Publication Cover
Applicable Analysis
An International Journal
Volume 101, 2022 - Issue 17
99
Views
1
CrossRef citations to date
0
Altmetric
Research Article

Solutions for a class of fractional Hamiltonian systems with exponential growth

Pages 6042-6058 | Received 28 Dec 2018, Accepted 25 Feb 2020, Published online: 27 Apr 2021

References

  • Ozawa T. On critical cases of Sobolev's inequalities. J Funct Anal. 1995;127:259–269.
  • Applebaum D. Lévy processes-from probability to finance quantum groups. Notices Am Math Soc. 2004;51:1336–1347.
  • Caffarelli LA. Non-local diffusions, drifts and games. In: Nonlinear partial differential equations. Heidelberg: Springer; 2012. p. 37–52. (Abel symposium; 7).
  • Laskin N. Fractional quantum mechanics and Lévy path integrals. Phys Lett A. 2000;268:298–305.
  • Laskin N. Fractional Schrödinger equation. Phys Rev E. 2002;66:056108.
  • Chang X, Wang Z-Q. Ground state of scalar field equations involving a fractional Laplacian with general nonlinearity. Nonlinearity. 2013;26:479–494.
  • Cheng M. Bound state for the fractional Schrödinger equation with unbounded potential. J Math Phys. 2012;53:043507.
  • do Ó JM, Miyagaki OH, Squassina M. Critical and subcritical fractional problems with vanishing potentials. Commun Contemp Math. 2016;18:1550063.
  • Dipierro S, Palatucci G, Valdinoci E. Existence and symmetry results for a Schrödinger type problem involving the fractional Laplacian. Matematiche. 2013;68:201–216.
  • Felmer P, Quaas A, Tan J. Positive solutions of nonlinear Schrödinger equation with the fractional Laplacian. Proc R Soc Edinb A Math. 2012;142:1237–1262.
  • Shang X, Zhang J, Yang Y. On fractional Schödinger equation in RN with critical growth. J Math Phys. 2013;54:Article ID 121502, 19 pp.
  • Secchi S. Ground state solutions for nonlinear fractional Schrödinger equations in Rn. J Math Phys. 2013;54:031501.
  • de Albuquerque JC, Araújo YLR, Clemente R. Existence of bound and ground states for a class of Kirchhoff–Schrödinger equations involving critical Trudinger–Moser growth. Math Meth Appl Sci. 2019;42:806–820.
  • de Souza M, Araújo YLR. On nonlinear perturbations of a periodic fractional Schrödinger equation with critical exponential growth. Math Nachr. 2016;289:610–625.
  • de Souza M, Araújo YLR. Semilinear elliptic equations for the fractional Laplacian involving critical exponential growth. Math Methods Appl Sci. 2017;40:1757–1772.
  • do Ó JM, Miyagaki OH, Squassina M. Ground states of nonlocal scalar field equations with Trudinger–Moser critical nonlinearity. Topol Methods Nonlinear Anal. 2016;48:477–492.
  • Guo Z, Luo S, Zou W. On critical systems involving fractional Laplacian. J Math Anal Appl. 2017;446:681–706.
  • He X, Squassina M, Zou W. The Nehari manifold for fractional systems involving critical nonlinearities. Commun Pure Appl Anal. 2016;15:1285–1308.
  • Lu DF, Peng SJ. On the positive vector solutions for nonlinear fractional Laplacian system with linear coupling. Discrete Contin Dys Syst. 2017;37:3327–3352.
  • Wang Q. Positive least energy solutions of fractional Laplacian systems with critical exponent. Electron J Differ Equ. 2016;150:1–16.
  • Xiang M, Zhang B, Wei Z. Existence of solutions to a class of quasilinear Schrödinger system involving the fractional p-Laplacian. Electron J Qual Theory Differ Equ. 2016;107:1–15.
  • Zhen MD, He JC, Xu HY. Critical system involving fractional Laplacian. Commun Pure Appl Anal. 2019;1:237–253.
  • do Ó JM, de Albuquerque JC. Coupled elliptic systems involving the square root of the Laplacian and Trudinger–Moser critical growth. Differ Integral Equ. 2018;31:403–434.
  • do Ó JM, de Albuquerque JC. Positive ground state of coupled systems of Schrödinger equations in R2 involving critical exponential growth. Math Methods Appl Sci. 2017;40:6864–6879.
  • do Ó JM, de Albuquerque JC. On coupled systems of nonlinear Schrödinger equations with critical exponential growth. Appl Anal. 2018;97:1000–1015.
  • Giacomoni J, Pawan KM, Sreenadh K. Critical growth fractional elliptic systems with exponential nonlinearity. Nonlinear Anal: Theory Methods Appl. 2016;136:117–135.
  • Giacomoni J, Pawan KM, do Ó JM. Nonautonomous fractional Hamiltonian system with critical exponential; 2018. arXiv:1811.04368v1
  • de Figueiredo DG, do Ó JM, Ruf B. Critical and subcritical elliptic systems in dimension two. Indiana Univ Math J. 2004;53:1037–1054.
  • de Souza M, do Ó JM. Hamiltonian elliptic systems in R2 with subcritical and critical exponential growth. Ann Mate Pura Appl. 2016;95:935–956.
  • de Souza M, Melo WG. On a class of Hamiltonian systems with Trudinger–Moser nonlinearities. Adv Differ Equ. 2015;20:233–258.
  • de Souza M. On a singular Hamiltonian elliptic systems involving critical growth in dimension two. Commun Pure Appl Anal. 2012;11:1859–1874.
  • de Figueiredo DG, do Ó JM, Ruf B. Elliptic equations and systems with critical Trudinger–Moser nonlinearities. Discrete Cont Dyn Syst. 2011;30:455–476.
  • Rabinowitz PH. Minimax methods in critical point theory with applications to differential equations. Providence (RI): AMS; 1986. (CBMS regional conference series in mathematics; vol. 65).
  • Rabinowitz PH. Some global results for nonlinear eigenvalue problems. J Funct Anal. 1971;7:487–513.
  • Rabinowitz PH. Periodic solutions of a Hamiltonian system on a prescribed energy surface. J Differ Equ. 1979;33:363–52.
  • Kozono H, Sato T, Wadade H. Upper bound of the best constant of a Trudinger–Moser inequality and its application to a Gagliardo–Nirenberg inequality. Indiana Univ Math J. 2006;55:1951–1974.
  • Moser J. A sharp form of an inequality by N. Trudinger. Indiana Univ Math J. 1970/71;20:1077–1092.
  • Trudinger NS. On the embedding into Orlicz spaces and some applications. J Math Mech. 1967;17:473–484.
  • Adimurthi. Existence of positive solutions of the semilinear Dirichlet problem with critical growth for the n-Laplacian. Ann Scuola Norm Sup Pisa Cl Sci. 1990;17:393–413.
  • Cao DM. Nontrivial solution of semilinear elliptic equation with critical exponent in R2. Commun Partial Differ Equ. 1992;17:407–435.
  • de Figueiredo DG, Miyagaki OH, Ruf B. Elliptic equations in R2 with nonlinearities in the critical growth range. Calc Var Partial Differ Equ. 1995;3:139–153.
  • Servadei R, Valdinoci E. Weak and viscosity solutions of the fractional Laplace equation. Publ Mat. 2014;58:133–154.
  • Martinazzi L. Fractional Adams–Moser–Trudinger type inequalities. Nonlinear Anal. 2015;127:263–278.

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.