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High energy solutions for the superlinear Schrödinger-Maxwell equations. (English) Zbl 1175.35142

Summary: We study the existence of infinitely many large energy solutions for the superlinear Schrödinger-Maxwell equations
\[ -\Delta u+V(x)u+\varphi u= f(x,u) \quad\text{in }\mathbb R^3, \qquad -\Delta\varphi=u^2 \quad\text{in }\mathbb R^3, \]
via the Fountain theorem in critical point theory.

MSC:

35Q60 PDEs in connection with optics and electromagnetic theory
78A25 Electromagnetic theory (general)
78M30 Variational methods applied to problems in optics and electromagnetic theory
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[1] Benci, V.; Fortunato, D., An eigenvalue problem for the Schrödinger-Maxwell equations, Topol. Methods Nonlinear Anal., 11, 2, 283-293 (1998) · Zbl 0926.35125
[2] Azzollini, A.; Pomponio, A., Ground state solutions for the nonlinear Schrödinger-Maxwell equations, J. Math. Anal. Appl., 345, 1, 90-108 (2008) · Zbl 1147.35091
[3] Coclite, G. M., A multiplicity result for the nonlinear Schrödinger-Maxwell equations, Commun. Appl. Anal., 7, 2-3, 417-423 (2003) · Zbl 1085.81510
[4] Coclite, G. M., A multiplicity result for the Schrödinger-Maxwell equations with negative potential, Ann. Polon. Math., 79, 1, 21-30 (2002) · Zbl 1130.35333
[5] D’Aprile, T.; Mugnai, D., Solitary waves for nonlinear Klein-Gordon-Maxwell and Schrödinger-Maxwell equations, Proc. Roy. Soc. Edinburgh Sect. A, 134, 5, 893-906 (2004) · Zbl 1064.35182
[6] Kikuchi, H., On the existence of a solution for elliptic system related to the Maxwell-Schrödinger equations, Nonlinear Anal., 67, 5, 1445-1456 (2007) · Zbl 1119.35085
[7] Ruiz, D., The Schrödinger-Poisson equation under the effect of a nonlinear local term, J. Funct. Anal., 237, 2, 655-674 (2006) · Zbl 1136.35037
[8] Salvatore, A., Multiple solitary waves for a non-homogeneous Schrödinger-Maxwell system in \(R^3\), Adv. Nonlinear Stud., 6, 2, 157-169 (2006) · Zbl 1229.35065
[9] Zhao, L.-G.; Zhao, F.-K., Positive solutions for Schrödinger-Poisson equations with a critical exponent, Nonlinear Anal., 70, 6, 2150-2164 (2009) · Zbl 1156.35374
[10] Bartolo, P.; Benci, V.; Fortunato, D., Abstract critical point theorems and applications to some nonlinear problems with strong resonance at infinity, Nonlinear Anal., 7, 9, 981-1012 (1983) · Zbl 0522.58012
[11] Wang, Z.-P.; Zhou, H.-S., Positive solution for a nonlinear stationary Schrödinger-Poisson system in \(R^3\), Discrete Contin. Dyn. Syst., 18, 4, 809-816 (2007) · Zbl 1133.35427
[12] Zhao, L.-G.; Zhao, F.-K., On the existence of solutions for the Schrödinger-Poisson equations, J. Math. Anal. Appl., 346, 1, 155-169 (2008) · Zbl 1159.35017
[13] Ackermann, N., A nonlinear superposition principle and multibump solutions of periodic Schrödinger equations, J. Funct. Anal., 234, 2, 277-320 (2006) · Zbl 1126.35057
[14] D’Aprile, T.; Mugnai, D., Non-existence results for the coupled Klein-Gordon-Maxwell equations, Adv. Nonlinear Stud., 4, 3, 307-322 (2004) · Zbl 1142.35406
[15] Ambrosetti, A.; Rabinowitz, P. H., Dual variational methods in critical point theory and applications, J. Funct. Anal., 14, 349-381 (1973) · Zbl 0273.49063
[16] Rabinowitz, P. H., (Minimax Methods in Critical Point Theory with Applications to Differential Equations. Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Regional Conference Series in Mathematics, vol.65 (1986), American Mathematical Society: American Mathematical Society Providence, RI), Published for the Conference Board of the Mathematical Sciences, Washington, DC · Zbl 0609.58002
[17] Bartsch, T., Infinitely many solutions of a symmetric Dirichlet problem, Nonlinear Anal., 20, 10, 1205-1216 (1993) · Zbl 0799.35071
[18] Bartsch, T.; Willem, M., On an elliptic equation with concave and convex nonlinearities, Proc. Amer. Math. Soc., 123, 11, 3555-3561 (1995) · Zbl 0848.35039
[19] Struwe, M., Variational methods, (Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems (2000), Springer-Verlag: Springer-Verlag Berlin) · Zbl 1371.49038
[20] Willem, M., Minimax Theorems (1996), Birkhäuser Boston, Inc.: Birkhäuser Boston, Inc. Boston, MA · Zbl 0856.49001
[21] Bartsch, T.; Wang, Z. Q., Existence and multiplicity results for some superlinear elliptic problems on \(R^N\), Comm. Partial Differential Equations, 20, 9-10, 1725-1741 (1995) · Zbl 0837.35043
[22] Strauss, W. A., Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55, 2, 149-162 (1977) · Zbl 0356.35028
[23] Benci, V.; Fortunato, D., The nonlinear Klein-Gordon equation coupled with the Maxwell equations, Proceedings of the Third World Congress of Nonlinear Analysts, Part 9 (Catania, 2000). Proceedings of the Third World Congress of Nonlinear Analysts, Part 9 (Catania, 2000), Nonlinear Anal., 47, 9, 6065-6072 (2001) · Zbl 1042.78500
[24] Cassani, D., Existence and non-existence of solitary waves for the critical Klein-Gordon equation coupled with Maxwell’s equations, Nonlinear Anal., 58, 7-8, 733-747 (2004) · Zbl 1057.35041
[25] Zou, W.-M.; Schechter, M., Critical Point Theory and its Applications (2006), Springer: Springer New York · Zbl 1186.35042
[26] Benci, V.; Fortunato, D.; Masiello, A.; Pisani, L., Solitons and the electromagnetic field, Math. Z., 232, 1, 73-102 (1999) · Zbl 0930.35168
[27] Gilbarg, D.; Trudinger, N. S., Elliptic Partial Differential Equations of Second Order (2001), Springer-Verlag: Springer-Verlag Berlin · Zbl 0691.35001
[28] Evans, L. C., Partial Differential Equations (1998), American Mathematical Society: American Mathematical Society Providence, RI
[29] Benmlih, K., Stationary solutions for a Schrödinger-Poisson system in \(R^3\), Proceedings of the 2002 Fez Conference on Partial Differential Equations. Proceedings of the 2002 Fez Conference on Partial Differential Equations, Electron. J. Differ. Equ. Conf., 9, 65-76 (2002), Southwest Texas State Univ., San Marcos, TX · Zbl 1109.35373
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