# American Institute of Mathematical Sciences

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

 1 Università di Milano Bicocca, Dipartimento di Matematica e Applicazioni, Via Cozzi 53, 20125 Milano 2 Università degli Studi del Piemonte Orientale, Viale Teresa Michel 11, 15121 Alessandria, Italy

Received  July 2011 Revised  December 2011 Published  June 2012

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.
Citation: Veronica Felli, Alberto Ferrero, Susanna Terracini. On the behavior at collisions of solutions to Schrödinger equations with many-particle and cylindrical potentials. Discrete & Continuous Dynamical Systems, 2012, 32 (11) : 3895-3956. doi: 10.3934/dcds.2012.32.3895
##### References:
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Anal., 163 (2002), 259-293. doi: 10.1007/s002050200201.  Google Scholar [6] H. Baum and A. Juhl, "Conformal Differential Geometry. Q-Curvature and Conformal Holonomy," Oberwolfach Seminars, 40, Birkhäuser Verlag, 2010.  Google Scholar [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.  Google Scholar [8] H. Brézis and T. Kato, Remarks on the Schrödinger operator with singular complex potentials, J. Math. Pures Appl. (9), 58 (1979), 137-151.  Google Scholar [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.  Google Scholar [10] L. Caffarelli, R. Kohn and L. Nirenberg, First order interpolation inequalities with weights, Compositio Math., 53 (1984), 259-275.  Google Scholar [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.  Google Scholar [12] J. Chabrowski, A. Szulkin and M. Willem, Schrödinger equation with multiparticle potential and critical nonlinearity, Topol. Meth. Nonl. Anal., 34 (2009), 201-211.  Google Scholar [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.  Google Scholar [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.  Google Scholar [15] T. 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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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [28] W. Hunziker and I. Sigal, The quantum $N$-body problem, J. Math. Phys., 41 (2000), 3448-3510. doi: 10.1063/1.533319.  Google Scholar [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.  Google Scholar [30] M. Lesch, "Operators of Fuchs Type, Conical Singularities, and Asymptotic Methods," Teubner Texts in Mathematics, 136, B. G. Teubner Verlagsgesellschaft mbH, Stuttgart, 1997.  Google Scholar [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.  Google Scholar [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.  Google Scholar [33] V. G. Maz'ja, "Sobolev Spaces," Springer Series in Soviet Mathematics, Springer-Verlag, Berlin, 1985.  Google Scholar [34] R. Mazzeo, Elliptic theory of differential edge operators. I, Comm. Partial Differential Equations, 16 (1991), 1615-1664. doi: 10.1080/03605309108820815.  Google Scholar [35] R. Mazzeo, Regularity for the singular Yamabe problem, Indiana Univ. Math. J., 40 (1991), 1277-1299. doi: 10.1512/iumj.1991.40.40057.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [41] M. Struwe, "Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems," Springer-Verlag, Berlin, 1990.  Google Scholar [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.  Google Scholar [43] Z.-Q. Wang and M. Zhu, Hardy inequalities with boundary terms,, Electron. J. Differential Equations, 2003 ().   Google Scholar [44] T. H. Wolff, A property of measures in $\R^ N$ and an application to unique continuation, Geom. Funct. Anal., 2 (1992), 225-284.  Google Scholar

show all references

##### References:
 [1] B. Abdellaoui, V. Felli and I. Peral, Some remarks on systems of elliptic equations doubly critical in the whole $\R^N$, Calc. Var. Partial Differential Equations, 34 (2009), 97-137. doi: 10.1007/s00526-008-0177-2.  Google Scholar [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.  Google Scholar [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.  Google Scholar [4] M. Badiale and S. Rolando, Elliptic problems with singular potential and double-power nonlinearity, Mediterr. J. Math., 2 (2005), 417-436.  Google Scholar [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.  Google Scholar [6] H. Baum and A. Juhl, "Conformal Differential Geometry. Q-Curvature and Conformal Holonomy," Oberwolfach Seminars, 40, Birkhäuser Verlag, 2010.  Google Scholar [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.  Google Scholar [8] H. Brézis and T. Kato, Remarks on the Schrödinger operator with singular complex potentials, J. Math. Pures Appl. (9), 58 (1979), 137-151.  Google Scholar [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.  Google Scholar [10] L. Caffarelli, R. Kohn and L. Nirenberg, First order interpolation inequalities with weights, Compositio Math., 53 (1984), 259-275.  Google Scholar [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.  Google Scholar [12] J. Chabrowski, A. Szulkin and M. Willem, Schrödinger equation with multiparticle potential and critical nonlinearity, Topol. Meth. Nonl. Anal., 34 (2009), 201-211.  Google Scholar [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.  Google Scholar [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.  Google Scholar [15] T. Duyckaerts, Inégalités de résolvante pour l'opérateur de Schrödinger avec potentiel multipolaire critique, Bulletin Bull. Soc. Math. France, 134 (2006), 201-239.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [28] W. Hunziker and I. Sigal, The quantum $N$-body problem, J. Math. Phys., 41 (2000), 3448-3510. doi: 10.1063/1.533319.  Google Scholar [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.  Google Scholar [30] M. Lesch, "Operators of Fuchs Type, Conical Singularities, and Asymptotic Methods," Teubner Texts in Mathematics, 136, B. G. Teubner Verlagsgesellschaft mbH, Stuttgart, 1997.  Google Scholar [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.  Google Scholar [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.  Google Scholar [33] V. G. Maz'ja, "Sobolev Spaces," Springer Series in Soviet Mathematics, Springer-Verlag, Berlin, 1985.  Google Scholar [34] R. Mazzeo, Elliptic theory of differential edge operators. I, Comm. Partial Differential Equations, 16 (1991), 1615-1664. doi: 10.1080/03605309108820815.  Google Scholar [35] R. Mazzeo, Regularity for the singular Yamabe problem, Indiana Univ. Math. J., 40 (1991), 1277-1299. doi: 10.1512/iumj.1991.40.40057.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [41] M. Struwe, "Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems," Springer-Verlag, Berlin, 1990.  Google Scholar [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.  Google Scholar [43] Z.-Q. Wang and M. Zhu, Hardy inequalities with boundary terms,, Electron. J. Differential Equations, 2003 ().   Google Scholar [44] T. H. Wolff, A property of measures in $\R^ N$ and an application to unique continuation, Geom. Funct. Anal., 2 (1992), 225-284.  Google Scholar
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