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Groundstates and radial solutions to nonlinear Schrödinger–Poisson–Slater equations at the critical frequency

Bibliographic reference Mercuri, Carlo ; Moroz, Vitaly ; Van Schaftingen, Jean. Groundstates and radial solutions to nonlinear Schrödinger–Poisson–Slater equations at the critical frequency. In: Calculus of Variations and Partial Differential Equations, Vol. 55, no.6, p. 58 (2016)
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  1. Ackermann Nils, On a periodic Schr�dinger equation with nonlocal superlinear part, 10.1007/s00209-004-0663-y
  2. Ackermann Nils, A nonlinear superposition principle and multibump solutions of periodic Schrödinger equations, 10.1016/j.jfa.2005.11.010
  3. Adams, D.R., Hedberg, L.I.: Function spaces and potential theory. Grundlehren der Mathematischen Wissenschaften. 314, Springer (1996)
  4. Ambrosetti Antonio, On Schrödinger-Poisson Systems, 10.1007/s00032-008-0094-z
  5. Ambrosetti Antonio, Rabinowitz Paul H, Dual variational methods in critical point theory and applications, 10.1016/0022-1236(73)90051-7
  6. Bao Weizhu, Mauser N.J., Stimming H.P., Effective One Particle Quantum Dynamics of Electrons: A Numerical Study of the Schrodinger-Poisson-Xalpha Model, 10.4310/cms.2003.v1.n4.a8
  7. Bellazzini Jacopo, Frank Rupert L., Visciglia Nicola, Maximizers for Gagliardo–Nirenberg inequalities and related non-local problems, 10.1007/s00208-014-1046-2
  8. Bellazzini Jacopo, Ghimenti Marco, Ozawa Tohru, Sharp lower bounds for Coulomb energy, 10.4310/mrl.2016.v23.n3.a2
  9. Benci Vieri, Fortunato Donato, An eigenvalue problem for the Schrödinger-Maxwell equations, 10.12775/tmna.1998.019
  10. Benedek A., Panzone R., The space $L\sp{p}$ , with mixed norm, 10.1215/s0012-7094-61-02828-9
  11. Boas Jr. R. P., Some uniformly convex spaces, 10.1090/s0002-9904-1940-07207-6
  12. Bogachev Vladimir I., Measure Theory, ISBN:9783540345138, 10.1007/978-3-540-34514-5
  13. Bokanowski Olivier, López José L., Soler Juan, On an Exchange Interaction Model for Quantum Transport: The Schrödinger–Poisson–Slater System, 10.1142/s0218202503002969
  14. Bonheure Denis, Mercuri Carlo, Embedding theorems and existence results for nonlinear Schrödinger–Poisson systems with unbounded and vanishing potentials, 10.1016/j.jde.2011.04.010
  15. Brezis, H.: Functional analysis, Sobolev spaces and partial differential equations, Universitext. Springer, New York (2011)
  16. Brezis Haim, Lieb Elliott, A Relation Between Pointwise Convergence of Functions and Convergence of Functionals, 10.2307/2044999
  17. Carleson, L.: Selected problems on exceptional sets. Van Nostrand Mathematical Studies, No. 13. Van Nostrand, Princeton, Toronto, London (1967)
  19. D'Aprile Teresa, Mugnai Dimitri, Solitary waves for nonlinear Klein–Gordon–Maxwell and Schrödinger–Maxwell equations, 10.1017/s030821050000353x
  20. Day Mahlon M., Some more uniformly convex spaces, 10.1090/s0002-9904-1941-07499-9
  21. Del Pino Manuel, Dolbeault Jean, Best constants for Gagliardo–Nirenberg inequalities and applications to nonlinear diffusions, 10.1016/s0021-7824(02)01266-7
  22. Di Cosmo Jonathan, Van Schaftingen Jean, Semiclassical stationary states for nonlinear Schrödinger equations under a strong external magnetic field, 10.1016/j.jde.2015.02.016
  23. Duoandikoetxea Javier, Fractional integrals on radial functions with applications to weighted inequalities, 10.1007/s10231-011-0237-7
  24. Gagliardo, E.: Proprietà di alcune classi di funzioni in più variabili. Ricerche Mat. 7, 102–137 (1958)
  25. Gilbarg David, Trudinger Neil S., Elliptic Partial Differential Equations of Second Order, ISBN:9783540411604, 10.1007/978-3-642-61798-0
  26. Fröhlich Jürg, Lieb Elliott H., Loss Michael, Stability of coulomb systems with magnetic fields : I. The one-electron atom, 10.1007/bf01211593
  27. Ianni, I., Ruiz, D.: Ground and bound states for a static Schrödinger–Poisson–Slater problem. Commun. Contemp. Math. 14(1), 1250003, 22 (2012)
  28. Koskela, M.: Some generalizations of Clarkson’s inequalities. Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz., pp. 634–677, pp. 89–93 (1979)
  29. Lebedev, N.N.: Special functions and their applications, translated by Silverman RA. Prentice–Hall, Englewood Cliffs (1965)
  30. Le Bris, C., Lions, P.L.: From atoms to crystals: a mathematical journey. Bull. Am. Math. Soc. (N.S.). 42(3), 291–363 (2005)
  31. Lieb, E.H.: Sharp constants in the Hardy–Littlewood–Sobolev and related inequalities. Ann. Math. (118), 349–374 (1983)
  32. Lieb, E.H., Loss, M.: Analysis, 2nd edn. Graduate Studies in Mathematics, vol. 14. American Mathematical Society, Providence, RI (2001)
  33. Lions P.L., Some remarks on Hartree equation, 10.1016/0362-546x(81)90016-x
  34. Lions P.L., The concentration-compactness principle in the Calculus of Variations. The locally compact case, part 1. * *Mp denotes the Marcinkiewicz space or weak Lp space, 10.1016/s0294-1449(16)30428-0
  35. Lions P. L., Solutions of Hartree-Fock equations for Coulomb systems, 10.1007/bf01205672
  36. Maligranda Lech, Sabourova Natalia, On Clarkson's inequality in the real case, 10.1002/mana.200610552
  37. Mauser N.J., The Schrödinger-Poisson-Xα equation, 10.1016/s0893-9659(01)80038-0
  38. Maźya, V.: Sobolev spaces with applications to elliptic partial differential equations, 2nd edn. Grundlehren der Mathematischen Wissenschaften, vol. 342. Springer, Heidelberg (2011)
  39. Mercuri, C.: Positive solutions of nonlinear Schrod̈inger–Poisson systems with radial potentials vanishing at infinity. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 19(3), 211–227 (2008)
  40. Merle F, Peletier L.A, Asymptotic behaviour of positive solutions of elliptic equations with critical and supercritical growth. II. The nonradial case, 10.1016/0022-1236(92)90070-y
  41. Milman, D.: On some criteria for the regularity of spaces of type (B). C. R. (Doklady) Acad. Sci. U.R.S.S. 20, 243–246 (1938)
  42. Moroz Vitaly, Van Schaftingen Jean, Groundstates of nonlinear Choquard equations: Existence, qualitative properties and decay asymptotics, 10.1016/j.jfa.2013.04.007
  43. Ni Wei-Ming, , 10.1512/iumj.1982.31.31056
  44. Nirenberg, L.: On elliptic partial differential equations. Ann. Scuola Norm. Sup. Pisa. 13(3), 115–162 (1959)
  45. Ohtsuka Makoto, Capacité d’Ensembles de Cantor Généralisés, 10.1017/s0027763000002038
  46. du Plessis, N.: An introduction to potential theory. University Mathematical Monographs, vol. 7. Oliver and Boyd, Edinburgh (1970)
  47. Rubin B. S., One-dimensional representation, inversion, and certain properties of the Riesz potentials of radial functions, 10.1007/bf01157392
  48. Ruiz David, The Schrödinger–Poisson equation under the effect of a nonlinear local term, 10.1016/j.jfa.2006.04.005
  49. Ruiz David, On the Schrödinger–Poisson–Slater System: Behavior of Minimizers, Radial and Nonradial Cases, 10.1007/s00205-010-0299-5
  50. Slater J. C., A Simplification of the Hartree-Fock Method, 10.1103/physrev.81.385
  51. Stein, E.M.: Singular integrals and differentiability properties of functions. Princeton Mathematical Series, vl. 30. Princeton University Press, Princeton (1970)
  52. Strauss Walter A., Existence of solitary waves in higher dimensions, 10.1007/bf01626517
  54. Su Jiabao, Wang Zhi-Qiang, Willem Michel, Weighted Sobolev embedding with unbounded and decaying radial potentials, 10.1016/j.jde.2007.03.018
  55. Thim Johan, Asymptotics and inversion of Riesz potentials through decomposition in radial and spherical parts, 10.1007/s10231-014-0465-8
  56. Trudinger, N.S.: Linear elliptic operators with measurable coefficients. Ann. Scuola Norm. Sup. Pisa. 27(3), 265–308 (1973)
  57. Van Schaftingen Jean, Interpolation inequalities between Sobolev and Morrey–Campanato spaces: A common gateway to concentration-compactness and Gagliardo–Nirenberg interpolation inequalities, 10.4171/pm/1947
  58. Willem Michel, Minimax Theorems, ISBN:9781461286738, 10.1007/978-1-4612-4146-1
  59. Willem Michel, Functional Analysis, ISBN:9781461470038, 10.1007/978-1-4614-7004-5
  60. Yang Minbo, Wei Yuanhong, Existence and multiplicity of solutions for nonlinear Schrödinger equations with magnetic field and Hartree type nonlinearities, 10.1016/j.jmaa.2013.02.062
  61. Yosida Kôsaku, Functional Analysis, ISBN:9783540586548, 10.1007/978-3-642-61859-8