Generalized Schrödinger-Poisson type systems

Previous Article
A new class of $(H^k,1)$-rectifiable subsets of metric spaces
CPAA Home
This Issue
Next Article
Local existence of strong solutions to the three dimensional compressible MHD equations with partial viscosity

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
Related Papers

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 and 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.
[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.
[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.
[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.
[5]	V. Benci and D. Fortunato, An eigenvalue problem for the Schrödinger-Maxwell equations, Topol. Methods Nonlinear Anal., 11 (1998), 283-293.
[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.
[7]	T. D'Aprile and D. Mugnai, Non-existence results for the coupled Klein-Gordon-Maxwell equations, Adv. Nonlinear Stud., 4 (2004), 307-322.
[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.
[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.
[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.
[11]	H. Kikuchi, Existence and stability of standing waves for Schrödinger-Poisson-Slater equation, Adv. Nonlinear Stud., 7 (2007), 403-437.
[12]	E. H. Lieb, Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation,, Studies in Appl. Math., 57 (): 93.
[13]	P. L. Lions, The Choquard equation and related questions, Nonlinear Anal., 4 (1980), 1063-1072. doi: 10.1016/0362-546X(80)90016-4.
[14]	D. Mugnai, The Schrödinger-Poisson system with positive potential, Comm. Partial Differential Equations, 36 (2011), 1099-1117. doi: 10.1080/03605302.2011.558551.
[15]	L. Pisani and G. Siciliano, Neumann condition in the Schrödinger-Maxwell system, Topol. Methods Nonlinear Anal., 29 (2007), 251-264.
[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.
[17]	S. I. Pohožaev, On the eigenfunctions of the equation $\Delta u+\l f(u)=0$, Dokl. Akad. Nauk SSSR, 165 (1965), 36-39.
[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.
[19]	D. Ruiz and G. Siciliano, A note on the Schrödinger-Poisson-Slater equation on bounded domains, Adv. Nonlinear Stud., 8 (2008), 179-190.
[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.
[21]	M. Struwe, "Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems,'' 4th edition, Springer-Verlag, Berlin, 2008.

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.
[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.
[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.
[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.
[5]	V. Benci and D. Fortunato, An eigenvalue problem for the Schrödinger-Maxwell equations, Topol. Methods Nonlinear Anal., 11 (1998), 283-293.
[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.
[7]	T. D'Aprile and D. Mugnai, Non-existence results for the coupled Klein-Gordon-Maxwell equations, Adv. Nonlinear Stud., 4 (2004), 307-322.
[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.
[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.
[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.
[11]	H. Kikuchi, Existence and stability of standing waves for Schrödinger-Poisson-Slater equation, Adv. Nonlinear Stud., 7 (2007), 403-437.
[12]	E. H. Lieb, Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation,, Studies in Appl. Math., 57 (): 93.
[13]	P. L. Lions, The Choquard equation and related questions, Nonlinear Anal., 4 (1980), 1063-1072. doi: 10.1016/0362-546X(80)90016-4.
[14]	D. Mugnai, The Schrödinger-Poisson system with positive potential, Comm. Partial Differential Equations, 36 (2011), 1099-1117. doi: 10.1080/03605302.2011.558551.
[15]	L. Pisani and G. Siciliano, Neumann condition in the Schrödinger-Maxwell system, Topol. Methods Nonlinear Anal., 29 (2007), 251-264.
[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.
[17]	S. I. Pohožaev, On the eigenfunctions of the equation $\Delta u+\l f(u)=0$, Dokl. Akad. Nauk SSSR, 165 (1965), 36-39.
[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.
[19]	D. Ruiz and G. Siciliano, A note on the Schrödinger-Poisson-Slater equation on bounded domains, Adv. Nonlinear Stud., 8 (2008), 179-190.
[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.
[21]	M. Struwe, "Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems,'' 4th edition, Springer-Verlag, Berlin, 2008.

[1]	Christopher Grumiau, Marco Squassina, Christophe Troestler. On the Mountain-Pass algorithm for the quasi-linear Schrödinger equation. Discrete and Continuous Dynamical Systems - B, 2013, 18 (5) : 1345-1360. doi: 10.3934/dcdsb.2013.18.1345
[2]	Kaimin Teng, Xian Wu. Concentration of bound states for fractional Schrödinger-Poisson system via penalization methods. Communications on Pure and Applied Analysis, 2022, 21 (4) : 1157-1187. doi: 10.3934/cpaa.2022014
[3]	Dmitry Glotov, P. J. McKenna. Numerical mountain pass solutions of Ginzburg-Landau type equations. Communications on Pure and Applied Analysis, 2008, 7 (6) : 1345-1359. doi: 10.3934/cpaa.2008.7.1345
[4]	Li Cai, Fubao Zhang. The Brezis-Nirenberg type double critical problem for a class of Schrödinger-Poisson equations. Electronic Research Archive, 2021, 29 (3) : 2475-2488. doi: 10.3934/era.2020125
[5]	Xu Zhang, Shiwang Ma, Qilin Xie. Bound state solutions of Schrödinger-Poisson system with critical exponent. Discrete and Continuous Dynamical Systems, 2017, 37 (1) : 605-625. doi: 10.3934/dcds.2017025
[6]	Marius Ghergu, Gurpreet Singh. On a class of mixed Choquard-Schrödinger-Poisson systems. Discrete and Continuous Dynamical Systems - S, 2019, 12 (2) : 297-309. doi: 10.3934/dcdss.2019021
[7]	Pierre-Damien Thizy. Schrödinger-Poisson systems in $4$-dimensional closed manifolds. Discrete and Continuous Dynamical Systems, 2016, 36 (4) : 2257-2284. doi: 10.3934/dcds.2016.36.2257
[8]	Zhi Chen, Xianhua Tang, Ning Zhang, Jian Zhang. Standing waves for Schrödinger-Poisson system with general nonlinearity. Discrete and Continuous Dynamical Systems, 2019, 39 (10) : 6103-6129. doi: 10.3934/dcds.2019266
[9]	Chunhua Wang, Jing Yang. Positive solutions for a nonlinear Schrödinger-Poisson system. Discrete and Continuous Dynamical Systems, 2018, 38 (11) : 5461-5504. doi: 10.3934/dcds.2018241
[10]	Daniele Cassani, Luca Vilasi, Jianjun Zhang. Concentration phenomena at saddle points of potential for Schrödinger-Poisson systems. Communications on Pure and Applied Analysis, 2021, 20 (4) : 1737-1754. doi: 10.3934/cpaa.2021039
[11]	Mengyao Chen, Qi Li, Shuangjie Peng. Bound states for fractional Schrödinger-Poisson system with critical exponent. Discrete and Continuous Dynamical Systems - S, 2021, 14 (6) : 1819-1835. doi: 10.3934/dcdss.2021038
[12]	Qian Shen, Na Wei. Stability of ground state for the Schrödinger-Poisson equation. Journal of Industrial and Management Optimization, 2021, 17 (5) : 2805-2816. doi: 10.3934/jimo.2020095
[13]	Rong Cheng, Jun Wang. Existence of ground states for Schrödinger-Poisson system with nonperiodic potentials. Discrete and Continuous Dynamical Systems - B, 2022 doi: 10.3934/dcdsb.2021317
[14]	Xinsheng Du, Qi Li, Zengqin Zhao, Gen Li. Bound state solutions for fractional Schrödinger-Poisson systems. Mathematical Foundations of Computing, 2022, 5 (1) : 57-66. doi: 10.3934/mfc.2021023
[15]	Caixia Chen, Aixia Qian. Multiple positive solutions for the Schrödinger-Poisson equation with critical growth. Mathematical Foundations of Computing, 2022, 5 (2) : 113-128. doi: 10.3934/mfc.2021036
[16]	Dorota Bors. Application of Mountain Pass Theorem to superlinear equations with fractional Laplacian controlled by distributed parameters and boundary data. Discrete and Continuous Dynamical Systems - B, 2018, 23 (1) : 29-43. doi: 10.3934/dcdsb.2018003
[17]	Ian Schindler, Kyril Tintarev. Mountain pass solutions to semilinear problems with critical nonlinearity. Conference Publications, 2007, 2007 (Special) : 912-919. doi: 10.3934/proc.2007.2007.912
[18]	Mingzheng Sun, Jiabao Su, Leiga Zhao. Infinitely many solutions for a Schrödinger-Poisson system with concave and convex nonlinearities. Discrete and Continuous Dynamical Systems, 2015, 35 (1) : 427-440. doi: 10.3934/dcds.2015.35.427
[19]	Claudianor O. Alves, Minbo Yang. Existence of positive multi-bump solutions for a Schrödinger-Poisson system in $\mathbb{R}^{3}$. Discrete and Continuous Dynamical Systems, 2016, 36 (11) : 5881-5910. doi: 10.3934/dcds.2016058
[20]	Margherita Nolasco. Breathing modes for the Schrödinger-Poisson system with a multiple--well external potential. Communications on Pure and Applied Analysis, 2010, 9 (5) : 1411-1419. doi: 10.3934/cpaa.2010.9.1411

2020 Impact Factor: 1.916

Tools

Metrics

Other articles
by authors

on AIMS
Antonio Azzollini Pietro d’Avenia Valeria Luisi
on Google Scholar
Antonio Azzollini Pietro d’Avenia Valeria Luisi

[Back to Top]