## Revista Matemática Iberoamericana

### Cluster solutions for the Schrödinger-Poisson-Slater problem around a local minimum of the potential

#### Abstract

In this paper we consider the system in $\mathbb{R}^3$ $$\left\{ \begin{array}{l} -\varepsilon^2 \Delta u + V(x)u + \phi(x)u = u^p, \\ -\Delta \phi = u^2, \end{array} \right.$$ 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.

#### Article information

Source
Rev. Mat. Iberoamericana, Volume 27, Number 1 (2011), 253-271.

Dates
First available in Project Euclid: 4 February 2011

https://projecteuclid.org/euclid.rmi/1296828834

Mathematical Reviews number (MathSciNet)
MR2815737

Zentralblatt MATH identifier
1216.35024

#### Citation

Ruiz, David; Vaira, Giusi. Cluster solutions for the Schrödinger-Poisson-Slater problem around a local minimum of the potential. Rev. Mat. Iberoamericana 27 (2011), no. 1, 253--271. https://projecteuclid.org/euclid.rmi/1296828834

#### References

• Ambrosetti, A., Badiale, M. and Cingolani, S.: Semiclassical states of nonlinear Schödinger equations. Arch. Rational Mech. Anal. 140 (1997), no. 3, 285-300.
• Ambrosetti, A. and Malchiodi, A.: Perturbation methods and semilinear elliptic problems on $\mathbbR^n$. Progress in Mathematics 240. Birkhäuser Verlag, Basel, 2006.
• Ambrosetti, A. and Ruiz, D.: Multiple bound states for the Schrödinger-Poisson problem. Commun. Contemp. Math. 10 (2008), no. 3, 391-404.
• Azzollini, A. and Pomponio, A.: Ground state solutions for the nonlinear Schrödinger-Maxwell equations. J. Math. Anal. Appl. 345 (2008), no. 1, 90-108.
• Benci, V. and Fortunato, D.: An eigenvalue problem for the Schrödinger-Maxwell equations. Topol. Methods Nonlinear Anal. 11 (1998), no. 2, 283-293.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• Kikuchi, H.: Existence and orbital stability of standing waves for nonlinear Schrödinger equations via the variational method. Doctoral Thesis.
• 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.
• 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.
• Li, Y.: On a singularly perturbed elliptic equation. Adv. Differential Equations 2 (1997), no. 6, 955-980.
• Mauser, N.J.: The Schrödinger-Poisson-X$\alpha$ equation. Appl. Math. Lett. 14 (2001), no. 6, 759-763.
• Pisani, L. and Siciliano, G.: Neumann condition in the Schrödinger-Maxwell system. Topol. Methods Nonlinear Anal. 29 (2007), no. 2, 251-264.
• Ruiz, D.: The Schrödinger-Poisson equation under the effect of a nonlinear local term. J. Funct. Anal. 237 (2006), no. 2, 655-674.
• Slater, J.C.: A simplification of the Hartree-Fock method. Phys. Review 81 (1951), 385-390.
• Sánchez, O. and Soler, J.: Long-time dynamics of the Schrödinger-Poisson-Slater system. J. Statistical Physics 114 (2004), 179-204.
• 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.
• 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.