In this paper, we study the nonautonomous Klein-Gordon-Maxwell system with logarithmic nonlinearity. We obtain the existence of nontrivial solution for this system by logarithmic Sobolev inequality and variational method.
Cite this paper
Shi, Q. (2022). Existence of Nontrivial Solution for Klein-Gordon-Maxwell System with Logarithmic Nonlinearity. Open Access Library Journal, 9, e9120. doi: http://dx.doi.org/10.4236/oalib.1109120.
Benci, V. and Fortunato, D. (2001) The Nonlinear Klein-Gordon Equation Coupled with the Maxwell Equations. Nonlinear Analysis: Theory, Methods & Applications, 47, 6065-6072. https://doi.org/10.1016/S0362-546X(01)00688-5
Benci, V. and Fortunato, D. (2002) Solitary Waves of the Nonlinear Klein-Gordon Equation Coupled with the Maxwell Equations. Reviews in Mathematical Physics, 14, 409-420. https://doi.org/10.1142/S0129055X02001168
D’Aprile, T. and Mugnai, D. (2004) Solitary Waves for Nonlinear Klein-Gordon-Maxwell and Schrödinger-Maxwell Equations. Proceedings of the Royal Society of Edinburgh Section A, 134, 893-906.
https://doi.org/10.1017/S030821050000353X
D’Aprile, T. and Mugnai, D. (2004) Non-Existence Results for the Coupled Klein-Gordon-Maxwell Equations. Advanced Nonlinear Studies, 4, 307-322.
https://doi.org/10.1515/ans-2004-0305
Azzollini, A. and Pomponio, A. (2010) Ground State Solutions for the Nonlinear Klein-Gordon-Maxwell Equations. Topological Methods in Nonlinear Analysis, 35, 33-42.
Cassani, D. (2004) Existence and Non-Existence of Solitary Waves for the Critical Klein-Gordon Equation Coupled with Maxwell’s Equations. Nonlinear Analysis, 58, 733-747. https://doi.org/10.1016/j.na.2003.05.001
Liu, X.-Q. and Tang, C.-L. (2022) Infinitely Many Solutions and Concentration of Ground State Solutions for the Klein-Gordon-Maxwell System. Journal of Mathematical Analysis and Applications, 505, Article ID: 125521.
https://doi.org/10.1016/j.jmaa.2021.125521
Gan, C.L., Xiao, T. and Zhang, Q.F. (2020) Existence Result for Fractional Klein-Gordon-Maxwell System with Quasicritical Potential Vanishing at Infinity. Journal of Applied Mathematics and Physics, 8, 1318-1327.
https://doi.org/10.4236/jamp.2020.87101
Carriao, P.C., Cunha, P.L. and Miyagaki, O.H. (2012) Positive Ground State Solutions for the Critical Klein-Gordon-Maxwell System with Potentials. Nonlinear Analysis: Theory, Methods & Applications, 75, 4068-4078.
https://doi.org/10.1016/j.na.2012.02.023
Che, G.F. and Chen, H.B. (2017) Existence and Multiplicity of Nontrivial Solutions for Klein-Gordon-Maxwell System with a Parameter. Journal of the Korean Mathematical Society, 54, 1015-1030. https://doi.org/10.4134/JKMS.j160344
Chen, S.T. and Tang, X.H. (2018) Infinitely Many Solutions and Least Energy Solutions for Klein-Gordon-Maxwell Systems with General Superlinear Nonlinearity. Computers & Mathematics with Applications, 75, 3358-3366.
https://doi.org/10.1016/j.camwa.2018.02.004
Ding, L. and Li, L. (2014) Infinitely Many Standing Wave Solutions for the Non-linear Klein-Gordon-Maxwell System with Sign-Changing Potential. Computers & Mathematics with Applications, 68, 589-595.
https://doi.org/10.1016/j.camwa.2014.07.001
d’Avenia, P., Pisani, L. and Siciliano, G. (2008) Klein-Gordon-Maxwell System in a Bounded Domain. Discrete and Continuous Dynamical Systems, 26, 135-149.
Wu, Y.H., Ge, B. and Miyagaki, O.H. (2019) Existence Results for the Klein-Gordon-Maxwell System in Rotationally Symmetric Bounded Domains. Zeitschrift für Analysis und ihre Anwendungen, 38, 209-229. https://doi.org/10.4171/ZAA/1635
Guo, L.F., Sun, Y. and Shi, G.N. (2022) Ground States for Fractional Nonlocal Equations with Logarithmic Nonlinearity. Opuscula Mathematica, 42, 157-178.
https://doi.org/10.7494/OpMath.2022.42.2.157
Ji, C. and Szulkin, A. (2016) A Logarithmic Schrödinger Equation with Asymptotic Conditions on the Potential. Journal of Mathematical Analysis and Applications, 437, 241-254. https://doi.org/10.1016/j.jmaa.2015.11.071
Peng, L.Y., Suo, H.M., Wu, D.K., Feng, H.X. and Lei, C.Y. (2021) Multiple Positive Solutions for a Logarithmic Schrödinger-Poisson System with Singular Nonelinearity. Electronic Journal of Qualitative Theory of Differential Equations, 2021, 1-15.
https://doi.org/10.14232/ejqtde.2021.1.90
Squassina, M. and Szulkin, A. (2015) Multiple Solutions to Logarithmic Schrödinger Equations with Periodic Potential. Calculus of Variations and Partial Differential Equations, 54, 585-597. https://doi.org/10.1007/s00526-014-0796-8
Zhang, Y.Y., Yang, Y. and Liang, S.H. (2022) Least Energy Sign-Changing Solu-tion for n-Laplacian Problem with Logarithmic and Exponential Nonlinearities. Journal of Mathematical Analysis and Applications, 505, Article ID: 125432.
https://doi.org/10.1016/j.jmaa.2021.125432
d’Avenia, P., Pisani, L. and Siciliano, G. (2009) Dirichlet and Neumann Problems for Klein-Gordon-Maxwell Systems. Nonlinear Analysis: Theory, Methods & Applications, 71, e1985-e1995. https://doi.org/10.1016/j.na.2009.02.111
Ambrosetti, A. and Rabinowitz, P.H. (1973) Dual Variational Methods in Critical Point Theory and Applications. Journal of Functional Analysis, 14, 349-381.
https://doi.org/10.1016/0022-1236(73)90051-7