References
- Bisci GM, Radulescu V, Servadei R. Variational methods for nonlocal fractional problems, 162. Cambridge: Cambridge University Press; 2016. Encyclopedia of Mathematics and its Application.
- Di Nezza E, Palatucci G, Valdinoci E. Hitchhikers guide to the fractional Sobolev spaces. Bull Sci Math. 2012;136:521–573.
- Bucur C, Valdinoci E. Nonlocal diffusion and applications, Bologna: Springer Unione Matematica Italiana; 2016. Lecture Notes of the Unione Matematica Italiana.
- Viswanathan GM, Afanasyev V, Buldyrev SV, et al. Lévy flight search patterns of wandering albatrosses. Nature. 1996;381(6581):413–415.
- Caffarelli L, Silvestre L. An extension problem related to the fractional Laplacian. Comm Partial Differ Equ. 2007;32:1245–1260.
- Bertoin J. Lévy processes, 121. Cambridge: Cambridge University Press; 1996. Cambridge Tracts in Mathematics.
- Laskin N. Fractional quantum mechanics and Lévy path integrals. Phys Lett A. 2000;268:298–305.
- Giammetta AR. Fractional Schrödinger–Poisson–Slater system in one dimension, e-print arXiv:1405.2796v1.
- Liu W. Existence of multi-bump solutions for the fractional Schrödinger–Poisson system. J Math Phys. 2016;57:091502.
- Teng K. Existence of ground state solutions for the nonlinear fractional Schrödinger–Poisson system with critical Sobolev exponent. J Differ Equ. 2016;261:3061–3106.
- Lions PL. Solutions of Hartree–Fock equations for Coulomb systems. Comm Math Phys. 1984;109:33–97.
- Benci V, Fortunato D. An eigenvalue problem for the Schrödinger–Maxwell equations. Topol Methods Nonlinear Anal. 1998;11:283–293.
- Benci V, Fortunato D. Solitary waves of the nonlinear Klein–Gordon equation coupled with Maxwell equations. Rev Math Phys. 2002;14:409–420.
- Azzollini A, Pomponio A. Ground state solutions for the nonlinear Schrödinger–Maxwell equations. J Math Anal Appl. 2008;345:90–108.
- Cerami G, Vaira G. Positive solutions for some non-autonomous Schrödinger–Poisson systems. J Differ Equ. 2010;248:521–543.
- D'Avenia P. Non-radially symmetric solutions of nonlinear Schrödinger equation coupled with Maxwell equations. Adv Nonlinear Stud. 2002;2:177–192.
- Ianni I. Sign-changing radial solutions for the Schrödinger–Poisson–Slater problem. Topol Methods Nonlinear Anal. 2013;41:365–385.
- Ianni I, Vaira G. Non-radial sign-changing solutions for the Schrödinger–Poisson problem in the semiclassical limit. Nonlinear Differ Equ Appl Nodea. 2012;22(4):741–776.
- Wang Z, Zhou H. Positive solutions for a nonlinear stationary Schrödinger–Poisson system in R3. Discrete Contin Dyn Syst. 2007;18(4):809–816.
- Zhao L, Zhao F. Positive solutions for Schrödinger–Poisson equations with critical exponent. Nonlinear Anal. 2009;70:2150–2164.
- Ianni I, Vaira G. On concentration of positive bound states for the Schrödinger–Poisson problem with potential. Adv Nonlinear Stud. 2008;8:573–595.
- He X. Multiplicity and concentration of positive solutions for the Schrödinger–Poisson equations. Z Angew Math Phys. 2011;62:869–889.
- Wang J, Tian L, Xu J, Existence of multiple solutions for Schrödinger–Poisson systems with critical growth. Z Angew Math Phys. 2015;66:2441–2471.
- He X, Zou W. Existence and concentration of ground states for Schrödinger–Poisson equations with critical growth. J Math Phys. 2012;53(2):143–162.
- Wang J, Tian L, Xu J, et al. Existence and concentration of positive solutions for semilinear Schrödinger–Poisson systems in R3. Calc Var Partial Differ Equ. 2013;48(1):275–276.
- Zhao L, Liu H, Zhao F. Existence and concentration of solutions for the Schrödinger–Poisson equations with steep well potential. J Differ Equ. 2013;255(1):1–23.
- Jeanjean L, Tanaka K. A positive solution for an nonlinear Schrödinger equation on RN. Indiana Univ Math J. 2005;54:443–464.
- Yu Y, Zhao F, Zhao L. The concentration behavior of ground state solutions for a fractional Schrödinger–Poisson system. Calc Var Partial Differ Equ. 2017;56(116):1–25.
- Yu Y, Zhao F, Zhao L. The existence and multiplicity of solutions of a fractional Schrödinger–Poisson system with critical growth. Sci China Ser A. 2018;61(6):1039–1062.
- Dipierro S, Valdinoci E, Medina M. Fractional elliptic problems with critical growth in the whole of , Pisa: Edizionidella Normalew; 2017. Lecture Normale Superiore.
- Yang Z, Yu Y, Zhao F. Concentration behavior of ground state solutions for a fractional Schrödinger–Poisson system involving critical exponent. Commun Contemp Math. (2018) 1850027.
- Dávila 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.
- Dávila J, Del Pino M, Wei J. Concentrating standing waves for the fractional nonlinear Schrödinger equation. J Differ Equ. 2014;256(2):858–892.
- Liu Z, Zhang J. Multiplicity and concentration of positive solutions for the fractional Schrödinger–Poisson system with critical growth. ESAIM Contol Optim Calc Var. 2017;23(4):1515–1542.
- Zhang J, do ó JM, Squassina M. Fractional Schrödinger–Poisson systems with a general subcritical or critical nonlinearity. Adv Nonlinear Stud. 2016;16(1):15–30.
- Szulkin A, Weth T. Ground state solutions for some indefinite variational problems. J Funct Anal. 2009;257:3802–3822.
- Szulkin A, Weth T. The method of Nehari manifold, handbook of nonconvex analysis and applications. In: Gao D.Y. and Motreanu D. editors. Boston: International Press; 2010. 597–632.
- Huang W, Tang X. Ground-state solutions for asymptotically cubic Schrödinger–Maxwell equations. Mediterr J Math. 2016;13(5):3469–3481.
- Fang X. A positive solution for asymptotically cubic quasilinear Schrödinger equations. Commun Pure Appl Anal. 2019;18(1):51–64.
- Choquard P, Wagner J. On a class of implicit solutions of the continuity and Euler's equations for 1D systems with long range interactions. Phys D. 2005;40:230–248.
- Choquard P, Stubbe J. The one-dimensional Schrödinger–Newton equations. Lett Math Phys. 2007;81:177–184.
- Choquard P, Stubbe J, Vuffray M. Stationary solutions of the Schrödinger–Newton model an ODE approach. Differ Integral Equ. 2008;21:665–679.
- Cingolani S, Weth T. On the Schrödinger–Poisson system. Ann Inst H Poincaré Anal Non Linéaire. 2016;33:169–197.
- Stubbe J. Bound states of two-dimensional Schrödinger–Newton equation, arXiv: 0807.4059v1.2008.
- Bahri A, Li YY. On a min–max procedure for the existence of positive solution for certain scalar field equations in RN. Rev Mat Iberoam. 1990;6:1–16.
- Willem M. Minimax theorems, 24. Boston, MA: Birkhäuser; 1996. Progress in Nonlinear Differential Equations and their Applications.
- Cerami G, Passaseo D. The effect of concentrating potentials in some singularly perturbed problems. Calc Var Partial Differ Equ. 2003;17(3):257–281.
- Rabinowitz PH. Minimax methods in critical point theory with application to differential equations. Vol. 65, CBMS Reg Conf Ser Math, Providence, RI: American Mathematical Society; 1986.