On the behavior at collisions of solutions to Schrödinger equations with many-particle and cylindrical potentials

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November 2012, 32(11): 3895-3956. doi: 10.3934/dcds.2012.32.3895

On the behavior at collisions of solutions to Schrödinger equations with many-particle and cylindrical potentials

Veronica Felli ^1, , Alberto Ferrero ^2, and Susanna Terracini ^1,

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

Received July 2011 Revised December 2011 Published June 2012

Abstract
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The asymptotic behavior of solutions to Schrödinger equations with singular homogeneous potentials is investigated. Through an Almgren type monotonicity formula and separation of variables, we describe the exact asymptotics near the singularity of solutions to at most critical semilinear elliptic equations with cylindrical and quantum multi-body singular potentials. Furthermore, by an iterative Brezis-Kato procedure, pointwise upper estimate are derived.

Keywords: Hardy's inequality, singular cylindrical potentials, Quantum $N$-body problem, Schrödinger operators..

Mathematics Subject Classification: Primary: 35J10, 35B40, 35J60; Secondary: 81V7.

Citation: Veronica Felli, Alberto Ferrero, Susanna Terracini. On the behavior at collisions of solutions to Schrödinger equations with many-particle and cylindrical potentials. Discrete and Continuous Dynamical Systems, 2012, 32 (11) : 3895-3956. doi: 10.3934/dcds.2012.32.3895

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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