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On the behavior at collisions of solutions to Schrödinger equations with many-particle and cylindrical potentials
1. | Università di Milano Bicocca, Dipartimento di Matematica e Applicazioni, Via Cozzi 53, 20125 Milano |
2. | Università degli Studi del Piemonte Orientale, Viale Teresa Michel 11, 15121 Alessandria, Italy |
References:
[1] |
B. Abdellaoui, V. Felli and I. Peral, Some remarks on systems of elliptic equations doubly critical in the whole $\mathbb{R}^{N}$, Calc. Var. Partial Differential Equations, 34 (2009), 97-137.
doi: 10.1007/s00526-008-0177-2. |
[2] |
F. J. Almgren, Jr., $Q$ valued functions minimizing Dirichlet's integral and the regularity of area minimizing rectifiable currents up to codimension two, Bull. Amer. Math. Soc. (N. S.), 8 (1983), 327-328. |
[3] |
M. Badiale, V. Benci and S. Rolando, A nonlinear elliptic equation with singular potential and applications to nonlinear field equations, J. Eur. Math. Soc., 9 (2007), 355-381. |
[4] |
M. Badiale and S. Rolando, Elliptic problems with singular potential and double-power nonlinearity, Mediterr. J. Math., 2 (2005), 417-436. |
[5] |
M. Badiale and G. Tarantello, A Sobolev-Hardy inequalitywith applications to a nonlinear elliptic equation arising inastrophysics, Arch. Ration. Mech. Anal., 163 (2002), 259-293.
doi: 10.1007/s002050200201. |
[6] |
H. Baum and A. Juhl, "Conformal Differential Geometry. Q-Curvature and Conformal Holonomy," Oberwolfach Seminars, 40, Birkhäuser Verlag, 2010. |
[7] |
R. Bosi, J. Dolbeault and M. J. Esteban, Estimates for the optimal constants in multipolar Hardy inequalities for Schrödinger and Dirac operators, Commun. Pure Appl. Anal., 7 (2008), 533-562. |
[8] |
H. Brézis and T. Kato, Remarks on the Schrödinger operator with singular complex potentials, J. Math. Pures Appl. (9), 58 (1979), 137-151. |
[9] |
V. S. Buslaev and S. B. Levin, Asymptotic behavior of the eigenfunctions of the many-particle Schrödinger operator. I. One-dimensional particles, in "Spectral Theory of Differential Operators," Amer. Math. Soc. Transl. Ser. 2, 225, Amer. Math. Soc., Providence, RI, (2008), 55-71. |
[10] |
L. Caffarelli, R. Kohn and L. Nirenberg, First order interpolation inequalities with weights, Compositio Math., 53 (1984), 259-275. |
[11] |
F. Catrina and Z.-Q. Wang, On the Caffarelli-Kohn-Nirenberg inequalities: Sharp constants, existence (and nonexistence), and symmetry of extremal functions, Comm. Pure Appl. Math., 54 (2001), 229-258.
doi: 10.1002/1097-0312(200102)54:2<229::AID-CPA4>3.0.CO;2-I. |
[12] |
J. Chabrowski, A. Szulkin and M. Willem, Schrödinger equation with multiparticle potential and critical nonlinearity, Topol. Meth. Nonl. Anal., 34 (2009), 201-211. |
[13] |
S.-Y. A. Chang, Conformal invariants and partial differential equations, Bull. Amer. Math. Soc. (N.S.), 42 (2005), 365-393.
doi: 10.1090/S0273-0979-05-01058-X. |
[14] |
J. Chen, Multiple positive solutions for a semilinear equation with prescribed singularity, J. Math. Anal. Appl., 305 (2005), 140-157.
doi: 10.1016/j.jmaa.2004.10.057. |
[15] |
T. Duyckaerts, Inégalités de résolvante pour l'opérateur de Schrödinger avec potentiel multipolaire critique, Bulletin Bull. Soc. Math. France, 134 (2006), 201-239. |
[16] |
H. Egnell, Elliptic boundary value problems with singular coefficients and critical nonlinearities, Indiana Univ. Math. J., 38 (1989), 235-251.
doi: 10.1512/iumj.1989.38.38012. |
[17] |
V. Felli, A. Ferrero and S. Terracini, Asymptotic behavior of solutions to Schrödinger equations near an isolated singularity of the electromagnetic potential, Journal of the European Mathematical Society, 13 (2011), 119-174.
doi: 10.4171/JEMS/246. |
[18] |
V. Felli, A. Ferrero and S. Terracini, A note on local asymptotics of solutions to singular elliptic equations via monotonicity methods, Milan J. Math., (2012), DOI 10.1007/s00032-012-0174-y. |
[19] |
V. Felli, E. M. Marchini and S. Terracini, On Schrödinger operators with multipolar inverse-square potentials, Journal of Functional Analysis, 250 (2007), 265-316.
doi: 10.1016/j.jfa.2006.10.019. |
[20] |
V. Felli, E. M. Marchini and S. Terracini, On the behavior of solutions to Schrödinger equations with dipole type potentials near the singularity, Discrete Contin. Dynam. Systems, 21 (2008), 91-119.
doi: 10.3934/dcds.2008.21.91. |
[21] |
V. Felli, E. M. Marchini and S. Terracini, On Schrödinger operators with multisingular inverse-square anisotropic potentials, Indiana Univ. Math. Journal, 58 (2009), 617-676.
doi: 10.1512/iumj.2009.58.3471. |
[22] |
V. Felli and M. Schneider, A note on regularity of solutions to degenerate elliptic equations ofCaffarelli-Kohn-Nirenberg type, Adv. Nonlinear Stud., 3 (2003), 431-443. |
[23] |
V. Felli and S. Terracini, Elliptic equations with multi-singular inverse-square potentials and critical nonlinearity, Comm. Partial Differential Equations, 31 (2006), 469-495.
doi: 10.1080/03605300500394439. |
[24] |
A. Ferrero and F. Gazzola, Existence of solutions for singular critical growth semilinear elliptic equations, J. Differential Equations, 177 (2001), 494-522.
doi: 10.1006/jdeq.2000.3999. |
[25] |
J. García Azorero and I. Peral Alonso, Hardy Inequalities and some critical elliptic and parabolic problems, J. Diff. Equations, 144 (1998), 441-476.
doi: 10.1006/jdeq.1997.3375. |
[26] |
N. Garofalo and F.-H. Lin, Monotonicity properties of variational integrals, $A_p$ weights and unique continuation, Indiana Univ. Math. J., 35 (1986), 245-268.
doi: 10.1512/iumj.1986.35.35015. |
[27] |
M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof, A. Laptev and J. Tidblom, Many-particle Hardy inequalities, J. Lond. Math. Soc. (2), 77 (2008), 99-114.
doi: 10.1112/jlms/jdm091. |
[28] |
W. Hunziker and I. Sigal, The quantum $N$-body problem, J. Math. Phys., 41 (2000), 3448-3510.
doi: 10.1063/1.533319. |
[29] |
E. Jannelli, The role played by space dimension in elliptic critical problems, J. Differential Equations, 156 (1999), 407-426.
doi: 10.1006/jdeq.1998.3589. |
[30] |
M. Lesch, "Operators of Fuchs Type, Conical Singularities, and Asymptotic Methods," Teubner Texts in Mathematics, 136, B. G. Teubner Verlagsgesellschaft mbH, Stuttgart, 1997. |
[31] |
E. H. Lieb and W. E. Thirring, Gravitational collapse in quantum mechanics with relativistic kinetic energy, Ann. Physics, 155 (1984), 494-512.
doi: 10.1016/0003-4916(84)90010-1. |
[32] |
G. Mancini, I. Fabbri and K. Sandeep, Classification of solutions of a critical Hardy-Sobolev operator, J. Differential Equations, 224 (2006), 258-276.
doi: 10.1016/j.jde.2005.07.001. |
[33] |
V. G. Maz'ja, "Sobolev Spaces," Springer Series in Soviet Mathematics, Springer-Verlag, Berlin, 1985. |
[34] |
R. Mazzeo, Elliptic theory of differential edge operators. I, Comm. Partial Differential Equations, 16 (1991), 1615-1664.
doi: 10.1080/03605309108820815. |
[35] |
R. Mazzeo, Regularity for the singular Yamabe problem, Indiana Univ. Math. J., 40 (1991), 1277-1299.
doi: 10.1512/iumj.1991.40.40057. |
[36] |
R. Musina, Ground state solutions of a critical problem involving cylindrical weights, Nonlin. Anal., 68 (2008), 3972-3986.
doi: 10.1016/j.na.2007.04.034. |
[37] |
Y. Pinchover, On positive Liouville theorems and asymptotic behavior of solutions of Fuchsian type elliptic operators, Ann. Inst. H. Poincaré Anal. Non Linéaire, 11 (1994), 313-341. |
[38] |
P. Pucci and J. Serrin, A general variational identity, Indiana Univ. Math. J., 35 (1986), 681-703.
doi: 10.1512/iumj.1986.35.35036. |
[39] |
S. Secchi, D. Smets and M. Willem, Remarks on a Hardy-Sobolev inequality, C. R. Math. Acad. Sci. Paris, 336 (2003), 811-815.
doi: 10.1016/S1631-073X(03)00202-4. |
[40] |
D. Smets, Nonlinear Schrödinger equations withHardy potential and critical nonlinearities, Trans. AMS, 357 (2005), 2909-2938.
doi: 10.1090/S0002-9947-04-03769-9. |
[41] |
M. Struwe, "Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems," Springer-Verlag, Berlin, 1990. |
[42] |
S. Terracini, On positive entire solutions to a class of equations with singular coefficient and critical exponent, Adv. Diff. Equa., 1 (1996), 241-264. |
[43] |
Z.-Q. Wang and M. Zhu, Hardy inequalities with boundary terms, Electron. J. Differential Equations, 2003, 8 pp. |
[44] |
T. H. Wolff, A property of measures in $\R^ N$ and an application to unique continuation, Geom. Funct. Anal., 2 (1992), 225-284. |
show all references
References:
[1] |
B. Abdellaoui, V. Felli and I. Peral, Some remarks on systems of elliptic equations doubly critical in the whole $\mathbb{R}^{N}$, Calc. Var. Partial Differential Equations, 34 (2009), 97-137.
doi: 10.1007/s00526-008-0177-2. |
[2] |
F. J. Almgren, Jr., $Q$ valued functions minimizing Dirichlet's integral and the regularity of area minimizing rectifiable currents up to codimension two, Bull. Amer. Math. Soc. (N. S.), 8 (1983), 327-328. |
[3] |
M. Badiale, V. Benci and S. Rolando, A nonlinear elliptic equation with singular potential and applications to nonlinear field equations, J. Eur. Math. Soc., 9 (2007), 355-381. |
[4] |
M. Badiale and S. Rolando, Elliptic problems with singular potential and double-power nonlinearity, Mediterr. J. Math., 2 (2005), 417-436. |
[5] |
M. Badiale and G. Tarantello, A Sobolev-Hardy inequalitywith applications to a nonlinear elliptic equation arising inastrophysics, Arch. Ration. Mech. Anal., 163 (2002), 259-293.
doi: 10.1007/s002050200201. |
[6] |
H. Baum and A. Juhl, "Conformal Differential Geometry. Q-Curvature and Conformal Holonomy," Oberwolfach Seminars, 40, Birkhäuser Verlag, 2010. |
[7] |
R. Bosi, J. Dolbeault and M. J. Esteban, Estimates for the optimal constants in multipolar Hardy inequalities for Schrödinger and Dirac operators, Commun. Pure Appl. Anal., 7 (2008), 533-562. |
[8] |
H. Brézis and T. Kato, Remarks on the Schrödinger operator with singular complex potentials, J. Math. Pures Appl. (9), 58 (1979), 137-151. |
[9] |
V. S. Buslaev and S. B. Levin, Asymptotic behavior of the eigenfunctions of the many-particle Schrödinger operator. I. One-dimensional particles, in "Spectral Theory of Differential Operators," Amer. Math. Soc. Transl. Ser. 2, 225, Amer. Math. Soc., Providence, RI, (2008), 55-71. |
[10] |
L. Caffarelli, R. Kohn and L. Nirenberg, First order interpolation inequalities with weights, Compositio Math., 53 (1984), 259-275. |
[11] |
F. Catrina and Z.-Q. Wang, On the Caffarelli-Kohn-Nirenberg inequalities: Sharp constants, existence (and nonexistence), and symmetry of extremal functions, Comm. Pure Appl. Math., 54 (2001), 229-258.
doi: 10.1002/1097-0312(200102)54:2<229::AID-CPA4>3.0.CO;2-I. |
[12] |
J. Chabrowski, A. Szulkin and M. Willem, Schrödinger equation with multiparticle potential and critical nonlinearity, Topol. Meth. Nonl. Anal., 34 (2009), 201-211. |
[13] |
S.-Y. A. Chang, Conformal invariants and partial differential equations, Bull. Amer. Math. Soc. (N.S.), 42 (2005), 365-393.
doi: 10.1090/S0273-0979-05-01058-X. |
[14] |
J. Chen, Multiple positive solutions for a semilinear equation with prescribed singularity, J. Math. Anal. Appl., 305 (2005), 140-157.
doi: 10.1016/j.jmaa.2004.10.057. |
[15] |
T. Duyckaerts, Inégalités de résolvante pour l'opérateur de Schrödinger avec potentiel multipolaire critique, Bulletin Bull. Soc. Math. France, 134 (2006), 201-239. |
[16] |
H. Egnell, Elliptic boundary value problems with singular coefficients and critical nonlinearities, Indiana Univ. Math. J., 38 (1989), 235-251.
doi: 10.1512/iumj.1989.38.38012. |
[17] |
V. Felli, A. Ferrero and S. Terracini, Asymptotic behavior of solutions to Schrödinger equations near an isolated singularity of the electromagnetic potential, Journal of the European Mathematical Society, 13 (2011), 119-174.
doi: 10.4171/JEMS/246. |
[18] |
V. Felli, A. Ferrero and S. Terracini, A note on local asymptotics of solutions to singular elliptic equations via monotonicity methods, Milan J. Math., (2012), DOI 10.1007/s00032-012-0174-y. |
[19] |
V. Felli, E. M. Marchini and S. Terracini, On Schrödinger operators with multipolar inverse-square potentials, Journal of Functional Analysis, 250 (2007), 265-316.
doi: 10.1016/j.jfa.2006.10.019. |
[20] |
V. Felli, E. M. Marchini and S. Terracini, On the behavior of solutions to Schrödinger equations with dipole type potentials near the singularity, Discrete Contin. Dynam. Systems, 21 (2008), 91-119.
doi: 10.3934/dcds.2008.21.91. |
[21] |
V. Felli, E. M. Marchini and S. Terracini, On Schrödinger operators with multisingular inverse-square anisotropic potentials, Indiana Univ. Math. Journal, 58 (2009), 617-676.
doi: 10.1512/iumj.2009.58.3471. |
[22] |
V. Felli and M. Schneider, A note on regularity of solutions to degenerate elliptic equations ofCaffarelli-Kohn-Nirenberg type, Adv. Nonlinear Stud., 3 (2003), 431-443. |
[23] |
V. Felli and S. Terracini, Elliptic equations with multi-singular inverse-square potentials and critical nonlinearity, Comm. Partial Differential Equations, 31 (2006), 469-495.
doi: 10.1080/03605300500394439. |
[24] |
A. Ferrero and F. Gazzola, Existence of solutions for singular critical growth semilinear elliptic equations, J. Differential Equations, 177 (2001), 494-522.
doi: 10.1006/jdeq.2000.3999. |
[25] |
J. García Azorero and I. Peral Alonso, Hardy Inequalities and some critical elliptic and parabolic problems, J. Diff. Equations, 144 (1998), 441-476.
doi: 10.1006/jdeq.1997.3375. |
[26] |
N. Garofalo and F.-H. Lin, Monotonicity properties of variational integrals, $A_p$ weights and unique continuation, Indiana Univ. Math. J., 35 (1986), 245-268.
doi: 10.1512/iumj.1986.35.35015. |
[27] |
M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof, A. Laptev and J. Tidblom, Many-particle Hardy inequalities, J. Lond. Math. Soc. (2), 77 (2008), 99-114.
doi: 10.1112/jlms/jdm091. |
[28] |
W. Hunziker and I. Sigal, The quantum $N$-body problem, J. Math. Phys., 41 (2000), 3448-3510.
doi: 10.1063/1.533319. |
[29] |
E. Jannelli, The role played by space dimension in elliptic critical problems, J. Differential Equations, 156 (1999), 407-426.
doi: 10.1006/jdeq.1998.3589. |
[30] |
M. Lesch, "Operators of Fuchs Type, Conical Singularities, and Asymptotic Methods," Teubner Texts in Mathematics, 136, B. G. Teubner Verlagsgesellschaft mbH, Stuttgart, 1997. |
[31] |
E. H. Lieb and W. E. Thirring, Gravitational collapse in quantum mechanics with relativistic kinetic energy, Ann. Physics, 155 (1984), 494-512.
doi: 10.1016/0003-4916(84)90010-1. |
[32] |
G. Mancini, I. Fabbri and K. Sandeep, Classification of solutions of a critical Hardy-Sobolev operator, J. Differential Equations, 224 (2006), 258-276.
doi: 10.1016/j.jde.2005.07.001. |
[33] |
V. G. Maz'ja, "Sobolev Spaces," Springer Series in Soviet Mathematics, Springer-Verlag, Berlin, 1985. |
[34] |
R. Mazzeo, Elliptic theory of differential edge operators. I, Comm. Partial Differential Equations, 16 (1991), 1615-1664.
doi: 10.1080/03605309108820815. |
[35] |
R. Mazzeo, Regularity for the singular Yamabe problem, Indiana Univ. Math. J., 40 (1991), 1277-1299.
doi: 10.1512/iumj.1991.40.40057. |
[36] |
R. Musina, Ground state solutions of a critical problem involving cylindrical weights, Nonlin. Anal., 68 (2008), 3972-3986.
doi: 10.1016/j.na.2007.04.034. |
[37] |
Y. Pinchover, On positive Liouville theorems and asymptotic behavior of solutions of Fuchsian type elliptic operators, Ann. Inst. H. Poincaré Anal. Non Linéaire, 11 (1994), 313-341. |
[38] |
P. Pucci and J. Serrin, A general variational identity, Indiana Univ. Math. J., 35 (1986), 681-703.
doi: 10.1512/iumj.1986.35.35036. |
[39] |
S. Secchi, D. Smets and M. Willem, Remarks on a Hardy-Sobolev inequality, C. R. Math. Acad. Sci. Paris, 336 (2003), 811-815.
doi: 10.1016/S1631-073X(03)00202-4. |
[40] |
D. Smets, Nonlinear Schrödinger equations withHardy potential and critical nonlinearities, Trans. AMS, 357 (2005), 2909-2938.
doi: 10.1090/S0002-9947-04-03769-9. |
[41] |
M. Struwe, "Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems," Springer-Verlag, Berlin, 1990. |
[42] |
S. Terracini, On positive entire solutions to a class of equations with singular coefficient and critical exponent, Adv. Diff. Equa., 1 (1996), 241-264. |
[43] |
Z.-Q. Wang and M. Zhu, Hardy inequalities with boundary terms, Electron. J. Differential Equations, 2003, 8 pp. |
[44] |
T. H. Wolff, A property of measures in $\R^ N$ and an application to unique continuation, Geom. Funct. Anal., 2 (1992), 225-284. |
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