# American Institute of Mathematical Sciences

May  2018, 38(5): 2333-2348. doi: 10.3934/dcds.2018096

## Improved results for Klein-Gordon-Maxwell systems with general nonlinearity

 School of Mathematics and Statistics, Central South University, Changsha, 410083 Hunan, China

* Corresponding author

Received  September 2017 Revised  December 2017 Published  March 2018

Fund Project: This work is partially supported by the Hunan Provincial Innovation Foundation for Postgraduate (No: CX2017B041) and the National Natural Science Foundation of China (No: 11571370).

This paper is concerned with the following Klein-Gordon-Maxwell system
 \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.
where
 $0 < ω≤ m_0$
and
 $f∈ \mathcal{C}(\mathbb{R}, \mathbb{R})$
. By introducing some new tricks, we prove that the above system has 1) a ground state solution in the case when
 $0 < ω < m_0$
and
 $f$
is superlinear at infinity; 2) a nontrivial solution in the zero mass case, i.e.
 $ω = m_0$
and
 $f$
is super-quadratic at infinity. These results improve the related ones in the literature.
Citation: Sitong Chen, Xianhua Tang. Improved results for Klein-Gordon-Maxwell systems with general nonlinearity. Discrete and Continuous Dynamical Systems, 2018, 38 (5) : 2333-2348. doi: 10.3934/dcds.2018096
##### 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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##### 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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