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

References:
[1]

Z. Angew. Math. Phys., 65 (2014), 1153-1166. doi: 10.1007/s00033-013-0376-3.  Google Scholar

[2]

J. Functional Analysis, 14 (1973), 349-381. doi: 10.1016/0022-1236(73)90051-7.  Google Scholar

[3]

J. Math. Pures Appl. (9), 55 (1976), 269-296.  Google Scholar

[4]

Commun. Pure Appl. Anal., 12 (2013), 867-879. doi: 10.3934/cpaa.2013.12.867.  Google Scholar

[5]

Adv. Nonlinear Stud., 12 (2012), 717-735.  Google Scholar

[6]

Rev. Math. Phys., 14 (2002), 409-420. doi: 10.1142/S0129055X02001168.  Google Scholar

[7]

Adv. Nonlinear Stud., 8 (2008), 327-352.  Google Scholar

[8]

Nonlinear Anal., 72 (2010), 2031-2046. doi: 10.1016/j.na.2009.10.004.  Google Scholar

[9]

Adv. Nonlinear Anal., 3 (2014), s1-s18. doi: 10.1515/anona-2013-0032.  Google Scholar

[10]

J. Partial Differential Equations, 19 (2006), 208-217.  Google Scholar

[11]

Comm. Pure Appl. Math., 36 (1983), 437-477. doi: 10.1002/cpa.3160360405.  Google Scholar

[12]

Comm. Pure Appl. Math., 42 (1989), 271-297. doi: 10.1002/cpa.3160420304.  Google Scholar

[13]

Mediterr. J. Math., 3 (2006), 483-493. doi: 10.1007/s00009-006-0092-8.  Google Scholar

[14]

Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 773-793. doi: 10.1016/j.anihpc.2006.06.005.  Google Scholar

[15]

Commun. Contemp. Math., 12 (2010), 831-869. doi: 10.1142/S0219199710004007.  Google Scholar

[16]

Princeton University Press, Princeton, NJ, 2004. doi: 10.1007/BF01158557.  Google Scholar

[17]

Mathematische Annalen, 362 (2015), 839-886. doi: 10.1007/s00208-014-1145-0.  Google Scholar

[18]

Classics in Mathematics, Springer-Verlag, Berlin, 2001, Reprint of the 1998 edition.  Google Scholar

[19]

2nd edition, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2011.  Google Scholar

[20]

Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Zürich, 2014. doi: 10.4171/134.  Google Scholar

[21]

J. Reine Angew. Math., 667 (2012), 221-248.  Google Scholar

[22]

J. Math. Pures Appl. (9), 76 (1997), 859-881. doi: 10.1016/S0021-7824(97)89975-8.  Google Scholar

[23]

Calc. Var. Partial Differential Equations, 47 (2013), 25-54. doi: 10.1007/s00526-012-0509-0.  Google Scholar

[24]

Adv. Nonlinear Stud., 8 (2008), 573-595.  Google Scholar

[25]

Commun. Contemp. Math., 1 (1999), 1-50. doi: 10.1142/S021919979900002X.  Google Scholar

[26]

Topol. Methods Nonlinear Anal., 29 (2007), 251-264.  Google Scholar

[27]

in Dynamic systems and applications, Dynamic, Atlanta, GA, 5 (2008), 402-406.  Google Scholar

[28]

Adv. Nonlinear Stud., 8 (2008), 179-190.  Google Scholar

[29]

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

[30]

Communications on Pure and Applied Analysis, 14 (2015), 1097-1125. doi: 10.3934/cpaa.2015.14.1097.  Google Scholar

[31]

Archiv der Math., 104 (2015), 485-490. doi: 10.1007/s00013-015-0763-4.  Google Scholar

[32]

Ann. Scuola Norm. Sup. Pisa (3), 22 (1968), 265-274.  Google Scholar

[33]

Discrete Contin. Dyn. Syst., 16 (2006), 657-688. doi: 10.3934/dcds.2006.16.657.  Google Scholar

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