-
Previous Article
Positive solutions for quasilinear Schrödinger equations in $\mathbb{R}^N$
- CPAA Home
- This Issue
-
Next Article
Exponential boundary stabilization for nonlinear wave equations with localized damping and nonlinear boundary condition
Multiple positive solutions for Schrödinger-Poisson system in $\mathbb{R}^{3}$ involving concave-convex nonlinearities with critical exponent
School of Mathematics and Statistics, Southwest University, Chongqing, 400715, China |
$\begin{equation*}\begin{cases}-Δ u-l(x)φ u=λ h(x)|u|^{q-2}u+|u|^{4}u,\ \text{in}\ \mathbb{R}^{3}, \\-Δφ=l(x)u^{2},\ \text{in}\ \mathbb{R}^{3},\end{cases}\end{equation*}$ |
$1 < q < 2 $ |
$λ>0 $ |
$ l$ |
$h $ |
$λ^{*}>0 $ |
$λ∈(0,λ^{*}) $ |
References:
[1] |
C. O. Alves, F. J. S. A. Corra and G. M. Figueiredo,
On a class of nonlocal elliptic problems with critical growth, Differential Equation and Applications, 23 (2010), 409-417.
doi: 10.7153/dea-02-25. |
[2] |
C. O. Alves, J. V. Goncalves and O. H. Miyagaki,
Multiple positive solutions for semilinear elliptic equations in $\mathbb{R}^{N}$ involving critical exponents, Nonlinear Anal., 34 (1998), 593-615.
doi: 10.1016/S0362-546X(97)00555-5. |
[3] |
A. Ambrosetti, J. G. Azorero and I. Peral,
Elliptic variational problems in $\mathbb{R}^{N}$ with critical growth, J. Differential Equations, 168 (2000), 10-32.
doi: 10.1006/jdeq.2000.3875. |
[4] |
A. Ambrosetti, H. Brezis and G. Cerami,
Combined effects of concave and convex nonlinear in some elliptic problems, J. Funct. Anal., 122 (1994), 519-543.
doi: 10.1006/jfan.1994.1078. |
[5] |
A. Ambrosetti and P. Rabinowitz,
Dual variational methods in critical point theory and applications, J. Funct. Anal., 41 (1973), 349-381.
|
[6] |
A. Ambrosetti and D. Ruiz,
Multiple bound states for the Schr$\ddot{\mathrm{o}}$dinger-Poisson problem, Commun. Contemp. Math., 10 (2008), 391-404.
doi: 10.1142/S021919970800282X. |
[7] |
A. Azzollini and A. Pomponio, Grond state solutions for the nonlininear Schrödinger-Maxwell equations J. Math. Anal. Appl. 345 (2008), 90-108
doi: 10.1016/j.jmaa.2008.03.057. |
[8] |
V. Benci and D. Fortunato,
An eigenvalue problem for the Schrödinger-Maxwell equations, Topol. Methods Nonlinear Anal., 11 (1998), 282-293.
|
[9] |
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. |
[10] |
R. Benguria, H. Brézis and E. H. Lieb,
The Thomas-Ferim-von Weizsäcker theory of atoms and moleculars, Comm. Math. Phys., 79 (1981), 167-180.
|
[11] |
H. Brézis and E. H. Lieb,
A relation between pointwise conergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), 486-490.
doi: 10.2307/2044999. |
[12] |
H. Brézis and L. Nirenberg,
Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure. Appl. Math., 36 (1983), 437-477.
|
[13] |
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 generalmolecular system, Comm. Partial Differential Equations, 17 (1992), 1051-1110.
doi: 10.1080/03605309208820878. |
[14] |
G. Cerimi 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] |
J. Chabrowski,
Concentration-compactness principle at infinity and semilinear elliptic equations involving critical and subcritical Sobolev exponents, Calc. Var. Partial Differential Equations, 3 (1995), 493-512.
doi: 10.1007/BF01187898. |
[16] |
G. M. Coclite,
A Multiplicity result for the nonlinear Schrödinger-Maxwell equations, Commun. Appl. Anal., 7 (2003), 417-423.
|
[17] |
L. Huang and E. M. Rocha,
A positive solution of a Schrödinger-Poisson system with critical exponent, Communications in Mathematical Analysis, 15 (2013), 29-43.
|
[18] |
L. Huang, E. M. Rocha and J. Chen,
Two positive solutions of a class of Schrödinger-Poisson system with indefinite nonlinearity, J. Differential Equations, 255 (2013), 2463-2483.
doi: 10.1016/j.jde.2013.06.022. |
[19] |
L. Huang, E. M. Rocha and J. Chen,
Positive and sign-changing solutions of a Schrödinger-Poisson system involving a critical nonlinearity, J. Math. Anal. Appl., 408 (2013), 55-69.
doi: 10.1016/j.jmaa.2013.05.071. |
[20] |
E. H. Lieb,
Thomas-Fermi and related theories and molecules, Rev. Modern Phys., 53 (1981), 603-641.
doi: 10.1103/RevModPhys.53.603. |
[21] |
P. L. Lions,
The concentration-compactness principle in the calculus of variations. The limit case, part 1, Rev. Mat. Iberoamericana, 1 (1985), 145-201.
doi: 10.4171/RMI/6. |
[22] |
P. L. Lions,
The concentration-compactness principle in the calculus of variations. The limit case, part 2, Rev. Mat. Iberoamericana, 1 (1985), 45-121.
doi: 10.4171/RMI/12. |
[23] |
J. J. Nie and X. Wu,
Exsistence and muitilicity of non-trivial solutions for Schrödinger-Kirchhoff-type equations with radial potential, Nonlinear Anal., 75 (2012), 3470-3479.
doi: 10.1016/j.na.2012.01.004. |
[24] |
M. Reed and B. Simon,
Methods of Modern Mathematical Physics Vols. Elsevier (Singapore) Pte Ltd, 2003. |
[25] |
G. Talenti,
Best constant in Sobolev inequality, Ann. Math., 110 (1976), 353-372.
doi: 10.1007/BF02418013. |
[26] |
G. Vaira,
Ground states for Schrödinger-Poisson type systems, Ricerche mat., 60 (2011), 263-297.
doi: 10.1007/s11587-011-0109-x. |
[27] |
G. Vaira,
Existence of bounded states for Schrödinger-Poisson type systems, S. I. S. S. A., 251 (2012), 112-146.
|
[28] |
M. Willem,
Minimax Theorems Birkhäuser, Boston, 1996.
doi: 10.1007/978-1-4612-4146-1. |
[29] |
Y. -P. Gao, S. -L. Yu and C. -L. Tang, On positive ground state solution to the Schrödinger-Poisson system with the negative non-local term,
Electron. J. Differential Equations 118 (2015), 11 pp. |
[30] |
L. Zhao and F. Zhao,
Positive solutions for Schrödinger-Poisson equations with a critical exponent, Nonlinear Anal., 70 (2009), 2150-2164.
doi: 10.1016/j.na.2008.02.116. |
[31] |
V. I. Bogachev,
Measure Theory Springer, Berlin, 2007.
doi: 10.1007/978-3-540-34514-5. |
[32] |
Stationary solutions for a Schrodinger-Poisson system in R3, in Proceedings of the 2002 Fez Conference on Partial Differential Equations, Electron. J. Differ. Equ. Conf. , 9 (2002), 65-76. Southwest Texas State Univ. , San Marcos, TX. |
show all references
References:
[1] |
C. O. Alves, F. J. S. A. Corra and G. M. Figueiredo,
On a class of nonlocal elliptic problems with critical growth, Differential Equation and Applications, 23 (2010), 409-417.
doi: 10.7153/dea-02-25. |
[2] |
C. O. Alves, J. V. Goncalves and O. H. Miyagaki,
Multiple positive solutions for semilinear elliptic equations in $\mathbb{R}^{N}$ involving critical exponents, Nonlinear Anal., 34 (1998), 593-615.
doi: 10.1016/S0362-546X(97)00555-5. |
[3] |
A. Ambrosetti, J. G. Azorero and I. Peral,
Elliptic variational problems in $\mathbb{R}^{N}$ with critical growth, J. Differential Equations, 168 (2000), 10-32.
doi: 10.1006/jdeq.2000.3875. |
[4] |
A. Ambrosetti, H. Brezis and G. Cerami,
Combined effects of concave and convex nonlinear in some elliptic problems, J. Funct. Anal., 122 (1994), 519-543.
doi: 10.1006/jfan.1994.1078. |
[5] |
A. Ambrosetti and P. Rabinowitz,
Dual variational methods in critical point theory and applications, J. Funct. Anal., 41 (1973), 349-381.
|
[6] |
A. Ambrosetti and D. Ruiz,
Multiple bound states for the Schr$\ddot{\mathrm{o}}$dinger-Poisson problem, Commun. Contemp. Math., 10 (2008), 391-404.
doi: 10.1142/S021919970800282X. |
[7] |
A. Azzollini and A. Pomponio, Grond state solutions for the nonlininear Schrödinger-Maxwell equations J. Math. Anal. Appl. 345 (2008), 90-108
doi: 10.1016/j.jmaa.2008.03.057. |
[8] |
V. Benci and D. Fortunato,
An eigenvalue problem for the Schrödinger-Maxwell equations, Topol. Methods Nonlinear Anal., 11 (1998), 282-293.
|
[9] |
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. |
[10] |
R. Benguria, H. Brézis and E. H. Lieb,
The Thomas-Ferim-von Weizsäcker theory of atoms and moleculars, Comm. Math. Phys., 79 (1981), 167-180.
|
[11] |
H. Brézis and E. H. Lieb,
A relation between pointwise conergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), 486-490.
doi: 10.2307/2044999. |
[12] |
H. Brézis and L. Nirenberg,
Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure. Appl. Math., 36 (1983), 437-477.
|
[13] |
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 generalmolecular system, Comm. Partial Differential Equations, 17 (1992), 1051-1110.
doi: 10.1080/03605309208820878. |
[14] |
G. Cerimi 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] |
J. Chabrowski,
Concentration-compactness principle at infinity and semilinear elliptic equations involving critical and subcritical Sobolev exponents, Calc. Var. Partial Differential Equations, 3 (1995), 493-512.
doi: 10.1007/BF01187898. |
[16] |
G. M. Coclite,
A Multiplicity result for the nonlinear Schrödinger-Maxwell equations, Commun. Appl. Anal., 7 (2003), 417-423.
|
[17] |
L. Huang and E. M. Rocha,
A positive solution of a Schrödinger-Poisson system with critical exponent, Communications in Mathematical Analysis, 15 (2013), 29-43.
|
[18] |
L. Huang, E. M. Rocha and J. Chen,
Two positive solutions of a class of Schrödinger-Poisson system with indefinite nonlinearity, J. Differential Equations, 255 (2013), 2463-2483.
doi: 10.1016/j.jde.2013.06.022. |
[19] |
L. Huang, E. M. Rocha and J. Chen,
Positive and sign-changing solutions of a Schrödinger-Poisson system involving a critical nonlinearity, J. Math. Anal. Appl., 408 (2013), 55-69.
doi: 10.1016/j.jmaa.2013.05.071. |
[20] |
E. H. Lieb,
Thomas-Fermi and related theories and molecules, Rev. Modern Phys., 53 (1981), 603-641.
doi: 10.1103/RevModPhys.53.603. |
[21] |
P. L. Lions,
The concentration-compactness principle in the calculus of variations. The limit case, part 1, Rev. Mat. Iberoamericana, 1 (1985), 145-201.
doi: 10.4171/RMI/6. |
[22] |
P. L. Lions,
The concentration-compactness principle in the calculus of variations. The limit case, part 2, Rev. Mat. Iberoamericana, 1 (1985), 45-121.
doi: 10.4171/RMI/12. |
[23] |
J. J. Nie and X. Wu,
Exsistence and muitilicity of non-trivial solutions for Schrödinger-Kirchhoff-type equations with radial potential, Nonlinear Anal., 75 (2012), 3470-3479.
doi: 10.1016/j.na.2012.01.004. |
[24] |
M. Reed and B. Simon,
Methods of Modern Mathematical Physics Vols. Elsevier (Singapore) Pte Ltd, 2003. |
[25] |
G. Talenti,
Best constant in Sobolev inequality, Ann. Math., 110 (1976), 353-372.
doi: 10.1007/BF02418013. |
[26] |
G. Vaira,
Ground states for Schrödinger-Poisson type systems, Ricerche mat., 60 (2011), 263-297.
doi: 10.1007/s11587-011-0109-x. |
[27] |
G. Vaira,
Existence of bounded states for Schrödinger-Poisson type systems, S. I. S. S. A., 251 (2012), 112-146.
|
[28] |
M. Willem,
Minimax Theorems Birkhäuser, Boston, 1996.
doi: 10.1007/978-1-4612-4146-1. |
[29] |
Y. -P. Gao, S. -L. Yu and C. -L. Tang, On positive ground state solution to the Schrödinger-Poisson system with the negative non-local term,
Electron. J. Differential Equations 118 (2015), 11 pp. |
[30] |
L. Zhao and F. Zhao,
Positive solutions for Schrödinger-Poisson equations with a critical exponent, Nonlinear Anal., 70 (2009), 2150-2164.
doi: 10.1016/j.na.2008.02.116. |
[31] |
V. I. Bogachev,
Measure Theory Springer, Berlin, 2007.
doi: 10.1007/978-3-540-34514-5. |
[32] |
Stationary solutions for a Schrodinger-Poisson system in R3, in Proceedings of the 2002 Fez Conference on Partial Differential Equations, Electron. J. Differ. Equ. Conf. , 9 (2002), 65-76. Southwest Texas State Univ. , San Marcos, TX. |
[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] |
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 |
[3] |
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 |
[4] |
Denis Bonheure, Silvia Cingolani, Simone Secchi. Concentration phenomena for the Schrödinger-Poisson system in $ \mathbb{R}^2 $. Discrete and Continuous Dynamical Systems - S, 2021, 14 (5) : 1631-1648. doi: 10.3934/dcdss.2020447 |
[5] |
Kaimin Teng, Xian Wu. Concentration of bound states for fractional Schrödinger-Poisson system via penalization methods. Communications on Pure and Applied Analysis, 2022, 21 (4) : 1157-1187. doi: 10.3934/cpaa.2022014 |
[6] |
Yong-Yong Li, Yan-Fang Xue, Chun-Lei Tang. Ground state solutions for asymptotically periodic modified Schr$ \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 |
[7] |
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 |
[8] |
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 |
[9] |
Xiaoping Chen, Chunlei Tang. Least energy sign-changing solutions for Schrödinger-Poisson system with critical growth. Communications on Pure and Applied Analysis, 2021, 20 (6) : 2291-2312. doi: 10.3934/cpaa.2021077 |
[10] |
Yao Du, Jiabao Su, Cong Wang. On the critical Schrödinger-Poisson system with $ p $-Laplacian. Communications on Pure and Applied Analysis, 2022, 21 (4) : 1329-1342. doi: 10.3934/cpaa.2022020 |
[11] |
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 |
[12] |
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 |
[13] |
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 |
[14] |
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 |
[15] |
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 |
[16] |
Peng Gao, Yong Li. Averaging principle for the Schrödinger equations†. Discrete and Continuous Dynamical Systems - B, 2017, 22 (6) : 2147-2168. doi: 10.3934/dcdsb.2017089 |
[17] |
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 |
[18] |
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 |
[19] |
Vincenzo Ambrosio. Concentration phenomena for critical fractional Schrödinger systems. Communications on Pure and Applied Analysis, 2018, 17 (5) : 2085-2123. doi: 10.3934/cpaa.2018099 |
[20] |
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 |
2020 Impact Factor: 1.916
Tools
Metrics
Other articles
by authors
[Back to Top]