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Multiple critical points for a class of nonlinear functionals. (English) Zbl 1230.58014
The authors prove a multiplicity result concerning the critical points of a class of functionals involving local and nonlocal nonlinearities. The result is applied to the nonlinear Schrödinger-Maxwell system in $\Bbb R^{3}$ and to the nonlinear elliptic Kirchhoff equation in $\Bbb R^{N}$ assuming on the local nonlinearity the general hypotheses introduced by Berestycki and Lions.

58E50Applications of variational methods in infinite-dimensional spaces
58E05Abstract critical point theory
35J20Second order elliptic equations, variational methods
35J60Nonlinear elliptic equations
Full Text: DOI
[1] Alves C.O., Correa F.J.S.A., Ma T.F.: Positive solutions for a quasilinear elliptic equation of Kirchhoff type. Comput. Math. Appl. 49, 85--93 (2005) · Zbl 1130.35045 · doi:10.1016/j.camwa.2005.01.008
[2] Ambrosetti A., Ruiz D.: Multiple bound states for the Schrödinger-Poisson problem. Commun. Contemp. Math. 10, 391--404 (2008) · Zbl 1188.35171 · doi:10.1142/S021919970800282X
[3] Azzollini, A.: Concentration and compactness in nonlinear Schrödinger-Poisson system with a general nonlinearity. J. Differ. Equ (to appear) · Zbl 1197.35096
[4] Azzollini, A.: The elliptic Kirchhoff equation in $${$\backslash$mathbb{R}\^N}$$ perturbed by a local nonlinearity (preprint) · Zbl 1321.35016
[5] Azzollini, A., d’Avenia, P., Pomponio, A.: On the Schrödinger-Maxwell equations under the effect of a general nonlinear term. Ann. Inst. H. Poincaré Anal. Non Linéaire 27, 779--791 · Zbl 1187.35231
[6] Azzollini A., Pomponio A.: Ground state solutions for the nonlinear Schrödinger-Maxwell equations. J. Math. Anal. Appl. 345, 90--108 (2008) · Zbl 1147.35091 · doi:10.1016/j.jmaa.2008.03.057
[7] Benci V., Fortunato D.: An eigenvalue problem for the Schrödinger-Maxwell equations. Topol. Methods Nonlinear Anal. 11, 283--293 (1998) · Zbl 0926.35125
[8] 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
[9] Berestycki H., Lions P.L.: Nonlinear scalar field equations. II. Existence of infinitely many solutions. Arch. Ration. Mech. Anal. 82, 347--375 (1983) · Zbl 0556.35046
[10] Berti M., Bolle P.: Periodic solutions of nonlinear wave equations with general nonlinearities. Comm. Math. Phys. 243, 315--328 (2003) · Zbl 1072.35015 · doi:10.1007/s00220-003-0972-8
[11] Coclite G.M.: A multiplicity result for the nonlinear Schrödinger-Maxwell equations. Commun. Appl. Anal. 7, 417--423 (2003) · Zbl 1085.81510
[12] D’Aprile T., Mugnai D.: Solitary waves for nonlinear Klein-Gordon-Maxwell and Schrödinger-Maxwell equations. Proc. Roy. Soc. Edinb. Sect. A 134, 893--906 (2004) · Zbl 1064.35182 · doi:10.1017/S030821050000353X
[13] d’Avenia P.: Non-radially symmetric solutions of nonlinear Schrödinger equation coupled with Maxwell equations. Adv. Nonlinear Stud. 2, 177--192 (2002)
[14] He X., Zou W.: Infinitely many positive solutions for Kirchhoff-type problems. Nonlinear Anal. 70, 1407--1414 (2009) · Zbl 1157.35382 · doi:10.1016/j.na.2008.02.021
[15] Hirata, J., Ikoma, N., Tanaka, K.: Nonlinear scalar field equations in $${$\backslash$mathbb{R}\^N}$$ : mountain pass and symmetric mountain pass approaches (preprint) · Zbl 1203.35106
[16] Jeanjean L.: Existence of solutions with prescribed norm for semilinear elliptic equations. Nonlinear Anal. 28, 1633--1659 (1997) · Zbl 0877.35091 · doi:10.1016/S0362-546X(96)00021-1
[17] Jeanjean L.: On the existence of bounded Palais-Smale sequences and application to a Landesman- Lazer-type problem set on $${$\backslash$mathbb{R}\^N}$$ . Proc. R. Soc. Edinb., Sect. A, Math. 129, 787--809 (1999) · Zbl 0935.35044 · doi:10.1017/S0308210500013147
[18] Jeanjean L., Le Coz S.: An existence and stability result for standing waves of nonlinear Schrödinger equations. Adv. Differ. Equ. 11, 813--840 (2006) · Zbl 1155.35095
[19] Jiang Y., Zhou H.S.: Bound states for a stationary nonlinear Schrödinger-Poisson system with sign-changing potential in $${$\backslash$mathbb{R}3}$$ . Acta Math. Sci. 29, 1095--1104 (2009) · Zbl 1212.35450
[20] Jin J., Wu X.: Infinitely many radial solutions for Kirchhoff-type problems in $${$\backslash$mathbb{R}\^N}$$ . J. Math. Anal. Appl. 369, 564--574 (2010) · Zbl 1196.35221 · doi:10.1016/j.jmaa.2010.03.059
[21] Kikuchi H.: On the existence of a solution for elliptic system related to the Maxwell-Schrödinger equations. Nonlinear Anal. Theory Methods Appl. 67, 1445--1456 (2007) · Zbl 1119.35085 · doi:10.1016/j.na.2006.07.029
[22] Kikuchi H.: Existence and stability of standing waves for Schrödinger-Poisson-Slater equation. Adv. Nonlinear Stud. 7, 403--437 (2007) · Zbl 1133.35013
[23] Kirchhoff G.: Mechanik. Teubner, Leipzig (1883)
[24] Ma T.F.: Remarks on an elliptic equation of Kirchhoff type. Nonlinear Anal. 63, e1967--e1977 (2005) · Zbl 1224.35140 · doi:10.1016/j.na.2005.03.021
[25] Mao A., Zhang Z.: Sign-changing and multiple solutions of Kirchhoff type problems without the P.S. condition. Nonlinear Anal. 70, 1275--1287 (2009) · Zbl 1160.35421 · doi:10.1016/j.na.2008.02.011
[26] Perera K., Zhang Z.T.: Nontrivial solutions of Kirchhoff-type problems via the Yang index. J. Differ. Equ. 221, 246--255 (2006) · Zbl 05013580 · doi:10.1016/j.jde.2005.03.006
[27] Ricceri B.: On an elliptic Kirchhoff-type problem depending on two parameters. J. Glob. Opt. 46, 543--549 (2010) · Zbl 1192.49007 · doi:10.1007/s10898-009-9438-7
[28] Ruiz D.: The Schrödinger-Poisson equation under the effect of a nonlinear local term. J. Funct. Anal. 237, 655--674 (2006) · Zbl 1136.35037 · doi:10.1016/j.jfa.2006.04.005
[29] Strauss W.A.: Existence of solitary waves in higher dimensions. Comm. Math. Phys. 55, 149--162 (1977) · Zbl 0356.35028 · doi:10.1007/BF01626517
[30] Struwe M.: On the evolution of harmonic mappings of Riemannian surfaces. Comment. Math. Helv. 60, 558--581 (1985) · Zbl 0595.58013 · doi:10.1007/BF02567432
[31] Wang Z., Zhou H.S.: Positive solution for a nonlinear stationary Schrödinger-Poisson system in $${$\backslash$mathbb{R}3}$$ . Discrete Contin. Dyn. Syst. 18, 809--816 (2007) · Zbl 1189.35350 · doi:10.3934/dcds.2007.18.121
[32] Zhang Z.T., Perera K.: Sign changing solutions of Kirchhoff type problems via invariant sets of descent flow. J. Math. Anal. Appl. 317, 456--463 (2006) · Zbl 1100.35008 · doi:10.1016/j.jmaa.2005.06.102
[33] 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 · doi:10.1016/j.jmaa.2008.04.053