Skip to main content Accessibility help
×
Home

ON A CLASS OF ELLIPTIC SYSTEM OF SCHRÖDINGER–POISSON TYPE

  • LUCAS C. F. FERREIRA (a1), EVERALDO S. MEDEIROS (a2) and MARCELO MONTENEGRO (a3)

Abstract

In this paper we prove existence and qualitative properties of solutions for a nonlinear elliptic system arising from the coupling of the nonlinear Schrödinger equation with the Poisson equation. We use a contraction map approach together with estimates of the Bessel potential used to rewrite the system in an integral form.

    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      ON A CLASS OF ELLIPTIC SYSTEM OF SCHRÖDINGER–POISSON TYPE
      Available formats
      ×

      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

      ON A CLASS OF ELLIPTIC SYSTEM OF SCHRÖDINGER–POISSON TYPE
      Available formats
      ×

      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

      ON A CLASS OF ELLIPTIC SYSTEM OF SCHRÖDINGER–POISSON TYPE
      Available formats
      ×

Copyright

Corresponding author

References

Hide All
[1]Ambrosetti, A. and Ruiz, D., ‘Multiple bound states for the Schrödinger–Poisson problem’, Commun. Contemp. Math. 10 (2008), 391404.
[2]Azzollini, A., ‘Concentration and compactness in nonlinear Schrödinger–Poisson system with a general nonlinearity’, J. Differential Equations 249 (2010), 17461763.
[3]Azzollini, A., D’Avenia, P. and Pomponio, A., ‘On the Schrödinger–Maxwell equations under the effect of a general nonlinear term’, Ann. Inst. H. Poincaré Anal. Non Linéaire 27 (2010), 779791.
[4]Azzollini, A., Pisani, L. and Pomponio, A., ‘Improved estimates and a limit case for the electrostatic Klein–Gordon–Maxwell system’, Proc. Roy. Soc. Edinburgh Sect. A 141 (2011), 449463.
[5]Azzollini, A. and Pomponio, A., ‘Ground state solutions for the nonlinear Schrödinger–Maxwell equations’, J. Math. Anal. Appl. 345 (2008), 90108.
[6]Benci, V. and Fortunato, D., ‘An eigenvalue problem for the Schrödinger–Maxwell equations’, Topol. Methods Nonlinear Anal. 11 (1998), 283293.
[7]Benci, V. and Fortunato, D., ‘Solitary waves of the nonlinear Klein–Gordon equation coupled with Maxwell equations’, Rev. Math. Phys. 14 (2002), 409420.
[8]Benguria, R., Brezis, H. and Lieb, E.-H., ‘The Thomas–Fermi–von Weizsäcker theory of atoms and molecules’, Comm. Math. Phys. 79 (1981), 167180.
[9]Candela, A. M. and Salvatore, A., ‘Multiple solitary waves for nonhomogeneous Schrödinger–Maxwell equations’, Mediterr. J. Math. 3 (2006), 483493.
[10]Carrillo, J. A., Ferreira, L. C. F. and Precioso, J. C., ‘A mass-transportation approach to a one dimensional fluid mechanics model with nonlocal velocity’, Adv. Math. 231 (2012), 306327.
[11]Cassani, D., ‘Existence and nonexistence of solitary waves for the critical Klein–Gordon equation coupled with Maxwell’s equations’, Nonlinear Anal. 58 (2004), 733747.
[12]Catto, I. and Lions, P.-L., ‘Binding of atoms and stability of molecules in Hartree and Thomas–Fermi type theories. Part 1: A necessary and sufficient condition for the stability of general molecular systems’, Comm. Partial Differential Equations 17 (1992), 10511110.
[13]Cerami, G. and Vaira, G., ‘Positive solutions for some nonautonomous Schrödinger–Poisson systems’, J. Differential Equations 248 (2010), 521543.
[14]Chen, S.-J. and Tang, C.-L., ‘Multiple solutions for nonhomogeneous Schrödinger–Maxwell and Klein–Gordon–Maxwell equations on ℝ3’, NoDEA Nonlinear Differential Equations Appl. 17 (2010), 559574.
[15]Coclite, G. M., ‘A multiplicity result for the nonlinear Schrödinger–Maxwell equations’, Commun. Appl. Anal. 7 (2003), 417423.
[16]D’Aprile, T. and Mugnai, D., ‘Nonexistence results for the coupled Klein–Gordon–Maxwell equations’, Adv. Nonlinear Stud. 4 (2004), 307322.
[17]D’Aprile, T. and Mugnai, D., ‘Solitary waves for nonlinear Klein–Gordon–Maxwell and Schrödinger–Maxwell equations’, Proc. Roy. Soc. Edinburgh Sect. A 134 (2004), 893906.
[18]D’Avenia, P., ‘Nonradially symmetric solutions of nonlinear Schödinger equation coupled with Maxwell equations’, Adv. Nonlinear Stud. 2 (2002), 177192.
[19]Ferreira, L. C. F. and Montenegro, M., ‘Existence and asymptotic behavior for elliptic equations with singular anisotropic potentials’, J. Differential Equations 250 (2011), 20452063.
[20]Ferreira, L. C. F. and Montenegro, M., ‘A Fourier approach for nonlinear equations with singular data’, Israel J. Math. 193(1) (2013), 83107.
[21]Ferreira, L. C. F., Medeiros, E. S. and Montenegro, M., ‘On the Laplace equation with a supercritical nonlinear Robin boundary condition in the half-space’, Calc. Var. Partial Differential Equations 47(3–4) (2013), 667682.
[22]Ferreira, L. C. F., Medeiros, E. S. and Montenegro, M., ‘A class of elliptic equations in anisotropic spaces’, Ann. Mat. Pura Appl. 192(4) (2013), 539552.
[23]Grafakos, L., Classical and Modern Fourier Analysis (Pearson Education, Upper Saddle River, NJ, 2004).
[24]Kikuchi, H., ‘On the existence of a solution for elliptic system related to the Maxwell–Schödinger equations’, Nonlinear Anal. 67 (2007), 14451456.
[25]Lieb, E. H., ‘Thomas–Fermi and related theories and molecules’, Rev. Modern Phys. 53 (1981), 603641.
[26]Lieb, E. H. and Simon, B., ‘The Thomas–Fermi theory of atoms, molecules and solids’, Adv. Math. 23 (1977), 22116.
[27]Lions, P.-L., ‘Solutions of Hartree–Fock equations for Coulomb systems’, Comm. Math. Phys. 109 (1984), 3397.
[28]Markowich, P., Ringhofer, C. and Schmeiser, C., Semiconductor Equations (Springer, New York, 1990).
[29]Mugnai, D., ‘The Schrödinger–Poisson system with positive potential’, Comm. Partial Differential Equations 36 (2011), 10991117.
[30]Ruiz, D., ‘Semiclassical states for coupled Schrödinger–Maxwell equations: concentration around a sphere’, Math. Models Methods Appl. Sci. 15 (2005), 141164.
[31]Ruiz, D., ‘The Schrödinger–Poisson equation under the effect of a nonlinear local term’, J. Funct. Anal. 237 (2006), 655674.
[32]Ruiz, D., ‘On the Schrödinger–Poisson–Slater system: behavior of minimizers, radial and nonradial cases’, Arch. Ration. Mech. Anal. 198 (2010), 349368.
[33]Salvatore, A., ‘Multiple solitary waves for a nonhomogeneous Schrödinger–Maxwell system in ℝ3’, Adv. Nonlinear Stud. 6 (2006), 157169.
[34]Siciliano, G., ‘Multiple positive solutions for a Schrödinger–Poisson–Slater system’, J. Math. Anal. Appl. 365 (2010), 288299.
[35]Stein, E. S., Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, 30 (Princeton University Press, Princeton, NJ, 1970).
[36]Sun, J., Chen, H. and Nieto, J. J., ‘On ground state solutions for some nonautonomous Schrödinger–Poisson systems’, J. Differential Equations 252 (2012), 33653380.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

Keywords

MSC classification

ON A CLASS OF ELLIPTIC SYSTEM OF SCHRÖDINGER–POISSON TYPE

  • LUCAS C. F. FERREIRA (a1), EVERALDO S. MEDEIROS (a2) and MARCELO MONTENEGRO (a3)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed