Publication Cover
Applicable Analysis
An International Journal
Volume 97, 2018 - Issue 7
357
Views
6
CrossRef citations to date
0
Altmetric
Articles

Existence of positive ground state solutions for fractional Schrödinger equations with a general nonlinearity

&
Pages 1154-1171 | Received 04 Oct 2016, Accepted 08 Mar 2017, Published online: 30 Mar 2017

References

  • Di Nezza E , Palatucci G , Valdinoci E . Hitchhiker’s guide to the fractional Sobolev spaces. Bull Sci Math. 2012;136:521–573.
  • Laskin N . Fractional Schrödinger equation. Phy Rev E. 2014;66:05618.
  • Ackermann N . Lévy processes-from probability theory to finance and quantum groups. Not Am Math Soc. 2004;51:1320–1331.
  • Strauss W . Existence of solitary waves in higher dimensions. Commun Math Phys. 1977;55:149–162.
  • Lions P . The concentration compactness principle in the calculus of variations: the locally compact case. Parts 1. Ann Inst H Poincar Anal Non Linaire. 1984;1:109–145.
  • Lions P. The concentration compactness principle in the calculus of variations: the locally compact case. Part 2. Ann Inst H Poincar Anal Non Linaire. 1984;2:223–283.
  • Rabinowitz P . On a class of nonlinear Schrödinger equations. Z Angew Math Phys. 1992;43:270–291.
  • Wang X . On concentration of positive bound states of nonlinear Schrödinger equations. Commun Math Phys. 1993;153:229–244.
  • Cabré X , Tan J . Positive solutions of nonlinear problems involving the square root of the Laplacian. Adv Math. 2010;224:2052–2093.
  • Servadei R , Valdinoci E . The Brezis–Nirenberg result for the fractional Laplacian. Trans Am Math Soc. 2015;367:67–102.
  • Dipierro S , Palatucci G , Valdinoci E . Existence and symmetry results for a Schrödinger type problem involving the fractional Laplacian. Le Matematiche. 2013;68:201–216.
  • Felmer P , Wang Y . Radial symmetry of positive solutions to equations involving the fractional Laplacian. Commun Contemp Math. 2014;16:1350023. 24p.
  • Frank R , Lenzmann E . Uniqueness and non degeneracy of ground states for (−Δ) s +Q−Q α+1 in ℝ. Acta Math. 2013;210:261–318.
  • Coti Zelati V , Nolasco M . Existence of ground states for nonlinear, pseudo-relativistic Schrödinger equations. Rend Lincei Mat Appl. 2011;22:51–72.
  • Cheng M . Bound state for the fractional Schrödinger equation with unbounded potential. J Math Phys. 2012;53:043507.
  • He X , Squassina M , Zou W . The Nehari manifold for fractional systems involving critical nonlinearities. Commun Pure Appl Anal. 2016;15:1285–1308.
  • Yang L , Liu Z . Multiplicity and concentration of solutions for fractional Schrödinger equation with sublinear perturbation and steep potential well. Comput Math Appl. 2016;72:1629–1640.
  • Shang X , Zhang J . On fractional Schrödinger equations in ℝ N with critical growth. J Math Phys. 2013;54:121502.
  • Fall M , Mahmoudi F , Valdinoci E . Ground states and concentration phenomena for the fractional Schrödinger equation. Nonlinearity. 2015;28:1937–1961.
  • Liu Z , Zhang J . Multiplicity and concentration of positive solutions for the fractional Schrödinger–Poisson systems with critical growth, ESAIM: Control. Optim Calc Variations. 2016, 27 pages. DOI:10.1051/cocv/2016063
  • Liu Z , Squassina M , Zhang J . Ground states for fractional Kirchhoff equations with critical nonlinearity in low dimension. 2016. arXiv:1612.07914.
  • Felmer P , Quaas A , Tan J . Positive solutions of the nonlinear Schrödinger equation with the fractional Laplacian. Proc Roy Soc Edinburgh Sect A. 2012;142:1237–1262.
  • Secchi S . Ground state solutions for nonlinear fractional Schrödinger equations in ℝ N . J Math Phys. 2013;54:031501.
  • Secchi S . On fractional Schrödinger equations in ℝ N without the Ambrosetti–Rabinowitz condition. Topol Methods Nonlinear Anal. 2016;47:19–41.
  • Chang X . Ground state solutions of asymptotically linear fractional Schrödinger equations. J Math Phys. 2013;54:061504.
  • Chang X . Ground states of some fractional Schrödinger equations on ℝ N . Proc Edinb Math Soc. 2015;58:305–321.
  • Zhang JJ , Marcos do Ó J , Squassina M . Fractional Schrödinger–Poisson systems with a general subcritical or critical nonlinearity. Adv Nonlinear Stud. 2016;16:15–30.
  • Chang X , Wang Z . Ground state of scalar field equations involving a fractional Laplacian with general nonlinearity. Nonlinearity. 2013;26:479–494.
  • Berestycki H , Lions P. Nonlinear scalar field equations I Existence of a ground state. Arch Ration Mech Anal. 1983;82:313–345.
  • Berestycki H , Lions P . Nonlinear scalar field equations II Existence of infinitely many solutions. Arch Ration Mech Anal. 1983;82:347–375.
  • Jeanjean L , Tanaka K . A remark on least energy solutions in ℝ N . Proc Am Math Soc. 2003;131:2399–408.
  • Jeanjean L . On the existence of bounded Palais–Smale sequence and application to a Landesman–Lazer type problem set on ℝ N . Proc Roy Soc Edinburgh Sect A. 1999;129:787–809.
  • Lions L . Symétrie et compacité dans les espaces de Sobolev [Symmetry and compactness of Sobolev spaces]. J Funct Anal. 1982;49:315–334.
  • Leonori T , Peral I , Primo A , et al . Basic estimates for solutions of a class of nonlocal elliptic and parabolic equations. Discrete Contin Dyn Syst. 2015;35:60316068.
  • Barrios B , Colorado E , Servadei R , et al . A critical fractional equation with concave-convex power nonlinearities. Ann Inst H Poincar Anal Non Linaire. 2015;32:875–900.
  • Dipierro S , Montoro L , Peral I , et al . Qualitative properties of positive solutions to nonlocal critical problems involving the Hardy–Leray potential. Calc Var Partial Differ Equ. 2016;55:29.
  • Brezis H , Lieb E . A relation between pointwise convergence of functions and convergence of functionals. Proc Am Math Soc. 1983;8:486–490.
  • Liu Z , Guo S . Existence of positive solutions for Kirchhoff-type problems. Nonlinear Anal. 2015;120:1–13.
  • Dipierro S , Medina M , Valdinoci E . Fractional elliptic problems with critical growth in the whole of ℝ N . 2016. arXiv:1506.01748v1.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.