-
Previous Article
Global existence and energy decay of solutions for a wave equation with non-constant delay and nonlinear weights
- ERA Home
- This Issue
-
Next Article
The anisotropic integrability logarithmic regularity criterion for the 3D MHD equations
A note on sign-changing solutions for the Schrödinger Poisson system
College of Mathematics and Computing Science, Hunan University of Science and Technology, Xiangtan, Hunan 411201, China |
$ \left\{\begin{array}{lll} -\Delta u+u+\lambda\phi(x) u = f(u)&\quad &x\in \mathbb{R}^3, \\ -\Delta \phi = u^2, \ \lim\limits_{|x|\to\infty} \phi(x) = 0&\quad &x\in \mathbb{R}^3, \end{array}\right. $ |
$ \lambda>0 $ |
$ f $ |
$ f $ |
$ f(u) = |u|^{p-2}u $ |
$ p\in(3, 4). $ |
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. |
[2] |
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. |
[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. |
[4] |
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. |
[5] |
I. Ianni, Sign-changing radial solutions for the Schrödinger-Poisson-Slater problem, Topol. Methods Nonlinear Anal., 41 (2013), 365-385.https://projecteuclid.org/euclid.tmna/1461245483 |
[6] |
S. Kim and J. Seok, On nodal solutions of the nonlinear Schrödinger-Poisson equations, Commun. Contemp. Math., 14 (2012), 16 pp.
doi: 10.1142/S0219199712500411. |
[7] |
E. H. Lieb and M. Loss, Analysis, Second edition, Vol. 14, Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2001.
doi: 10.1090/gsm/014. |
[8] |
Z. Liu, Z.-Q. Wang and J. Zhang,
Infinitely many sign-changing solutions for the nonlinear Schrödinger-Poisson system, Ann. Mat. Pura Appl., 195 (2016), 775-794.
doi: 10.1007/s10231-015-0489-8. |
[9] |
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. |
[10] |
D. Ruiz,
On the Schrödinger-Poisson-Slater system: Behavior of minimizers, radial and nonradial cases, Arch. Ration. Mech. Anal., 198 (2010), 349-368.
doi: 10.1007/s00205-010-0299-5. |
[11] |
J. Seok,
On nonlinear Schrödinger-Poisson equations with general potentials, J. Math. Anal. Appl., 401 (2013), 672-681.
doi: 10.1016/j.jmaa.2012.12.054. |
[12] |
W. A. Strauss,
Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55 (1977), 149-162.
doi: 10.1007/BF01626517. |
[13] |
Z. Wang and H.-S. Zhou,
Sign-changing solutions for the nonlinear Schrödinger-Poisson system in $\Bbb{R}^3$, Calc. Var. Partial Differential Equations, 52 (2015), 927-943.
doi: 10.1007/s00526-014-0738-5. |
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. |
[2] |
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. |
[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. |
[4] |
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. |
[5] |
I. Ianni, Sign-changing radial solutions for the Schrödinger-Poisson-Slater problem, Topol. Methods Nonlinear Anal., 41 (2013), 365-385.https://projecteuclid.org/euclid.tmna/1461245483 |
[6] |
S. Kim and J. Seok, On nodal solutions of the nonlinear Schrödinger-Poisson equations, Commun. Contemp. Math., 14 (2012), 16 pp.
doi: 10.1142/S0219199712500411. |
[7] |
E. H. Lieb and M. Loss, Analysis, Second edition, Vol. 14, Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2001.
doi: 10.1090/gsm/014. |
[8] |
Z. Liu, Z.-Q. Wang and J. Zhang,
Infinitely many sign-changing solutions for the nonlinear Schrödinger-Poisson system, Ann. Mat. Pura Appl., 195 (2016), 775-794.
doi: 10.1007/s10231-015-0489-8. |
[9] |
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. |
[10] |
D. Ruiz,
On the Schrödinger-Poisson-Slater system: Behavior of minimizers, radial and nonradial cases, Arch. Ration. Mech. Anal., 198 (2010), 349-368.
doi: 10.1007/s00205-010-0299-5. |
[11] |
J. Seok,
On nonlinear Schrödinger-Poisson equations with general potentials, J. Math. Anal. Appl., 401 (2013), 672-681.
doi: 10.1016/j.jmaa.2012.12.054. |
[12] |
W. A. Strauss,
Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55 (1977), 149-162.
doi: 10.1007/BF01626517. |
[13] |
Z. Wang and H.-S. Zhou,
Sign-changing solutions for the nonlinear Schrödinger-Poisson system in $\Bbb{R}^3$, Calc. Var. Partial Differential Equations, 52 (2015), 927-943.
doi: 10.1007/s00526-014-0738-5. |
[1] |
Jiaquan Liu, Xiangqing Liu, Zhi-Qiang Wang. Sign-changing solutions for a parameter-dependent quasilinear equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1779-1799. doi: 10.3934/dcdss.2020454 |
[2] |
Denis Bonheure, Silvia Cingolani, Simone Secchi. Concentration phenomena for the Schrödinger-Poisson system in $ \mathbb{R}^2 $. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1631-1648. doi: 10.3934/dcdss.2020447 |
[3] |
Kuan-Hsiang Wang. An eigenvalue problem for nonlinear Schrödinger-Poisson system with steep potential well. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021030 |
[4] |
Yanqin Fang, Jihui Zhang. Multiplicity of solutions for the nonlinear Schrödinger-Maxwell system. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1267-1279. doi: 10.3934/cpaa.2011.10.1267 |
[5] |
Wentao Huang, Jianlin Xiang. Soliton solutions for a quasilinear Schrödinger equation with critical exponent. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1309-1333. doi: 10.3934/cpaa.2016.15.1309 |
[6] |
Yimin Zhang, Youjun Wang, Yaotian Shen. Solutions for quasilinear Schrödinger equations with critical Sobolev-Hardy exponents. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1037-1054. doi: 10.3934/cpaa.2011.10.1037 |
[7] |
Sandrine Anthoine, Jean-François Aujol, Yannick Boursier, Clothilde Mélot. Some proximal methods for Poisson intensity CBCT and PET. Inverse Problems & Imaging, 2012, 6 (4) : 565-598. doi: 10.3934/ipi.2012.6.565 |
[8] |
Diana Keller. Optimal control of a linear stochastic Schrödinger equation. Conference Publications, 2013, 2013 (special) : 437-446. doi: 10.3934/proc.2013.2013.437 |
[9] |
Shu-Yu Hsu. Existence and properties of ancient solutions of the Yamabe flow. Discrete & Continuous Dynamical Systems - A, 2018, 38 (1) : 91-129. doi: 10.3934/dcds.2018005 |
[10] |
Zhouxin Li, Yimin Zhang. Ground states for a class of quasilinear Schrödinger equations with vanishing potentials. Communications on Pure & Applied Analysis, 2021, 20 (2) : 933-954. doi: 10.3934/cpaa.2020298 |
[11] |
Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1693-1716. doi: 10.3934/dcdss.2020450 |
[12] |
Amit Goswami, Sushila Rathore, Jagdev Singh, Devendra Kumar. Analytical study of fractional nonlinear Schrödinger equation with harmonic oscillator. Discrete & Continuous Dynamical Systems - S, 2021 doi: 10.3934/dcdss.2021021 |
[13] |
Pavel I. Naumkin, Isahi Sánchez-Suárez. Asymptotics for the higher-order derivative nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021028 |
[14] |
Sergi Simon. Linearised higher variational equations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4827-4854. doi: 10.3934/dcds.2014.34.4827 |
[15] |
Ying Yang. Global classical solutions to two-dimensional chemotaxis-shallow water system. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2625-2643. doi: 10.3934/dcdsb.2020198 |
[16] |
Valeria Chiado Piat, Sergey S. Nazarov, Andrey Piatnitski. Steklov problems in perforated domains with a coefficient of indefinite sign. Networks & Heterogeneous Media, 2012, 7 (1) : 151-178. doi: 10.3934/nhm.2012.7.151 |
[17] |
Y. Latushkin, B. Layton. The optimal gap condition for invariant manifolds. Discrete & Continuous Dynamical Systems - A, 1999, 5 (2) : 233-268. doi: 10.3934/dcds.1999.5.233 |
[18] |
Christopher Bose, Rua Murray. Minimum 'energy' approximations of invariant measures for nonsingular transformations. Discrete & Continuous Dynamical Systems - A, 2006, 14 (3) : 597-615. doi: 10.3934/dcds.2006.14.597 |
[19] |
Matthias Erbar, Jan Maas. Gradient flow structures for discrete porous medium equations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (4) : 1355-1374. doi: 10.3934/dcds.2014.34.1355 |
[20] |
Xue-Ping Luo, Yi-Bin Xiao, Wei Li. Strict feasibility of variational inclusion problems in reflexive Banach spaces. Journal of Industrial & Management Optimization, 2020, 16 (5) : 2495-2502. doi: 10.3934/jimo.2019065 |
Impact Factor: 0.263
Tools
Article outline
[Back to Top]