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Articles

An existence result for singular nonlocal fractional Kirchhoff–Schrödinger–Poisson system

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Pages 1817-1846 | Received 01 Jun 2020, Accepted 04 Mar 2021, Published online: 22 Mar 2021

References

  • Bisci GM, Rădulescu V, Servadei R. Variational methods for nonlocal fractional problems. Cambridge: Cambridge University Press; 2016. (Encyclopedia of Mathematics and its Applications; 162).
  • Cont R. Financial modelling with jump processes. Chapman and Hall/CRC; 2003. (Chapman and Hall/CRC Financial Mathematics Series).
  • Laskin N. Fractional quantum mechanics and Lévy path integrals. Phys Lett A. 2000;268(4):298–305.
  • Laskin N. Fractional Schrödinger equation. Phys Rev E. 2002;66(5):056108.
  • Valdinoci E. From the long jump random walk to the fractional laplacian. Bol Soc Esp Mat Apl SeMA. 2009;49:33–44.
  • Callegari A, Nachman A. Some singular nonlinear differential equations arising in boundary layer theory. J Math Anal Appl. 1978;64:96–105.
  • Bertoin J. Lévy processes, volume 121 of cambridge tracts in mathematics, Cambridge University Press; 1998. (Cambridge Tracts in Mathematics; 121) .
  • Caffarelli L, Salsa S, Silvestre L. Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian. Invent Math. 2008;171:425–461.
  • Buades A, Coll B, Morel JM. Neighborhood filters and PDEs. Numer Math. 2006;105:1–34.
  • Kindermann S, Osher S, Jones PW. Deblurring and denoising of images by nonlocal functionals. Multiscale Model Simul. 2005;4:1091-–1115.
  • Di Nezza E, Palatucci G, Valdinoci E. Hitchhiker's guide to the fractional Sobolev spaces. Bull Sci Math. 2012;136:521–573.
  • Klafter J, Lim SC, Metzler R. Fractional dynamics. Recent advances. Hackensack (NJ):World Scientific; 2012.
  • Laskin N. Principles of fractional quantum mechanics. Fractional dynamics: recent advances. Hackensack (NJ): World Sci. Publ.; 2012. p. 393–427.
  • Benci V, Fortunato D. An eigenvalue problem for the Schrödinger Maxwell equations. Topol Methods Nonlinear Anal. 1998;11(2):283–293.
  • Benci V, Fortunato D. Solitary waves of the nonlinear Klein–Gordon equation coupled with the Maxwell equations. Rev Math Phys. 2002;14(4):409–420.
  • Davila J, Del Pino M, Dipierro S, et al. Concentration phenomena for the nonlocal Schrödinger equation with Dirichlet datum. Anal PDE. 2015;8(5):1165–1235.
  • Ambrosetti A, Ruiz D. Multiple bound states for the Schrödinger–Poisson problem. Commun Contemp Math. 2008;10(3):391–404.
  • Azzollini A, Luisi V. Generalized Schrödinger–Poisson type systems. Commun Pure Appl Anal. 2013;12(2):867–879.
  • Teng K. Existence of ground state solutions for the nonlinear fractional Schrödinger–Poisson system with critical Sobolev exponent. J Differ Eq. 2016;261(6):3061–3106.
  • Kirchhoff G. Mechanik. Teubner: Leipzig;1883.
  • Fiscella A, Valdinoci E. A critical Kirchhoff type problem involving a nonlocal operator. Nonlinear Anal. 2014;94:156–170.
  • Lazer AC, McKenna PJ. On a singular nonlinear elliptic boundary-value problem. Proc Am Math Soc. 1991;111(3):721–730.
  • Sun Y, Zhang D. The role of the power 3 for elliptic equations with negative exponents. Calculus Var Partial Differ Eq. 2014;49(3-4):909–922.
  • Nachman A, Callegari A. A nonlinear singular boundary value problem in the theory of pseudoplastic fluids. SIAM J Appl Math. 1980;38(2):275–281.
  • Canino A, Montoro L, Sciunzi B, et al. Nonlocal problems with singular nonlinearity. Bull Des Sci Math. 2017;141(3):223–250.
  • Fang Y. Existence, uniqueness of positive solution to a fractional Laplacians with singular nonlinearity. arXiv preprint arXiv:1403.3149; 2014.
  • Boccardo L, Orsina L. Semilinear elliptic equations with singular nonlinearities. Calc Var Partial Differ Equ. 2010;37(3/4):363–380.
  • Crandall MG, Rabinowitz PH, Tartar L. On a Dirichlet problem with a singular nonlinearity. Commun Partial Differ Eq. 1977;2(2):193–222.
  • Ghosh S, Choudhuri D. Existence of infinitely many solutions for a nonlocal elliptic PDE involving singularity. Positivity. 2020;24(2):463–479.
  • Ghosh S, Choudhuri D, Giri RK. Singular nonlocal problem involving measure data. Bull Brazilian Math Soc New Series. 2019;50(1):187–209.
  • Giacomoni J, Saoudi K. Multiplicity of positive solutions for a singular and critical problem. Nonlinear Anal: Theory, Methods Appl. 2009;71(9):4060–4077.
  • Lei CY, Liao JF, Tang CL. Multiple positive solutions for Kirchhoff type of problems with singularity and critical exponents. J Math Anal Appl. 2015;421(1):521–538.
  • Liao JF, Zhang P, Liu J, et al. Existence and multiplicity of positive solutions for a class of Kirchhoff type problems with singularity. J Math Anal Appl. 2015;430(2):1124–1148.
  • Liu X, Sun Y. Multiple positive solutions for Kirchhoff type problems with singularity. Commun Pure Appl Anal. 2013;12:721–733.
  • Oliva F, Petitta F. On singular elliptic equations with measure sources. ESAIM Control Optim Calculus Var. 2016;22:289–308.
  • Saoudi K. A critical fractional elliptic equation with singular nonlinearities. Fract Calculus Appl Anal. 2017;20(6):1507–1530.
  • Saoudi K, Ghosh S, Choudhuri D. Multiplicity and Hölder regularity of solutions for a nonlocal elliptic PDE involving singularity. J Math Phys. 2019;60(10):07517.
  • Sun Y. Compatibility phenomena in singular problems. Proc R Soc Edinburgh Sect A. 2013;143(6):1321–1330.
  • Sun Y, Wu S, Long Y. Combined effects of singular and superlinear nonlinearities in some singular boundary value problems. J Differ Eq. 2001;176(2):511–531.
  • Yang H. Multiplicity and asymptotic behavior of positive solutions for a singular semilinear elliptic problem. J Differ Equ. 2003;189(2):487–512.
  • Fiscella A. A fractional Kirchhoff problem involving a singular term and a critical nonlinearity. Adv Nonlinear Anal. 2019;8:645–660.
  • Fiscella A, Mishra PK. The Nehari manifold for fractional Kirchhoff problems involving singular and critical terms. Nonlinear Anal. 2019;186:6–32.
  • Lü D. A note on Kirchhoff-type equations with Hartree-type nonlinearities. Nonlinear Anal. 2014;99:35-–48.
  • Li F, Zhang Q. Existence of positive solutions to the Schrödinger–Poisson system without compactness conditions. J Math Anal Appl. 2013;401(2):754–762.
  • Li F, Song Z, Zhang Q. Existence and uniqueness results for Kirchhoff–Schrödinger–Poisson system with general singularity. Appl Anal. 2017;96(16):2906–2916.
  • Liao JF, Ke XF, Lei CY, et al. A uniqueness result for Kirchhoff type problems with singularity. Appl Math Lett. 2016;59:24–30.
  • Zhang Q. Existence, uniqueness and multiplicity of positive solutions for Schrödinger–Poisson system with singularity. J Math Anal Appl. 2016;437(1):160–180.
  • Zhang Q. Existence of positive solution to Kirchhoff–Schrödinger–Poisson system with strong singular term. J Math Phys. 2019;60:041504.
  • Zhao G, Zhu X, Li Y. Existence of infinitely many solutions to a class of Kirchhoff–Schrödinger–Poisson. Appl Math Comput. 2015;256:572–581.
  • Ruiz D. The Schrödinger–Poisson equation under the effect of a nonlinear local term. J Funct Anal. 2006;237(2):655–674.
  • Clark D. A variant of the Lusternik–Schnirelman theory. Indiana Uni Math J. 1972;22(1):65–74.
  • Ambrosetti A, Rabinowitz PH. Dual variational methods in critical point theory and applications. J Funct Anal. 1973;14(4):349–381.
  • 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.
  • Zhang B, Bisci GM, Servadei R. Superlinear nonlocal fractional problems with infinitely many solutions. Nonlinearity. 2015;28(7):2247–2264.
  • Zhang B, Fiscella A, Liang S. Infinitely many solutions for critical degenerate Kirchhoff type equations involving the fractional p-Laplacian. Appl Math Optim. 2019;80:63–80.
  • Figueiredo G, Bisci GM, Servadei R. On a fractional Kirchhoff-type equation via Krasnoselskii's genus. Asymptot Anal. 2015;94:347–361.
  • Li W, Rădulescu V, Zhang B. Infinitely many solutions for fractional Kirchhoff–Schrödinger–Poisson systems. J Math Phys. 2019;60:011506.
  • Li L, Zhong X. Infinitely many small solutions for the Kirchhoff equation with local sublinear nonlinearities. J Math Anal Appl. 2016;435(1):955–967.
  • Servadei R, Valdinoci E. Mountain pass solutions for non-local elliptic operators. J Math Anal Appl. 2012;389(2):887–898.
  • Brasco L, Parini E. The second eigenvalue of the fractional p-laplacian. Adv Calculus Var. 2016;9(4):323–355.
  • Willem M. Minimax theorems. Vol. 24 of Progress in Nonlinear Differential Equations and their Applications. Boston, MA: Birkhäuser Boston, Inc.; 1996.

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