J Math Anal Appl. 1989;140:363–373. doi: 10.1016/0022-247X(89)90070-X
  • Bonanno G, D'Aguì G, Winkert P. Sturm-Liouville equations involving discontinuous nonlinearities. Minimax Theory Appl. 2016;1(1:125–143.
  • Chang KC. The obstacle problem and partial differential equations with discontinuous nonlinearities. Commun Pure Appl Math. 1978;33(2):117–146. doi: 10.1002/cpa.3160330203
  • Chang KC. Variational methods for nondifferentiable functionals and their applications to partial differential equations. J Math Anal. 1981;80:102–129. doi: 10.1016/0022-247X(81)90095-0
  • Clarke FH. Optimization and nonsmooth analysis. New York (NY): John Wiley & Sons; 1983.
  • Clarke FH. Generalized gradients and applications. Trans Amer Math Soc. 1975;265:247–262. doi: 10.1090/S0002-9947-1975-0367131-6
  • Figueiredo GM, dos Santos GGC. Existence of positive solution for Kirchhoff type problem with critical discontinuous nonlinearity. Ann Acad Sci Fenn Math. 2019;44:987–1002. doi: 10.5186/aasfm.2019.4453
  • Gazzola F, Radulescu VD. A nonsmooth critical point theory approach to some nonlinear elliptic equations in RN. Differ Int Equ. 2000;13:47–60.
  • Grossinho MR, Tersian SA. An introduction to minimax theorems and their applications to differential equations. Dordrecht: Kluwer Academic Publishers; 2001.
  • Rădulescu VD. Mountain pass theorems for non-differentiable functions and applications. Proc Jpn Acad. 1993;69(Ser.A):193–198. doi: 10.3792/pjaa.69.193
  • dos Santos GG, Figueiredo GM. Solutions for a Kirchhoff equation with critical Caffarelli-Kohn-Nirenberg growth and discontinuous nonlinearity. Z Angew Math Phys. 2018;69:75. doi: 10.1007/s00033-018-0966-1
  • Chang KC. On the multiple solutions of the elliptic differential equations with discontinuous nonlinear terms. Sci Sinica. 1978;21:139–158.
  • Fragala I, Gazzola F, Kawohl B. Existence and nonexistence results for anisotropic quasilinear elliptic equations. Ann Inst H Poincaré Anal Non Linéaire. 2004;21:715–734. doi: 10.1016/j.anihpc.2003.12.001
  • El Hamidi A, Rakotoson JM. Extremal functions for the anisotropic Sobolev inequalities. Ann I H Poincarsé AN. 2007;24:741–756. doi: 10.1016/j.anihpc.2006.06.003
  • Afrouzi GA, Mirzapour M, Radulescu VD. Variational analysis of anisotropic Schröndinger equations without Ambrosetti-Rabinowitz-type condition. Z Angew Math Phys. 2018;69(9):9–17. doi: 10.1007/s00033-017-0900-y
  • Alves CO, El Hamidi A. Existence of solution for a anisotropic equation with critical exponent. Difer Int Equ. 2008;21:25–40.
  • Bahrouni A, Radulescu VD, Repovs DD. A weighted anisotropic variant of the Caffarelli-Kohn-Nirenberg inequality and applications. Nonlinearity. 2018;31(4):1516–1534. doi: 10.1088/1361-6544/aaa5dd
  • Cherfils L, Miranville A, Peng S. Higher-order anisotropic models in phase separation. Adv Nonlinear Anal. 2019;8(1):278–302. doi: 10.1515/anona-2016-0137
  • Di Castro A, Montefusco E. Nonlinear eigenvalues for anisotropic quasilinear degenerate elliptic equations. Nonlinear Anal. 2009;70:4093–4105. doi: 10.1016/j.na.2008.06.001
  • Della Pietra F, di Blasio G, Gavitone N. Sharp estimates on the first Dirichlet eigenvalue of nonlinear elliptic operators via maximum principle. Adv Nonlinear Anal. 2020;9(1):278–291. doi: 10.1515/anona-2017-0281
  • Figueiredo GM, Silva JRS. A critical anisotropic problem with discontinuous nonlinearities. Nonlinear Anal-Real World Appl. 2019;47:364–372. doi: 10.1016/j.nonrwa.2018.11.008
  • Molica Bisci G, Repovš D. On sequences of solutions for discrete anisotropic equations. Expo Math. 2014;32(3):284–295. doi: 10.1016/j.exmath.2013.12.001
  • Bear J. Dynamics of fluids in porous media. New York (NY): American Elsevier; 1972.
  • Bendahmane M, Karlsen KH. Renormalized solutions of an anisotropic reaction-diffusion advection system with L1 data. Commun Pure Appl Anal. 2006;5(4):733–762. doi: 10.3934/cpaa.2006.5.733
  • Bendahmane M, Langlais M, Saad M. On some anisotropic reaction-diffusion systems with L1 data modeling the propagation of an epidemic disease. Nonlinear Anal. 2003;54(4):617–636. doi: 10.1016/S0362-546X(03)00090-7
  • Hajiaboli MR. An anisotropic fourth-order diffusion filter for image noise removal. Int J Comput Vis. 2011;92(2):177–191. doi: 10.1007/s11263-010-0330-1
  • Stuart C. Boundary-value problems with discontinuous non-linearities. Berlin: Springer-Verlag; 1976. (Vol. 564 of lecture notes in math).
  • Stuart C. Differential equations with discontinuous nonlinearities. Arch Rational Mech Anal. 1976/1977;63:59–75. doi: 10.1007/BF00280142
  • Gasìnski L, Papageorgiou NS. Nonsmooth critical point theory and nonlinear boundary value problems. Boca Raton (FL): Chapman and Hall/CRC; 2005.
  • Troisi M. Teoremi di inclusione per spazi di Sobolev non isotropi. Ricerche Mat. 1969;18:3–24.
  • Wheeden R, Zygmund A. Measure and integral. New York (NY): Dekker; 1977.
  • Gilbarg D, Trudinger NS. Elliptic partial differential equations of second order. Berlin: Springer-Verlag; 1983.

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