Existence and multiplicity results for critical anisotropic Kirchhoff-type problems with nonlocal nonlinearities

References

• El Hamidi A, Rakotoson JM. Extremal functions for the anisotropic Sobolev inequalities. Ann I H Poincaré AN. 2007;24:741–756.
   Web of Science ®Google Scholar
• Rabinowitz PH. Minimax methods in critical point theory with applications to differential equations. Providence RI: American Mathematical Society; 1986. (CBMS Reg Conf Ser Math; 65).
   Google Scholar
• Kajikiya R. A critical-point theorem related to the symmetric mountain-pass lemma and its applications to elliptic equations. J Funct Anal. 2005;225(2):352–370.
   Web of Science ®Google Scholar
• Bebernes JW, Talaga P. Nonlocal problems modelling shear banding. Nonlinear Anal. 1996;3:79–103.
   Google Scholar
• Furter J, Grinfeld M. Local vs nonlocal interactions in population dynamics. J Math Biol. 1989;27:65–80.
   Web of Science ®Google Scholar
• Enguiça R, Sanchez L. Radial solutions for a nonlocal boundary value problem. Bound Value Probl. 2006;1:32950.Art. ID 32950, 18 pp.
   Google Scholar
• Corrêa FJSA, Delgado M, Suárez A. Some nonlinear heterogeneous problems with nonlocal reaction term. Adv Differ Equ. 2011;16:623–641.
   Web of Science ®Google Scholar
• 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.
   Web of Science ®Google Scholar
• Gomes JM, Sanchez L. On a variational approach to some non-local boundary value problems. Appl Anal. 2005;84:909–925.
   Google Scholar
• Corrêa FJSA, Costa ACR. On a bi-nonlocal p(x)−Kirchhoff equation via Krasnoselskii's genus. Math Meth Appl Sci. 2014;38:87–93.
   Web of Science ®Google Scholar
• 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.
   Web of Science ®Google Scholar
• 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, 2005. p. 195–206.
   Google Scholar
• 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.
   Web of Science ®Google Scholar
• Kirchhoff G. Mechanik. Leipzig: Teubner; 1883.
   Google Scholar
• Alves CO, Corrêa FJSA, Ma TF. Positive solutions for a quasi-linear elliptic equation of Kirchhoff type. Comput Math Appl. 2005;49:85–93.
   Web of Science ®Google Scholar
• Ma TF. Remarks on an elliptic equation of Kirchhoff type. Nonlinear Anal. 2005;63(5–7):e1967–e1977.
   Web of Science ®Google Scholar
• Ambrosetti A, Arcoya D. Positive solutions of elliptic Kirchhoff equations. Adv Nonlinear Stud. 2017;17(1):3–15.
   Web of Science ®Google Scholar
• Ambrosio V. Concentration phenomena for a class of fractional Kirchhoff equations in RN with general nonlinearities. Nonlinear Anal. 2020;195:111761. 39 pp.
   Google Scholar
• Ambrosio V. Infinitely many small energy solutions for a fractional Kirchhoff equation involving sublinear nonlinearities. Proceedings of the International Conference “Two nonlinear days in Urbino 2017”, Electron. J. Differ. Equ. Conf.; Vol. 25, San Marcos, TX: Texas State Univ.San Marcos, Dept. Math.; 2018. p. 1–13.
   Google Scholar
• Ambrosio V. Infinitely many periodic solutions for a class of fractional Kirchhoff problems. Monatsh Math. 2019;190(4):615–639.
   Web of Science ®Google Scholar
• Ambrosio V, Fiscella A, Isernia T. Infinitely many solutions for fractional Kirchhoff-Sobolev-Hardy critical problems. Electron J Qual Theory Differ Equ. 2019;25: 1–13.13 pp.
   Google Scholar
• Ambrosio V, Isernia T. A multiplicity result for a fractional Kirchhoff equation in RN with a general nonlinearity. Commun Contemp Math. 2018;20(5):1750054. 17 pp.
   Google Scholar
• Ambrosio V, Isernia T. Concentration phenomena for a fractional Schrödinger-Kirchhoff type problem. Math Meth Appl Sci. 2018;41(2):615–645.
   Web of Science ®Google Scholar
• Ambrosio V, Servadei R. Supercritical fractional Kirchhoff type problems. Fract Calc Appl Anal. 2019;22(5):1351–1377.
   Web of Science ®Google Scholar
• Chen C. Infinitely many solutions for N-Kirchhoff equation with critical exponential growth in RN. Mediterr J Math. 2018;15(1):1909.Art. 4: .
   Web of Science ®Google Scholar
• Figueiredo GM. Existence of positive solution for a Kirchhoff problem type with critical growth via truncation argument. J Math Anal Appl. 2013;401:706–713.
   Web of Science ®Google Scholar
• Figueiredo GM, Molica Bisci G, Servadei R. On a fractional Kirchhoff-type equation via Krasnoselskii's genus. Asymptot Anal. 2015;94:347–361.
   Web of Science ®Google Scholar
• Fiscella A, Valdinoci E. A critical Kirchhoff type problem involving a nonlocal operator. Nonlinear Anal. 2014;94:156–170.
   Web of Science ®Google Scholar
• Fragala I, Gazzola F, Kawohl B. Existence and nonexistence results for anisotropic quasilinear elliptic quations. Ann Inst H Poincar Anal Non Linéaire. 2004;21:715–734.
   Web of Science ®Google Scholar
• Furtado MF, Zanata HR. Multiple solutions for a Kirchhoff equation with nonlinearity having arbitrary growth. Bull Aust Math Soc. 2017;96(1):98–109.
   Web of Science ®Google Scholar
• Hssini ELM, Isouli L, Haddaoui M. Existence results for a Kirchhoff type equation in Orlicz-Sobolev spaces. Adv Pure Appl Math. 2017;8(3):197–208.
   Web of Science ®Google Scholar
• Hurtado EJ, Miyagaki OH, Rodrigues RS. Existence and asymptotic behaviour for a Kirchhoff type equation with variable critical growth exponent. Milan J Math. 2017;85(1):71–102.
   Web of Science ®Google Scholar
• Molica Bisci G. Fractional equations with bounded primitive. Appl Math Lett. 2014;27:53–58.
   Web of Science ®Google Scholar
• Naimen D. On the critical problem of Kirchhoff type elliptic equations in dimension four. J Differ Equ. 2014;257:1168–1193.
   Web of Science ®Google Scholar
• dos Santos GCG, Figueiredo GM. Solutions for a Kirchhoff equation with critical Caffarelli-Kohn-Nirenberg growth and discontinuous nonlinearity. Z Angew Math Phys. 2018;69–75. Available From: https://doi.org/10.1007/s00033-018-0966-1
   Google Scholar
• dos Santos GCG, Figueiredo GM. Existence of positive solution for Kirchhoff type problem with critical discontinuous nonlinearity. Ann Academiae Scientiarum Fennicae Math. 2019;44:987–1002.
   Web of Science ®Google Scholar
• Zhisu L, Shangjiang G, Fang Y. Positive solutions of Kirchhoff type elliptic equations in R4 with critical growth. Math Nachr. 2017;290(2–3):367–381.
   Web of Science ®Google Scholar
• Zhongrui S, Shujun W. Existence of solutions for Kirchhoff type problems in Musielak-Orlicz-Sobolev spaces. J Math Anal Appl. 2016;436(2):1002–1016.
   Web of Science ®Google Scholar
• Afrouzi GA, Mirzapour M, Rădulescu VD. Variational analysis of anisotropic Schrödinger equations without Ambrosetti-Rabinowitz-type condition. Z Angew Math Phys. 2018;69(1):541.
   Web of Science ®Google Scholar
• Afrouzi GA, Mirzapour M, Rădulescu VD. Qualitative analysis of solutions for a class of anisotropic elliptic equations with variable exponent. Proc Edinb Math Soc. 2016;59(3):541–557.
   Web of Science ®Google Scholar
• Afrouzi GA, Mirzapour M. Existence and multiplicity of solutions for nonlocal p(x)-Laplacian problem. Taiwan J Math. 2014;18:219–236.
   Web of Science ®Google Scholar
• Alves CO, El Hamidi A. Existence of solution for a anisotropic equation with critical exponent. Differ Integral Equ. 2008;21:25–40.
   Web of Science ®Google Scholar
• Avci M. On a nonlocal problem involving the generalized anisotropic p→(.)-Laplace operator. An Univ Craiova Ser Mat Inform. 2016;43(2):259–272.
   Google Scholar
• Di Castro A. Elliptic problems for some anisotropic operators. University of Rome ‘Sapienza’, academic year 2008/2009.
   Google Scholar
• Di Castro A, Montefusco E. Nonlinear eigenvalues for anisotropic quasilinear degenerate elliptic equations. Nonlinear Anal. 2009;70:4093–4105.
   Web of Science ®Google Scholar
• Figueiredo GM, Silva JRS. A critical anisotropic problem with discontinuous nonlinearities. Nonlinear Anal-Real World Appl. 2019;47:364–372.
   Web of Science ®Google Scholar
• Mihăilescu M, Pucci P, Rădulescu V. Eigenvalue problems for anisotropic quasilinear elliptic equations with variable exponent. J Math Anal Appl. 2008;340:687–698.
   Web of Science ®Google Scholar
• Mihăilescu M, Rădulescu V, Tersian S. Eigenvalue problems for anisotropic discrete boundary value problems. J Differ Equ Appl. 2009;15:557–567.
   Web of Science ®Google Scholar
• Molica Bisci G, Repovš D. On sequences of solutions for discrete anisotropic equations: Expo. Math. 2014;32(3):284–295.
   Google Scholar
• Rădulescu V, Repovš D. Partial differential equations with variable exponents: variational methods and qualitative analysis. Boca Raton, FL: CRC Press; 2015.
   Google Scholar
• Repovš D. Infinitely many symmetric solutions for anisotropic problems driven by nonhomogeneous operators. Discrete Contin Dyn Syst Ser S. 2019;12(2):401–411.
   Web of Science ®Google Scholar
• Antontsev SN, Shmarev S. Energy Methods for Free Boundary Problems. Progress in nonlinear differential equations and their applications. Vol. 48, Boston, MA: Birkhauser Boston; 2002.
   Google Scholar
• Bear J. Dynamics of Fluids in Porous Media. New York: American Elsevier; 1972.
   Google Scholar
• 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.
   Web of Science ®Google Scholar
• 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.
   Web of Science ®Google Scholar
• Hajiaboli MR. An anisotropic fourth-order diffusion for image noise removal. Int J Comput Vis. 2011;92(2):177–191.
   Web of Science ®Google Scholar
• Avci M. Solutions of an anisotropic nonlocal problem involving variable exponent. Adv Nonlinear Anal. 2013;2(3):325–338.
   Web of Science ®Google Scholar
• Bensedik A. On existence results for an anisotropic elliptic equation of Kirchhoff-type by a monotonicity method. Funkcialaj Ekvacioj. 2014;57(3):489–502.
   Web of Science ®Google Scholar
• Chung NT. Multiple solutions for an anisotropic elliptic equation of Kirchhoff type in bounded domain. Results in Nonlinear Anal. 2018;1(3):116–127.
   Google Scholar
• Lei CY. Multiple positive solutions for a class of nonlocal problems involving a sign-changing potential. Electron J Differ Equ. 2017;09:1–8.
   Google Scholar
• Zhang J, Zhang Z. Existence of nontrivial solution for a nonlocal problem with subcritical nonlinearity. Adv Differ Equ. 2018;359: Availabel From: https://doi.org/10.1186/s13662-018-1823-4.
   Google Scholar
• Devillanova G, Solimini S. Some remarks on profile decomposition theorems. Adv Nonlinear Stud. 2016;16(4):384.
   Web of Science ®Google Scholar
• Devillanova G. Multiscale weak compactness in metric spaces. J Elliptic Parabol Equ. 2016;2(1–2):131–144.
   Web of Science ®Google Scholar
• Krasnoselskii MA. Topological Methods in the Theory of Nonlinear Integral Equations. New York: Mac Millan; 1964.
   Google Scholar
• Garcia Azorero J, Peral Alonso I. Multiplicity of solutions for elliptic problems with critical exponent or with a nonsymmetric term. Trans Amer Math Soc. 1991;323:877–895.
   Web of Science ®Google Scholar
• Lions PL. The concentration-compactness principle in the calculus of variations. The limit case, Rev Mat Iberoamericana. 1985;1:145–201.
   Google Scholar
• Chabrowski J. Variational methods for potential operator equations with applications to nonlinear elliptic equations. Berlin-New York: Walter de Gruyter; 1997.
   Google Scholar