October  2019, 39(10): 5867-5889. doi: 10.3934/dcds.2019257

Ground state solutions of Nehari-Pohozaev type for the planar Schrödinger-Poisson system with general nonlinearity

1. 

School of Mathematics and Statistics, Central South University, Changsha 410083, Hunan, China

2. 

Department of Mathematics, College of William and Mary, Williamsburg, Virginia, 23187-8795, USA

* Corresponding author: Junping Shi

Received  October 2018 Revised  March 2019 Published  July 2019

Fund Project: This work is partially supported by the National Natural Science Foundation of China (No: 11571370)

It is shown that the planar Schrödinger-Poisson system with a general nonlinear interaction function has a nontrivial solution of mountain-pass type and a ground state solution of Nehari-Pohozaev type. The conditions on the nonlinear functions are much weaker and flexible than previous ones, and new variational and analytic techniques are used in the proof.

Citation: 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
References:
[1]

A. Ambrosetti and D. Ruiz, Multiple bound states for the Schrödinger-Poisson problem, Commun. Contemp. Math., 10 (2008), 391-404. doi: 10.1142/S021919970800282X. Google Scholar

[2]

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. Google Scholar

[3]

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. Google Scholar

[4]

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

[5]

R. BenguriaH. Brezis and E. Lieb, The Thomas-Fermi-von Weizsäcker theory of atoms and molecules, Comm. Math. Phys., 79 (1981), 167-180. doi: 10.1007/BF01942059. Google Scholar

[6]

D. BonheureS. Cingolani and J. Van Schaftingen, The logarithmic Choquard equation: Sharp asymptotics and nondegeneracy of the groundstate, J. Functional Analysis, 272 (2017), 5255-5281. doi: 10.1016/j.jfa.2017.02.026. Google Scholar

[7]

I. Catto and P. Lions, Binding of atoms and stability of molecules in hartree and thomas-fermi type theories, Comm. Partial Differential Equations, 18 (1993), 1149-1159. doi: 10.1080/03605309308820967. Google Scholar

[8]

G. Cerami and J. 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

[9]

S. Chen and X. H. Tang, Ground state sign-changing solutions for a class of Schrödinger-Poisson type problems in $\mathbb R^3$, Z. Angew. Math. Phys., 67 (2016), Art. 102, 18 pp. doi: 10.1007/s00033-016-0695-2. Google Scholar

[10]

S. Chen and X. H. Tang, Improved results for Klein-Gordon-Maxwell systems with general nonlinearity, Discrete Contin. Dyn. Syst.-A, 38 (2018), 2333-2348. doi: 10.3934/dcds.2018096. Google Scholar

[11]

S. Chen and X. H. Tang, Ground state solutions of Schrödinger-Poisson systems with variable potential and convolution nonlinearity, J. Math. Anal. Appl., 473 (2019), 87-111. doi: 10.1016/j.jmaa.2018.12.037. Google Scholar

[12]

S. Chen and X. H. Tang, Geometrically distinct solutions for Klein-Gordon-Maxwell systems with super-linear nonlinearities, Appl. Math. Lett., 90 (2019), 188-193. doi: 10.1016/j.aml.2018.11.007. Google Scholar

[13]

S. Cingolani and T. Weth, On the planar Schrödinger-Poisson system, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 169-197. doi: 10.1016/j.anihpc.2014.09.008. Google Scholar

[14]

G. Coclite, A multiplicity result for the nonlinear Schrödinger-Maxwell equations, Commun. Appl. Anal., 7 (2003), 417-423. Google Scholar

[15]

T. D'Aprile and D. Mugnai, Solitary waves for nonlinear Klein-Gordon-Maxwell and Schrödinger-Maxwell equations, Proc. Roy. Soc. Edinburgh Sect. A, 134 (2004), 893-906. doi: 10.1017/S030821050000353X. Google Scholar

[16]

M. Du and T. Weth, Ground states and high energy solutions of the planar Schrödinger-Poisson system, Nonlinearity, 30 (2017), 3492-3515. doi: 10.1088/1361-6544/aa7eac. Google Scholar

[17]

L. Jeanjean, On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer-type problem set on $\mathbb R^N$, Proc. Roy. Soc. Edinburgh Sect. A, 129 (1999), 787-809. doi: 10.1017/S0308210500013147. Google Scholar

[18]

Y. Jiang and H. Zhou, Schrödinger-Poisson system with steep potential well, J. Differential Equations, 251 (2011), 582-608. doi: 10.1016/j.jde.2011.05.006. Google Scholar

[19]

F. Li, Y. Li and J. Shi, Existence of positive solutions to Schrödinger-Poisson type systems with critical exponent, Commun. Contemp. Math., 16 (2014), 1450036, 28pp. doi: 10.1142/S0219199714500369. Google Scholar

[20]

G. Li and C. Wang, The existence of a nontrivial solution to a nonlinear elliptic problem of linking type without the Ambrosetti-Rabinowitz condition, Ann. Acad. Sci. Fenn. Math., 36 (2011), 461-480. doi: 10.5186/aasfm.2011.3627. Google Scholar

[21]

Y. Li, F. Li and J. Shi, Existence and multiplicity of positive solutions to Schrödinger-Poisson type systems with critical nonlocal term, Calc. Var. Partial Differential Equations, 56 (2017), Art. 134, 17 pp. doi: 10.1007/s00526-017-1229-2. Google Scholar

[22]

E. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev inequality and related inequalities, Ann. of Math., 118 (1983), 349-374. doi: 10.2307/2007032. Google Scholar

[23]

E. Lieb and M. Loss, Analysis, vol. 14 of Graduate Studies in Mathematics, 2nd ed., American Mathematical Society, Providence, RI, 2001. doi: 10.1090/gsm/014. Google Scholar

[24]

E. H. Lieb, Thomas-Fermi and related theories of atoms and molecules, Rev. Modern Phys., 53 (1981), 603-641. doi: 10.1103/RevModPhys.53.603. Google Scholar

[25]

P. Lions, The concentration-compactness principle in the calculus of variations. the locally compact case. Ⅰ & Ⅱ, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 109–145,223–283. doi: 10.1016/S0294-1449(16)30422-X. Google Scholar

[26]

P. Lions, Solutions of Hartree-Fock equations for Coulomb systems, Comm. Math. Phys., 109 (1987), 33-97. doi: 10.1007/BF01205672. Google Scholar

[27]

P. A. Markowich, C. A. Ringhofer and C. Schmeiser, Semiconductor Equations, Springer-Verlag, Vienna, 1990. doi: 10.1007/978-3-7091-6961-2. Google Scholar

[28]

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. Google Scholar

[29]

J. Stubbe, Bound states of two-dimensional Schrödinger-Newton equations, eprint, arXiv: 0807.4059.Google Scholar

[30]

J. SunH. Chen and J. Nieto, On ground state solutions for some non-autonomous Schrödinger-Maxwell systems, J. Differential Equations, 252 (2012), 3365-3380. doi: 10.1016/j.jde.2011.12.007. Google Scholar

[31]

J. Sun and S. Ma, Ground state solutions for some Schrödinger-Poisson systems with periodic potentials, J. Differential Equations, 260 (2016), 2119-2149. doi: 10.1016/j.jde.2015.09.057. Google Scholar

[32]

X. Tang, Non-Nehar manifold method for asymptotically periodic Schrödinger equation, Sci. China Math., 58 (2015), 715-728. doi: 10.1007/s11425-014-4957-1. Google Scholar

[33]

X. Tang and X. Lin, Existence of ground state solutions of Nehari-Pankov type to Schrödinger systems, Sci. China Math, (2018), 1–22. doi: 10.1007/s11425-017-9332-3. Google Scholar

[34]

X. Tang and S. Chen, Ground state solutions of Nehari-Pohozaev type for Kirchhoff-type problems with general potentials, Calc. Var. Partial Differential Equations, 56 (2017), Art. 110, 25 pp. doi: 10.1007/s00526-017-1214-9. Google Scholar

[35]

X. Tang and S. Chen, Ground state solutions of Nehari-Pohozaev type for Schrödinger-Poisson problems with general potentials, Disc. Contin. Dyn. Syst.-A, 37 (2017), 4973-5002. doi: 10.3934/dcds.2017214. Google Scholar

[36]

X. Tang and B. Cheng, Ground state sign-changing solutions for Kirchhoff type problems in bounded domains, J. Differential Equations, 261 (2016), 2384-2402. doi: 10.1016/j.jde.2016.04.032. Google Scholar

[37]

J. WangL. TianJ. Xu and F. Zhang, Existence and concentration of positive solutions for semilinear Schrödinger-Poisson systems in $\mathbb R^3$, Calc. Var. Partial Differential Equations, 48 (2013), 243-273. doi: 10.1007/s00526-012-0548-6. Google Scholar

[38]

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. Google Scholar

[39]

Z. Wang and H. Zhou, Sign-changing solutions for the nonlinear Schrödinger-Poisson system in $\mathbb R^3$, Calc. Var. Partial Differential Equations, 52 (2015), 927-943. doi: 10.1007/s00526-014-0738-5. Google Scholar

[40]

M. Willem, Minimax theorems, Progress in Nonlinear Differential Equations and their Applications, 24, Birkhäuser Boston Inc., Boston, MA, 1996. doi: 10.1007/978-1-4612-4146-1. Google Scholar

[41]

L. Zhao and F. Zhao, On the existence of solutions for the Schrödinger-Poisson equations, J. Math. Anal. Appl., 346 (2008), 155-169. doi: 10.1016/j.jmaa.2008.04.053. Google Scholar

show all references

References:
[1]

A. Ambrosetti and D. Ruiz, Multiple bound states for the Schrödinger-Poisson problem, Commun. Contemp. Math., 10 (2008), 391-404. doi: 10.1142/S021919970800282X. Google Scholar

[2]

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. Google Scholar

[3]

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. Google Scholar

[4]

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

[5]

R. BenguriaH. Brezis and E. Lieb, The Thomas-Fermi-von Weizsäcker theory of atoms and molecules, Comm. Math. Phys., 79 (1981), 167-180. doi: 10.1007/BF01942059. Google Scholar

[6]

D. BonheureS. Cingolani and J. Van Schaftingen, The logarithmic Choquard equation: Sharp asymptotics and nondegeneracy of the groundstate, J. Functional Analysis, 272 (2017), 5255-5281. doi: 10.1016/j.jfa.2017.02.026. Google Scholar

[7]

I. Catto and P. Lions, Binding of atoms and stability of molecules in hartree and thomas-fermi type theories, Comm. Partial Differential Equations, 18 (1993), 1149-1159. doi: 10.1080/03605309308820967. Google Scholar

[8]

G. Cerami and J. 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

[9]

S. Chen and X. H. Tang, Ground state sign-changing solutions for a class of Schrödinger-Poisson type problems in $\mathbb R^3$, Z. Angew. Math. Phys., 67 (2016), Art. 102, 18 pp. doi: 10.1007/s00033-016-0695-2. Google Scholar

[10]

S. Chen and X. H. Tang, Improved results for Klein-Gordon-Maxwell systems with general nonlinearity, Discrete Contin. Dyn. Syst.-A, 38 (2018), 2333-2348. doi: 10.3934/dcds.2018096. Google Scholar

[11]

S. Chen and X. H. Tang, Ground state solutions of Schrödinger-Poisson systems with variable potential and convolution nonlinearity, J. Math. Anal. Appl., 473 (2019), 87-111. doi: 10.1016/j.jmaa.2018.12.037. Google Scholar

[12]

S. Chen and X. H. Tang, Geometrically distinct solutions for Klein-Gordon-Maxwell systems with super-linear nonlinearities, Appl. Math. Lett., 90 (2019), 188-193. doi: 10.1016/j.aml.2018.11.007. Google Scholar

[13]

S. Cingolani and T. Weth, On the planar Schrödinger-Poisson system, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 169-197. doi: 10.1016/j.anihpc.2014.09.008. Google Scholar

[14]

G. Coclite, A multiplicity result for the nonlinear Schrödinger-Maxwell equations, Commun. Appl. Anal., 7 (2003), 417-423. Google Scholar

[15]

T. D'Aprile and D. Mugnai, Solitary waves for nonlinear Klein-Gordon-Maxwell and Schrödinger-Maxwell equations, Proc. Roy. Soc. Edinburgh Sect. A, 134 (2004), 893-906. doi: 10.1017/S030821050000353X. Google Scholar

[16]

M. Du and T. Weth, Ground states and high energy solutions of the planar Schrödinger-Poisson system, Nonlinearity, 30 (2017), 3492-3515. doi: 10.1088/1361-6544/aa7eac. Google Scholar

[17]

L. Jeanjean, On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer-type problem set on $\mathbb R^N$, Proc. Roy. Soc. Edinburgh Sect. A, 129 (1999), 787-809. doi: 10.1017/S0308210500013147. Google Scholar

[18]

Y. Jiang and H. Zhou, Schrödinger-Poisson system with steep potential well, J. Differential Equations, 251 (2011), 582-608. doi: 10.1016/j.jde.2011.05.006. Google Scholar

[19]

F. Li, Y. Li and J. Shi, Existence of positive solutions to Schrödinger-Poisson type systems with critical exponent, Commun. Contemp. Math., 16 (2014), 1450036, 28pp. doi: 10.1142/S0219199714500369. Google Scholar

[20]

G. Li and C. Wang, The existence of a nontrivial solution to a nonlinear elliptic problem of linking type without the Ambrosetti-Rabinowitz condition, Ann. Acad. Sci. Fenn. Math., 36 (2011), 461-480. doi: 10.5186/aasfm.2011.3627. Google Scholar

[21]

Y. Li, F. Li and J. Shi, Existence and multiplicity of positive solutions to Schrödinger-Poisson type systems with critical nonlocal term, Calc. Var. Partial Differential Equations, 56 (2017), Art. 134, 17 pp. doi: 10.1007/s00526-017-1229-2. Google Scholar

[22]

E. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev inequality and related inequalities, Ann. of Math., 118 (1983), 349-374. doi: 10.2307/2007032. Google Scholar

[23]

E. Lieb and M. Loss, Analysis, vol. 14 of Graduate Studies in Mathematics, 2nd ed., American Mathematical Society, Providence, RI, 2001. doi: 10.1090/gsm/014. Google Scholar

[24]

E. H. Lieb, Thomas-Fermi and related theories of atoms and molecules, Rev. Modern Phys., 53 (1981), 603-641. doi: 10.1103/RevModPhys.53.603. Google Scholar

[25]

P. Lions, The concentration-compactness principle in the calculus of variations. the locally compact case. Ⅰ & Ⅱ, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 109–145,223–283. doi: 10.1016/S0294-1449(16)30422-X. Google Scholar

[26]

P. Lions, Solutions of Hartree-Fock equations for Coulomb systems, Comm. Math. Phys., 109 (1987), 33-97. doi: 10.1007/BF01205672. Google Scholar

[27]

P. A. Markowich, C. A. Ringhofer and C. Schmeiser, Semiconductor Equations, Springer-Verlag, Vienna, 1990. doi: 10.1007/978-3-7091-6961-2. Google Scholar

[28]

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. Google Scholar

[29]

J. Stubbe, Bound states of two-dimensional Schrödinger-Newton equations, eprint, arXiv: 0807.4059.Google Scholar

[30]

J. SunH. Chen and J. Nieto, On ground state solutions for some non-autonomous Schrödinger-Maxwell systems, J. Differential Equations, 252 (2012), 3365-3380. doi: 10.1016/j.jde.2011.12.007. Google Scholar

[31]

J. Sun and S. Ma, Ground state solutions for some Schrödinger-Poisson systems with periodic potentials, J. Differential Equations, 260 (2016), 2119-2149. doi: 10.1016/j.jde.2015.09.057. Google Scholar

[32]

X. Tang, Non-Nehar manifold method for asymptotically periodic Schrödinger equation, Sci. China Math., 58 (2015), 715-728. doi: 10.1007/s11425-014-4957-1. Google Scholar

[33]

X. Tang and X. Lin, Existence of ground state solutions of Nehari-Pankov type to Schrödinger systems, Sci. China Math, (2018), 1–22. doi: 10.1007/s11425-017-9332-3. Google Scholar

[34]

X. Tang and S. Chen, Ground state solutions of Nehari-Pohozaev type for Kirchhoff-type problems with general potentials, Calc. Var. Partial Differential Equations, 56 (2017), Art. 110, 25 pp. doi: 10.1007/s00526-017-1214-9. Google Scholar

[35]

X. Tang and S. Chen, Ground state solutions of Nehari-Pohozaev type for Schrödinger-Poisson problems with general potentials, Disc. Contin. Dyn. Syst.-A, 37 (2017), 4973-5002. doi: 10.3934/dcds.2017214. Google Scholar

[36]

X. Tang and B. Cheng, Ground state sign-changing solutions for Kirchhoff type problems in bounded domains, J. Differential Equations, 261 (2016), 2384-2402. doi: 10.1016/j.jde.2016.04.032. Google Scholar

[37]

J. WangL. TianJ. Xu and F. Zhang, Existence and concentration of positive solutions for semilinear Schrödinger-Poisson systems in $\mathbb R^3$, Calc. Var. Partial Differential Equations, 48 (2013), 243-273. doi: 10.1007/s00526-012-0548-6. Google Scholar

[38]

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. Google Scholar

[39]

Z. Wang and H. Zhou, Sign-changing solutions for the nonlinear Schrödinger-Poisson system in $\mathbb R^3$, Calc. Var. Partial Differential Equations, 52 (2015), 927-943. doi: 10.1007/s00526-014-0738-5. Google Scholar

[40]

M. Willem, Minimax theorems, Progress in Nonlinear Differential Equations and their Applications, 24, Birkhäuser Boston Inc., Boston, MA, 1996. doi: 10.1007/978-1-4612-4146-1. Google Scholar

[41]

L. Zhao and F. Zhao, On the existence of solutions for the Schrödinger-Poisson equations, J. Math. Anal. Appl., 346 (2008), 155-169. doi: 10.1016/j.jmaa.2008.04.053. Google Scholar

Table 1.  Examples of nonlinear functions $ f(u) $ satisfying conditions in Theorems 1.1 and 1.2. Here $ b_0 = \frac{q(p-2)}{(q-1)(3-q)}\left[\frac{(p-1)(p-3)}{p(q-2)}\right]^{\frac{q-2}{p-2}}[2(p-q)]^{\frac{q-p}{p-2}}. $
$ f(u) $ (F4) (F5) (F6)
$ f_1(u)=|u|^{p-2}u $ $ 3\le p $ $ 2 <p\le 3 $ $ 3\le p $
$ f_2(u)=\left(|u|^{p-2}+b|u|^{q-2}\right)u $ $ 2<q<3< p $ $ 2<q<p\le 3 $ $ 2<q<3\le p $
$ 0\le b\le b_0 $
$ f_3(u)=u\left[1-\frac{1}{\ln(e+u^2)}\right] $ YES YES NO
$ f_4(u)=u\ln (1+u^2) $ NO YES NO
$ f_5(u)=|u|u\ln (1+u^2) $ YES NO YES
$ f_6(u)=3|u|u\ln\left(1+u^2\right)+\frac{2|u|^3u}{1+u^2} $ YES NO YES
$ f_7(u)=4u^3\int_{0}^{u}|s|^{1+\sin s}s\mathrm{d}s+|u|^{5+\sin u}u $ YES NO NO
$ f(u) $ (F4) (F5) (F6)
$ f_1(u)=|u|^{p-2}u $ $ 3\le p $ $ 2 <p\le 3 $ $ 3\le p $
$ f_2(u)=\left(|u|^{p-2}+b|u|^{q-2}\right)u $ $ 2<q<3< p $ $ 2<q<p\le 3 $ $ 2<q<3\le p $
$ 0\le b\le b_0 $
$ f_3(u)=u\left[1-\frac{1}{\ln(e+u^2)}\right] $ YES YES NO
$ f_4(u)=u\ln (1+u^2) $ NO YES NO
$ f_5(u)=|u|u\ln (1+u^2) $ YES NO YES
$ f_6(u)=3|u|u\ln\left(1+u^2\right)+\frac{2|u|^3u}{1+u^2} $ YES NO YES
$ f_7(u)=4u^3\int_{0}^{u}|s|^{1+\sin s}s\mathrm{d}s+|u|^{5+\sin u}u $ YES NO NO
[1]

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

[2]

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

[3]

Xianhua Tang, Sitong Chen. Ground state solutions of Nehari-Pohozaev type for Schrödinger-Poisson problems with general potentials. Discrete & Continuous Dynamical Systems - A, 2017, 37 (9) : 4973-5002. doi: 10.3934/dcds.2017214

[4]

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

[5]

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

[6]

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

[7]

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

[8]

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

[9]

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

[10]

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

[11]

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 & Applied Analysis, 2016, 15 (1) : 103-125. doi: 10.3934/cpaa.2016.15.103

[12]

Zhanping Liang, Yuanmin Song, Fuyi Li. Positive ground state solutions of a quadratically coupled schrödinger system. Communications on Pure & Applied Analysis, 2017, 16 (3) : 999-1012. doi: 10.3934/cpaa.2017048

[13]

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

[14]

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

[15]

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

[16]

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

[17]

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

[18]

Xiaoyan Lin, Yubo He, Xianhua Tang. Existence and asymptotic behavior of ground state solutions for asymptotically linear Schrödinger equation with inverse square potential. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1547-1565. doi: 10.3934/cpaa.2019074

[19]

Alex H. Ardila. Stability of ground states for logarithmic Schrödinger equation with a $δ^{\prime}$-interaction. Evolution Equations & Control Theory, 2017, 6 (2) : 155-175. doi: 10.3934/eect.2017009

[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: 1.143

Metrics

  • PDF downloads (87)
  • HTML views (134)
  • Cited by (0)

Other articles
by authors

[Back to Top]