Generalized Schrödinger-Poisson type systems

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March 2013, 12(2): 867-879. doi: 10.3934/cpaa.2013.12.867

Generalized Schrödinger-Poisson type systems

Antonio Azzollini ^1, , Pietro d’Avenia ^2, and Valeria Luisi ^3,

1.	Dipartimento di Matematica, Informatica ed Economia, Università degli Studi della Basilicata, Via dell'Ateneo Lucano 10, I-85100 Potenza, Italy
2.	Dipartimento di Matematica, Politecnico di Bari, Via Orabona, 4, I-70125 Bari
3.	Dipartimento di Matematica, Universita degli Studi di Bari, Via E. Orabona 4, 70125 Bari, Italy

Received September 2011 Revised June 2012 Published September 2012

Abstract
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In this paper we study the boundary value problem \begin{eqnarray*} -\Delta u+ \varepsilon q\Phi f(u)=\eta|u|^{p-1}u \quad in \quad \Omega, \\ - \Delta \Phi=2 qF(u) \quad in \quad \Omega, \\ u=\Phi=0 \quad on \quad \partial \Omega, \end{eqnarray*} where $\Omega \subset R^3$ is a smooth bounded domain, $1 < p < 5$, $\varepsilon,\eta= \pm 1$, $q>0$, $f: R\to R$ is a continuous function and $F$ is the primitive of $f$ such that $F(0)=0.$ We provide existence and multiplicity results assuming on $f$ a subcritical growth condition. The critical case is also considered and existence and nonexistence results are proved.

Keywords: variational methods, mountain pass., Schrödinger-Poisson equations.

Mathematics Subject Classification: Primary: 35J55, 35J60; Secondary: 35J2.

Citation: Antonio Azzollini, Pietro d’Avenia, Valeria Luisi. Generalized Schrödinger-Poisson type systems. Communications on Pure & Applied Analysis, 2013, 12 (2) : 867-879. doi: 10.3934/cpaa.2013.12.867

References:

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[10]	L. Jeanjean and S. Le Coz, An existence and stability result for standing waves of nonlinear Schrödinger equations,, Adv. Differential Equations, 11 (2006), 813. Google Scholar
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[13]	P. L. Lions, The Choquard equation and related questions,, Nonlinear Anal., 4 (1980), 1063. doi: 10.1016/0362-546X(80)90016-4. Google Scholar
[14]	D. Mugnai, The Schrödinger-Poisson system with positive potential,, Comm. Partial Differential Equations, 36 (2011), 1099. doi: 10.1080/03605302.2011.558551. Google Scholar
[15]	L. Pisani and G. Siciliano, Neumann condition in the Schrödinger-Maxwell system,, Topol. Methods Nonlinear Anal., 29 (2007), 251. Google Scholar
[16]	L. Pisani and G. Siciliano, Note on a Schrödinger-Poisson system in a bounded domain,, Appl. Math. Lett., 21 (2008), 521. doi: 10.1016/j.aml.2007.06.005. Google Scholar
[17]	S. I. Pohožaev, On the eigenfunctions of the equation $\Delta u+\l f(u)=0$,, Dokl. Akad. Nauk SSSR, 165 (1965), 36. Google Scholar
[18]	D. Ruiz, The Schrödinger-Poisson equation under the effect of a nonlinear local term,, J. Funct. Anal., 237 (2006), 655. doi: 10.1016/j.jfa.2006.04.005. Google Scholar
[19]	D. Ruiz and G. Siciliano, A note on the Schrödinger-Poisson-Slater equation on bounded domains,, Adv. Nonlinear Stud., 8 (2008), 179. Google Scholar
[20]	G. Siciliano, Multiple positive solutions for a Schrödinger-Poisson-Slater system,, J. Math. Anal. Appl., 365 (2010), 288. doi: 10.1016/j.jmaa.2009.10.061. Google Scholar
[21]	M. Struwe, "Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems,'', 4$^{th}$ edition, (2008). Google Scholar

show all references

References:

[1]	A. Ambrosetti, On Schrödinger-Poisson systems,, Milan J. Math., 76 (2008), 257. doi: 10.1007/s00032-008-0094-z. Google Scholar
[2]	A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications,, J. Functional Analysis, 14 (1973), 349. doi: 10.1016/0022-1236(73)90051-7. Google Scholar
[3]	A. Azzollini and P. d'Avenia, On a system involving a critically growing nonlinearity,, J. Math. Anal. Appl., 387 (2012), 433. doi: 10.1016/j.jmaa.2011.09.012. Google Scholar
[4]	A. Azzollini, P. d'Avenia and A. Pomponio, On the Schrödinger-Maxwell equations under the effect of a general nonlinear term,, Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire, 27 (2010), 779. doi: 10.1016/j.anihpc.2009.11.012. Google Scholar
[5]	V. Benci and D. Fortunato, An eigenvalue problem for the Schrödinger-Maxwell equations,, Topol. Methods Nonlinear Anal., 11 (1998), 283. Google Scholar
[6]	M. Berti and P. Bolle, Periodic solutions of nonlinear wave equations with general nonlinearities,, Comm. Math. Phys., 243 (2003), 315. doi: 10.1007/s00220-003-0972-8. Google Scholar
[7]	T. D'Aprile and D. Mugnai, Non-existence results for the coupled Klein-Gordon-Maxwell equations,, Adv. Nonlinear Stud., 4 (2004), 307. Google Scholar
[8]	P. d'Avenia, L. Pisani and G. Siciliano, Dirichlet and Neumann problems for Klein-Gordon-Maxwell systems,, Nonlinear Anal., 71 (2009). doi: 10.1016/j.na.2009.02.111. Google Scholar
[9]	P. d'Avenia, L. Pisani and G. Siciliano, Klein-Gordon-Maxwell systems in a bounded domain,, Discrete Contin. Dyn. Syst., 26 (2010), 135. doi: 10.3934/dcds.2010.26.135. Google Scholar
[10]	L. Jeanjean and S. Le Coz, An existence and stability result for standing waves of nonlinear Schrödinger equations,, Adv. Differential Equations, 11 (2006), 813. Google Scholar
[11]	H. Kikuchi, Existence and stability of standing waves for Schrödinger-Poisson-Slater equation,, Adv. Nonlinear Stud., 7 (2007), 403. Google Scholar
[12]	E. H. Lieb, Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation,, Studies in Appl. Math., 57 (): 93. Google Scholar
[13]	P. L. Lions, The Choquard equation and related questions,, Nonlinear Anal., 4 (1980), 1063. doi: 10.1016/0362-546X(80)90016-4. Google Scholar
[14]	D. Mugnai, The Schrödinger-Poisson system with positive potential,, Comm. Partial Differential Equations, 36 (2011), 1099. doi: 10.1080/03605302.2011.558551. Google Scholar
[15]	L. Pisani and G. Siciliano, Neumann condition in the Schrödinger-Maxwell system,, Topol. Methods Nonlinear Anal., 29 (2007), 251. Google Scholar
[16]	L. Pisani and G. Siciliano, Note on a Schrödinger-Poisson system in a bounded domain,, Appl. Math. Lett., 21 (2008), 521. doi: 10.1016/j.aml.2007.06.005. Google Scholar
[17]	S. I. Pohožaev, On the eigenfunctions of the equation $\Delta u+\l f(u)=0$,, Dokl. Akad. Nauk SSSR, 165 (1965), 36. Google Scholar
[18]	D. Ruiz, The Schrödinger-Poisson equation under the effect of a nonlinear local term,, J. Funct. Anal., 237 (2006), 655. doi: 10.1016/j.jfa.2006.04.005. Google Scholar
[19]	D. Ruiz and G. Siciliano, A note on the Schrödinger-Poisson-Slater equation on bounded domains,, Adv. Nonlinear Stud., 8 (2008), 179. Google Scholar
[20]	G. Siciliano, Multiple positive solutions for a Schrödinger-Poisson-Slater system,, J. Math. Anal. Appl., 365 (2010), 288. doi: 10.1016/j.jmaa.2009.10.061. Google Scholar
[21]	M. Struwe, "Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems,'', 4$^{th}$ edition, (2008). Google Scholar

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