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Topological obstructions to dominated splitting for ergodic translations on the higher dimensional torus
On the concentration of semiclassical states for nonlinear Dirac equations
School of Mathematics and Statistics, Central South University, Changsha 410083, Hunan, China |
$\begin{equation*}-i\varepsilonα·\nabla w+aβ w+V(x)w = g(|w|)w, \ x∈ \mathbb{R}^3, \ {\rm for}\ w∈ H^1(\mathbb R^3, \mathbb C^4), \end{equation*}$ |
$a > 0$ |
$α = (α_1, α_2, α_3)$ |
$α_1, α_2, α_3$ |
$β$ |
$4×4$ |
$V$ |
$g$ |
$C^1$ |
$g$ |
$\varepsilon \to 0$ |
References:
[1] |
A. Ambrosetti, A. Malchiodi and W. M. Ni,
Singularly perturbed elliptic equations with symmetry: existence of solutions concentrating on spheres Ⅰ, Comm. Math. Phys., 235 (2003), 427-466.
doi: 10.1007/s00220-003-0811-y. |
[2] |
T. Bartsch and Y. H. Ding,
Solutions of nonlinear Dirac equations, J. Differential Equations, 226 (2006), 210-249.
doi: 10.1016/j.jde.2005.08.014. |
[3] |
J. Byeon and Z.-Q. Wang,
Standing waves with a critical frequency for nonlinear Schrödinger equations, Arch. Rational Mech. Anal., 165 (2002), 295-316.
doi: 10.1007/s00205-002-0225-6. |
[4] |
J. Byeon and Z.-Q. Wang,
Standing waves with a critical frequency for nonlinear Schrödinger equations. Ⅱ, Calc. Var. Partial Differential Equations, 18 (2003), 207-219.
doi: 10.1007/s00526-002-0191-8. |
[5] |
R. Dautray and J. L. Lions,
Mathematical Analysis and Numerical Methods for Science and Technology, vol. 3, Springer, Berlin, 1990. |
[6] |
P. D'Avenia, A. Pomponio and D. Ruiz,
Semiclassical states for the onlinear Schrödinger equation on saddle points of the potential via variational methods, J. Funct. Anal., 262 (2012), 4600-4633.
doi: 10.1016/j.jfa.2012.03.009. |
[7] |
M. Del Pino and P. Felmer,
Semi-classical states for nonlinear Schrödinger equations, J. Funct. Anal., 149 (1997), 245-265.
doi: 10.1006/jfan.1996.3085. |
[8] |
M. Del Pino and P. Felmer,
Multi-peak bound states for nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Lináire, 15 (1998), 127-149.
doi: 10.1016/S0294-1449(97)89296-7. |
[9] |
M. Del Pino, M. Kowalczyk and J. C. Wei,
Concentration on curves for nonlinear Schrödinger equations, Comm. Pure Appl. Math., 60 (2007), 113-146.
doi: 10.1002/cpa.20135. |
[10] |
Y. H. Ding,
Variational Methods for Strongly Indefinite Problems, Interdiscip. Math. Sci., vol. 7. World Scientific Publ., Singapore, 2007.
doi: 10.1142/9789812709639. |
[11] |
Y. H. Ding,
Semi-classical ground states concentrating on the nonlinear potential for a Dirac equation, J. Differential Equations, 249 (2010), 1015-1034.
doi: 10.1016/j.jde.2010.03.022. |
[12] |
Y. H. Ding, C. Lee and B. Ruf,
On semiclassical states of a nonlinear Dirac equation, Proc. Roy. Soc. Edinburgh Sect. A, 143 (2013), 765-790.
doi: 10.1017/S0308210511001752. |
[13] |
Y. H. Ding and X. Y. Liu,
Semi-classical limits of ground states of a nonlinear Dirac equation, J. Differential Equations, 252 (2012), 4962-4987.
doi: 10.1016/j.jde.2012.01.023. |
[14] |
Y. H. Ding and B. Ruf,
Solutions of a nonlinear Dirac equation with external fields, Arch. Ration. Mech. Anal., 190 (2008), 57-82.
doi: 10.1007/s00205-008-0163-z. |
[15] |
Y. H. Ding and B. Ruf,
Existence and concentration of semiclassical solutions for Dirac equations with critical nonlinearities, SIAM J. Math. Anal., 44 (2012), 3755-3785.
doi: 10.1137/110850670. |
[16] |
Y. H. Ding and J. C. Wei,
Stationary states of nonlinear Dirac equations with general potentials, J. Math. Phys., 20 (2008), 1007-1032.
doi: 10.1142/S0129055X0800350X. |
[17] |
Y. H. Ding, J. C. Wei and T. Xu, Existence and concentration of semi-classical solutions for a nonlinear Maxwell-Dirac system,
J. Math. Phys., 54 (2013), 061505, 33pp.
doi: 10.1063/1.4811541. |
[18] |
Y. H. Ding and T. Xu,
Localized concentration of semi-classical states for nonlinear Dirac equations, Arch. Rational Mech. Anal., 216 (2015), 415-447.
doi: 10.1007/s00205-014-0811-4. |
[19] |
M.J. Esteban, M. Lewin and E. Séré,
Variational methods in relativistic quantum mechanics, Bull. Amer. Math. Soc. (N.S.), 45 (2008), 535-593.
doi: 10.1090/S0273-0979-08-01212-3. |
[20] |
M. J. Esteban and E. Séré,
Stationary states of the nonlinear Dirac equation: A variational approach, Commun. Math. Phys., 171 (1995), 323-350.
doi: 10.1007/BF02099273. |
[21] |
G. M. Figueiredo and M. T. O. Pimenta,
Existence of ground state solutions to Dirac equations with vanishing potentials at infinity, J. Differential Equations, 262 (2017), 486-505.
doi: 10.1016/j.jde.2016.09.034. |
[22] |
A. Floer and A. Weistein,
Nonspreading wave pockets for the cubic Schrödinger equation with a bounded potential, J. Funct. Anal., 69 (1986), 397-408.
doi: 10.1016/0022-1236(86)90096-0. |
[23] |
D. Gilbarg and N. Trudinger,
Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin Heidelberg New York, 1997. |
[24] |
L. Jeanjean and K. Tanaka,
Singularly perturbed elliptic problems with superlinear or asymptotically linear nonlinearities, Calc. Var. Partial Differential Equations, 21 (2004), 287-318.
doi: 10.1007/s00526-003-0261-6. |
[25] |
X. Kang and J. C. Wei,
On interacting bumps of semi-classical states of nonlinear Schrödinger equations, Adv. Differ. Equ., 5 (2000), 899-928.
|
[26] |
Y. Y. Li,
On a singularly perturbed elliptic equation, Adv. Diff. Equ., 2 (1997), 955-980.
|
[27] |
P. L. Lions,
The concentration-compactness principle in the calculus of variations. The locally compact case, parts 1 and 2, Ann. Inst. H. Poincaré Anual. Non Linéair, 1 (1984), 109-145,223-283.
doi: 10.1016/S0294-1449(16)30422-X. |
[28] |
S. B. Liu,
On superlinear Schrödinger equations with periodic potential, Calc. Var. Partial Differential Equations, 45 (2012), 1-9.
doi: 10.1007/s00526-011-0447-2. |
[29] |
F. Merle,
Existence of stationary states for nonlinear Dirac equations, J. Differential Equations, 74 (1988), 50-68.
doi: 10.1016/0022-0396(88)90018-6. |
[30] |
Y. G. Oh,
Existence of semiclassical bound states of nonlinear Schrödinger equations with potentials of the class $(V)_a$, Comm. Partial. Diff. Eq., 13 (1988), 1499-1519.
doi: 10.1080/03605308808820585. |
[31] |
A. Pankov,
On decay of solutions to nonlinear Schrödinger equations, Proc. Amer. Math. Soc., 136 (2008), 2565-2570.
doi: 10.1090/S0002-9939-08-09484-7. |
[32] |
P. H. Rabinowitz,
On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270-292.
doi: 10.1007/BF00946631. |
[33] |
A. Szulkin and T. Weth,
Ground state solutions for some indefinite variational problems, J. Funct. Anal., 257 (2009), 3802-3822.
doi: 10.1016/j.jfa.2009.09.013. |
[34] |
X. Wang,
On concentration of positive bound states of nonlinear Schrödinger equations, Comm. Math. Phys., 153 (1993), 229-244.
doi: 10.1007/BF02096642. |
[35] |
X. Wang and B. Zeng,
On concentration of positive bound states of nonlinear Schrödinger equations with competing potential functions, SIAM J. Math. Anal., 28 (1997), 633-655.
doi: 10.1137/S0036141095290240. |
[36] |
J. Zhang, X. Tang and W. Zhang, Ground state solutions for nonperiodic Dirac equation with superquadratic nonlinearity,
J. Math. Phys., 54 (2013), 101502, 10pp.
doi: 10.1063/1.4824132. |
[37] |
J. Zhang, X. Tang and W. Zhang,
On ground state solutions for superlinear Dirac equation, Acta Math. Sci. Ser. B Engl. Ed., 34 (2014), 840-850.
doi: 10.1016/S0252-9602(14)60054-0. |
show all references
References:
[1] |
A. Ambrosetti, A. Malchiodi and W. M. Ni,
Singularly perturbed elliptic equations with symmetry: existence of solutions concentrating on spheres Ⅰ, Comm. Math. Phys., 235 (2003), 427-466.
doi: 10.1007/s00220-003-0811-y. |
[2] |
T. Bartsch and Y. H. Ding,
Solutions of nonlinear Dirac equations, J. Differential Equations, 226 (2006), 210-249.
doi: 10.1016/j.jde.2005.08.014. |
[3] |
J. Byeon and Z.-Q. Wang,
Standing waves with a critical frequency for nonlinear Schrödinger equations, Arch. Rational Mech. Anal., 165 (2002), 295-316.
doi: 10.1007/s00205-002-0225-6. |
[4] |
J. Byeon and Z.-Q. Wang,
Standing waves with a critical frequency for nonlinear Schrödinger equations. Ⅱ, Calc. Var. Partial Differential Equations, 18 (2003), 207-219.
doi: 10.1007/s00526-002-0191-8. |
[5] |
R. Dautray and J. L. Lions,
Mathematical Analysis and Numerical Methods for Science and Technology, vol. 3, Springer, Berlin, 1990. |
[6] |
P. D'Avenia, A. Pomponio and D. Ruiz,
Semiclassical states for the onlinear Schrödinger equation on saddle points of the potential via variational methods, J. Funct. Anal., 262 (2012), 4600-4633.
doi: 10.1016/j.jfa.2012.03.009. |
[7] |
M. Del Pino and P. Felmer,
Semi-classical states for nonlinear Schrödinger equations, J. Funct. Anal., 149 (1997), 245-265.
doi: 10.1006/jfan.1996.3085. |
[8] |
M. Del Pino and P. Felmer,
Multi-peak bound states for nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Lináire, 15 (1998), 127-149.
doi: 10.1016/S0294-1449(97)89296-7. |
[9] |
M. Del Pino, M. Kowalczyk and J. C. Wei,
Concentration on curves for nonlinear Schrödinger equations, Comm. Pure Appl. Math., 60 (2007), 113-146.
doi: 10.1002/cpa.20135. |
[10] |
Y. H. Ding,
Variational Methods for Strongly Indefinite Problems, Interdiscip. Math. Sci., vol. 7. World Scientific Publ., Singapore, 2007.
doi: 10.1142/9789812709639. |
[11] |
Y. H. Ding,
Semi-classical ground states concentrating on the nonlinear potential for a Dirac equation, J. Differential Equations, 249 (2010), 1015-1034.
doi: 10.1016/j.jde.2010.03.022. |
[12] |
Y. H. Ding, C. Lee and B. Ruf,
On semiclassical states of a nonlinear Dirac equation, Proc. Roy. Soc. Edinburgh Sect. A, 143 (2013), 765-790.
doi: 10.1017/S0308210511001752. |
[13] |
Y. H. Ding and X. Y. Liu,
Semi-classical limits of ground states of a nonlinear Dirac equation, J. Differential Equations, 252 (2012), 4962-4987.
doi: 10.1016/j.jde.2012.01.023. |
[14] |
Y. H. Ding and B. Ruf,
Solutions of a nonlinear Dirac equation with external fields, Arch. Ration. Mech. Anal., 190 (2008), 57-82.
doi: 10.1007/s00205-008-0163-z. |
[15] |
Y. H. Ding and B. Ruf,
Existence and concentration of semiclassical solutions for Dirac equations with critical nonlinearities, SIAM J. Math. Anal., 44 (2012), 3755-3785.
doi: 10.1137/110850670. |
[16] |
Y. H. Ding and J. C. Wei,
Stationary states of nonlinear Dirac equations with general potentials, J. Math. Phys., 20 (2008), 1007-1032.
doi: 10.1142/S0129055X0800350X. |
[17] |
Y. H. Ding, J. C. Wei and T. Xu, Existence and concentration of semi-classical solutions for a nonlinear Maxwell-Dirac system,
J. Math. Phys., 54 (2013), 061505, 33pp.
doi: 10.1063/1.4811541. |
[18] |
Y. H. Ding and T. Xu,
Localized concentration of semi-classical states for nonlinear Dirac equations, Arch. Rational Mech. Anal., 216 (2015), 415-447.
doi: 10.1007/s00205-014-0811-4. |
[19] |
M.J. Esteban, M. Lewin and E. Séré,
Variational methods in relativistic quantum mechanics, Bull. Amer. Math. Soc. (N.S.), 45 (2008), 535-593.
doi: 10.1090/S0273-0979-08-01212-3. |
[20] |
M. J. Esteban and E. Séré,
Stationary states of the nonlinear Dirac equation: A variational approach, Commun. Math. Phys., 171 (1995), 323-350.
doi: 10.1007/BF02099273. |
[21] |
G. M. Figueiredo and M. T. O. Pimenta,
Existence of ground state solutions to Dirac equations with vanishing potentials at infinity, J. Differential Equations, 262 (2017), 486-505.
doi: 10.1016/j.jde.2016.09.034. |
[22] |
A. Floer and A. Weistein,
Nonspreading wave pockets for the cubic Schrödinger equation with a bounded potential, J. Funct. Anal., 69 (1986), 397-408.
doi: 10.1016/0022-1236(86)90096-0. |
[23] |
D. Gilbarg and N. Trudinger,
Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin Heidelberg New York, 1997. |
[24] |
L. Jeanjean and K. Tanaka,
Singularly perturbed elliptic problems with superlinear or asymptotically linear nonlinearities, Calc. Var. Partial Differential Equations, 21 (2004), 287-318.
doi: 10.1007/s00526-003-0261-6. |
[25] |
X. Kang and J. C. Wei,
On interacting bumps of semi-classical states of nonlinear Schrödinger equations, Adv. Differ. Equ., 5 (2000), 899-928.
|
[26] |
Y. Y. Li,
On a singularly perturbed elliptic equation, Adv. Diff. Equ., 2 (1997), 955-980.
|
[27] |
P. L. Lions,
The concentration-compactness principle in the calculus of variations. The locally compact case, parts 1 and 2, Ann. Inst. H. Poincaré Anual. Non Linéair, 1 (1984), 109-145,223-283.
doi: 10.1016/S0294-1449(16)30422-X. |
[28] |
S. B. Liu,
On superlinear Schrödinger equations with periodic potential, Calc. Var. Partial Differential Equations, 45 (2012), 1-9.
doi: 10.1007/s00526-011-0447-2. |
[29] |
F. Merle,
Existence of stationary states for nonlinear Dirac equations, J. Differential Equations, 74 (1988), 50-68.
doi: 10.1016/0022-0396(88)90018-6. |
[30] |
Y. G. Oh,
Existence of semiclassical bound states of nonlinear Schrödinger equations with potentials of the class $(V)_a$, Comm. Partial. Diff. Eq., 13 (1988), 1499-1519.
doi: 10.1080/03605308808820585. |
[31] |
A. Pankov,
On decay of solutions to nonlinear Schrödinger equations, Proc. Amer. Math. Soc., 136 (2008), 2565-2570.
doi: 10.1090/S0002-9939-08-09484-7. |
[32] |
P. H. Rabinowitz,
On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270-292.
doi: 10.1007/BF00946631. |
[33] |
A. Szulkin and T. Weth,
Ground state solutions for some indefinite variational problems, J. Funct. Anal., 257 (2009), 3802-3822.
doi: 10.1016/j.jfa.2009.09.013. |
[34] |
X. Wang,
On concentration of positive bound states of nonlinear Schrödinger equations, Comm. Math. Phys., 153 (1993), 229-244.
doi: 10.1007/BF02096642. |
[35] |
X. Wang and B. Zeng,
On concentration of positive bound states of nonlinear Schrödinger equations with competing potential functions, SIAM J. Math. Anal., 28 (1997), 633-655.
doi: 10.1137/S0036141095290240. |
[36] |
J. Zhang, X. Tang and W. Zhang, Ground state solutions for nonperiodic Dirac equation with superquadratic nonlinearity,
J. Math. Phys., 54 (2013), 101502, 10pp.
doi: 10.1063/1.4824132. |
[37] |
J. Zhang, X. Tang and W. Zhang,
On ground state solutions for superlinear Dirac equation, Acta Math. Sci. Ser. B Engl. Ed., 34 (2014), 840-850.
doi: 10.1016/S0252-9602(14)60054-0. |
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