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Unique continuation properties for relativistic Schrödinger operators with a singular potential
Symmetry of components, Liouville-type theorems and classification results for some nonlinear elliptic systems
1. | LAMFA, CNRS UMR 7352, Université de Picardie Jules Verne, 33 rue Saint-Leu, 80039 Amiens |
References:
[1] |
G. Bianchi, Non-existence of positive solutions to semilinear equations on $\mathbbR^n$ or $\mathbbR^n_+$ through the method of moving planes,, Comm. Partial Diff. Eqns., 22 (1997), 1671.
doi: 10.1080/03605309708821315. |
[2] |
H. Brezis, Semilinear equations in $R^N$ without condition at infinity,, Appl. Math. Optim., 12 (1984), 271.
doi: 10.1007/BF01449045. |
[3] |
L. A. Caffarelli, B. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth,, Comm. Pure Appl. Math., 42 (1989), 271.
doi: 10.1002/cpa.3160420304. |
[4] |
C. Cortàzar, M. Elgueta and P. Felmer, On a semilinear elliptic problem in $\mathbbR^N$ with a non-Lipschitzian non-linearity,, Advances in Differential Equations, 1 (1996), 199.
|
[5] |
E. N. Dancer and Y. Du, Some remarks on Liouville type results for quasilinear elliptic equations,, Proc. Amer. Math. Soc., 131 (2003), 1891.
doi: 10.1090/S0002-9939-02-06733-3. |
[6] |
A. Farina, Liouville-type theorems for elliptic problems,, in Handbook of Differential Equations, (2007), 61.
doi: 10.1016/S1874-5733(07)80005-2. |
[7] |
D. J. Frantzeskakis, Dark solitons in atomic Bose-Einstein condensates: From theory to experiments,, J. Phys. A, 43 (2010).
doi: 10.1088/1751-8113/43/21/213001. |
[8] |
B. Gidas, W.-M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $R^n$,, Adv. Math. Supp. Stud., 7a (1981), 369.
|
[9] |
F. Gazzola, J. Serrin and M. Tang, Existence of ground states and free boundary problems for quasilinear elliptic operators,, Advances in Differential Equations, 5 (2000), 1.
|
[10] |
B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations,, Comm. Partial Differential Equations, 6 (1981), 883.
doi: 10.1080/03605308108820196. |
[11] |
B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations,, Comm. Pure Appl. Math., 34 (1981), 525.
doi: 10.1002/cpa.3160340406. |
[12] |
T. Kato, Schrödinger operators with singular potentials,, Proceedings of the International Symposium on Partial Differential Equations and the Geometry of Normed Linear Spaces (Jerusalem, 13 (1972), 135.
doi: 10.1007/BF02760233. |
[13] |
J. B. Keller, On solutions of $ \Delta u = f (u)$,, Comm. Pure Appl. Math., 10 (1957), 503.
doi: 10.1002/cpa.3160100402. |
[14] |
Yu. S. Kivshar and B. Luther-Davies, Dark optical solitons: Physics and applications,, Phys. Rep., 298 (1998), 81.
doi: 10.1016/S0370-1573(97)00073-2. |
[15] |
T.-C. Lin and J.-C. Wei, Symbiotic bright solitary wave solutions of coupled nonlinear Schrödinger equations,, Nonlinearity, 19 (2006), 2755.
doi: 10.1088/0951-7715/19/12/002. |
[16] |
L. Ma and L. Zhao, Uniqueness of ground states of some coupled nonlinear Schrödinger systems and their application,, J. Differential Equations, 245 (2008), 2551.
doi: 10.1016/j.jde.2008.04.008. |
[17] |
E. Mitidieri and S. I. Pohozaev, A priori estimates and blow-up of solutions of nonlinear partial differential equations and inequalities,, Proc. Steklov Inst. Math., 234 (2001), 1.
|
[18] |
R. Osserman, On the inequality $\Delta u \ge f(u)$,, Pacific J. Math., 7 (1957), 1641.
|
[19] |
V. M. Perez-Garcia and J. B. Beitia, Symbiotic solitons in heteronuclear multicomponent Bose-Einstein condensates,, Phys. Rev. A, 72 (2005).
doi: 10.1103/PhysRevA.72.033620. |
[20] |
P. Pucci and J. Serrin, Uniqueness of ground states for quasilinear elliptic operators,, Indiana Univ. Math. J., 47 (1998), 501.
doi: 10.1512/iumj.1998.47.1517. |
[21] |
P. Quittner and P. Souplet, Symmetry of components for semilinear elliptic systems,, SIAM J. Math. Anal., 44 (2012), 2545.
doi: 10.1137/11085428X. |
[22] |
J. Serrin and M. Tang, Uniqueness of ground states for quasilinear elliptic equations,, Indiana Univ. Math. J., 49 (2000), 897.
doi: 10.1512/iumj.2000.49.1893. |
[23] |
J. Serrin and H. Zou, Symmetry of ground states of quasilinear elliptic equations,, Arch. Rational Mech. Anal., 148 (1999), 265.
doi: 10.1007/s002050050162. |
show all references
References:
[1] |
G. Bianchi, Non-existence of positive solutions to semilinear equations on $\mathbbR^n$ or $\mathbbR^n_+$ through the method of moving planes,, Comm. Partial Diff. Eqns., 22 (1997), 1671.
doi: 10.1080/03605309708821315. |
[2] |
H. Brezis, Semilinear equations in $R^N$ without condition at infinity,, Appl. Math. Optim., 12 (1984), 271.
doi: 10.1007/BF01449045. |
[3] |
L. A. Caffarelli, B. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth,, Comm. Pure Appl. Math., 42 (1989), 271.
doi: 10.1002/cpa.3160420304. |
[4] |
C. Cortàzar, M. Elgueta and P. Felmer, On a semilinear elliptic problem in $\mathbbR^N$ with a non-Lipschitzian non-linearity,, Advances in Differential Equations, 1 (1996), 199.
|
[5] |
E. N. Dancer and Y. Du, Some remarks on Liouville type results for quasilinear elliptic equations,, Proc. Amer. Math. Soc., 131 (2003), 1891.
doi: 10.1090/S0002-9939-02-06733-3. |
[6] |
A. Farina, Liouville-type theorems for elliptic problems,, in Handbook of Differential Equations, (2007), 61.
doi: 10.1016/S1874-5733(07)80005-2. |
[7] |
D. J. Frantzeskakis, Dark solitons in atomic Bose-Einstein condensates: From theory to experiments,, J. Phys. A, 43 (2010).
doi: 10.1088/1751-8113/43/21/213001. |
[8] |
B. Gidas, W.-M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $R^n$,, Adv. Math. Supp. Stud., 7a (1981), 369.
|
[9] |
F. Gazzola, J. Serrin and M. Tang, Existence of ground states and free boundary problems for quasilinear elliptic operators,, Advances in Differential Equations, 5 (2000), 1.
|
[10] |
B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations,, Comm. Partial Differential Equations, 6 (1981), 883.
doi: 10.1080/03605308108820196. |
[11] |
B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations,, Comm. Pure Appl. Math., 34 (1981), 525.
doi: 10.1002/cpa.3160340406. |
[12] |
T. Kato, Schrödinger operators with singular potentials,, Proceedings of the International Symposium on Partial Differential Equations and the Geometry of Normed Linear Spaces (Jerusalem, 13 (1972), 135.
doi: 10.1007/BF02760233. |
[13] |
J. B. Keller, On solutions of $ \Delta u = f (u)$,, Comm. Pure Appl. Math., 10 (1957), 503.
doi: 10.1002/cpa.3160100402. |
[14] |
Yu. S. Kivshar and B. Luther-Davies, Dark optical solitons: Physics and applications,, Phys. Rep., 298 (1998), 81.
doi: 10.1016/S0370-1573(97)00073-2. |
[15] |
T.-C. Lin and J.-C. Wei, Symbiotic bright solitary wave solutions of coupled nonlinear Schrödinger equations,, Nonlinearity, 19 (2006), 2755.
doi: 10.1088/0951-7715/19/12/002. |
[16] |
L. Ma and L. Zhao, Uniqueness of ground states of some coupled nonlinear Schrödinger systems and their application,, J. Differential Equations, 245 (2008), 2551.
doi: 10.1016/j.jde.2008.04.008. |
[17] |
E. Mitidieri and S. I. Pohozaev, A priori estimates and blow-up of solutions of nonlinear partial differential equations and inequalities,, Proc. Steklov Inst. Math., 234 (2001), 1.
|
[18] |
R. Osserman, On the inequality $\Delta u \ge f(u)$,, Pacific J. Math., 7 (1957), 1641.
|
[19] |
V. M. Perez-Garcia and J. B. Beitia, Symbiotic solitons in heteronuclear multicomponent Bose-Einstein condensates,, Phys. Rev. A, 72 (2005).
doi: 10.1103/PhysRevA.72.033620. |
[20] |
P. Pucci and J. Serrin, Uniqueness of ground states for quasilinear elliptic operators,, Indiana Univ. Math. J., 47 (1998), 501.
doi: 10.1512/iumj.1998.47.1517. |
[21] |
P. Quittner and P. Souplet, Symmetry of components for semilinear elliptic systems,, SIAM J. Math. Anal., 44 (2012), 2545.
doi: 10.1137/11085428X. |
[22] |
J. Serrin and M. Tang, Uniqueness of ground states for quasilinear elliptic equations,, Indiana Univ. Math. J., 49 (2000), 897.
doi: 10.1512/iumj.2000.49.1893. |
[23] |
J. Serrin and H. Zou, Symmetry of ground states of quasilinear elliptic equations,, Arch. Rational Mech. Anal., 148 (1999), 265.
doi: 10.1007/s002050050162. |
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