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Synchronized and ground-state solutions to a coupled Schrödinger system
Multiple non-radially symmetrical nodal solutions for the Schrödinger system with positive quasilinear term
1. | School of Mathematics and Statistics, Fujian Normal University, Qishan Campus, Fuzhou 350117, China |
2. | FJKLMAA and Center for Applied Mathematics of Fujian Province(FJNU), Fuzhou, 350117, China |
$ \mathbb R^{N}(N\geq3) $ |
$ \left\{\begin{aligned} &-\Delta u+A(x)u+\frac{k}{2}\triangle(u^{2})u = \frac{2\alpha }{\alpha+\beta}|u|^{\alpha-2}u|v|^{\beta},\\ &-\Delta v+Bv+\frac{k}{2}\triangle(v^{2})v = \frac{2\beta}{\alpha+\beta}|u|^{\alpha}|v|^{\beta-2}v,\\ & u(x)\to 0,\ \ v(x)\to 0\ \ \hbox{as}\ |x|\to \infty,\end{aligned}\right. $ |
$ \alpha,\beta>1 $ |
$ 2<\alpha+\beta<2^* = \frac{2N}{N-2} $ |
$ k >0 $ |
$ \xi\geq2 $ |
$ 2\xi $ |
References:
[1] |
C. Alves, Y. Wang and Y. Shen,
Soliton solutions for a class of quasilinear Schrödinger equations with a parameter, J. Differ. Equ., 259 (2015), 318-343.
doi: 10.1016/j.jde.2015.02.030. |
[2] |
T. Bartsch and Z. Wang,
Existence and multiplicity results for superlinear elliptic problems on $\mathbb R^{N}$, Commun. Partial Differ. Equ., 20 (1995), 1725-1741.
doi: 10.1080/03605309508821149. |
[3] |
H. Berestycki and P. Lion,
Nonlinear Scalar field equations, Ⅰ, Arch. Ration. Mech. Anal., 82 (1983), 313-346.
doi: 10.1007/BF00250555. |
[4] |
G. Bonnaud, H. Brandi, C. Manus, G. Mainfray and T. Lehner,
Relativistic and ponderomotive self-focusing of a laser beam in a radially inhomogeneous plasma. Ⅰ: paraxial approximation, Phys. Fluids, 5 (1993), 3539-3550.
|
[5] |
A. Borovskii and A. Galkin,
Dynamical modulation of an ultrashort high-intensity laser pulse in matter, J. Exp. Theor. Phys., 77 (1993), 562-573.
|
[6] |
A. Bouard, N. Hayashi and J. Saut,
Global existence of small solutions to a relativistic nonlinear Schrödinger equation, Commun. Math. Phys., 189 (1997), 73-105.
doi: 10.1007/s002200050191. |
[7] |
H. Brézis and E. Lieb,
A relation between pointwise convergence of function and convergence of functional, Proc. Amer. Math. Soc., 88 (1983), 486-490.
doi: 10.2307/2044999. |
[8] |
L. Brüll and H. Lange,
Solitary waves for quasilinear Schrödinger equations, Expos. Math., 4 (1986), 278-288.
|
[9] |
M. Colin and L. Jeanjean,
Solutions for a quasilinear Schrödinger equations: a dual approach, Nonlinear Anal., 56 (2004), 213-226.
doi: 10.1016/j.na.2003.09.008. |
[10] |
D. Costa and Z. Wang,
Multiplicity results for a class of superlinear elliptic problems, Proc. Amer. Math. Soc., 133 (2005), 787-794.
doi: 10.1090/S0002-9939-04-07635-X. |
[11] |
Y. Deng, S. Peng and J. Wang,
Nodal soliton solutions for generalized quasilinear Schrödinger equations, J. Math. Phys., 55 (2014), 051501.
doi: 10.1063/1.4874108. |
[12] |
Y. Deng, S. Peng and J. Wang,
Nodal soliton solutions for quasilinear Schrödinger equations with critical exponent, J. Math. Phys., 54 (2013), 011504.
doi: 10.1063/1.4774153. |
[13] |
J. Gamboa and J. Zhou,
Antisymmetric solutions for a class of quasilinear defocusing Schrödinger equations, Electron. J. Qual. Theory Differ. Equ., 16 (2020), 1-18.
doi: 10.14232/ejqtde.2020.1.16. |
[14] |
D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, Berlin, 2001. |
[15] |
C. Huang and G. Jia,
Existence of positive solutions for supercritical quasilinear Schrödinger elliptic equations, J. Math. Anal. Appl., 472 (2019), 705-727.
doi: 10.1016/j.jmaa.2018.11.048. |
[16] |
S. Kurihura,
Large-amplitude quasi-solitons in superfluids films, J. Phys. Soc. Japan, 50 (1981), 3262-3267.
doi: 10.1143/JPSJ.50.3801. |
[17] |
P. Lions, The concentration-compactness principle in the calculus of variations, Part 1-2, Ann. Inst. H. Poincaré., 1 (1984), 109-145 and 223–283. |
[18] |
H. Liu,
Positive solution for a quasilinear elliptic equation involving critical or supercritical exponent, J. Math. Phys., 57 (2016), 159-180.
doi: 10.1063/1.4947109. |
[19] |
J. Liu and Z. Wang,
Soliton solutions for quasilinear Schrödinger equations Ⅱ, J. Differ. Equ., 187 (2003), 473-493.
doi: 10.1016/S0022-0396(02)00064-5. |
[20] |
J. Liu, Y. Wang and Z. Wang,
Solutions for quasilinear Schrödinger equations via the Nehari method, Commun. Partial Differ. Equ., 29 (2004), 879-901.
doi: 10.1081/PDE-120037335. |
[21] |
V. Moroz and J. Schaftingen,
Ground states of nonlinear Choquard equations: Existence, qualitative properties and decay asymptotics, J. Funct. Anal., 265 (2013), 153-184.
doi: 10.1016/j.jfa.2013.04.007. |
[22] |
M. Poppenberg, K. Schmitt and Z. Wang,
On the existence of soliton solutions to quasilinear Schrödinger equations, Calc. Var. Partial Differ. Equ., 14 (2002), 329-344.
doi: 10.1007/s005260100105. |
[23] |
P. Rabinowitz,
On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270-291.
doi: 10.1007/BF00946631. |
[24] |
B. Ritchie,
Relativistic self-focusing and channel formation in laser-plasma interactions, Phys. Rev. E., 50 (1994), 687-689.
|
[25] |
E. Silva and G. Vieira,
Quasilinear asymptotically periodic Schrödinger equations with critical growth, Calc. Var. Partial Differ. Equ., 39 (2010), 1-33.
doi: 10.1007/s00526-009-0299-1. |
[26] |
A. Szulkin and S. Waliullah,
Sign-changing and symmetry-breaking solutions to singular problems, Complex Var. Theory Appl., 57 (2012), 1191-1208.
doi: 10.1080/17476933.2010.504849. |
[27] |
Y. Wang,
Multiplicity of solutions for singular quasilinear Schrödinger equations with critical exponents, J. Math. Anal. Appl., 458 (2018), 1027-1043.
doi: 10.1016/j.jmaa.2017.10.015. |
[28] |
M. Willem, Minimax Theorems, Birkhäuser, 1996.
doi: 10.1007/978-1-4612-4146-1. |
[29] |
W. Zhang and X. Liu,
Infinitely many sign-changing solutions for a quasilinear elliptic equation in $\mathbb R^{N}$, J. Math. Anal. Appl., 427 (2015), 722-740.
doi: 10.1016/j.jmaa.2015.02.070. |
show all references
References:
[1] |
C. Alves, Y. Wang and Y. Shen,
Soliton solutions for a class of quasilinear Schrödinger equations with a parameter, J. Differ. Equ., 259 (2015), 318-343.
doi: 10.1016/j.jde.2015.02.030. |
[2] |
T. Bartsch and Z. Wang,
Existence and multiplicity results for superlinear elliptic problems on $\mathbb R^{N}$, Commun. Partial Differ. Equ., 20 (1995), 1725-1741.
doi: 10.1080/03605309508821149. |
[3] |
H. Berestycki and P. Lion,
Nonlinear Scalar field equations, Ⅰ, Arch. Ration. Mech. Anal., 82 (1983), 313-346.
doi: 10.1007/BF00250555. |
[4] |
G. Bonnaud, H. Brandi, C. Manus, G. Mainfray and T. Lehner,
Relativistic and ponderomotive self-focusing of a laser beam in a radially inhomogeneous plasma. Ⅰ: paraxial approximation, Phys. Fluids, 5 (1993), 3539-3550.
|
[5] |
A. Borovskii and A. Galkin,
Dynamical modulation of an ultrashort high-intensity laser pulse in matter, J. Exp. Theor. Phys., 77 (1993), 562-573.
|
[6] |
A. Bouard, N. Hayashi and J. Saut,
Global existence of small solutions to a relativistic nonlinear Schrödinger equation, Commun. Math. Phys., 189 (1997), 73-105.
doi: 10.1007/s002200050191. |
[7] |
H. Brézis and E. Lieb,
A relation between pointwise convergence of function and convergence of functional, Proc. Amer. Math. Soc., 88 (1983), 486-490.
doi: 10.2307/2044999. |
[8] |
L. Brüll and H. Lange,
Solitary waves for quasilinear Schrödinger equations, Expos. Math., 4 (1986), 278-288.
|
[9] |
M. Colin and L. Jeanjean,
Solutions for a quasilinear Schrödinger equations: a dual approach, Nonlinear Anal., 56 (2004), 213-226.
doi: 10.1016/j.na.2003.09.008. |
[10] |
D. Costa and Z. Wang,
Multiplicity results for a class of superlinear elliptic problems, Proc. Amer. Math. Soc., 133 (2005), 787-794.
doi: 10.1090/S0002-9939-04-07635-X. |
[11] |
Y. Deng, S. Peng and J. Wang,
Nodal soliton solutions for generalized quasilinear Schrödinger equations, J. Math. Phys., 55 (2014), 051501.
doi: 10.1063/1.4874108. |
[12] |
Y. Deng, S. Peng and J. Wang,
Nodal soliton solutions for quasilinear Schrödinger equations with critical exponent, J. Math. Phys., 54 (2013), 011504.
doi: 10.1063/1.4774153. |
[13] |
J. Gamboa and J. Zhou,
Antisymmetric solutions for a class of quasilinear defocusing Schrödinger equations, Electron. J. Qual. Theory Differ. Equ., 16 (2020), 1-18.
doi: 10.14232/ejqtde.2020.1.16. |
[14] |
D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, Berlin, 2001. |
[15] |
C. Huang and G. Jia,
Existence of positive solutions for supercritical quasilinear Schrödinger elliptic equations, J. Math. Anal. Appl., 472 (2019), 705-727.
doi: 10.1016/j.jmaa.2018.11.048. |
[16] |
S. Kurihura,
Large-amplitude quasi-solitons in superfluids films, J. Phys. Soc. Japan, 50 (1981), 3262-3267.
doi: 10.1143/JPSJ.50.3801. |
[17] |
P. Lions, The concentration-compactness principle in the calculus of variations, Part 1-2, Ann. Inst. H. Poincaré., 1 (1984), 109-145 and 223–283. |
[18] |
H. Liu,
Positive solution for a quasilinear elliptic equation involving critical or supercritical exponent, J. Math. Phys., 57 (2016), 159-180.
doi: 10.1063/1.4947109. |
[19] |
J. Liu and Z. Wang,
Soliton solutions for quasilinear Schrödinger equations Ⅱ, J. Differ. Equ., 187 (2003), 473-493.
doi: 10.1016/S0022-0396(02)00064-5. |
[20] |
J. Liu, Y. Wang and Z. Wang,
Solutions for quasilinear Schrödinger equations via the Nehari method, Commun. Partial Differ. Equ., 29 (2004), 879-901.
doi: 10.1081/PDE-120037335. |
[21] |
V. Moroz and J. Schaftingen,
Ground states of nonlinear Choquard equations: Existence, qualitative properties and decay asymptotics, J. Funct. Anal., 265 (2013), 153-184.
doi: 10.1016/j.jfa.2013.04.007. |
[22] |
M. Poppenberg, K. Schmitt and Z. Wang,
On the existence of soliton solutions to quasilinear Schrödinger equations, Calc. Var. Partial Differ. Equ., 14 (2002), 329-344.
doi: 10.1007/s005260100105. |
[23] |
P. Rabinowitz,
On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270-291.
doi: 10.1007/BF00946631. |
[24] |
B. Ritchie,
Relativistic self-focusing and channel formation in laser-plasma interactions, Phys. Rev. E., 50 (1994), 687-689.
|
[25] |
E. Silva and G. Vieira,
Quasilinear asymptotically periodic Schrödinger equations with critical growth, Calc. Var. Partial Differ. Equ., 39 (2010), 1-33.
doi: 10.1007/s00526-009-0299-1. |
[26] |
A. Szulkin and S. Waliullah,
Sign-changing and symmetry-breaking solutions to singular problems, Complex Var. Theory Appl., 57 (2012), 1191-1208.
doi: 10.1080/17476933.2010.504849. |
[27] |
Y. Wang,
Multiplicity of solutions for singular quasilinear Schrödinger equations with critical exponents, J. Math. Anal. Appl., 458 (2018), 1027-1043.
doi: 10.1016/j.jmaa.2017.10.015. |
[28] |
M. Willem, Minimax Theorems, Birkhäuser, 1996.
doi: 10.1007/978-1-4612-4146-1. |
[29] |
W. Zhang and X. Liu,
Infinitely many sign-changing solutions for a quasilinear elliptic equation in $\mathbb R^{N}$, J. Math. Anal. Appl., 427 (2015), 722-740.
doi: 10.1016/j.jmaa.2015.02.070. |
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