×

zbMATH — the first resource for mathematics

Solutions to the critical Klein-Gordon-Maxwell system with external potential. (English) Zbl 1375.35156
Summary: In this paper, we consider the critical Klein-Gordon-Maxwell system with external potential. When the potential well is steep, by using the penalization technique and the elliptic estimate of solutions, we prove the existence and concentration phenomenon of solutions. When the potential well may be not steep, we can also obtain the existence of solutions.
MSC:
35J47 Second-order elliptic systems
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Ambrosetti, A.; Rabinowitz, P. H., Dual variational methods in critical point theory and applications, J. Funct. Anal., 14, 349-381, (1973) · Zbl 0273.49063
[2] Azzollini, A.; Pisani, L.; Pomponio, A., Improved estimates and a limit case for the electrostatic Klein-Gordon-Maxwell system, Proc. Roy. Soc. Edinburgh Sect. A, 141, 449-463, (2011) · Zbl 1231.35244
[3] 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
[4] Bartsch, T.; Pankov, A.; Wang, Z.-Q., Nonlinear schrodinger equations with steep potential well, Commun. Contemp. Math., 3, 549-569, (2001) · Zbl 1076.35037
[5] Bartsch, T.; Wang, Z.-Q., Existence and multiplicity results for some superlinear elliptic problems on \(\mathbb{R}^N\), Comm. Partial Differential Equations, 20, 1725-1741, (1995) · Zbl 0837.35043
[6] Bartsch, T.; Wang, Z.-Q., Multiple positive solutions for a nonlinear Schrödinger equation, Z. Angew. Math. Phys., 51, 366-384, (2000) · Zbl 0972.35145
[7] 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
[8] Brezis, H.; Nirenberg, L., Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math., 36, 437-477, (1983) · Zbl 0541.35029
[9] 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
[10] Carrião, P.; Cunha, P.; Miyagaki, O., Positive ground state solutions for the critical Klein-Gordon-Maxwell system with potentials, Nonlinear Anal., 75, 4068-4078, (2012) · Zbl 1238.35109
[11] 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
[12] Chen, S.; Song, S., Multiple solutions for nonhomogeneous Klein-Gordon-Maxwell equations on \(\mathbb{R}^3\), Nonlinear Anal. Real World Appl., 22, 259-271, (2015) · Zbl 1306.35024
[13] Clapp, M.; Ding, Y. H., Positive solutions of a Schrödinger equation with critical nonlinearity, Z. Angew. Math. Phys., 55, 592-605, (2004) · Zbl 1060.35130
[14] D’Aprile, T.; Mugnai, D., Solitary waves for nonlinear Klein-Gordon-Maxwell and schrodinger-Maxwell equations, Proc. Roy. Soc. Edinburgh Sect. A, 134, 893-906, (2004) · Zbl 1064.35182
[15] 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
[16] del Pino, M.; Felmer, P., Local mountain pass for semilinear elliptic problems in unbounded domains, Calc. Var. Partial Differential Equations, 4, 121-137, (1996) · Zbl 0844.35032
[17] Ding, Y. H.; Tanaka, K., Multiplicity of positive solutions of a nonlinear schrodinger equation, Manuscripta Math., 112, 109-135, (2003) · Zbl 1038.35114
[18] Gilbarg, D.; Trudinger, N. S., Elliptic partial differential equations of second order, (1998), Springer-Verlag New York · Zbl 0691.35001
[19] Guo, Y.; Tang, Z., Sign changing bump solutions for Schrödinger equations involving critical growth and indefinite potential wells, J. Differential Equations, 259, 6038-6071, (2015) · Zbl 1326.35347
[20] He, X., Multiplicity of solutions for a nonlinear Klein-Gordon-Maxwell system, Acta Appl. Math., 130, 237-250, (2014) · Zbl 1301.35175
[21] Jiang, Y. S.; Zhou, H. S., Schrödinger-Poisson system with steep potential well, J. Differential Equations, 251, 582-608, (2011) · Zbl 1233.35086
[22] Li, L.; Tang, C., Infinitely many solutions for a nonlinear Klein-Gordon-Maxwell system, Nonlinear Anal., 110, 157-169, (2014) · Zbl 1301.35136
[23] Schechter, M., A variation of the mountain pass lemma and applications, J. Lond. Math. Soc., 44, 491-502, (1991) · Zbl 0756.35032
[24] Sun, J.; Wu, T. F., Ground state solutions for an indefinite Kirchhoff type problem with steep potential well, J. Differential Equations, 256, 1771-1792, (2014) · Zbl 1288.35219
[25] Wang, F., Solitary waves for the Klein-Gordon-Maxwell system with critical exponent, Nonlinear Anal., 74, 827-835, (2011) · Zbl 1204.35159
[26] Willem, M., Minimax theorems, (1996), Birkhäuser Boston · Zbl 0856.49001
[27] Zhang, X.; Ma, S., Multi-bump solutions of Schrödinger-Poisson equations with steep potential well, Z. Angew. Math. Phys., 66, 1615-1631, (2015) · Zbl 1321.35045
[28] Zhao, L.; Liu, H.; Zhao, F., Existence and concentration of solutions for the Schrödinger-Poisson equations with steep well potential, J. Differential Equations, 255, 1-23, (2013) · Zbl 1286.35103
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.