On the Schrödinger–Maxwell equations under the effect of a general nonlinear term

Azzollini, A.; d'Avenia, P.; Pomponio, A.

Annales de l'I.H.P. Analyse non linéaire, Tome 27 (2010) no. 2, pp. 779-791

In this paper we prove the existence of a nontrivial solution to the nonlinear Schrödinger–Maxwell equations in ${ℝ}^3$, assuming on the nonlinearity the general hypotheses introduced by Berestycki and Lions.

Dans cet article on démontre l'existence d'une solution non-banale et positive pour les équations non-linéaires de Schrödinger–Maxwell dans ${ℝ}^3$ en supposant que le terme non-linéaire satisfait les hypothèses introduites par Berestycki et Lions.

[1] A. Ambrosetti, D. Ruiz, Multiple bound states for the Schrödinger–Poisson problem, Commun. Contemp. Math. 10 (2008), 391-404 | MR | Zbl

[2] A. Azzollini, A. Pomponio, Ground state solutions for the nonlinear Schrödinger–Maxwell equations, J. Math. Anal. Appl. 345 (2008), 90-108 | MR | Zbl

[3] A. Azzollini, A. Pomponio, On the Schrödinger equation in ${ℝ}^n$ under the effect of a general nonlinear term, Indiana Univ. Math. J. 58 (2009), 1361-1378 | MR | Zbl

[4] V. Benci, D. Fortunato, An eigenvalue problem for the Schrödinger–Maxwell equations, Topol. Methods Nonlinear Anal. 11 (1998), 283-293 | MR | Zbl

[5] V. Benci, D. Fortunato, Existence of hylomorphic solitary waves in Klein–Gordon and in Klein–Gordon–Maxwell equations, Rend. Accad. Lincei 20 (2009), 243-279 | MR | Zbl

[6] V. Benci, D. Fortunato, A. Masiello, L. Pisani, Solitons and the electromagnetic field, Math. Z. 232 (1999), 73-102 | MR | Zbl

[7] H. Berestycki, P.L. Lions, Nonlinear scalar field equations. I. Existence of a ground state, Arch. Ration. Mech. Anal. 82 (1983), 313-345 | MR | Zbl

[8] H. Berestycki, P.L. Lions, Nonlinear scalar field equations. II. Existence of infinitely many solutions, Arch. Ration. Mech. Anal. 82 (1983), 347-375 | MR | Zbl

[9] G.M. Coclite, A multiplicity result for the linear Schrödinger–Maxwell equations with negative potential, Ann. Polon. Math. 79 (2002), 21-30 | MR | Zbl

[10] G.M. Coclite, A multiplicity result for the nonlinear Schrödinger–Maxwell equations, Commun. Appl. Anal. 7 (2003), 417-423 | MR | Zbl

[11] G.M. Coclite, V. Georgiev, Solitary waves for Maxwell–Schrödinger equations, Electron. J. Differential Equations 94 (2004), 1-31 | MR | EuDML | Zbl

[12] T. D'Aprile, D. Mugnai, Solitary waves for nonlinear Klein–Gordon–Maxwell and Schrödinger–Maxwell equations, Proc. Roy. Soc. Edinburgh Sect. A 134 (2004), 893-906 | MR | Zbl

[13] T. D'Aprile, D. Mugnai, Non-existence results for the coupled Klein–Gordon–Maxwell equations, Adv. Nonlinear Stud. 4 (2004), 307-322 | MR | Zbl

[14] P. D'Avenia, Non-radially symmetric solutions of nonlinear Schrödinger equation coupled with Maxwell equations, Adv. Nonlinear Stud. 2 (2002), 177-192 | MR | Zbl

[15] L. Jeanjean, On the existence of bounded Palais–Smale sequences and application to a Landesman–Lazer-type problem set on ${ℝ}^n$, Proc. Roy. Soc. Edinburgh Sect. A Math. 129 (1999), 787-809 | MR | Zbl

[16] L. Jeanjean, Local condition insuring bifurcation from the continuous spectrum, Math. Z. 232 (1999), 651-664 | MR | Zbl

[17] L. Jeanjean, S. Le Coz, An existence and stability result for standing waves of nonlinear Schrödinger equations, Adv. Differential Equations 11 (2006), 813-840 | MR | Zbl

[18] L. Jeanjean, K. Tanaka, A positive solution for a nonlinear Schrödinger equation on ${ℝ}^n$, Indiana Univ. Math. J. 54 (2005), 443-464 | MR | Zbl

[19] Y. Jiang, H.S. Zhou, Bound states for a stationary nonlinear Schrödinger–Poisson system with sign-changing potential in ${ℝ}^n$, preprint | MR

[20] H. Kikuchi, On the existence of a solution for elliptic system related to the Maxwell–Schrödinger equations, Nonlinear Anal. Theory Methods Appl. 67 (2007), 1445-1456 | MR | Zbl

[21] H. Kikuchi, Existence and stability of standing waves for Schrödinger–Poisson–Slater equation, Adv. Nonlinear Stud. 7 (2007), 403-437 | MR | Zbl

[22] L. Pisani, G. Siciliano, Neumann condition in the Schrödinger–Maxwell system, Topol. Methods Nonlinear Anal. 29 (2007), 251-264 | MR | Zbl

[23] A. Pomponio, S. Secchi, A note on coupled nonlinear Schrödinger systems under the effect of general nonlinearities, preprint | MR

[24] D. Ruiz, The Schrödinger–Poisson equation under the effect of a nonlinear local term, J. Func. Anal. 237 (2006), 655-674 | MR | Zbl

[25] W.A. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys. 55 (1977), 149-162 | MR | Zbl

[26] M. Struwe, On the evolution of harmonic mappings of Riemannian surfaces, Comment. Math. Helv. 60 (1985), 558-581 | MR | EuDML | Zbl

[27] Z. Wang, H.S. Zhou, Positive solution for a nonlinear stationary Schrödinger–Poisson system in ${ℝ}^3$, Discrete Contin. Dyn. Syst. 18 (2007), 809-816 | MR | Zbl

[28] L. Zhao, F. Zhao, On the existence of solutions for the Schrödinger–Poisson equations, J. Math. Anal. Appl. 346 (2008), 155-169 | MR | Zbl