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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

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:
[1]

A. Ambrosetti, On Schrödinger-Poisson systems, Milan J. Math., 76 (2008), 257-274. 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), 349-381. 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), 433-438. doi: 10.1016/j.jmaa.2011.09.012.  Google Scholar

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A. Azzollini, P. d'Avenia and A. Pomponio, On the Schrödinger-Maxwell equations under the effect of a general nonlinear term, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 779-791. 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), 283-293.  Google Scholar

[6]

M. Berti and P. Bolle, Periodic solutions of nonlinear wave equations with general nonlinearities, Comm. Math. Phys., 243 (2003), 315-328. 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), 307-322.  Google Scholar

[8]

P. d'Avenia, L. Pisani and G. Siciliano, Dirichlet and Neumann problems for Klein-Gordon-Maxwell systems, Nonlinear Anal., 71 (2009), e1985-e1995. 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), 135-149. 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), 813-840.  Google Scholar

[11]

H. Kikuchi, Existence and stability of standing waves for Schrödinger-Poisson-Slater equation, Adv. Nonlinear Stud., 7 (2007), 403-437.  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), 1063-1072. 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), 1099-1117. 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), 251-264.  Google Scholar

[16]

L. Pisani and G. Siciliano, Note on a Schrödinger-Poisson system in a bounded domain, Appl. Math. Lett., 21 (2008), 521-528. 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), 36-39.  Google Scholar

[18]

D. Ruiz, The Schrödinger-Poisson equation under the effect of a nonlinear local term, J. Funct. Anal., 237 (2006), 655-674. 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), 179-190.  Google Scholar

[20]

G. Siciliano, Multiple positive solutions for a Schrödinger-Poisson-Slater system, J. Math. Anal. Appl., 365 (2010), 288-299. doi: 10.1016/j.jmaa.2009.10.061.  Google Scholar

[21]

M. Struwe, "Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems,'' 4th edition, Springer-Verlag, Berlin, 2008.  Google Scholar

show all references

References:
[1]

A. Ambrosetti, On Schrödinger-Poisson systems, Milan J. Math., 76 (2008), 257-274. 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), 349-381. 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), 433-438. 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. Poincaré Anal. Non Linéaire, 27 (2010), 779-791. 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), 283-293.  Google Scholar

[6]

M. Berti and P. Bolle, Periodic solutions of nonlinear wave equations with general nonlinearities, Comm. Math. Phys., 243 (2003), 315-328. 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), 307-322.  Google Scholar

[8]

P. d'Avenia, L. Pisani and G. Siciliano, Dirichlet and Neumann problems for Klein-Gordon-Maxwell systems, Nonlinear Anal., 71 (2009), e1985-e1995. 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), 135-149. 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), 813-840.  Google Scholar

[11]

H. Kikuchi, Existence and stability of standing waves for Schrödinger-Poisson-Slater equation, Adv. Nonlinear Stud., 7 (2007), 403-437.  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), 1063-1072. 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), 1099-1117. 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), 251-264.  Google Scholar

[16]

L. Pisani and G. Siciliano, Note on a Schrödinger-Poisson system in a bounded domain, Appl. Math. Lett., 21 (2008), 521-528. 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), 36-39.  Google Scholar

[18]

D. Ruiz, The Schrödinger-Poisson equation under the effect of a nonlinear local term, J. Funct. Anal., 237 (2006), 655-674. 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), 179-190.  Google Scholar

[20]

G. Siciliano, Multiple positive solutions for a Schrödinger-Poisson-Slater system, J. Math. Anal. Appl., 365 (2010), 288-299. doi: 10.1016/j.jmaa.2009.10.061.  Google Scholar

[21]

M. Struwe, "Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems,'' 4th edition, Springer-Verlag, Berlin, 2008.  Google Scholar

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