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KdV-like solitary waves in two-dimensional FPU-lattices
Improved results for Klein-Gordon-Maxwell systems with general nonlinearity
School of Mathematics and Statistics, Central South University, Changsha, 410083 Hunan, China |
$\left\{ \begin{align} &-\vartriangle u+\left[ m_{0}^{2}-{{(\omega +\phi )}^{2}} \right]u = f(u),\ \ \ \ \text{in}\ \ {{\mathbb{R}}^{3}}, \\ &\vartriangle \phi = (\omega +\phi ){{u}^{2}},\ \ \ \ \text{in}\ \ {{\mathbb{R}}^{3}}, \\ \end{align} \right.$ |
$0 < ω≤ m_0$ |
$f∈ \mathcal{C}(\mathbb{R}, \mathbb{R})$ |
$0 < ω < m_0$ |
$f$ |
$ω = m_0$ |
$f$ |
References:
[1] |
A. Azzollini, V. Benci, T. D'Aprile and D. Fortunato,
Existence of static solutions of the semilinear Maxwell equations, Ricerche di Matematica, 55 (2006), 283-297.
|
[2] |
A. Azzollini and A. Pomponio,
Ground state solutions for the nonlinear Klein-Gordon-Maxwell equations, Topol. Methods Nonlinear Anal., 35 (2010), 33-42.
|
[3] |
A. Azzollini, L. Pisani and A. Pomponio,
Improved estimates and a limit case for the electrostatic Klein-Gordon-Maxwell system, Proc. Roy. Soc. Edinburgh Sect. A, 141 (2011), 449-463.
doi: 10.1017/S0308210509001814. |
[4] |
V. Benci and D. Fortunato,
The nonlinear Klein-Gordon equation coupled with the Maxwell equations, Nonlinear Anal., 47 (2001), 6065-6072.
doi: 10.1016/S0362-546X(01)00688-5. |
[5] |
V. Benci and D. Fortunato,
Solitary waves of the nonlinear Klein-Gordon equation coupled with the Maxwell equations, Rev. Math. Phys., 14 (2002), 409-420.
doi: 10.1142/S0129055X02001168. |
[6] |
H. Berestycki and P. L. Lions,
Nonlinear scalar field equations, I -Existence of a ground state, Arch. Rational Mech. Anal., 82 (1983), 313-345.
|
[7] |
P. Carrião, P. Cunha and O. Miyagaki,
Positive ground state solutions for the critical Klein-Gordon-Maxwell system with potentials, Nonlinear Anal., 75 (2012), 4068-4078.
doi: 10.1016/j.na.2012.02.023. |
[8] |
D. Cassani,
Existence and non-existence of solitary waves for the critical Klein-Gordon equation coupled with Maxwell' s equations, Nonlinear Anal., 58 (2004), 733-747.
doi: 10.1016/j.na.2003.05.001. |
[9] |
S. T. Chen and X. H. Tang, Ground state sign-changing solutions for a class of Schrödinger-Poisson type problems in $\mathbb{R}^3$,
Z. Angew. Math. Phys. , 67 (2016), Art. 102, 18 pp. |
[10] |
S. T. Chen and X. H. Tang,
Nehari type ground state solutions for asymptotically periodic Schrödinger-Poisson systems, Taiwan. J. Math., 21 (2017), 363-383.
doi: 10.11650/tjm/7784. |
[11] |
S. T. Chen and X. H. Tang,
Ground state sign-changing solutions for asymptotically cubic or super-cubic Schrödinger-Poisson systems without compact condition, Comput. Math. Appl., 74 (2017), 446-458.
doi: 10.1016/j.camwa.2017.04.031. |
[12] |
P. L. Cunha,
Subcritical and supercritical Klein-Gordon-Maxwell equations without Ambrosetti-Rabinowitz condition, Differ. Integral Equ., 27 (2014), 387-399.
|
[13] |
T. D'Aprile and D. Mugnai,
Solitary waves for nonlinear Klein-Gordon-Maxwell and Schrödinger-Maxwell equations, Proc. R. Soc. Edinb. Sect. A, 134 (2004), 893-906.
doi: 10.1017/S030821050000353X. |
[14] |
T. D'Aprile and D. Mugnai,
Non-existence results for the coupled Klein-Gordon-Maxwell equations, Adv. Nonlinear Stud., 4 (2004), 307-322.
|
[15] |
L. Ding and L. Li,
Infinitely many standing wave solutions for the nonlinear Klein-Gordon-Maxwell system with sign-changing potential, Comput. Math. Appl., 68 (2014), 589-595.
doi: 10.1016/j.camwa.2014.07.001. |
[16] |
L. Jeanjean,
Existence of solutions with prescribed norm for semilinear elliptic equations, Nonlinear Anal., 28 (1997), 1633-1659.
doi: 10.1016/S0362-546X(96)00021-1. |
[17] |
L. Jeanjean,
On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer-type problem set on $\mathbb{R}^N$, Proc. R. Soc. Edinb., Sect. A, Math., 129 (1999), 787-809.
doi: 10.1017/S0308210500013147. |
[18] |
W. Jeong and J. Seok,
On perturbation of a functional with the mountain pass geometry, Calc. Var. Partial Differential Equations, Calc. Var. Partial Differential Equations, 49 (2014), 649-668.
doi: 10.1007/s00526-013-0595-7. |
[19] |
G. B. Li and C. Wang,
The existence of a nontrivial solution to a nonlinear elliptic problem of linking type without the Ambrosetti-Rabinowitz condition, Ann. Acad. Sci. Fenn. Math., 36 (2011), 461-480.
doi: 10.5186/aasfm.2011.3627. |
[20] |
L. Li and C. L. Tang,
Infinitely many solutions for a nonlinear Klein-Gordon-Maxwell system, Nonlinear Anal., 110 (2014), 157-169.
doi: 10.1016/j.na.2014.07.019. |
[21] |
P. L. Lions,
The concentration-compactness principle in the calculus of variations. The locally compact case. Ⅰ & Ⅱ, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 109-145.
doi: 10.1016/S0294-1449(16)30428-0. |
[22] |
D. D. Qin, Y. B. He and X. H. Tang,
Ground state solutions for Kirchhoff type equations with asymptotically 4-linear nonlinearity, Comput. Math. Appl., 71 (2016), 1524-1536.
doi: 10.1016/j.camwa.2016.02.037. |
[23] |
M. Struwe,
The existence of surfaces of constant mean curvature with free boundaries, Acta Math., 160 (1988), 19-64.
doi: 10.1007/BF02392272. |
[24] |
X. H. Tang and B. T. Cheng,
Ground state sign-changing solutions for Kirchhoff type problems in bounded domains, J. Differential Equations, 261 (2016), 2384-2402.
doi: 10.1016/j.jde.2016.04.032. |
[25] |
X. H. Tang and S. T. Chen,
Ground state solutions of Nehari-Pohozaev type for Schrödinger-Poisson problems with general potentials, Disc. Contin. Dyn. Syst., 37 (2017), 4973-5002.
doi: 10.3934/dcds.2017214. |
[26] |
X. H. Tang and S. T. Chen, Ground state solutions of Nehari-Pohozaev type for Kirchhoff-type problems with general potentials,
Calc. Var. Partial Differential Equations, 56 (2017), Art. 110, 25 pp. |
[27] |
F. Wang,
Ground-state solutions for the electrostatic nonlinear Klein-Gordon-Maxwell system, Nonlinear Anal., 74 (2011), 4796-4803.
doi: 10.1016/j.na.2011.04.050. |
[28] |
M. Willem,
Minimax Theorems, Birkhäuser, Boston, 1996.
doi: 10.1007/978-1-4612-4146-1. |
[29] |
L. Zhang, X. H. Tang and Y. Chen,
Infinitely many solutions for a class of perturbed elliptic equations with nonlocal operators, Commun. Pur. Appl. Anal., 16 (2017), 823-842.
doi: 10.3934/cpaa.2017039. |
[30] |
J. Zhang, W. Zhang and X. L. Xie,
Existence and concentration of semiclassical solutions for Hamiltonian elliptic system, Commun. Pure Appl. Anal., 15 (2016), 599-622.
doi: 10.3934/cpaa.2016.15.599. |
[31] |
J. Zhang, W. Zhang and X. H. Tang,
Ground state solutions for Hamiltonian elliptic system with inverse square potential, Discrete Contin. Dyn. Syst., 37 (2017), 4565-4583.
doi: 10.3934/dcds.2017195. |
show all references
References:
[1] |
A. Azzollini, V. Benci, T. D'Aprile and D. Fortunato,
Existence of static solutions of the semilinear Maxwell equations, Ricerche di Matematica, 55 (2006), 283-297.
|
[2] |
A. Azzollini and A. Pomponio,
Ground state solutions for the nonlinear Klein-Gordon-Maxwell equations, Topol. Methods Nonlinear Anal., 35 (2010), 33-42.
|
[3] |
A. Azzollini, L. Pisani and A. Pomponio,
Improved estimates and a limit case for the electrostatic Klein-Gordon-Maxwell system, Proc. Roy. Soc. Edinburgh Sect. A, 141 (2011), 449-463.
doi: 10.1017/S0308210509001814. |
[4] |
V. Benci and D. Fortunato,
The nonlinear Klein-Gordon equation coupled with the Maxwell equations, Nonlinear Anal., 47 (2001), 6065-6072.
doi: 10.1016/S0362-546X(01)00688-5. |
[5] |
V. Benci and D. Fortunato,
Solitary waves of the nonlinear Klein-Gordon equation coupled with the Maxwell equations, Rev. Math. Phys., 14 (2002), 409-420.
doi: 10.1142/S0129055X02001168. |
[6] |
H. Berestycki and P. L. Lions,
Nonlinear scalar field equations, I -Existence of a ground state, Arch. Rational Mech. Anal., 82 (1983), 313-345.
|
[7] |
P. Carrião, P. Cunha and O. Miyagaki,
Positive ground state solutions for the critical Klein-Gordon-Maxwell system with potentials, Nonlinear Anal., 75 (2012), 4068-4078.
doi: 10.1016/j.na.2012.02.023. |
[8] |
D. Cassani,
Existence and non-existence of solitary waves for the critical Klein-Gordon equation coupled with Maxwell' s equations, Nonlinear Anal., 58 (2004), 733-747.
doi: 10.1016/j.na.2003.05.001. |
[9] |
S. T. Chen and X. H. Tang, Ground state sign-changing solutions for a class of Schrödinger-Poisson type problems in $\mathbb{R}^3$,
Z. Angew. Math. Phys. , 67 (2016), Art. 102, 18 pp. |
[10] |
S. T. Chen and X. H. Tang,
Nehari type ground state solutions for asymptotically periodic Schrödinger-Poisson systems, Taiwan. J. Math., 21 (2017), 363-383.
doi: 10.11650/tjm/7784. |
[11] |
S. T. Chen and X. H. Tang,
Ground state sign-changing solutions for asymptotically cubic or super-cubic Schrödinger-Poisson systems without compact condition, Comput. Math. Appl., 74 (2017), 446-458.
doi: 10.1016/j.camwa.2017.04.031. |
[12] |
P. L. Cunha,
Subcritical and supercritical Klein-Gordon-Maxwell equations without Ambrosetti-Rabinowitz condition, Differ. Integral Equ., 27 (2014), 387-399.
|
[13] |
T. D'Aprile and D. Mugnai,
Solitary waves for nonlinear Klein-Gordon-Maxwell and Schrödinger-Maxwell equations, Proc. R. Soc. Edinb. Sect. A, 134 (2004), 893-906.
doi: 10.1017/S030821050000353X. |
[14] |
T. D'Aprile and D. Mugnai,
Non-existence results for the coupled Klein-Gordon-Maxwell equations, Adv. Nonlinear Stud., 4 (2004), 307-322.
|
[15] |
L. Ding and L. Li,
Infinitely many standing wave solutions for the nonlinear Klein-Gordon-Maxwell system with sign-changing potential, Comput. Math. Appl., 68 (2014), 589-595.
doi: 10.1016/j.camwa.2014.07.001. |
[16] |
L. Jeanjean,
Existence of solutions with prescribed norm for semilinear elliptic equations, Nonlinear Anal., 28 (1997), 1633-1659.
doi: 10.1016/S0362-546X(96)00021-1. |
[17] |
L. Jeanjean,
On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer-type problem set on $\mathbb{R}^N$, Proc. R. Soc. Edinb., Sect. A, Math., 129 (1999), 787-809.
doi: 10.1017/S0308210500013147. |
[18] |
W. Jeong and J. Seok,
On perturbation of a functional with the mountain pass geometry, Calc. Var. Partial Differential Equations, Calc. Var. Partial Differential Equations, 49 (2014), 649-668.
doi: 10.1007/s00526-013-0595-7. |
[19] |
G. B. Li and C. Wang,
The existence of a nontrivial solution to a nonlinear elliptic problem of linking type without the Ambrosetti-Rabinowitz condition, Ann. Acad. Sci. Fenn. Math., 36 (2011), 461-480.
doi: 10.5186/aasfm.2011.3627. |
[20] |
L. Li and C. L. Tang,
Infinitely many solutions for a nonlinear Klein-Gordon-Maxwell system, Nonlinear Anal., 110 (2014), 157-169.
doi: 10.1016/j.na.2014.07.019. |
[21] |
P. L. Lions,
The concentration-compactness principle in the calculus of variations. The locally compact case. Ⅰ & Ⅱ, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 109-145.
doi: 10.1016/S0294-1449(16)30428-0. |
[22] |
D. D. Qin, Y. B. He and X. H. Tang,
Ground state solutions for Kirchhoff type equations with asymptotically 4-linear nonlinearity, Comput. Math. Appl., 71 (2016), 1524-1536.
doi: 10.1016/j.camwa.2016.02.037. |
[23] |
M. Struwe,
The existence of surfaces of constant mean curvature with free boundaries, Acta Math., 160 (1988), 19-64.
doi: 10.1007/BF02392272. |
[24] |
X. H. Tang and B. T. Cheng,
Ground state sign-changing solutions for Kirchhoff type problems in bounded domains, J. Differential Equations, 261 (2016), 2384-2402.
doi: 10.1016/j.jde.2016.04.032. |
[25] |
X. H. Tang and S. T. Chen,
Ground state solutions of Nehari-Pohozaev type for Schrödinger-Poisson problems with general potentials, Disc. Contin. Dyn. Syst., 37 (2017), 4973-5002.
doi: 10.3934/dcds.2017214. |
[26] |
X. H. Tang and S. T. Chen, Ground state solutions of Nehari-Pohozaev type for Kirchhoff-type problems with general potentials,
Calc. Var. Partial Differential Equations, 56 (2017), Art. 110, 25 pp. |
[27] |
F. Wang,
Ground-state solutions for the electrostatic nonlinear Klein-Gordon-Maxwell system, Nonlinear Anal., 74 (2011), 4796-4803.
doi: 10.1016/j.na.2011.04.050. |
[28] |
M. Willem,
Minimax Theorems, Birkhäuser, Boston, 1996.
doi: 10.1007/978-1-4612-4146-1. |
[29] |
L. Zhang, X. H. Tang and Y. Chen,
Infinitely many solutions for a class of perturbed elliptic equations with nonlocal operators, Commun. Pur. Appl. Anal., 16 (2017), 823-842.
doi: 10.3934/cpaa.2017039. |
[30] |
J. Zhang, W. Zhang and X. L. Xie,
Existence and concentration of semiclassical solutions for Hamiltonian elliptic system, Commun. Pure Appl. Anal., 15 (2016), 599-622.
doi: 10.3934/cpaa.2016.15.599. |
[31] |
J. Zhang, W. Zhang and X. H. Tang,
Ground state solutions for Hamiltonian elliptic system with inverse square potential, Discrete Contin. Dyn. Syst., 37 (2017), 4565-4583.
doi: 10.3934/dcds.2017195. |
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