Abstract
We study the following nonlinear fractional Schrödinger-Poisson system with critical growth:
where 0 < s, t < 1, 2s + 2t>3 and \(2_s^* = {6 \over {3 - 2s}}\) is the critical Sobolev exponent in ℝ3. Under some more general assumptions on f, we prove that (0.1) admits a nontrivial ground state solution by using a constrained minimization on a Nehari-Pohozaev manifold.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Ambrosetti A, Ruiz D. Multiple bound states for the Schrödinger-Poisson problem. Commun Contemp Math, 2008, 10: 391–404
Applebaum D. Lévy processes-from probability to finance quantum groups. Notices Amer Math Soc, 2004, 51: 1336–1347
Azzollini A, Pomponio A. Ground state solutions for the nonlinear Schrödinger-Maxwell equations. J Math Anal Appl, 2008, 345: 90–108
Benci V, Fortunato D. An eigenvalue problem for the Schrödinger-Maxwell equations. Topol Methods Nonlinear Anal, 1998, 11: 283–293
Biler P, Karch G, Woyczyhski W A. Critical nonlinearity exponent and self-similar asymptotics for Levy conservation laws. Ann Inst H Poincaré Anal Non Linéaire, 2001, 18: 613–637
Caffarelli L, Roquejoffre J M, Savin O. Non-local minimal surfaces. Comm Pure Appl Math, 2010, 63: 1111–1144
Caffarelli L, Silvestre L. An extension problem related to the fractional Laplacian. Comm Partial Differential Equations, 2007, 32: 1245–1260
Caffarelli L, Valdinoci E. Uniform estimates and limiting arguments for nonlocal minimal surfaces. Calc Var Partial Differential Equations, 2011, 41: 203–240
Chang X, Wang Z. Ground state of scalar field equations involving a fractional Laplacian with general nonlinearity. Nonlinearity, 2013, 26: 479–494
Cont R, Tankov P. Financial Modelling with Jump Processes. London/Boca Raton: Chapman & Hall/CRC Press, 2004
Cotsiolis A, Tavoularis N K. Best constants for Sobolev inequalities for higher order fractional derivatives. J Math Anal Appl, 2004, 295: 225–236
D’Aprile T, Mugnai D. Solitary waves for nonlinear Klein-Gordon-Maxwell and Schrödinger-Maxwell equations. Proc Roy Soc Edinburgh Sect A, 2004, 134: 893–906
D’Aprile T, Mugnai D. Non-existence results for the coupled Klein-Gordon-Maxwell equations. Adv Nonlinear Stud, 2004, 4: 307–322
Di Nezza E, Palatucci G, Valdinoci E. Hitchhiker’s guide to the fractional Sobolev spaces. Bull Sci Math, 2012, 136: 521–573
Dipierro S, Medina M, Valdinoci E. Fractional elliptic problems with critical growth in the whole of ℝn. Pisa: Edizioni della Normale, 2017
Dipierro S, Palatucci G, Valdinoci E. Existence and symmetry results for a Schröodinger type problem involving the fractional Laplacian. Matematiche (Catania), 2013, 68: 201–216
Duvaut G, Lions J L. Inequalities in Mechanics and Physics. Berlin: Springer-Verlag, 1976
Felmer P, Quaas A, Tan J. Positive solutions of nonlinear Schrödinger equation with the fractional Laplacian. Proc Roy Soc Edinburgh Sect A, 2012, 142: 1237–1262
He X, Zou W. Existence and concentration of ground states for Schrödinger-Poisson equations with critical growth. J Math Phys, 2012, 53: 023702
Huang L, Rocha E, Chen J. Two positive solutions of a class of Schrödinger-Poisson system with indefinite nonlinearity. J Differential Equations, 2013, 255: 2463–2483
Jiang Y, Zhou H. Schrödinger-Poisson system with steep potential well. J Differential Equations, 2011, 251: 582–608
Laskin N. Fractional quantum mechanics and Lévy path integrals. Phys Lett A, 2000, 268: 298–305
Laskin N. Fractional Schröodinger equation. Phys Rev E, 2002, 66: 56–108
Lions P L. Solutions of Hartree-Fock equations for Coulomb systems. Comm Math Phys, 1984, 109: 33–97
Liu Z, Zhang J. Multiplicity and concentration of positive solutions for the fractional Schröodinger-Poisson systems with critical growth. ESAIM Control Optim Calc Var, 2017, 23: 1515–1542
Milakis E, Silvestre L. Regularity for the nonlinear Signorini problem. Adv Math, 2008, 217: 1301–1312
Murcia E G, Siciliano G. Positive semiclassical states for a fractional Schrödinger-Poisson system. Differential Integral Equations, 2017, 30: 231–258
Ruiz D. The Schrödinger-Poisson equation under the effect of a nonlinear local term. J Funct Anal, 2006, 237: 655–674
Secchi S. Ground state solutions for nonlinear fractional Schrödinger equations in ℝN. J Math Phys, 2013, 54: 031501
Servadei R, Valdinoci E. The Brezis-Nirenberg result for the fractional Laplacian. Trans Amer Math Soc, 2015, 367: 67–102
Shang X, Zhang J. Ground states for fractional Schrödinger equations with critical growth. Nonlinearity, 2014, 27: 187–207
Silvestre L. Regularity of the obstacle problem for a fractional power of the Laplace operator. Comm Pure Appl Math, 2007, 60: 67–112
Sun J, Ma S. Ground state solutions for some Schrödinger-Poisson systems with periodic potentials. J Differential Equations, 2016, 260: 2119–2149
Tang X, Chen S. Ground state solutions of Nehari-Pohozaev type for Schröodinger-Poisson problems with general potentials. Discrete Contin Dyn Syst, 2017, 37: 4973–5002
Teng K. Ground state solutions for the nonlinear fractional Schrödinger-Poisson system. Appl Anal, 2019, 98: 1959–1996
Teng K. Existence of ground state solutions for the nonlinear fractional Schröodinger-Poisson system with critical Sobolev exponent. J Differential Equations, 2016, 261: 3061–3106
Teng K. Corrigendum to “Existence of ground state solutions for the nonlinear fractional Schrödinger-Poisson system with critical Sobolev exponent”. J Differential Equations, 2017, 262: 3132–3138
Willem M. Minimax Theorems. Basel: Birkhauser, 1996
Xu Y, Tan Z, Sun D. Multiplicity results for a nonlinear elliptic problem involving the fractional Laplacian. Acta Math Sci, 2016, 36B: 1793–1803
Zhang J. On the Schrödinger-Poisson equations with a general nonlinearity in the critical growth. Nonlinear Anal, 2012, 75: 6391–6401
Zhang J, do Ó J M, Squassina M. Fractional Schrödinger-Poisson systems with a general subcritical or critical nonlinearity. Adv Nonlinear Stud, 2016, 16: 15–30
Zhao L, Zhao F. On the existence of solutions for the Schröodinger-Poisson equations. J Math Anal Appl, 2008, 346: 155–169
Zhao L, Zhao F. Positive solutions for Schröodinger-Poisson equations with a critical exponent. Nonlinear Anal, 2009, 70: 2150–2164
Acknowledgements
The authors would like to thank Professor Yinbin Deng for useful suggestions and comments.
Author information
Authors and Affiliations
Corresponding author
Additional information
The first author was supported by the Science and Technology Project of Education Department in Jiangxi Province (GJJ180357) and the second author was supported by NSFC (11701178).
Rights and permissions
About this article
Cite this article
Huang, W., Wang, L. Ground state solutions of Nehari-Pohozaev type for a fractional Schrödinger-Poisson system with critical growth. Acta Math Sci 40, 1064–1080 (2020). https://doi.org/10.1007/s10473-020-0413-1
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10473-020-0413-1