Carrião, Paulo C.; Cunha, Patrícia L.; Miyagaki, Olímpio H. Positive ground state solutions for the critical Klein-Gordon-Maxwell system with potentials. (English) Zbl 1238.35109 Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 75, No. 10, 4068-4078 (2012). Summary: This paper deals with the Klein-Gordon-Maxwell system when the nonlinearity exhibits critical growth. We prove the existence of positive ground state solutions for this system when a periodic potential \(V\) is introduced. The method combines the minimization of the corresponding Euler-Lagrange functional on the Nehari manifold with the Brézis and Nirenberg technique. Cited in 1 ReviewCited in 30 Documents MSC: 35Q40 PDEs in connection with quantum mechanics 35Q61 Maxwell equations 35B09 Positive solutions to PDEs 35J50 Variational methods for elliptic systems Keywords:variational methods; ground state solutions; critical growth PDFBibTeX XMLCite \textit{P. C. Carrião} et al., Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 75, No. 10, 4068--4078 (2012; Zbl 1238.35109) Full Text: DOI arXiv References: [1] Benci, V.; Fortunato, D., Solitary waves of the nonlinear Klein-Gordon equation coupled with the Maxwell equations, Rev. Math. Phys., 14, 409-420 (2002) · Zbl 1037.35075 [2] D’Aprile, T.; Mugnai, D., Solitary waves for nonlinear Klein-Gordon-Maxwell and Schrodinger-Maxwell equations, Proc. R. Soc. Edinb. Sect. A, 134, 1-14 (2004) · Zbl 1064.35182 [3] Cassani, D., Existence and non-existence of solitary waves for the critical Klein-Gordon equation coupled with Maxwell’s equations, Nonlinear Anal., 58, 733-747 (2004) · Zbl 1057.35041 [4] Azzollini, A.; Pisani, L.; Pomponio, A., Improved estimates ans a limit case for the electrostatic Klein-Gordon-Maxwell system, Proc. R. Soc. Edinb. Sect. A, 141, 449-463 (2011) · Zbl 1231.35244 [5] Azzollini, A.; Pomponio, A., Ground state solutions for the nonlinear Klein-Gordon-Maxwell equations, Topol. Methods Nonlinear Anal., 35, 33-42 (2010) · Zbl 1203.35274 [6] D’Aprile, T.; Mugnai, D., Non-existence results for the coupled Klein-Gordon-Maxwell equations, Adv. Nonlinear Stud., 4, 307-322 (2004) · Zbl 1142.35406 [7] Georgiev, V.; Visciglia, N., Solitary waves for Klein-Gordon-Maxwell system with external Coulomb potential, J. Math. Pures Appl., 84, 957-983 (2005) · Zbl 1078.35098 [8] Azzollini, A.; Pomponio, A., Ground state solutions for the nonlinear Schrödinger-Maxwell equations, J. Math. Anal. Appl., 345, 90-108 (2008) · Zbl 1147.35091 [9] Berestycki, H.; Lions, P. L., Nonlinear scalar field equations. I. Existence of a ground state, Arch. Ration. Mech. Anal., 82, 313-345 (1983) · Zbl 0533.35029 [10] Li, Y.; Wang, Z.-Q.; Zeng, J., Ground states of nonlinear Schrödinger equations with potentials, Ann. I. H. Poincaré, 23, 829-837 (2006) · Zbl 1111.35079 [11] Zhao, L.; Zhao, F., On the existence of solutions for the Schrödinger-Poisson equations, J. Math. Anal. Appl., 346, 155-169 (2008) · Zbl 1159.35017 [12] Brézis, H.; Nirenberg, L., Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Commun. Pure Appl. Math., 36, 437-477 (1983) · Zbl 0541.35029 [13] Carrião, P.; Cunha, P.; Miyagaki, O., Existence results for the Klein-Gordon-Maxwell equations in higher dimensions with critical exponents, Commun. Pure Appl. Anal., 10, 709-718 (2011) · Zbl 1231.35247 [14] Miyagaki, O. H., On a class of semilinear elliptic problems in \(R^N\) with critical growth, Nonlinear Anal., 29, 7, 773-781 (1997) · Zbl 0877.35043 [15] Talenti, G., Best constants in Sobolev inequality, Ann. Mat. Pura Appl., 110, 353-372 (1976) · Zbl 0353.46018 [16] Ambrosetti, A.; Rabinowitz, P. H., Dual variational methods in critical point theory and applications, J. Funct. Anal., 149, 349-381 (1973) · Zbl 0273.49063 [17] Lions, P. L., The concentration-compactness principle in the calculus of variations, the locallly compact case. Part II, Ann Inst. Henri. Poincaré, 1, 223-283 (1984) · Zbl 0704.49004 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.