American Institute of Mathematical Sciences

• 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
September  2017, 16(5): 1587-1602. doi: 10.3934/cpaa.2017076

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

* Corresponding author: Chun-Lei Tang

Received  April 2016 Revised  January 2017 Published  May 2017

Fund Project: The second author is supported by National Natural Science Foundation of China(No. 11471267)

In this paper, we study the existence of multiple positive solutions of the following Schrödinger-Poisson system with critical exponent
 $\begin{equation*}\begin{cases}-Δ u-l(x)φ u=λ h(x)|u|^{q-2}u+|u|^{4}u,\ \text{in}\ \mathbb{R}^{3}, \\-Δφ=l(x)u^{2},\ \text{in}\ \mathbb{R}^{3},\end{cases}\end{equation*}$
where
 $1 < q < 2$
and
 $λ>0$
. Under some appropriate conditions on
 $l$
and
 $h$
, we show that there exists
 $λ^{*}>0$
such that the above problem has at least two positive solutions for each
 $λ∈(0,λ^{*})$
by using the Mountain Pass Theorem and Ekeland's Variational Principle.
Citation: 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 & Applied Analysis, 2017, 16 (5) : 1587-1602. doi: 10.3934/cpaa.2017076
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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [5] A. Ambrosetti and P. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 41 (1973), 349-381. Google Scholar [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. Google Scholar [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. Google Scholar [8] V. Benci and D. Fortunato, An eigenvalue problem for the Schrödinger-Maxwell equations, Topol. Methods Nonlinear Anal., 11 (1998), 282-293. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [12] H. Brézis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure. Appl. Math., 36 (1983), 437-477. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [16] G. M. Coclite, A Multiplicity result for the nonlinear Schrödinger-Maxwell equations, Commun. Appl. Anal., 7 (2003), 417-423. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [20] E. H. Lieb, Thomas-Fermi and related theories and molecules, Rev. Modern Phys., 53 (1981), 603-641. doi: 10.1103/RevModPhys.53.603. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [24] M. Reed and B. Simon, Methods of Modern Mathematical Physics Vols. Elsevier (Singapore) Pte Ltd, 2003.Google Scholar [25] G. Talenti, Best constant in Sobolev inequality, Ann. Math., 110 (1976), 353-372. doi: 10.1007/BF02418013. Google Scholar [26] G. Vaira, Ground states for Schrödinger-Poisson type systems, Ricerche mat., 60 (2011), 263-297. doi: 10.1007/s11587-011-0109-x. Google Scholar [27] G. Vaira, Existence of bounded states for Schrödinger-Poisson type systems, S. I. S. S. A., 251 (2012), 112-146. Google Scholar [28] M. Willem, Minimax Theorems Birkhäuser, Boston, 1996. doi: 10.1007/978-1-4612-4146-1. Google Scholar [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. Google Scholar [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. Google Scholar [31] V. I. Bogachev, Measure Theory Springer, Berlin, 2007. doi: 10.1007/978-3-540-34514-5. Google Scholar [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. Google Scholar

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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [5] A. Ambrosetti and P. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 41 (1973), 349-381. Google Scholar [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. Google Scholar [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. Google Scholar [8] V. Benci and D. Fortunato, An eigenvalue problem for the Schrödinger-Maxwell equations, Topol. Methods Nonlinear Anal., 11 (1998), 282-293. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [12] H. Brézis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure. Appl. Math., 36 (1983), 437-477. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [16] G. M. Coclite, A Multiplicity result for the nonlinear Schrödinger-Maxwell equations, Commun. Appl. Anal., 7 (2003), 417-423. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [20] E. H. Lieb, Thomas-Fermi and related theories and molecules, Rev. Modern Phys., 53 (1981), 603-641. doi: 10.1103/RevModPhys.53.603. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [24] M. Reed and B. Simon, Methods of Modern Mathematical Physics Vols. Elsevier (Singapore) Pte Ltd, 2003.Google Scholar [25] G. Talenti, Best constant in Sobolev inequality, Ann. Math., 110 (1976), 353-372. doi: 10.1007/BF02418013. Google Scholar [26] G. Vaira, Ground states for Schrödinger-Poisson type systems, Ricerche mat., 60 (2011), 263-297. doi: 10.1007/s11587-011-0109-x. Google Scholar [27] G. Vaira, Existence of bounded states for Schrödinger-Poisson type systems, S. I. S. S. A., 251 (2012), 112-146. Google Scholar [28] M. Willem, Minimax Theorems Birkhäuser, Boston, 1996. doi: 10.1007/978-1-4612-4146-1. Google Scholar [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. Google Scholar [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. Google Scholar [31] V. I. Bogachev, Measure Theory Springer, Berlin, 2007. doi: 10.1007/978-3-540-34514-5. Google Scholar [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. Google Scholar
 [1] Xu Zhang, Shiwang Ma, Qilin Xie. Bound state solutions of Schrödinger-Poisson system with critical exponent. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 605-625. doi: 10.3934/dcds.2017025 [2] 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 & Applied Analysis, 2019, 18 (5) : 2299-2324. doi: 10.3934/cpaa.2019104 [3] Christopher Grumiau, Marco Squassina, Christophe Troestler. On the Mountain-Pass algorithm for the quasi-linear Schrödinger equation. Discrete & Continuous Dynamical Systems - B, 2013, 18 (5) : 1345-1360. doi: 10.3934/dcdsb.2013.18.1345 [4] Chunhua Wang, Jing Yang. Positive solutions for a nonlinear Schrödinger-Poisson system. Discrete & Continuous Dynamical Systems - A, 2018, 38 (11) : 5461-5504. doi: 10.3934/dcds.2018241 [5] Zhi Chen, Xianhua Tang, Ning Zhang, Jian Zhang. Standing waves for Schrödinger-Poisson system with general nonlinearity. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 6103-6129. doi: 10.3934/dcds.2019266 [6] Peng Gao, Yong Li. Averaging principle for the Schrödinger equations†. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2147-2168. doi: 10.3934/dcdsb.2017089 [7] 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 [8] 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 [9] Mingzheng Sun, Jiabao Su, Leiga Zhao. Infinitely many solutions for a Schrödinger-Poisson system with concave and convex nonlinearities. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 427-440. doi: 10.3934/dcds.2015.35.427 [10] Margherita Nolasco. Breathing modes for the Schrödinger-Poisson system with a multiple--well external potential. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1411-1419. doi: 10.3934/cpaa.2010.9.1411 [11] Claudianor O. Alves, Minbo Yang. Existence of positive multi-bump solutions for a Schrödinger-Poisson system in $\mathbb{R}^{3}$. Discrete & Continuous Dynamical Systems - A, 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 & 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 & Applied Analysis, 2016, 15 (5) : 1743-1756. doi: 10.3934/cpaa.2016011 [14] Juntao Sun, Tsung-Fang Wu, Zhaosheng Feng. Non-autonomous Schrödinger-Poisson system in $\mathbb{R}^{3}$. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 1889-1933. doi: 10.3934/dcds.2018077 [15] Zhengping Wang, Huan-Song Zhou. Positive solution for a nonlinear stationary Schrödinger-Poisson system in $R^3$. Discrete & Continuous Dynamical Systems - A, 2007, 18 (4) : 809-816. doi: 10.3934/dcds.2007.18.809 [16] Sitong Chen, Xianhua Tang. Existence of ground state solutions for the planar axially symmetric Schrödinger-Poisson system. Discrete & Continuous Dynamical Systems - B, 2019, 24 (9) : 4685-4702. doi: 10.3934/dcdsb.2018329 [17] Sitong Chen, Junping Shi, Xianhua Tang. Ground state solutions of Nehari-Pohozaev type for the planar Schrödinger-Poisson system with general nonlinearity. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 5867-5889. doi: 10.3934/dcds.2019257 [18] Vincenzo Ambrosio. Concentration phenomena for critical fractional Schrödinger systems. Communications on Pure & Applied Analysis, 2018, 17 (5) : 2085-2123. doi: 10.3934/cpaa.2018099 [19] 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 & Applied Analysis, 2019, 18 (4) : 1663-1693. doi: 10.3934/cpaa.2019079 [20] Marius Ghergu, Gurpreet Singh. On a class of mixed Choquard-Schrödinger-Poisson systems. Discrete & Continuous Dynamical Systems - S, 2019, 12 (2) : 297-309. doi: 10.3934/dcdss.2019021

2018 Impact Factor: 0.925