-
Previous Article
Two-weight and three-weight linear codes constructed from Weil sums
- MFC Home
- This Issue
-
Next Article
Predictive analytics for 30-day hospital readmissions
Multiple positive solutions for the Schrödinger-Poisson equation with critical growth
1. | School of Mathematics and Computer Application Technology, Jining university, Shandong 273155, China |
2. | School of Mathematical Sciences, Qufu Normal University, Shandong 273165, China |
$ \left\{\begin{aligned} &-\triangle u + u + \phi u = u^{5}+\lambda g(u), &\hbox{in}\ \ \Omega, \\\ & -\triangle \phi = u^{2}, & \hbox{in}\ \ \Omega, \\\ & u, \phi = 0, & \hbox{on}\ \ \partial\Omega.\end{aligned}\right. $ |
$ \Omega $ |
$ \mathbb{R}^{3} $ |
$ \lambda>0 $ |
$ u^{5} $ |
References:
[1] |
C. O. Alves and M. A. Souto,
Existence of least energy nodal solution for a Schrödinger-Poisson system in bounded domains, Z. Angew. Math. Phys, 65 (2014), 1153-1166.
doi: 10.1007/s00033-013-0376-3. |
[2] |
C. O. Alves, M. A. Souto and S. H. M. Soares,
Schrödinger-Poisson equations without Ambrosetti-Rabinowitz condition, J. Math. Anal. Appl, 377 (2011), 584-592.
doi: 10.1016/j.jmaa.2010.11.031. |
[3] |
A. Ambrosetti,
On Schrödinger-Poisson systems, Milan J. Math., 76 (2008), 257-274.
doi: 10.1007/s00032-008-0094-z. |
[4] |
A. Azzollini and A. Pomponio,
Ground state solutions for the nonlinear Schrödinger-Maxwell equations, J. Math. Anal. Appl., 345 (2008), 90-108.
doi: 10.1016/j.jmaa.2008.03.057. |
[5] |
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. |
[6] |
I. Catto and P. L. Lions,
Binding of atoms and stability of molecules in Hartree and Thomas-Fermi type theories. Part 1: A necessary and sufficient condition for the stability of general molecular system, Comm. Partial Differential Equations, 17 (1992), 1051-1110.
|
[7] |
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. |
[8] |
S. Chen and C. Tang,
High energy solutions for the superlinear Schrödinger-Maxwell equations, Nonlinear Anal., 71 (2009), 4927-4934.
doi: 10.1016/j.na.2009.03.050. |
[9] |
H. Guo,
Nonexistence of least energy nodal solutions for Schrödinger-Poisson equation, Appl. Math. Lett., 68 (2017), 135-142.
doi: 10.1016/j.aml.2016.12.016. |
[10] |
L. Huang, E. M. Rocha and J. Chen,
On the Schrödinger-Poisson system with a general indefinite nonlinear, Nonlinear Anal., 28 (2016), 1-19.
doi: 10.1016/j.nonrwa.2015.09.001. |
[11] |
C. Y. Lei, G. S. Liu and L. T. Guo,
Multiple positive solutions for a Kirchhoff type problem with a critical nonlinearity, Nonlinear Anal. Real World Appl., 31 (2016), 343-355.
doi: 10.1016/j.nonrwa.2016.01.018. |
[12] |
H. Liu,
Positive solutions of an asymptotically periodic Schrödinger-Poisson system with critical exponent, Nonlinear Anal., 32 (2016), 198-212.
doi: 10.1016/j.nonrwa.2016.04.007. |
[13] |
Z. Liu and S. Guo,
On ground state solutions for the Schrödinger-Poisson equations with critical growth, J. Math. Anal. Appl., 412 (2014), 435-448.
doi: 10.1016/j.jmaa.2013.10.066. |
[14] |
A. Mao, L. Yang, A. Qian and S. Luan,
Existence and concentration of solutions of Schrödinger-Poisson system, Appl. Math. Lett, 68 (2017), 8-12.
doi: 10.1016/j.aml.2016.12.014. |
[15] |
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. |
[16] |
J. Sun,
Infinitely many solutions for a class of sublinear Schrödinger-Maxwell equations, J. Math. Anal. Appl., 390 (2012), 514-522.
doi: 10.1016/j.jmaa.2012.01.057. |
[17] |
J. Sun and T. Wu,
Multiplicity and concentration of homoclinic solutions for some second order Hamiltonian systems, Nonlinear Anal., 114 (2015), 105-115.
doi: 10.1016/j.na.2014.11.009. |
[18] |
M. Willem, Minimax Theorems, Birthäuser, Boston, 1996.
doi: 10.1007/978-1-4612-4146-1. |
show all references
References:
[1] |
C. O. Alves and M. A. Souto,
Existence of least energy nodal solution for a Schrödinger-Poisson system in bounded domains, Z. Angew. Math. Phys, 65 (2014), 1153-1166.
doi: 10.1007/s00033-013-0376-3. |
[2] |
C. O. Alves, M. A. Souto and S. H. M. Soares,
Schrödinger-Poisson equations without Ambrosetti-Rabinowitz condition, J. Math. Anal. Appl, 377 (2011), 584-592.
doi: 10.1016/j.jmaa.2010.11.031. |
[3] |
A. Ambrosetti,
On Schrödinger-Poisson systems, Milan J. Math., 76 (2008), 257-274.
doi: 10.1007/s00032-008-0094-z. |
[4] |
A. Azzollini and A. Pomponio,
Ground state solutions for the nonlinear Schrödinger-Maxwell equations, J. Math. Anal. Appl., 345 (2008), 90-108.
doi: 10.1016/j.jmaa.2008.03.057. |
[5] |
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. |
[6] |
I. Catto and P. L. Lions,
Binding of atoms and stability of molecules in Hartree and Thomas-Fermi type theories. Part 1: A necessary and sufficient condition for the stability of general molecular system, Comm. Partial Differential Equations, 17 (1992), 1051-1110.
|
[7] |
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. |
[8] |
S. Chen and C. Tang,
High energy solutions for the superlinear Schrödinger-Maxwell equations, Nonlinear Anal., 71 (2009), 4927-4934.
doi: 10.1016/j.na.2009.03.050. |
[9] |
H. Guo,
Nonexistence of least energy nodal solutions for Schrödinger-Poisson equation, Appl. Math. Lett., 68 (2017), 135-142.
doi: 10.1016/j.aml.2016.12.016. |
[10] |
L. Huang, E. M. Rocha and J. Chen,
On the Schrödinger-Poisson system with a general indefinite nonlinear, Nonlinear Anal., 28 (2016), 1-19.
doi: 10.1016/j.nonrwa.2015.09.001. |
[11] |
C. Y. Lei, G. S. Liu and L. T. Guo,
Multiple positive solutions for a Kirchhoff type problem with a critical nonlinearity, Nonlinear Anal. Real World Appl., 31 (2016), 343-355.
doi: 10.1016/j.nonrwa.2016.01.018. |
[12] |
H. Liu,
Positive solutions of an asymptotically periodic Schrödinger-Poisson system with critical exponent, Nonlinear Anal., 32 (2016), 198-212.
doi: 10.1016/j.nonrwa.2016.04.007. |
[13] |
Z. Liu and S. Guo,
On ground state solutions for the Schrödinger-Poisson equations with critical growth, J. Math. Anal. Appl., 412 (2014), 435-448.
doi: 10.1016/j.jmaa.2013.10.066. |
[14] |
A. Mao, L. Yang, A. Qian and S. Luan,
Existence and concentration of solutions of Schrödinger-Poisson system, Appl. Math. Lett, 68 (2017), 8-12.
doi: 10.1016/j.aml.2016.12.014. |
[15] |
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. |
[16] |
J. Sun,
Infinitely many solutions for a class of sublinear Schrödinger-Maxwell equations, J. Math. Anal. Appl., 390 (2012), 514-522.
doi: 10.1016/j.jmaa.2012.01.057. |
[17] |
J. Sun and T. Wu,
Multiplicity and concentration of homoclinic solutions for some second order Hamiltonian systems, Nonlinear Anal., 114 (2015), 105-115.
doi: 10.1016/j.na.2014.11.009. |
[18] |
M. Willem, Minimax Theorems, Birthäuser, Boston, 1996.
doi: 10.1007/978-1-4612-4146-1. |
[1] |
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 |
[2] |
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 |
[3] |
Yong-Yong Li, Yan-Fang Xue, Chun-Lei Tang. Ground state solutions for asymptotically periodic modified Schr$ \ddot{\mbox{o}} $dinger-Poisson system involving critical exponent. Communications on Pure and Applied Analysis, 2019, 18 (5) : 2299-2324. doi: 10.3934/cpaa.2019104 |
[4] |
Yu Su. Ground state solution of critical Schrödinger equation with singular potential. Communications on Pure and Applied Analysis, 2021, 20 (10) : 3347-3371. doi: 10.3934/cpaa.2021108 |
[5] |
Hangzhou Hu, Yuan Li, Dun Zhao. Ground state for fractional Schrödinger-Poisson equation in Coulomb-Sobolev space. Discrete and Continuous Dynamical Systems - S, 2021, 14 (6) : 1899-1916. doi: 10.3934/dcdss.2021064 |
[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] |
Yongpeng Chen, Yuxia Guo, Zhongwei Tang. Concentration of ground state solutions for quasilinear Schrödinger systems with critical exponents. Communications on Pure and Applied Analysis, 2019, 18 (5) : 2693-2715. doi: 10.3934/cpaa.2019120 |
[8] |
Kaimin Teng, Xiumei He. Ground state solutions for fractional Schrödinger equations with critical Sobolev exponent. Communications on Pure and Applied Analysis, 2016, 15 (3) : 991-1008. doi: 10.3934/cpaa.2016.15.991 |
[9] |
Lun Guo, Wentao Huang, Huifang Jia. Ground state solutions for the fractional Schrödinger-Poisson systems involving critical growth in $ \mathbb{R} ^{3} $. Communications on Pure and Applied Analysis, 2019, 18 (4) : 1663-1693. doi: 10.3934/cpaa.2019079 |
[10] |
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 |
[11] |
Xianhua Tang, Sitong Chen. Ground state solutions of Nehari-Pohozaev type for Schrödinger-Poisson problems with general potentials. Discrete and Continuous Dynamical Systems, 2017, 37 (9) : 4973-5002. doi: 10.3934/dcds.2017214 |
[12] |
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 |
[13] |
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 |
[14] |
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 |
[15] |
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 |
[16] |
Claudianor O. Alves, Geilson F. Germano. Existence of ground state solution and concentration of maxima for a class of indefinite variational problems. Communications on Pure and Applied Analysis, 2020, 19 (5) : 2887-2906. doi: 10.3934/cpaa.2020126 |
[17] |
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 |
[18] |
Yi He, Lu Lu, Wei Shuai. Concentrating ground-state solutions for a class of Schödinger-Poisson equations in $\mathbb{R}^3$ involving critical Sobolev exponents. Communications on Pure and Applied Analysis, 2016, 15 (1) : 103-125. doi: 10.3934/cpaa.2016.15.103 |
[19] |
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 |
[20] |
Miao-Miao Li, Chun-Lei Tang. Multiple positive solutions for Schrödinger-Poisson system in $\mathbb{R}^{3}$ involving concave-convex nonlinearities with critical exponent. Communications on Pure and Applied Analysis, 2017, 16 (5) : 1587-1602. doi: 10.3934/cpaa.2017076 |
Impact Factor:
Tools
Metrics
Other articles
by authors
[Back to Top]