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Infinitely many sign-changing solutions for the nonlinear Schrödinger-Poisson system. (English) Zbl 1341.35041

Summary: In this paper, we consider the following Schrödinger-Poisson system
\[ \begin{cases} -\Delta u+V(x)u+\phi u=f(u) &\quad \text{ in }\mathbb R^3,\\ -\Delta \phi =u^2&\quad \text{ in } \mathbb R^3. \end{cases} \]
We investigate the existence of multiple bound state solutions, in particular sign-changing solutions. By using the method of invariant sets of descending flow, we prove that this system has infinitely many sign-changing solutions. In particular, the nonlinear term includes the power-type nonlinearity \(f(u)=| u|^{p-2}u\) for the well-studied case \(p\in (4,6)\), and the less studied case \(p\in (3,4)\), and for the latter case, few existence results are available in the literature.

MSC:

35J47 Second-order elliptic systems
35J50 Variational methods for elliptic systems
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[1] Alves, C.O., Souto, M.A.S.: Existence of least energy nodal solution for a Schrödinger-Poisson system in bounded domains. Z. Angew. Math. Phys. 65, 1153-1166 (2014) · Zbl 1308.35262 · doi:10.1007/s00033-013-0376-3
[2] Ambrosetti, A.: On Schrödinger-Poisson systems. Milan J. Math. 76, 257-274 (2008) · Zbl 1181.35257 · doi:10.1007/s00032-008-0094-z
[3] Ambrosetti, A., Ruiz, R.: Multiple bound states for the Schrödinger-Poisson problem. Commun. Contemp. Math. 10, 391-404 (2008) · Zbl 1188.35171 · doi:10.1142/S021919970800282X
[4] 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 (2010) · Zbl 1187.35231 · doi:10.1016/j.anihpc.2009.11.012
[5] 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
[6] Bartsch, T., Liu, Z.: On a superlinear elliptic p-Laplacian equation. J. Differ. Equ. 198, 149-175 (2004) · Zbl 1087.35034 · doi:10.1016/j.jde.2003.08.001
[7] Bartsch, T., Liu, Z., Weth, T.: Sign-changing solutions of superlinear Schrödinger equations. Commun. Partial Differ. Equ. 29, 25-42 (2004) · Zbl 1140.35410 · doi:10.1081/PDE-120028842
[8] Bartsch, T., Liu, Z., Weth, T.: Nodal solutions of a p-Laplacian equation. Proc. Lond. Math. Soc. 91, 129-152 (2005) · Zbl 1162.35364 · doi:10.1112/S0024611504015187
[9] Bartsch, T., Pankov, A., Wang, Z.-Q.: Nonlinear Schrödinger equations with steep potential well. Commun. Contemp. Math. 3, 1-21 (2001) · Zbl 1076.35037 · doi:10.1142/S0219199701000494
[10] Bartsch, T., Wang, Z.-Q.: Existence and multiplicity results for some superlinear elliptic problems on \[R^N\] RN. Commun. Partial Differ. Equ. 20, 1725-1741 (1995) · Zbl 0837.35043 · doi:10.1080/03605309508821149
[11] Bartsch, T., Wang, Z.-Q.: On the existence of sign changing solutions for semilinear Dirichlet problems. Topol. Methods Nonlinear Anal. 7, 115-131 (1996) · Zbl 0903.58004
[12] Benci, V., Fortunato, D.: An eigenvalue problem for the Schrödinger-Maxwell equations. Topol. Methods Nonlinear Anal. 11, 283-293 (1998) · Zbl 0926.35125
[13] 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 · doi:10.1142/S0129055X02001168
[14] D’Aprile, T.: Non-radially symmetric solution of the nonlinear Schrödinger equation coupled with Maxwell equations. Adv. Nonlinear Stud. 2, 177-192 (2002) · Zbl 1007.35090
[15] D’Aprile, T., Wei, J.: Standing waves in the Maxwell-Schrödinger equation and an optimal configuration problem. Calc. Var. Partial Differ. Equ. 25, 105-137 (2006) · Zbl 1207.35129 · doi:10.1007/s00526-005-0342-9
[16] Ianni, I.: Sign-changing radial solutions for the Schrödinger-Poisson-Slater problem. Topol. Methods Nonlinear Anal. 41, 365-386 (2013) · Zbl 1330.35128
[17] Ianni, I., Vaira, G.: On concentration of positive bound states for the Schrödinger-Poisson system with potentials. Adv. Nonlinear Stud. 8, 573-595 (2008) · Zbl 1216.35138 · doi:10.1515/ans-2008-0305
[18] Ianni, I., Vaira, G.: A note on non-radial sign-changing solutions for the Schrödinger-Poisson problem in the semiclassical limit. In: Concentration Analysis and Applications to PDE Trends inMathematics, pp. 143-156 (2013) · Zbl 1283.35034
[19] Jeanjean, L.: On the existence of bounded Palais-Smale sequence and application to a Landesman-Lazer type problem set on \[{{\mathbb{R}}^N}\] RN. Proc. R. Soc. Edinb. Sect. A Math. 129, 787-809 (1999) · Zbl 0935.35044 · doi:10.1017/S0308210500013147
[20] Kim, S., Seok, J.: On nodal solutions of the nonlinear Schrödinger-Poisson equations. Commun. Contemp. Math. 14, 1250041 (2012) · Zbl 1263.35197 · doi:10.1142/S0219199712500411
[21] Li, G., Peng, S., Yan, S.: Infinitely many positive solutions for the nonlinear Schrödinger-Poisson system. Commun. Contemp. Math. 12, 1069-1092 (2010) · Zbl 1206.35082 · doi:10.1142/S0219199710004068
[22] Lieb, E., Loss, M.: Analysis, Graduate Studies in Mathematics, vol. 14. AMS, Providence (1997)
[23] Liu, J., Liu, X., Wang, Z.-Q.: Multiple mixed states of nodal solutions for nonlinear Schröinger systems. Calc. Var. Partial Differ. Equ. 52, 565-586 (2015). doi:10.1007/s00526-014-0724-y · Zbl 1311.35291 · doi:10.1007/s00526-014-0724-y
[24] Liu, Z., Sun, J.: Invariant sets of descending flow in critical point theory with applications to nonlinear differential equations. J. Differ. Equ. 172, 257-299 (2001) · Zbl 0995.58006 · doi:10.1006/jdeq.2000.3867
[25] Liu, Z., Wang, Z.-Q.: Sign-changing solutions of nonlinear elliptic equations. Front. Math. China 3, 221-238 (2008) · Zbl 1158.35370 · doi:10.1007/s11464-008-0014-0
[26] 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
[27] Ruiz, D.: On the Schrödinger-Poisson-Slater system: behavior of minimizers, radial and nonradial cases. Arch. Ration. Mech. Anal. 198, 349-368 (2010) · Zbl 1235.35232 · doi:10.1007/s00205-010-0299-5
[28] Vaira, G.: Ground states for Schrödinger-Poisson type systems. Ric. Mat. 2, 263-297 (2011) · Zbl 1261.35057 · doi:10.1007/s11587-011-0109-x
[29] Wang, Z., Zhou, H.: Positive solution for a nonlinear stationary Schrödinger-Poisson system in \[{\mathbb{R}}^3\] R3. Discrete Contin. Dyn. Syst. 18, 809-816 (2007) · Zbl 1133.35427 · doi:10.3934/dcds.2007.18.121
[30] Wang, Z., Zhou, H.: Sign-changing solutions for the nonlinear Schrödinger-Poisson system in \[{\mathbb{R}}^3\] R3. Calc. Var. Partial. Differ. Equ. 52, 927-943 (2015) · Zbl 1311.35300 · doi:10.1007/s00526-014-0738-5
[31] 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
[32] Zhao, L., Liu, H., Zhao, F.: Existence and concentration of solutions for the Schrödinger-Poisson equations with steep well potential. J. Differ. Equ. 255, 1-23 (2013) · Zbl 1286.35103 · doi:10.1016/j.jde.2013.03.005
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