Scaling properties of functionals and existence of constrained minimizers

Journal of Functional Analysis

Volumn 261, Issue 9, 2011, Pages 2486-2507

  Bellazzini, Jacopo ^a   Siciliano, Gaetano ^b  

^a UNIVERSITY OF SASSARI   (Italy)

^b UNIVERSITY OF SÃO PAULO   (Brazil)

Author keywords

Constrained minimization; Schr dinger Poisson equations; Standing waves; Subadditivity inequality

Indexed keywords

EID: 80052098615     PISSN: 00221236     EISSN: 10960783     Source Type: Journal    
DOI: 10.1016/j.jfa.2011.06.014     Document Type: Article

References (21)

1. Azzollini A., Pomponio A., d'Avenia P. On the Schrödinger-Maxwell equations under the effect of a general nonlinear term. Ann. Inst. H. Poincaré Anal. Non Linéaire 2010, 27(2):779-791.
2. Bellazzini J., Benci V., Ghimenti M., Micheletti A.M. On the existence of the fundamental eigenvalue of an elliptic problem in RN. Adv. Nonlinear Stud. 2007, 7:439-458.
3. Bellazzini J., Bonanno C. Nonlinear Schrödinger equations with strongly singular potentials. Proc. Roy. Soc. Edinburgh Sect. A 2010, 140:707-721.
4. Bellazzini J., Siciliano G. Stable standing waves for a class of nonlinear Schrödinger-Poisson equations. Z. Angew. Math. Phys. 2011, 62:267-280.
5. Bellazzini J., Visciglia N. On the orbital stability for a class of nonautonomous NLS. Indiana Univ. Math. J. 2010, 59(3):1211-1230.
6. Benci V., Fortunato D. Hylomorphic solitons on lattices. Discrete Contin. Dyn. Syst. Ser. A 2010, 28:875-897.
7. Brezis H., Lieb E. A relation between pointwise convergence of functions and convergence of functionals. Proc. Amer. Math. Soc. 1983, 88(3):486-490.
8. Catto I., Lions P.L. Binding of atoms and stability of molecules in Hartree and Thomas-Fermi type theories. I. A necessary and sufficient condition for the stability of general molecular systems. Comm. Partial Differential Equations 1992, 17(7-8):1051-1110.
9. Cazenave T., Lions P.L. Orbital stability of standing waves for some non linear Schrödinger equations. Comm. Math. Phys. 1982, 85:549-561.
10. D'Aprile T., Mugnai D. Solitary waves for nonlinear Klein-Gordon-Maxwell and Schrödinger-Maxwell equations. Proc. Roy. Soc. Edinburgh Sect. A 2004, 134(5):893-906.
11. Dobos J. The standard Cantor function is subadditive. Proc. Amer. Math. Soc. 1996, 124:3425-3426.
12. Grillakis M., Shatah J., Strauss W. Stability theory of solitary waves in the presence of symmetry. I. J. Funct. Anal. 1987, 74(1):160-197.
13. Jeanjean L., Squassina M. An approach to minimization under a constraint: the added mass technique. Calc. Var. Partial Differential Equations 2011, 41:511-534.
14. Kikuchi H. Existence and stability of standing waves for Schrödinger-Poisson-Slater equation. Adv. Nonlinear Stud. 2007, 7(3):403-437.
15. Lieb E.H. Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation. Stud. Appl. Math. 1977, 57(2):93-105.
16. Lieb E.H., Simon B. The Thomas-Fermi theory of atoms, molecules, and solids. Adv. Math. 1977, 23:22-116.
17. Lions P.L. The concentration-compactness principle in the calculus of variation. The locally compact case, part I and II. Ann. Inst. H. Poincaré Anal. Non Linéaire 1984, 1:109-145. and 223-283.
18. Lions P.L. Solutions of Hartree-Fock equations for Coulomb systems. Comm. Math. Phys. 1987, 109:33-97.
19. Ruiz D. The Schrödinger-Poisson equation under the effect of a nonlinear local term. J. Funct. Anal. 2006, 237(2):655-674.
20. Sanchez O., Soler J. Long time dynamics of the Schrödinger-Poisson-Slater system. J. Stat. Phys. 2004, 114:179-204.
21. Stuart C.A. Bifurcation for the Dirichlet problems without eigenvalues. Proc. Lond. Math. Soc. 1982, 45:169-192.