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On the self-dual Einstein-Maxwell-Higgs equation on compact surfaces
Construction of solutions for some localized nonlinear Schrödinger equations
Departamento de Matemática, Pontificia Universidad Católica de Chile, Av. Vicuña Mackenna 4860, Santiago, Chile |
$N$ |
$\begin{align*}-\varphi''+V(x)h'(|\varphi|^2)\varphi = λ \varphi,\end{align*}$ |
$V(x) = -χ_{[-1,1]} (x)$ |
$[-1,1]$ |
$h'$ |
$λ$ |
References:
[1] |
M. J. Ablowitz, B. Prinari and A. D. Trubatch, Discrete and Continuous Nonlinear Schrödinger Systems, London Mathematical Society Lecture Note Series, 302, Cambridge University Press, 2004. |
[2] |
A. Ambrosetti, A. Malchiodi and D. Ruiz,
Bound states of nonlinear Schrödinger equations with potentials vanishing at infinity, J. Anal. Math., 98 (2006), 317-348.
doi: 10.1007/BF02790279. |
[3] |
G. Arioli and A. Szulkin,
Semilinear Schrödinger equation in the presence of a magnetic field, Arch. Ration. Mech. Anal., 170 (2003), 277-295.
doi: 10.1007/s00205-003-0274-5. |
[4] |
J. Bourgain, On nonlinear Schrödinger equations in Les Relations Entre Les Mathématiques et la Physique Théorique, Inst. Hautes Études Sci., Bures-sur-Yvette, (1998), 11-21. |
[5] |
J. Bourgain, Global Solutions of Nonlinear Schrödinger Equations, American Mathematical Society Colloquium Publications, 46. American Mathematical Society, 1999.
doi: 10.1090/coll/046. |
[6] |
N. Burq and M. Zworski,
Instability for the Semiclassical Non-linear Schrödinger equation, Commun. Math. Phys., 260 (2005), 45-58.
doi: 10.1007/s00220-005-1402-x. |
[7] |
J. Byeon, L. Jeanjean and K. Tanaka,
Standing waves for nonlinear Schrödinger equations with a general nonlinearity: one and two dimensional cases, Comm. Partial Differential Equations, 33 (2008), 1113-1136.
doi: 10.1080/03605300701518174. |
[8] |
C. Cacciapuoti, D. Finco, D. Noja and A. Teta,
The NLS equation in dimension one with spatially concentrated nonlinearities: the point like limit, Lett. Math. Phys., 104 (2014), 1557-1570.
doi: 10.1007/s11005-014-0725-y. |
[9] |
R. Carretero-Gonzáaleza, J. D. Talley, C. Chong and B. A. Malomed,
Multistable solitons in the cubic-quintic discrete nonlinear Schrödinger equation, Physica D, 216 (2006), 77-89.
doi: 10.1016/j.physd.2006.01.022. |
[10] |
T. Cazenave,
Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, 10, American Mathematical Society, 2003.
doi: 10.1090/cln/010. |
[11] |
T. Chen and N. Pavlović,
The quintic NLS as the mean field limit of a boson gas with three-body interactions, J. Funct. Anal., 260 (2011), 959-997.
doi: 10.1016/j.jfa.2010.11.003. |
[12] |
H. Christianson, J. Marzuola, J. Metcalfe and M. Taylor,
Nonlinear Bound States on Weakly Homogeneous Spaces, Communications in Partial Differential Equations, 39 (2014), 34-97.
doi: 10.1080/03605302.2013.845044. |
[13] |
S. Cingolani, L. Jeanjean and K. Tanaka,
Multiplicity of positive solutions of nonlinear Schrödinger equations concentrating at a potential well, Calc. Var. Partial Differential Equations, 53 (2015), 413-439.
doi: 10.1007/s00526-014-0754-5. |
[14] |
S. Cingolani, L. Jeanjean and K. Tanaka,
Multiple complex-valued solutions for nonlinear magnetic Schrödinger equations, J. Fixed Point Theory Appl., 19 (2017), 37-66.
doi: 10.1007/s11784-016-0347-3. |
[15] |
D. G. Costa and H. Tehrani,
On a class of asymptotically linear elliptic problems in ${\mathbb R}^N$, J. Differential Equations, 173 (2001), 470-494.
doi: 10.1006/jdeq.2000.3944. |
[16] |
R. de Marchi and R. Ruviaro,
Existence of solutions for a nonperiodic semilinear Schrödinger equation, Complex Var. Elliptic Equ., 61 (2016), 1290-1302.
doi: 10.1080/17476933.2016.1167887. |
[17] |
F. Della Casa and A. Sacchetti,
Stationary states for non linear one-dimensional Schrödinger equations with singular potential, Phys. D, 219 (2006), 60-68.
doi: 10.1016/j.physd.2006.05.014. |
[18] |
S. Deng, D. Garrido and M. Musso,
Multiple blow-up solutions for an exponential nonlinearity with potential in ${\mathbb R}^2$, Nonlinear Anal., 119 (2015), 419-442.
doi: 10.1016/j.na.2014.10.034. |
[19] |
Y. Ding and Z. Wang,
Bound states of nonlinear Schrödinger equations with magnetic fields, Annali di Matematica, 190 (2011), 427-451.
doi: 10.1007/s10231-010-0157-y. |
[20] |
M. Fei and H. Yin,
Bound states of asymptotically linear Schrödinger equations with compactly supported potential, Pacific J. of Math., 261 (2013), 335-367.
doi: 10.2140/pjm.2013.261.335. |
[21] |
R. Fukuizumi and L. Jeanjean,
Stability of standing waves for a nonlinear Schrödinger equation with repulsive Dirac delta potential, Disc. and Cont. Dynamical Systems, 21 (2008), 121-136.
doi: 10.3934/dcds.2008.21.121. |
[22] |
F. Gazzola, J. Serrin and M. Tang,
Existence of ground states and free boundary problems for quasilinear elliptic operators, Adv. in Diff. Eq., 5 (2000), 1-30.
|
[23] |
J. F. Lam, B. Lippmann and F. Tappert, Self-trapped laser beams in plasma,
Physics of Fluids, 20 (1977), 1176.
doi: 10.1063/1.861679. |
[24] |
M. Lewin, Mean-field limit of Bose systems: Rigorous results, Proceedings from the International Congress of Mathematical Physics at Santiago de Chile, 2015. |
[25] |
M. Lewin, P. T. Nam and N. Rougerie,
Derivation of Hartree's theory for generic mean-field Bose systems, Adv. Math., 254 (2014), 570-621.
doi: 10.1016/j.aim.2013.12.010. |
[26] |
M. Lewin, P. T. Nam and N. Rougerie,
The mean-field approximation and the non-linear Schrödinger functional for trapped Bose gases, Trans. Amer. Math. Soc., 368 (2016), 6131-6157.
doi: 10.1090/tran/6537. |
[27] |
Z. Liu, J. Su and T. Weth,
Compactness results for Schrödinger equations with asymptotically linear terms, J. Differential Equations, 231 (2006), 501-512.
doi: 10.1016/j.jde.2006.05.007. |
[28] |
C. Liu, Z. Wangand and H.-S. Zhou,
Asymptotically linear Schrödinger equation with potential vanishing at infinity, J. Differential Equations, 245 (2008), 201-222.
doi: 10.1016/j.jde.2008.01.006. |
[29] |
K. W. Mahmud, J. N. Kutz and W. P. Reinhardt, Bose-Einstein condensates in a one-dimensional double square well: Analytical solutions of the nonlinear Schrödinger equation, Physical Review A, 66 (2002), 063607, 11 pages. |
[30] |
B. Malomed and D. Pelinovsky,
Persistence of the Thomas-Fermi approximation for ground states of the Gross-Pitaevskii equation supported by the nonlinear confinement, Applied Mathematics Letters, 40 (2015), 45-48.
doi: 10.1016/j.aml.2014.09.004. |
[31] |
M. I. Molina and C. A. Bustamante, The Attractive Nonlinear Delta-function Potential,
American Journal of Physics 70, 67 (2002); doi: http://dx.doi.org/10.1119/1.1417529. |
[32] |
A. R. Nahmod,
The nonlinear Schrödinger equation on tori: Integrating harmonic analysis, geometry, and probability, Bull. Amer. Math. Soc. (N.S.), 53 (2016), 57-91.
doi: 10.1090/bull/1516. |
[33] |
D. E. Pelinovsky,
Localization in Periodic Potentials. From Schr ödinger operators to the Gross-Pitaevskii equation (London Mathematical Society Lecture Note Series), 1st Ed., Cambridge University Press, 2011.
doi: 10.1017/CBO9780511997754. |
[34] |
P. Pucci and R. Servadei,
Existence, non-existence and regularity of radial ground states for p-Laplacian equations with singular weights, Ann. Inst. H. Poincaré, 25 (2008), 505-537.
doi: 10.1016/j.anihpc.2007.02.004. |
[35] |
J. J. Rasmussen and K. Rypdal,
Blow-up in nonlinear Schrödinger equations-I, A general review, Physica Scripta, 33 (1986), 481-497.
doi: 10.1088/0031-8949/33/6/001. |
[36] |
S. Sheng, F. Wang and T. An, Existence and multiplicity of positive bound states for Schrödinger equations,
Boundary Value Problems, 2013 (2013), 11pp.
doi: 10.1186/1687-2770-2013-271. |
[37] |
A. Soffer and M. I. Weinstein,
Selection of the ground state for nonlinear Schrödinger equations, Reviews in Mathematical Physics, 16 (2004), 977-1071.
doi: 10.1142/S0129055X04002175. |
[38] |
C. Sourdis,
On the existence of dark solitons of the defocusing cubic nonlinear Schrödinger equation with periodic inhomogeneous nonlinearity, Applied Mathematics Letters, 46 (2015), 123-126.
doi: 10.1016/j.aml.2015.02.018. |
[39] |
C. Sulem and P.-L. Sulem,
The Nonlinear Schrödinger Equation. Self-focusing and Wave Collapse (Applied Mathematical Sciences), Springer, 1999. |
[40] |
T. Tao,
Nonlinear Dispersive Equations. Local and Global Analysis (CBMS Regional Conference Series in Mathematics), American Mathematical Society, 2006.
doi: 10.1090/cbms/106. |
[41] |
T. Tao,
A global compact attractor for high-dimensional defocusing non-linear Schrödinger equations with potential, Dynamics of PDE, 5 (2008), 101-116.
doi: 10.4310/DPDE.2008.v5.n2.a1. |
[42] |
T. Tsai,
Asymptotic dynamics of nonlinear Schrödinger equations with many bound states, J. Od Diff. Eq., 192 (2003), 225-282.
doi: 10.1016/S0022-0396(03)00041-X. |
[43] |
C. E. Wayne and M. I. Weinstein,
Dynamics of Partial Differential Equations (Frontiers in Applied Dynamical Systems: Reviews and Tutorials), Springer, 2015.
doi: 10.1007/978-3-319-19935-1. |
[44] |
M. Willem,
Minimax Theorems (Progress in Nonlinear Differential Equations and Their Applications), Birkhäuser, 1996.
doi: 10.1007/978-1-4612-4146-1. |
show all references
References:
[1] |
M. J. Ablowitz, B. Prinari and A. D. Trubatch, Discrete and Continuous Nonlinear Schrödinger Systems, London Mathematical Society Lecture Note Series, 302, Cambridge University Press, 2004. |
[2] |
A. Ambrosetti, A. Malchiodi and D. Ruiz,
Bound states of nonlinear Schrödinger equations with potentials vanishing at infinity, J. Anal. Math., 98 (2006), 317-348.
doi: 10.1007/BF02790279. |
[3] |
G. Arioli and A. Szulkin,
Semilinear Schrödinger equation in the presence of a magnetic field, Arch. Ration. Mech. Anal., 170 (2003), 277-295.
doi: 10.1007/s00205-003-0274-5. |
[4] |
J. Bourgain, On nonlinear Schrödinger equations in Les Relations Entre Les Mathématiques et la Physique Théorique, Inst. Hautes Études Sci., Bures-sur-Yvette, (1998), 11-21. |
[5] |
J. Bourgain, Global Solutions of Nonlinear Schrödinger Equations, American Mathematical Society Colloquium Publications, 46. American Mathematical Society, 1999.
doi: 10.1090/coll/046. |
[6] |
N. Burq and M. Zworski,
Instability for the Semiclassical Non-linear Schrödinger equation, Commun. Math. Phys., 260 (2005), 45-58.
doi: 10.1007/s00220-005-1402-x. |
[7] |
J. Byeon, L. Jeanjean and K. Tanaka,
Standing waves for nonlinear Schrödinger equations with a general nonlinearity: one and two dimensional cases, Comm. Partial Differential Equations, 33 (2008), 1113-1136.
doi: 10.1080/03605300701518174. |
[8] |
C. Cacciapuoti, D. Finco, D. Noja and A. Teta,
The NLS equation in dimension one with spatially concentrated nonlinearities: the point like limit, Lett. Math. Phys., 104 (2014), 1557-1570.
doi: 10.1007/s11005-014-0725-y. |
[9] |
R. Carretero-Gonzáaleza, J. D. Talley, C. Chong and B. A. Malomed,
Multistable solitons in the cubic-quintic discrete nonlinear Schrödinger equation, Physica D, 216 (2006), 77-89.
doi: 10.1016/j.physd.2006.01.022. |
[10] |
T. Cazenave,
Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, 10, American Mathematical Society, 2003.
doi: 10.1090/cln/010. |
[11] |
T. Chen and N. Pavlović,
The quintic NLS as the mean field limit of a boson gas with three-body interactions, J. Funct. Anal., 260 (2011), 959-997.
doi: 10.1016/j.jfa.2010.11.003. |
[12] |
H. Christianson, J. Marzuola, J. Metcalfe and M. Taylor,
Nonlinear Bound States on Weakly Homogeneous Spaces, Communications in Partial Differential Equations, 39 (2014), 34-97.
doi: 10.1080/03605302.2013.845044. |
[13] |
S. Cingolani, L. Jeanjean and K. Tanaka,
Multiplicity of positive solutions of nonlinear Schrödinger equations concentrating at a potential well, Calc. Var. Partial Differential Equations, 53 (2015), 413-439.
doi: 10.1007/s00526-014-0754-5. |
[14] |
S. Cingolani, L. Jeanjean and K. Tanaka,
Multiple complex-valued solutions for nonlinear magnetic Schrödinger equations, J. Fixed Point Theory Appl., 19 (2017), 37-66.
doi: 10.1007/s11784-016-0347-3. |
[15] |
D. G. Costa and H. Tehrani,
On a class of asymptotically linear elliptic problems in ${\mathbb R}^N$, J. Differential Equations, 173 (2001), 470-494.
doi: 10.1006/jdeq.2000.3944. |
[16] |
R. de Marchi and R. Ruviaro,
Existence of solutions for a nonperiodic semilinear Schrödinger equation, Complex Var. Elliptic Equ., 61 (2016), 1290-1302.
doi: 10.1080/17476933.2016.1167887. |
[17] |
F. Della Casa and A. Sacchetti,
Stationary states for non linear one-dimensional Schrödinger equations with singular potential, Phys. D, 219 (2006), 60-68.
doi: 10.1016/j.physd.2006.05.014. |
[18] |
S. Deng, D. Garrido and M. Musso,
Multiple blow-up solutions for an exponential nonlinearity with potential in ${\mathbb R}^2$, Nonlinear Anal., 119 (2015), 419-442.
doi: 10.1016/j.na.2014.10.034. |
[19] |
Y. Ding and Z. Wang,
Bound states of nonlinear Schrödinger equations with magnetic fields, Annali di Matematica, 190 (2011), 427-451.
doi: 10.1007/s10231-010-0157-y. |
[20] |
M. Fei and H. Yin,
Bound states of asymptotically linear Schrödinger equations with compactly supported potential, Pacific J. of Math., 261 (2013), 335-367.
doi: 10.2140/pjm.2013.261.335. |
[21] |
R. Fukuizumi and L. Jeanjean,
Stability of standing waves for a nonlinear Schrödinger equation with repulsive Dirac delta potential, Disc. and Cont. Dynamical Systems, 21 (2008), 121-136.
doi: 10.3934/dcds.2008.21.121. |
[22] |
F. Gazzola, J. Serrin and M. Tang,
Existence of ground states and free boundary problems for quasilinear elliptic operators, Adv. in Diff. Eq., 5 (2000), 1-30.
|
[23] |
J. F. Lam, B. Lippmann and F. Tappert, Self-trapped laser beams in plasma,
Physics of Fluids, 20 (1977), 1176.
doi: 10.1063/1.861679. |
[24] |
M. Lewin, Mean-field limit of Bose systems: Rigorous results, Proceedings from the International Congress of Mathematical Physics at Santiago de Chile, 2015. |
[25] |
M. Lewin, P. T. Nam and N. Rougerie,
Derivation of Hartree's theory for generic mean-field Bose systems, Adv. Math., 254 (2014), 570-621.
doi: 10.1016/j.aim.2013.12.010. |
[26] |
M. Lewin, P. T. Nam and N. Rougerie,
The mean-field approximation and the non-linear Schrödinger functional for trapped Bose gases, Trans. Amer. Math. Soc., 368 (2016), 6131-6157.
doi: 10.1090/tran/6537. |
[27] |
Z. Liu, J. Su and T. Weth,
Compactness results for Schrödinger equations with asymptotically linear terms, J. Differential Equations, 231 (2006), 501-512.
doi: 10.1016/j.jde.2006.05.007. |
[28] |
C. Liu, Z. Wangand and H.-S. Zhou,
Asymptotically linear Schrödinger equation with potential vanishing at infinity, J. Differential Equations, 245 (2008), 201-222.
doi: 10.1016/j.jde.2008.01.006. |
[29] |
K. W. Mahmud, J. N. Kutz and W. P. Reinhardt, Bose-Einstein condensates in a one-dimensional double square well: Analytical solutions of the nonlinear Schrödinger equation, Physical Review A, 66 (2002), 063607, 11 pages. |
[30] |
B. Malomed and D. Pelinovsky,
Persistence of the Thomas-Fermi approximation for ground states of the Gross-Pitaevskii equation supported by the nonlinear confinement, Applied Mathematics Letters, 40 (2015), 45-48.
doi: 10.1016/j.aml.2014.09.004. |
[31] |
M. I. Molina and C. A. Bustamante, The Attractive Nonlinear Delta-function Potential,
American Journal of Physics 70, 67 (2002); doi: http://dx.doi.org/10.1119/1.1417529. |
[32] |
A. R. Nahmod,
The nonlinear Schrödinger equation on tori: Integrating harmonic analysis, geometry, and probability, Bull. Amer. Math. Soc. (N.S.), 53 (2016), 57-91.
doi: 10.1090/bull/1516. |
[33] |
D. E. Pelinovsky,
Localization in Periodic Potentials. From Schr ödinger operators to the Gross-Pitaevskii equation (London Mathematical Society Lecture Note Series), 1st Ed., Cambridge University Press, 2011.
doi: 10.1017/CBO9780511997754. |
[34] |
P. Pucci and R. Servadei,
Existence, non-existence and regularity of radial ground states for p-Laplacian equations with singular weights, Ann. Inst. H. Poincaré, 25 (2008), 505-537.
doi: 10.1016/j.anihpc.2007.02.004. |
[35] |
J. J. Rasmussen and K. Rypdal,
Blow-up in nonlinear Schrödinger equations-I, A general review, Physica Scripta, 33 (1986), 481-497.
doi: 10.1088/0031-8949/33/6/001. |
[36] |
S. Sheng, F. Wang and T. An, Existence and multiplicity of positive bound states for Schrödinger equations,
Boundary Value Problems, 2013 (2013), 11pp.
doi: 10.1186/1687-2770-2013-271. |
[37] |
A. Soffer and M. I. Weinstein,
Selection of the ground state for nonlinear Schrödinger equations, Reviews in Mathematical Physics, 16 (2004), 977-1071.
doi: 10.1142/S0129055X04002175. |
[38] |
C. Sourdis,
On the existence of dark solitons of the defocusing cubic nonlinear Schrödinger equation with periodic inhomogeneous nonlinearity, Applied Mathematics Letters, 46 (2015), 123-126.
doi: 10.1016/j.aml.2015.02.018. |
[39] |
C. Sulem and P.-L. Sulem,
The Nonlinear Schrödinger Equation. Self-focusing and Wave Collapse (Applied Mathematical Sciences), Springer, 1999. |
[40] |
T. Tao,
Nonlinear Dispersive Equations. Local and Global Analysis (CBMS Regional Conference Series in Mathematics), American Mathematical Society, 2006.
doi: 10.1090/cbms/106. |
[41] |
T. Tao,
A global compact attractor for high-dimensional defocusing non-linear Schrödinger equations with potential, Dynamics of PDE, 5 (2008), 101-116.
doi: 10.4310/DPDE.2008.v5.n2.a1. |
[42] |
T. Tsai,
Asymptotic dynamics of nonlinear Schrödinger equations with many bound states, J. Od Diff. Eq., 192 (2003), 225-282.
doi: 10.1016/S0022-0396(03)00041-X. |
[43] |
C. E. Wayne and M. I. Weinstein,
Dynamics of Partial Differential Equations (Frontiers in Applied Dynamical Systems: Reviews and Tutorials), Springer, 2015.
doi: 10.1007/978-3-319-19935-1. |
[44] |
M. Willem,
Minimax Theorems (Progress in Nonlinear Differential Equations and Their Applications), Birkhäuser, 1996.
doi: 10.1007/978-1-4612-4146-1. |
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