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Cluster solutions for the Schrödinger-Poisson-Slater problem around a local minimum of the potential. (English) Zbl 1216.35024

Summary: We consider the system in \(\mathbb{R}^3\)
\[ -\varepsilon^2\Delta u+ V(x)u+ \phi(x)u= u^p,\qquad -\Delta\phi= u^2, \]
for \(p\in (1,5)\). We prove the existence of multi-bump solutions whose bumps concentrate around a local minimum of the potential \(V(x)\). We point out that such solutions do not exist in the framework of the usual nonlinear Schrödinger equation.

MSC:

35J47 Second-order elliptic systems
35B40 Asymptotic behavior of solutions to PDEs
35J50 Variational methods for elliptic systems
35J10 Schrödinger operator, Schrödinger equation
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[1] Ambrosetti, A., Badiale, M. and Cingolani, S.: Semiclassical states of nonlinear Schödinger equations. Arch. Rational Mech. Anal. 140 (1997), no. 3, 285-300. · Zbl 0896.35042
[2] Ambrosetti, A. and Malchiodi, A.: Perturbation methods and semilinear elliptic problems on \(\mathbbR^n\). Progress in Mathematics 240 . Birkhäuser Verlag, Basel, 2006. · Zbl 1115.35004
[3] Ambrosetti, A. and Ruiz, D.: Multiple bound states for the Schrödinger-Poisson problem. Commun. Contemp. Math. 10 (2008), no. 3, 391-404. · Zbl 1188.35171
[4] Azzollini, A. and Pomponio, A.: Ground state solutions for the nonlinear Schrödinger-Maxwell equations. J. Math. Anal. Appl. 345 (2008), no. 1, 90-108. · Zbl 1147.35091
[5] Benci, V. and Fortunato, D.: An eigenvalue problem for the Schrödinger-Maxwell equations. Topol. Methods Nonlinear Anal. 11 (1998), no. 2, 283-293. · Zbl 0926.35125
[6] Bokanowski, O. and Mauser, N.J.: Local approximation for the Hartree-Fock exchange potential: a deformation approach. Math. Models Methods Appl. Sci. 9 (1999), no. 6, 941-961. · Zbl 0956.81097
[7] Bokanowski, O., López, J.L. and Soler, J.: On an exchange interaction model for quantum transport; the Schrödinger-Poisson-Slater term. Math. Models Methods Appl. Sci. 13 (2003), no. 10, 1397-1412. · Zbl 1073.35188
[8] Cornean, H., Hoke, K., Neidhardt, H., Racec, P.N. and Rehberg, J.: A Kohn-Sham system at zero temperature. J. Phys. A 41 (2008), no. 38, 385304, 21pp. · Zbl 1159.34057
[9] D’Aprile, T. and Mugnai, D.: Solitary waves for nonlinear Klein-Gordon-Maxwell and Schrödinger-Maxwell equations. Proc. Roy. Soc. Edinburgh Sect. A 134 (2004), no. 5, 893-906. · Zbl 1064.35182
[10] D’Aprile, T. and Mugnai, D.: Non-existence results for the coupled Klein-Gordon-Maxwell equations. Adv. Nonlinear Stud. 4 (2004), no. 3, 307-322. · Zbl 1142.35406
[11] D’Aprile, T. and Wei, J.: Standing waves in the Maxwell-Schrödinger equation and an optimal configuration problem. Calc. Var. Partial Differential Equations 25 (2006), no. 1, 105-137. · Zbl 1207.35129
[12] D’Aprile, T. and Wei, J.: Clustered solutions around harmonic centers to a coupled elliptic system. Ann. Inst. H. Poincaré Anal. Non Linéaire 24 (2007), no. 4, 605-628. · Zbl 1406.35053
[13] Del Pino, M. and Felmer, P.: Multi-peak bound states for nonlinear Schrödinger equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 15 (1998), no. 2, 127-149. · Zbl 0901.35023
[14] Gui, C.: Existence of multi-bump solutions for nonlinear Schrödinger equations via variational methods. Comm. Partial Differential Equations 21 (1996), no. 5-6, 787-820. · Zbl 0857.35116
[15] Kang, X. and Wei, J.: On interacting bumps of semi-classical states of nonlinear Schrödinger equations. Adv. Differential Equations 5 (2000), no. 7-9, 899-928. · Zbl 1217.35065
[16] Kikuchi, H.: On the existence of a solution for elliptic system related to the Maxwell-Schrödinger equations. Nonlinear Anal. 67 (2007), no. 5, 1445-1456. · Zbl 1119.35085
[17] Kikuchi, H.: Existence and orbital stability of standing waves for nonlinear Schrödinger equations via the variational method. Doctoral Thesis. · Zbl 1168.35042
[18] Kwong, M.K.: Uniqueness of positive solutions of \(\Delta u -u+u^p=0\) in \(\mathbbR^N\). Arch. Rational Mech. Anal. 105 (1989), no. 3, 243-266. · Zbl 0676.35032
[19] Ianni, I. and Vaira, G.: On concentration of positive bound states for the Schrödinger-Poisson problem with potentials. Adv. Nonlinear Stud. 8 (2008), no. 3, 573-595. · Zbl 1216.35138
[20] Li, Y.: On a singularly perturbed elliptic equation. Adv. Differential Equations 2 (1997), no. 6, 955-980. · Zbl 1023.35500
[21] Mauser, N.J.: The Schrödinger-Poisson-X\(\alpha\) equation. Appl. Math. Lett. 14 (2001), no. 6, 759-763. · Zbl 0990.81024
[22] Pisani, L. and Siciliano, G.: Neumann condition in the Schrödinger-Maxwell system. Topol. Methods Nonlinear Anal. 29 (2007), no. 2, 251-264. · Zbl 1157.35480
[23] Ruiz, D.: The Schrödinger-Poisson equation under the effect of a nonlinear local term. J. Funct. Anal. 237 (2006), no. 2, 655-674. · Zbl 1136.35037
[24] Slater, J.C.: A simplification of the Hartree-Fock method. Phys. Review 81 (1951), 385-390. · Zbl 0042.23202
[25] Sánchez, O. and Soler, J.: Long-time dynamics of the Schrödinger-Poisson-Slater system. J. Statistical Physics 114 (2004), 179-204. · Zbl 1060.82039
[26] Zhao, L. and Zhao, F.: On the existence of solutions for the Schrödinger-Poisson equations. J. Math. Anal. Appl. 346 (2008), no. 1, 155-169. · Zbl 1159.35017
[27] Wang, Z. and Zhou, H.S.: Positive solution for a nonlinear stationary Schrödinger-Poisson system in \(\mathbbR^3\). Discrete Contin. Dyn. Syst. 18 (2007), no. 4, 809-816. · Zbl 1133.35427
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