American Institute of Mathematical Sciences

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April  2016, 36(4): 2257-2284. doi: 10.3934/dcds.2016.36.2257

Schrödinger-Poisson systems in $4$-dimensional closed manifolds

Pierre-Damien Thizy 1,

1. 

Université de Cergy-Pontoise, CNRS, Département de Mathématiques, F-95000 Cergy-Pontoise

Received  January 2015 Revised  April 2015 Published  September 2015

We investigate existence, nonexistence and uniqueness of positive solutions of critical Schrödinger-Poisson systems in closed $4$-manifolds. In the process we provide a sharp criterion for the non-existence of resonant states.
Keywords: Nonlinear elliptic PDE's, Schrödinger-Poisson, solitary waves., non-resonant states, analysis on manifolds, critical systems.
Mathematics Subject Classification: Primary: 58J05; Secondary: 35J47, 35Q61, 35R01, 58J37, 81Q3.
Citation: Pierre-Damien Thizy. Schrödinger-Poisson systems in $4$-dimensional closed manifolds. Discrete & Continuous Dynamical Systems - A, 2016, 36 (4) : 2257-2284. doi: 10.3934/dcds.2016.36.2257
References:
[1]

C. O. Alves and M. A. S. Souto, Existence of least energy nodal solution for a Schrödinger-Poisson system in bounded domains,, Z. Angew. Math. Phys., 65 (2014), 1153.  doi: 10.1007/s00033-013-0376-3.  Google Scholar

[2]

A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications,, J. Functional Analysis, 14 (1973), 349.  doi: 10.1016/0022-1236(73)90051-7.  Google Scholar

[3]

T. Aubin, Équations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire,, J. Math. Pures Appl. (9), 55 (1976), 269.   Google Scholar

[4]

A. Azzollini, P. d'Avenia and V. Luisi, Generalized Schrödinger-Poisson type systems,, Commun. Pure Appl. Anal., 12 (2013), 867.  doi: 10.3934/cpaa.2013.12.867.  Google Scholar

[5]

V. Benci and C. Bonanno, Solitary waves and vortices in non-Abelian gauge theories with matter,, Adv. Nonlinear Stud., 12 (2012), 717.   Google Scholar

[6]

V. Benci and D. Fortunato, Solitary waves of the nonlinear Klein-Gordon equation coupled with the Maxwell equations,, Rev. Math. Phys., 14 (2002), 409.  doi: 10.1142/S0129055X02001168.  Google Scholar

[7]

V. Benci and D. Fortunato, Solitary waves in abelian gauge theories,, Adv. Nonlinear Stud., 8 (2008), 327.   Google Scholar

[8]

C. Bonanno, Existence and multiplicity of stable bound states for the nonlinear Klein-Gordon equation,, Nonlinear Anal., 72 (2010), 2031.  doi: 10.1016/j.na.2009.10.004.  Google Scholar

[9]

C. Bonanno, Solitons in gauge theories: Existence and dependence on the charge,, Adv. Nonlinear Anal., 3 (2014).  doi: 10.1515/anona-2013-0032.  Google Scholar

[10]

H. Brezis and Y. Li, Some nonlinear elliptic equations have only constant solutions,, J. Partial Differential Equations, 19 (2006), 208.   Google Scholar

[11]

H. Brézis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents,, Comm. Pure Appl. Math., 36 (1983), 437.  doi: 10.1002/cpa.3160360405.  Google Scholar

[12]

L. A. Caffarelli, B. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth,, Comm. Pure Appl. Math., 42 (1989), 271.  doi: 10.1002/cpa.3160420304.  Google Scholar

[13]

A. M. Candela and A. Salvatore, Multiple solitary waves for non-homogeneous Schrödinger-Maxwell equations,, Mediterr. J. Math., 3 (2006), 483.  doi: 10.1007/s00009-006-0092-8.  Google Scholar

[14]

G. M. Coclite and H. Holden, The Schrödinger-Maxwell system with Dirac mass,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 773.  doi: 10.1016/j.anihpc.2006.06.005.  Google Scholar

[15]

O. Druet and E. Hebey, Existence and a priori bounds for electrostatic Klein-Gordon-Maxwell systems in fully inhomogeneous spaces,, Commun. Contemp. Math., 12 (2010), 831.  doi: 10.1142/S0219199710004007.  Google Scholar

[16]

O. Druet, E. Hebey and F. Robert, Blow-up Theory for Elliptic PDEs in Riemannian Geometry, vol. 45 of Mathematical Notes,, Princeton University Press, (2004).  doi: 10.1007/BF01158557.  Google Scholar

[17]

O. Druet and B. Premoselli, Stability of the Einstein-Lichnerowicz constraints system,, Mathematische Annalen, 362 (2015), 839.  doi: 10.1007/s00208-014-1145-0.  Google Scholar

[18]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations Of Second Order,, Classics in Mathematics, (2001).   Google Scholar

[19]

Q. Han and F. Lin, Elliptic Partial Differential Equations, vol. 1 of Courant Lecture Notes in Mathematics,, 2nd edition, (2011).   Google Scholar

[20]

E. Hebey, Compactness and Stability for Nonlinear Elliptic Equations,, Zurich Lectures in Advanced Mathematics, (2014).  doi: 10.4171/134.  Google Scholar

[21]

E. Hebey and T. T. Truong, Static Klein-Gordon-Maxwell-Proca systems in 4-dimensional closed manifolds,, J. Reine Angew. Math., 667 (2012), 221.   Google Scholar

[22]

E. Hebey and M. Vaugon, Sobolev spaces in the presence of symmetries,, J. Math. Pures Appl. (9), 76 (1997), 859.  doi: 10.1016/S0021-7824(97)89975-8.  Google Scholar

[23]

E. Hebey and J. Wei, Schrödinger-Poisson systems in the 3-sphere,, Calc. Var. Partial Differential Equations, 47 (2013), 25.  doi: 10.1007/s00526-012-0509-0.  Google Scholar

[24]

I. Ianni and G. Vaira, On concentration of positive bound states for the Schrödinger-Poisson problem with potentials,, Adv. Nonlinear Stud., 8 (2008), 573.   Google Scholar

[25]

Y. Li and M. Zhu, Yamabe type equations on three-dimensional Riemannian manifolds,, Commun. Contemp. Math., 1 (1999), 1.  doi: 10.1142/S021919979900002X.  Google Scholar

[26]

L. Pisani and G. Siciliano, Neumann condition in the Schrödinger-Maxwell system,, Topol. Methods Nonlinear Anal., 29 (2007), 251.   Google Scholar

[27]

L. Pisani and G. Siciliano, Some results on the Schrödinger-Poisson system in a bounded domain,, in Dynamic systems and applications, 5 (2008), 402.   Google Scholar

[28]

D. Ruiz and G. Siciliano, A note on the Schrödinger-Poisson-Slater equation on bounded domains,, Adv. Nonlinear Stud., 8 (2008), 179.   Google Scholar

[29]

P.-D. Thizy, Blow-up for Schrödinger-Poisson critical systems in dimensions 4 and 5,, Preprint., ().   Google Scholar

[30]

P.-D. Thizy, Klein-Gordon-Maxwell equations in high dimensions,, Communications on Pure and Applied Analysis, 14 (2015), 1097.  doi: 10.3934/cpaa.2015.14.1097.  Google Scholar

[31]

P.-D. Thizy, Non resonant states for Schrödinger-Poisson critical systems in high dimension,, Archiv der Math., 104 (2015), 485.  doi: 10.1007/s00013-015-0763-4.  Google Scholar

[32]

N. S. Trudinger, Remarks concerning the conformal deformation of Riemannian structures on compact manifolds,, Ann. Scuola Norm. Sup. Pisa (3), 22 (1968), 265.   Google Scholar

[33]

P. Zhang and J. Sun, Clustered layers for the Schrödinger-Maxwell system on a ball,, Discrete Contin. Dyn. Syst., 16 (2006), 657.  doi: 10.3934/dcds.2006.16.657.  Google Scholar

show all references

References:
[1]

C. O. Alves and M. A. S. Souto, Existence of least energy nodal solution for a Schrödinger-Poisson system in bounded domains,, Z. Angew. Math. Phys., 65 (2014), 1153.  doi: 10.1007/s00033-013-0376-3.  Google Scholar

[2]

A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications,, J. Functional Analysis, 14 (1973), 349.  doi: 10.1016/0022-1236(73)90051-7.  Google Scholar

[3]

T. Aubin, Équations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire,, J. Math. Pures Appl. (9), 55 (1976), 269.   Google Scholar

[4]

A. Azzollini, P. d'Avenia and V. Luisi, Generalized Schrödinger-Poisson type systems,, Commun. Pure Appl. Anal., 12 (2013), 867.  doi: 10.3934/cpaa.2013.12.867.  Google Scholar

[5]

V. Benci and C. Bonanno, Solitary waves and vortices in non-Abelian gauge theories with matter,, Adv. Nonlinear Stud., 12 (2012), 717.   Google Scholar

[6]

V. Benci and D. Fortunato, Solitary waves of the nonlinear Klein-Gordon equation coupled with the Maxwell equations,, Rev. Math. Phys., 14 (2002), 409.  doi: 10.1142/S0129055X02001168.  Google Scholar

[7]

V. Benci and D. Fortunato, Solitary waves in abelian gauge theories,, Adv. Nonlinear Stud., 8 (2008), 327.   Google Scholar

[8]

C. Bonanno, Existence and multiplicity of stable bound states for the nonlinear Klein-Gordon equation,, Nonlinear Anal., 72 (2010), 2031.  doi: 10.1016/j.na.2009.10.004.  Google Scholar

[9]

C. Bonanno, Solitons in gauge theories: Existence and dependence on the charge,, Adv. Nonlinear Anal., 3 (2014).  doi: 10.1515/anona-2013-0032.  Google Scholar

[10]

H. Brezis and Y. Li, Some nonlinear elliptic equations have only constant solutions,, J. Partial Differential Equations, 19 (2006), 208.   Google Scholar

[11]

H. Brézis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents,, Comm. Pure Appl. Math., 36 (1983), 437.  doi: 10.1002/cpa.3160360405.  Google Scholar

[12]

L. A. Caffarelli, B. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth,, Comm. Pure Appl. Math., 42 (1989), 271.  doi: 10.1002/cpa.3160420304.  Google Scholar

[13]

A. M. Candela and A. Salvatore, Multiple solitary waves for non-homogeneous Schrödinger-Maxwell equations,, Mediterr. J. Math., 3 (2006), 483.  doi: 10.1007/s00009-006-0092-8.  Google Scholar

[14]

G. M. Coclite and H. Holden, The Schrödinger-Maxwell system with Dirac mass,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 773.  doi: 10.1016/j.anihpc.2006.06.005.  Google Scholar

[15]

O. Druet and E. Hebey, Existence and a priori bounds for electrostatic Klein-Gordon-Maxwell systems in fully inhomogeneous spaces,, Commun. Contemp. Math., 12 (2010), 831.  doi: 10.1142/S0219199710004007.  Google Scholar

[16]

O. Druet, E. Hebey and F. Robert, Blow-up Theory for Elliptic PDEs in Riemannian Geometry, vol. 45 of Mathematical Notes,, Princeton University Press, (2004).  doi: 10.1007/BF01158557.  Google Scholar

[17]

O. Druet and B. Premoselli, Stability of the Einstein-Lichnerowicz constraints system,, Mathematische Annalen, 362 (2015), 839.  doi: 10.1007/s00208-014-1145-0.  Google Scholar

[18]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations Of Second Order,, Classics in Mathematics, (2001).   Google Scholar

[19]

Q. Han and F. Lin, Elliptic Partial Differential Equations, vol. 1 of Courant Lecture Notes in Mathematics,, 2nd edition, (2011).   Google Scholar

[20]

E. Hebey, Compactness and Stability for Nonlinear Elliptic Equations,, Zurich Lectures in Advanced Mathematics, (2014).  doi: 10.4171/134.  Google Scholar

[21]

E. Hebey and T. T. Truong, Static Klein-Gordon-Maxwell-Proca systems in 4-dimensional closed manifolds,, J. Reine Angew. Math., 667 (2012), 221.   Google Scholar

[22]

E. Hebey and M. Vaugon, Sobolev spaces in the presence of symmetries,, J. Math. Pures Appl. (9), 76 (1997), 859.  doi: 10.1016/S0021-7824(97)89975-8.  Google Scholar

[23]

E. Hebey and J. Wei, Schrödinger-Poisson systems in the 3-sphere,, Calc. Var. Partial Differential Equations, 47 (2013), 25.  doi: 10.1007/s00526-012-0509-0.  Google Scholar

[24]

I. Ianni and G. Vaira, On concentration of positive bound states for the Schrödinger-Poisson problem with potentials,, Adv. Nonlinear Stud., 8 (2008), 573.   Google Scholar

[25]

Y. Li and M. Zhu, Yamabe type equations on three-dimensional Riemannian manifolds,, Commun. Contemp. Math., 1 (1999), 1.  doi: 10.1142/S021919979900002X.  Google Scholar

[26]

L. Pisani and G. Siciliano, Neumann condition in the Schrödinger-Maxwell system,, Topol. Methods Nonlinear Anal., 29 (2007), 251.   Google Scholar

[27]

L. Pisani and G. Siciliano, Some results on the Schrödinger-Poisson system in a bounded domain,, in Dynamic systems and applications, 5 (2008), 402.   Google Scholar

[28]

D. Ruiz and G. Siciliano, A note on the Schrödinger-Poisson-Slater equation on bounded domains,, Adv. Nonlinear Stud., 8 (2008), 179.   Google Scholar

[29]

P.-D. Thizy, Blow-up for Schrödinger-Poisson critical systems in dimensions 4 and 5,, Preprint., ().   Google Scholar

[30]

P.-D. Thizy, Klein-Gordon-Maxwell equations in high dimensions,, Communications on Pure and Applied Analysis, 14 (2015), 1097.  doi: 10.3934/cpaa.2015.14.1097.  Google Scholar

[31]

P.-D. Thizy, Non resonant states for Schrödinger-Poisson critical systems in high dimension,, Archiv der Math., 104 (2015), 485.  doi: 10.1007/s00013-015-0763-4.  Google Scholar

[32]

N. S. Trudinger, Remarks concerning the conformal deformation of Riemannian structures on compact manifolds,, Ann. Scuola Norm. Sup. Pisa (3), 22 (1968), 265.   Google Scholar

[33]

P. Zhang and J. Sun, Clustered layers for the Schrödinger-Maxwell system on a ball,, Discrete Contin. Dyn. Syst., 16 (2006), 657.  doi: 10.3934/dcds.2006.16.657.  Google Scholar

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