-
Previous Article
Orbital stability of peakons for a modified Camassa-Holm equation with higher-order nonlinearity
- DCDS Home
- This Issue
-
Next Article
Periodic solutions for the N-vortex problem via a superposition principle
Positive solutions for a nonlinear Schrödinger-Poisson system
1. | School of Mathematics and Statistics & Hubei Key Laboratory of Mathematical Sciences, Central China Normal University, Wuhan 430079, China |
2. | School of Science, Jiangsu University of Science and Technology, Zhenjiang 212003, China |
$\left\{\begin{array}{ll} -\Delta u+u+\epsilon K(x)\Phi(x)u = f(u),& x\in \mathbb{R}^{3} , \\ -\Delta \Phi = K(x)u^{2},\,\,& x\in \mathbb{R}^{3}, \\\end{array}\right.$ |
$K(x)$ |
$f(u)$ |
$\epsilon_{0}>0$ |
$0<\epsilon<\epsilon_{0}$ |
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 D. Ruiz,
Multiple bound states for the Schrödinger-Poisson problem, Commun. Contemp. Math., 10 (2008), 391-404.
doi: 10.1142/S021919970800282X. |
[3] |
W. Ao and J. Wei,
Infinitely many positive solutions for nonlinear equations with nonsymmetric potential, Calc. Var. Partial Differential Equ., 51 (2014), 761-798.
doi: 10.1007/s00526-013-0694-5. |
[4] |
W. Ao, J. Wei and J. Zeng,
An optimal bound on the number of interior spike solutions for
the Lin-Ni-Takagi problem, J. Funct. Anal., 265 (2013), 1324-1356.
doi: 10.1016/j.jfa.2013.06.016. |
[5] |
A. Azzollini and A. Pomponio,
Ground state solutions for the non-linear Schrödinger-Maxwell
equations, J. Math. Anal. Appl., 345 (2008), 90-108.
doi: 10.1016/j.jmaa.2008.03.057. |
[6] |
A. Azzollini and A. Pomponio, Ground state solutions for the non-linear Schrödinger-Maxwell equations with a singular potential, arXiv: 0706.1679[math.AP]. |
[7] |
A. Bahri and Y. Y. Li,
On a min-max procedure for the existence of a positive solution for certain scalar field equations in $ \mathbb{R}^{N}$, Rev. Mat. Iberoamericana, 6 (1990), 1-15.
doi: 10.4171/RMI/92. |
[8] |
V. Benci and D. Fortunato,
Solitary waves of the nonlinear Klein-Gordon equation coupled
with Maxwell equations, Rev. Math. Phys., 14 (2002), 409-420.
doi: 10.1142/S0129055X02001168. |
[9] |
V. Benci and D. Fortunato,
An eigenvalue problem for the Schrödinger-Maxwell equations, Topol. Methods Nonlinear Anal., 11 (1998), 283-293.
doi: 10.12775/TMNA.1998.019. |
[10] |
R. Benguria, H. Brézis and E. Lieb,
The Thomas-Fermi-Von Weizsäcker theory of atoms and molecules, Comm. Math. Phys., 79 (1981), 167-180.
doi: 10.1007/BF01942059. |
[11] |
D. Cassani,
Existence and non-existence of solitary waves for the critical Klein-Gordon equation coupled with Maxwell's equations, Nonlinear Anal., 58 (2004), 733-747.
doi: 10.1016/j.na.2003.05.001. |
[12] |
I. Catto and P. Loins,
Binding of atoms and stability of molecules in Hartree and Thomas-Fermi type theories, Ⅰ. A necessary and sufficient condition for the stability of general molecular systems, Comm. Partial diferential Equations, 17 (1992), 1051-1110.
doi: 10.1080/03605309208820878. |
[13] |
G. Cerami, D. Passaseo and S. Solimini,
Infinitely many positive solutions to some scalar field equations with non symmetric coefficients, Comm. Pure Appl. Math., 66 (2013), 372-413.
doi: 10.1002/cpa.21410. |
[14] |
G. Cerami and G. Vaira,
Positive solutions for some non-autonomous Schrödinger-Poisson systems, J. Differential Equations, 248 (2010), 521-543.
doi: 10.1016/j.jde.2009.06.017. |
[15] |
G. Coclite,
A multiplicity result for the nonlinear Schrödinger-Maxwell equations, Commun. Appl., 7 (2003), 417-423.
|
[16] |
E. N. Dancer,
On the uniqueness of the positive solution of a singularly perturbed problem, Rocky Mountain J. Math., 25 (1995), 957-975.
doi: 10.1216/rmjm/1181072198. |
[17] |
T. $ \acute{{\rm{D}}}$Aprile and D. Mugnai,
Non-existence results for the coupled Klein-Gordon-Maxwell
equations, Adv. Nonlinear Stud, 4 (2004), 307-322.
doi: 10.1515/ans-2004-0305. |
[18] |
T. $ \acute{{\rm{D}}}$Aprile and D. Mugnai,
Solitary waves for nonlinear Klein-Gordon-Maxwell and
Schrödinger-Maxwell equations, Proc. Roy. Soc. Edindurgh Sect., 134 (2004), 893-906.
doi: 10.1017/S030821050000353X. |
[19] |
T. $ \acute{{\rm{D}}}$Aprile and J. Wei,
On bound states concentration on spheres for the Maxwell-Schrödinger equations, SIAM J. Math. Anal., 37 (2005), 321-342.
doi: 10.1137/S0036141004442793. |
[20] |
M. Delpino, J. Wei and W. Yao,
Intermediate reduction method and infinitely many positive solutions of nonlinear Schrödinger equations with non-symmetric potentials, Calc. Var. Partial Differential Equations, 53 (2015), 473-523.
doi: 10.1007/s00526-014-0756-3. |
[21] |
B. Gidas, W. M. Ni, L. Nirernberg, Symmetry of positive solutions of nonlinear elliptic equations in $ \mathbb{R}^N$. In: Mathematical Analysis and Applications, part A, 369–402. Adv. in Math. Suppl. Stud., 7a, Academic Press, New York-London, 1981. |
[22] |
I. Ianni,
Solutions of the Schrödinger-Poisson system concentrating on spheres, part Ⅱ: Existence, Math. Models Methods Appl. Sci., 19 (2009), 877-910.
doi: 10.1142/S0218202509003656. |
[23] |
I. Ianni and G. Vaira,
Solutions of the Schrödinger-Poisson system concentrating on spheres, part Ⅰ: Necessary conditions, Math. Models Methods Appl. Sci., 19 (2009), 707-720.
doi: 10.1142/S0218202509003589. |
[24] |
I. Ianni and G. Vaira,
On concentration of positive bound states for the Schrödinger-Poisson system with potential, Adv. Nonlinear Stud., 8 (2008), 573-595.
doi: 10.1515/ans-2008-0305. |
[25] |
G. Li, S. Peng and S. Yan,
Infinitely many positive solutions for the nonlinear Schrödinger-Poisson system, Commun. Contemp. Math., 12 (2010), 1069-1092.
doi: 10.1142/S0219199710004068. |
[26] |
G. Li, S. Peng and C. Wang, Multi-bump solutions for the nonlinear Schrödinger-Poisson system, J. Math. Phys., 52 (2011), 053505, 19 pp.
doi: 10.1063/1.3585657. |
[27] |
F. Lin, W. Ni and J. Wei,
On the number of interior peak solutions for a singularly perturbed
Neumann problem, Comm. Pure Appl. Math., 60 (2007), 252-281.
doi: 10.1002/cpa.20139. |
[28] |
E. Lions and B. Simon,
The Thomas-Fermi theory of atoms, moleules and solids, Adv. Math., 23 (997), 22-116.
doi: 10.1016/0001-8708(77)90108-6. |
[29] |
P. Lions,
Solutions of Hartree-Fock equations for Coulomb systems, Comm. Math. Phys., 109 (1987), 33-97.
doi: 10.1007/BF01205672. |
[30] |
P. Markowich, C. Ringhofer and C. Schmeiser, Semiconductor Equations, Springer-Verlag, Vienna, 1990.
doi: 10.1007/978-3-7091-6961-2. |
[31] |
C. Mercuri,
Positive solutions of nonlinear Schrödinger-Poisson systems with radial potentials vanishing at infinity, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 19 (2008), 211-227.
doi: 10.4171/RLM/520. |
[32] |
M. Monica, F. Pacard and J. Wei,
Finite-energy sign-changing solutions with dihedral symmetry for the stationary nonlinear Schrödinger equation, J. Eur. Math. Soc. (JEMS), 14 (2012), 1923-1953.
doi: 10.4171/JEMS/351. |
[33] |
W. M. Ni and I. Takagi,
Locating the peaks of least energy solutions to a semilinear Neumann problem, Duke Math. J., 70 (1993), 247-281.
doi: 10.1215/S0012-7094-93-07004-4. |
[34] |
P. Poláčik,
Morse indices and bifurations of positive solutions of $ \Delta u +f(u)=0$ on $ \mathbb{R}^{N}$, Indiana Univ. Math. J., 50 (2001), 1407-1432.
doi: 10.1512/iumj.2001.50.1909. |
[35] |
D. Ruiz,
Semiclassical states for coupled Schrödinger-Maxwell equations: concentration
around a sphere, Math. Models Methods Appl. Sci., 15 (2005), 141-164.
doi: 10.1142/S0218202505003939. |
[36] |
D. Ruiz,
The nonlinear 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. |
[37] |
Z. Wang and H. Zhou,
Positive solution for a nonlinear stationary Schrödinger-Poisson system in $ {\mathbb{R}^{3}}$, Discrete Contin. Dyn. Syst., 18 (2007), 809-816.
doi: 10.3934/dcds.2007.18.809. |
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 D. Ruiz,
Multiple bound states for the Schrödinger-Poisson problem, Commun. Contemp. Math., 10 (2008), 391-404.
doi: 10.1142/S021919970800282X. |
[3] |
W. Ao and J. Wei,
Infinitely many positive solutions for nonlinear equations with nonsymmetric potential, Calc. Var. Partial Differential Equ., 51 (2014), 761-798.
doi: 10.1007/s00526-013-0694-5. |
[4] |
W. Ao, J. Wei and J. Zeng,
An optimal bound on the number of interior spike solutions for
the Lin-Ni-Takagi problem, J. Funct. Anal., 265 (2013), 1324-1356.
doi: 10.1016/j.jfa.2013.06.016. |
[5] |
A. Azzollini and A. Pomponio,
Ground state solutions for the non-linear Schrödinger-Maxwell
equations, J. Math. Anal. Appl., 345 (2008), 90-108.
doi: 10.1016/j.jmaa.2008.03.057. |
[6] |
A. Azzollini and A. Pomponio, Ground state solutions for the non-linear Schrödinger-Maxwell equations with a singular potential, arXiv: 0706.1679[math.AP]. |
[7] |
A. Bahri and Y. Y. Li,
On a min-max procedure for the existence of a positive solution for certain scalar field equations in $ \mathbb{R}^{N}$, Rev. Mat. Iberoamericana, 6 (1990), 1-15.
doi: 10.4171/RMI/92. |
[8] |
V. Benci and D. Fortunato,
Solitary waves of the nonlinear Klein-Gordon equation coupled
with Maxwell equations, Rev. Math. Phys., 14 (2002), 409-420.
doi: 10.1142/S0129055X02001168. |
[9] |
V. Benci and D. Fortunato,
An eigenvalue problem for the Schrödinger-Maxwell equations, Topol. Methods Nonlinear Anal., 11 (1998), 283-293.
doi: 10.12775/TMNA.1998.019. |
[10] |
R. Benguria, H. Brézis and E. Lieb,
The Thomas-Fermi-Von Weizsäcker theory of atoms and molecules, Comm. Math. Phys., 79 (1981), 167-180.
doi: 10.1007/BF01942059. |
[11] |
D. Cassani,
Existence and non-existence of solitary waves for the critical Klein-Gordon equation coupled with Maxwell's equations, Nonlinear Anal., 58 (2004), 733-747.
doi: 10.1016/j.na.2003.05.001. |
[12] |
I. Catto and P. Loins,
Binding of atoms and stability of molecules in Hartree and Thomas-Fermi type theories, Ⅰ. A necessary and sufficient condition for the stability of general molecular systems, Comm. Partial diferential Equations, 17 (1992), 1051-1110.
doi: 10.1080/03605309208820878. |
[13] |
G. Cerami, D. Passaseo and S. Solimini,
Infinitely many positive solutions to some scalar field equations with non symmetric coefficients, Comm. Pure Appl. Math., 66 (2013), 372-413.
doi: 10.1002/cpa.21410. |
[14] |
G. Cerami and G. Vaira,
Positive solutions for some non-autonomous Schrödinger-Poisson systems, J. Differential Equations, 248 (2010), 521-543.
doi: 10.1016/j.jde.2009.06.017. |
[15] |
G. Coclite,
A multiplicity result for the nonlinear Schrödinger-Maxwell equations, Commun. Appl., 7 (2003), 417-423.
|
[16] |
E. N. Dancer,
On the uniqueness of the positive solution of a singularly perturbed problem, Rocky Mountain J. Math., 25 (1995), 957-975.
doi: 10.1216/rmjm/1181072198. |
[17] |
T. $ \acute{{\rm{D}}}$Aprile and D. Mugnai,
Non-existence results for the coupled Klein-Gordon-Maxwell
equations, Adv. Nonlinear Stud, 4 (2004), 307-322.
doi: 10.1515/ans-2004-0305. |
[18] |
T. $ \acute{{\rm{D}}}$Aprile and D. Mugnai,
Solitary waves for nonlinear Klein-Gordon-Maxwell and
Schrödinger-Maxwell equations, Proc. Roy. Soc. Edindurgh Sect., 134 (2004), 893-906.
doi: 10.1017/S030821050000353X. |
[19] |
T. $ \acute{{\rm{D}}}$Aprile and J. Wei,
On bound states concentration on spheres for the Maxwell-Schrödinger equations, SIAM J. Math. Anal., 37 (2005), 321-342.
doi: 10.1137/S0036141004442793. |
[20] |
M. Delpino, J. Wei and W. Yao,
Intermediate reduction method and infinitely many positive solutions of nonlinear Schrödinger equations with non-symmetric potentials, Calc. Var. Partial Differential Equations, 53 (2015), 473-523.
doi: 10.1007/s00526-014-0756-3. |
[21] |
B. Gidas, W. M. Ni, L. Nirernberg, Symmetry of positive solutions of nonlinear elliptic equations in $ \mathbb{R}^N$. In: Mathematical Analysis and Applications, part A, 369–402. Adv. in Math. Suppl. Stud., 7a, Academic Press, New York-London, 1981. |
[22] |
I. Ianni,
Solutions of the Schrödinger-Poisson system concentrating on spheres, part Ⅱ: Existence, Math. Models Methods Appl. Sci., 19 (2009), 877-910.
doi: 10.1142/S0218202509003656. |
[23] |
I. Ianni and G. Vaira,
Solutions of the Schrödinger-Poisson system concentrating on spheres, part Ⅰ: Necessary conditions, Math. Models Methods Appl. Sci., 19 (2009), 707-720.
doi: 10.1142/S0218202509003589. |
[24] |
I. Ianni and G. Vaira,
On concentration of positive bound states for the Schrödinger-Poisson system with potential, Adv. Nonlinear Stud., 8 (2008), 573-595.
doi: 10.1515/ans-2008-0305. |
[25] |
G. Li, S. Peng and S. Yan,
Infinitely many positive solutions for the nonlinear Schrödinger-Poisson system, Commun. Contemp. Math., 12 (2010), 1069-1092.
doi: 10.1142/S0219199710004068. |
[26] |
G. Li, S. Peng and C. Wang, Multi-bump solutions for the nonlinear Schrödinger-Poisson system, J. Math. Phys., 52 (2011), 053505, 19 pp.
doi: 10.1063/1.3585657. |
[27] |
F. Lin, W. Ni and J. Wei,
On the number of interior peak solutions for a singularly perturbed
Neumann problem, Comm. Pure Appl. Math., 60 (2007), 252-281.
doi: 10.1002/cpa.20139. |
[28] |
E. Lions and B. Simon,
The Thomas-Fermi theory of atoms, moleules and solids, Adv. Math., 23 (997), 22-116.
doi: 10.1016/0001-8708(77)90108-6. |
[29] |
P. Lions,
Solutions of Hartree-Fock equations for Coulomb systems, Comm. Math. Phys., 109 (1987), 33-97.
doi: 10.1007/BF01205672. |
[30] |
P. Markowich, C. Ringhofer and C. Schmeiser, Semiconductor Equations, Springer-Verlag, Vienna, 1990.
doi: 10.1007/978-3-7091-6961-2. |
[31] |
C. Mercuri,
Positive solutions of nonlinear Schrödinger-Poisson systems with radial potentials vanishing at infinity, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 19 (2008), 211-227.
doi: 10.4171/RLM/520. |
[32] |
M. Monica, F. Pacard and J. Wei,
Finite-energy sign-changing solutions with dihedral symmetry for the stationary nonlinear Schrödinger equation, J. Eur. Math. Soc. (JEMS), 14 (2012), 1923-1953.
doi: 10.4171/JEMS/351. |
[33] |
W. M. Ni and I. Takagi,
Locating the peaks of least energy solutions to a semilinear Neumann problem, Duke Math. J., 70 (1993), 247-281.
doi: 10.1215/S0012-7094-93-07004-4. |
[34] |
P. Poláčik,
Morse indices and bifurations of positive solutions of $ \Delta u +f(u)=0$ on $ \mathbb{R}^{N}$, Indiana Univ. Math. J., 50 (2001), 1407-1432.
doi: 10.1512/iumj.2001.50.1909. |
[35] |
D. Ruiz,
Semiclassical states for coupled Schrödinger-Maxwell equations: concentration
around a sphere, Math. Models Methods Appl. Sci., 15 (2005), 141-164.
doi: 10.1142/S0218202505003939. |
[36] |
D. Ruiz,
The nonlinear 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. |
[37] |
Z. Wang and H. Zhou,
Positive solution for a nonlinear stationary Schrödinger-Poisson system in $ {\mathbb{R}^{3}}$, Discrete Contin. Dyn. Syst., 18 (2007), 809-816.
doi: 10.3934/dcds.2007.18.809. |
[1] |
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 |
[2] |
Kuan-Hsiang Wang. An eigenvalue problem for nonlinear Schrödinger-Poisson system with steep potential well. Communications on Pure and Applied Analysis, 2021, 20 (4) : 1497-1519. doi: 10.3934/cpaa.2021030 |
[3] |
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 |
[4] |
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 |
[5] |
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 |
[6] |
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 |
[7] |
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 |
[8] |
Sitong Chen, Wennian Huang, Xianhua Tang. Existence criteria of ground state solutions for Schrödinger-Poisson systems with a vanishing potential. Discrete and Continuous Dynamical Systems - S, 2021, 14 (9) : 3055-3066. doi: 10.3934/dcdss.2020339 |
[9] |
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 |
[10] |
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 |
[11] |
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 |
[12] |
Qiangchang Ju, Fucai Li, Hailiang Li. Asymptotic limit of nonlinear Schrödinger-Poisson system with general initial data. Kinetic and Related Models, 2011, 4 (3) : 767-783. doi: 10.3934/krm.2011.4.767 |
[13] |
Amna Dabaa, O. Goubet. Long time behavior of solutions to a Schrödinger-Poisson system in $R^3$. Communications on Pure and Applied Analysis, 2016, 15 (5) : 1743-1756. doi: 10.3934/cpaa.2016011 |
[14] |
Lirong Huang, Jianqing Chen. Existence and asymptotic behavior of bound states for a class of nonautonomous Schrödinger-Poisson system. Electronic Research Archive, 2020, 28 (1) : 383-404. doi: 10.3934/era.2020022 |
[15] |
Sitong Chen, Xianhua Tang. Existence of ground state solutions for the planar axially symmetric Schrödinger-Poisson system. Discrete and Continuous Dynamical Systems - B, 2019, 24 (9) : 4685-4702. doi: 10.3934/dcdsb.2018329 |
[16] |
Denis Bonheure, Silvia Cingolani, Simone Secchi. Concentration phenomena for the Schrödinger-Poisson system in $ \mathbb{R}^2 $. Discrete and Continuous Dynamical Systems - S, 2021, 14 (5) : 1631-1648. doi: 10.3934/dcdss.2020447 |
[17] |
Juntao Sun, Tsung-Fang Wu, Zhaosheng Feng. Non-autonomous Schrödinger-Poisson system in $\mathbb{R}^{3}$. Discrete and Continuous Dynamical Systems, 2018, 38 (4) : 1889-1933. doi: 10.3934/dcds.2018077 |
[18] |
Zhengping Wang, Huan-Song Zhou. Positive solution for a nonlinear stationary Schrödinger-Poisson system in $R^3$. Discrete and Continuous Dynamical Systems, 2007, 18 (4) : 809-816. doi: 10.3934/dcds.2007.18.809 |
[19] |
Xia Sun, Kaimin Teng. Positive bound states for fractional Schrödinger-Poisson system with critical exponent. Communications on Pure and Applied Analysis, 2020, 19 (7) : 3735-3768. doi: 10.3934/cpaa.2020165 |
[20] |
Sitong Chen, Junping Shi, Xianhua Tang. Ground state solutions of Nehari-Pohozaev type for the planar Schrödinger-Poisson system with general nonlinearity. Discrete and Continuous Dynamical Systems, 2019, 39 (10) : 5867-5889. doi: 10.3934/dcds.2019257 |
2020 Impact Factor: 1.392
Tools
Metrics
Other articles
by authors
[Back to Top]