# American Institute of Mathematical Sciences

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

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

 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.
Citation: Pierre-Damien Thizy. Schrödinger-Poisson systems in $4$-dimensional closed manifolds. Discrete and Continuous Dynamical Systems, 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), 1153-1166. doi: 10.1007/s00033-013-0376-3. [2] A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Functional Analysis, 14 (1973), 349-381. doi: 10.1016/0022-1236(73)90051-7. [3] T. Aubin, Équations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire, J. Math. Pures Appl. (9), 55 (1976), 269-296. [4] A. Azzollini, P. d'Avenia and V. Luisi, Generalized Schrödinger-Poisson type systems, Commun. Pure Appl. Anal., 12 (2013), 867-879. doi: 10.3934/cpaa.2013.12.867. [5] V. Benci and C. Bonanno, Solitary waves and vortices in non-Abelian gauge theories with matter, Adv. Nonlinear Stud., 12 (2012), 717-735. [6] V. Benci and D. Fortunato, Solitary waves of the nonlinear Klein-Gordon equation coupled with the Maxwell equations, Rev. Math. Phys., 14 (2002), 409-420. doi: 10.1142/S0129055X02001168. [7] V. Benci and D. Fortunato, Solitary waves in abelian gauge theories, Adv. Nonlinear Stud., 8 (2008), 327-352. [8] C. Bonanno, Existence and multiplicity of stable bound states for the nonlinear Klein-Gordon equation, Nonlinear Anal., 72 (2010), 2031-2046. doi: 10.1016/j.na.2009.10.004. [9] C. Bonanno, Solitons in gauge theories: Existence and dependence on the charge, Adv. Nonlinear Anal., 3 (2014), s1-s18. doi: 10.1515/anona-2013-0032. [10] H. Brezis and Y. Li, Some nonlinear elliptic equations have only constant solutions, J. Partial Differential Equations, 19 (2006), 208-217. [11] H. Brézis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math., 36 (1983), 437-477. doi: 10.1002/cpa.3160360405. [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), 271-297. doi: 10.1002/cpa.3160420304. [13] A. M. Candela and A. Salvatore, Multiple solitary waves for non-homogeneous Schrödinger-Maxwell equations, Mediterr. J. Math., 3 (2006), 483-493. doi: 10.1007/s00009-006-0092-8. [14] G. M. Coclite and H. Holden, The Schrödinger-Maxwell system with Dirac mass, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 773-793. doi: 10.1016/j.anihpc.2006.06.005. [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), 831-869. doi: 10.1142/S0219199710004007. [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, Princeton, NJ, 2004. doi: 10.1007/BF01158557. [17] O. Druet and B. Premoselli, Stability of the Einstein-Lichnerowicz constraints system, Mathematische Annalen, 362 (2015), 839-886. doi: 10.1007/s00208-014-1145-0. [18] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations Of Second Order, Classics in Mathematics, Springer-Verlag, Berlin, 2001, Reprint of the 1998 edition. [19] Q. Han and F. Lin, Elliptic Partial Differential Equations, vol. 1 of Courant Lecture Notes in Mathematics, 2nd edition, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2011. [20] E. Hebey, Compactness and Stability for Nonlinear Elliptic Equations, Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Zürich, 2014. doi: 10.4171/134. [21] E. Hebey and T. T. Truong, Static Klein-Gordon-Maxwell-Proca systems in 4-dimensional closed manifolds, J. Reine Angew. Math., 667 (2012), 221-248. [22] E. Hebey and M. Vaugon, Sobolev spaces in the presence of symmetries, J. Math. Pures Appl. (9), 76 (1997), 859-881. doi: 10.1016/S0021-7824(97)89975-8. [23] E. Hebey and J. Wei, Schrödinger-Poisson systems in the 3-sphere, Calc. Var. Partial Differential Equations, 47 (2013), 25-54. doi: 10.1007/s00526-012-0509-0. [24] I. Ianni and G. Vaira, On concentration of positive bound states for the Schrödinger-Poisson problem with potentials, Adv. Nonlinear Stud., 8 (2008), 573-595. [25] Y. Li and M. Zhu, Yamabe type equations on three-dimensional Riemannian manifolds, Commun. Contemp. Math., 1 (1999), 1-50. doi: 10.1142/S021919979900002X. [26] L. Pisani and G. Siciliano, Neumann condition in the Schrödinger-Maxwell system, Topol. Methods Nonlinear Anal., 29 (2007), 251-264. [27] L. Pisani and G. Siciliano, Some results on the Schrödinger-Poisson system in a bounded domain, in Dynamic systems and applications, Dynamic, Atlanta, GA, 5 (2008), 402-406. [28] D. Ruiz and G. Siciliano, A note on the Schrödinger-Poisson-Slater equation on bounded domains, Adv. Nonlinear Stud., 8 (2008), 179-190. [29] P.-D. Thizy, Blow-up for Schrödinger-Poisson critical systems in dimensions 4 and 5,, Preprint., (). [30] P.-D. Thizy, Klein-Gordon-Maxwell equations in high dimensions, Communications on Pure and Applied Analysis, 14 (2015), 1097-1125. doi: 10.3934/cpaa.2015.14.1097. [31] P.-D. Thizy, Non resonant states for Schrödinger-Poisson critical systems in high dimension, Archiv der Math., 104 (2015), 485-490. doi: 10.1007/s00013-015-0763-4. [32] N. S. Trudinger, Remarks concerning the conformal deformation of Riemannian structures on compact manifolds, Ann. Scuola Norm. Sup. Pisa (3), 22 (1968), 265-274. [33] P. Zhang and J. Sun, Clustered layers for the Schrödinger-Maxwell system on a ball, Discrete Contin. Dyn. Syst., 16 (2006), 657-688. doi: 10.3934/dcds.2006.16.657.

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), 1153-1166. doi: 10.1007/s00033-013-0376-3. [2] A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Functional Analysis, 14 (1973), 349-381. doi: 10.1016/0022-1236(73)90051-7. [3] T. Aubin, Équations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire, J. Math. Pures Appl. (9), 55 (1976), 269-296. [4] A. Azzollini, P. d'Avenia and V. Luisi, Generalized Schrödinger-Poisson type systems, Commun. Pure Appl. Anal., 12 (2013), 867-879. doi: 10.3934/cpaa.2013.12.867. [5] V. Benci and C. Bonanno, Solitary waves and vortices in non-Abelian gauge theories with matter, Adv. Nonlinear Stud., 12 (2012), 717-735. [6] V. Benci and D. Fortunato, Solitary waves of the nonlinear Klein-Gordon equation coupled with the Maxwell equations, Rev. Math. Phys., 14 (2002), 409-420. doi: 10.1142/S0129055X02001168. [7] V. Benci and D. Fortunato, Solitary waves in abelian gauge theories, Adv. Nonlinear Stud., 8 (2008), 327-352. [8] C. Bonanno, Existence and multiplicity of stable bound states for the nonlinear Klein-Gordon equation, Nonlinear Anal., 72 (2010), 2031-2046. doi: 10.1016/j.na.2009.10.004. [9] C. Bonanno, Solitons in gauge theories: Existence and dependence on the charge, Adv. Nonlinear Anal., 3 (2014), s1-s18. doi: 10.1515/anona-2013-0032. [10] H. Brezis and Y. Li, Some nonlinear elliptic equations have only constant solutions, J. Partial Differential Equations, 19 (2006), 208-217. [11] H. Brézis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math., 36 (1983), 437-477. doi: 10.1002/cpa.3160360405. [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), 271-297. doi: 10.1002/cpa.3160420304. [13] A. M. Candela and A. Salvatore, Multiple solitary waves for non-homogeneous Schrödinger-Maxwell equations, Mediterr. J. Math., 3 (2006), 483-493. doi: 10.1007/s00009-006-0092-8. [14] G. M. Coclite and H. Holden, The Schrödinger-Maxwell system with Dirac mass, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 773-793. doi: 10.1016/j.anihpc.2006.06.005. [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), 831-869. doi: 10.1142/S0219199710004007. [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, Princeton, NJ, 2004. doi: 10.1007/BF01158557. [17] O. Druet and B. Premoselli, Stability of the Einstein-Lichnerowicz constraints system, Mathematische Annalen, 362 (2015), 839-886. doi: 10.1007/s00208-014-1145-0. [18] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations Of Second Order, Classics in Mathematics, Springer-Verlag, Berlin, 2001, Reprint of the 1998 edition. [19] Q. Han and F. Lin, Elliptic Partial Differential Equations, vol. 1 of Courant Lecture Notes in Mathematics, 2nd edition, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2011. [20] E. Hebey, Compactness and Stability for Nonlinear Elliptic Equations, Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Zürich, 2014. doi: 10.4171/134. [21] E. Hebey and T. T. Truong, Static Klein-Gordon-Maxwell-Proca systems in 4-dimensional closed manifolds, J. Reine Angew. Math., 667 (2012), 221-248. [22] E. Hebey and M. Vaugon, Sobolev spaces in the presence of symmetries, J. Math. Pures Appl. (9), 76 (1997), 859-881. doi: 10.1016/S0021-7824(97)89975-8. [23] E. Hebey and J. Wei, Schrödinger-Poisson systems in the 3-sphere, Calc. Var. Partial Differential Equations, 47 (2013), 25-54. doi: 10.1007/s00526-012-0509-0. [24] I. Ianni and G. Vaira, On concentration of positive bound states for the Schrödinger-Poisson problem with potentials, Adv. Nonlinear Stud., 8 (2008), 573-595. [25] Y. Li and M. Zhu, Yamabe type equations on three-dimensional Riemannian manifolds, Commun. Contemp. Math., 1 (1999), 1-50. doi: 10.1142/S021919979900002X. [26] L. Pisani and G. Siciliano, Neumann condition in the Schrödinger-Maxwell system, Topol. Methods Nonlinear Anal., 29 (2007), 251-264. [27] L. Pisani and G. Siciliano, Some results on the Schrödinger-Poisson system in a bounded domain, in Dynamic systems and applications, Dynamic, Atlanta, GA, 5 (2008), 402-406. [28] D. Ruiz and G. Siciliano, A note on the Schrödinger-Poisson-Slater equation on bounded domains, Adv. Nonlinear Stud., 8 (2008), 179-190. [29] P.-D. Thizy, Blow-up for Schrödinger-Poisson critical systems in dimensions 4 and 5,, Preprint., (). [30] P.-D. Thizy, Klein-Gordon-Maxwell equations in high dimensions, Communications on Pure and Applied Analysis, 14 (2015), 1097-1125. doi: 10.3934/cpaa.2015.14.1097. [31] P.-D. Thizy, Non resonant states for Schrödinger-Poisson critical systems in high dimension, Archiv der Math., 104 (2015), 485-490. doi: 10.1007/s00013-015-0763-4. [32] N. S. Trudinger, Remarks concerning the conformal deformation of Riemannian structures on compact manifolds, Ann. Scuola Norm. Sup. Pisa (3), 22 (1968), 265-274. [33] P. Zhang and J. Sun, Clustered layers for the Schrödinger-Maxwell system on a ball, Discrete Contin. Dyn. Syst., 16 (2006), 657-688. doi: 10.3934/dcds.2006.16.657.
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