# American Institute of Mathematical Sciences

• 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
March  2020, 28(1): 195-203. doi: 10.3934/era.2020013

## 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

* Corresponding author: Hui Guo

Received  January 2020 Revised  February 2020 Published  March 2020

Fund Project: The first author is supported by Scientific Research Fund of Hunan Provincial Education Department (Grant No. 18C0293), and the second author is supported by Natural Science Foundation of Hunan Province (Grant No. 2018JJ3136) and Scientific Research Fund of Hunan Provincial Education Department (Grant No. 19C0781)

We consider the following nonlinear Schrödinger-Poisson system
 $\left\{\begin{array}{lll} -\Delta u+u+\lambda\phi(x) u = f(u)&\quad &x\in \mathbb{R}^3, \\ -\Delta \phi = u^2, \ \lim\limits_{|x|\to\infty} \phi(x) = 0&\quad &x\in \mathbb{R}^3, \end{array}\right.$
where
 $\lambda>0$
and
 $f$
is continuous. By combining delicate analysis and the method of invariant subsets of descending flow, we prove the existence and asymptotic behavior of infinitely many radial sign-changing solutions for odd
 $f$
. The nonlinearity covers the case of pure power-type nonlinearity
 $f(u) = |u|^{p-2}u$
with the less studied situation
 $p\in(3, 4).$
This result extends and complements the ones in [Z. Liu, Z. Q. Wang, and J. Zhang, Ann. Mat. Pura Appl., 2016] from the coercive potential case to the constant potential case.
Citation: Hui Guo, Tao Wang. A note on sign-changing solutions for the Schrödinger Poisson system. Electronic Research Archive, 2020, 28 (1) : 195-203. doi: 10.3934/era.2020013
##### 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] 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 [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] 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.  Google Scholar [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  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [12] W. A. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55 (1977), 149-162.  doi: 10.1007/BF01626517.  Google Scholar [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.  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] 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 [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] 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.  Google Scholar [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  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [12] W. A. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55 (1977), 149-162.  doi: 10.1007/BF01626517.  Google Scholar [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.  Google Scholar
 [1] Fangyi Qin, Jun Wang, Jing Yang. Infinitely many positive solutions for Schrödinger-poisson systems with nonsymmetry potentials. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021054 [2] 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 [3] Mengyao Chen, Qi Li, Shuangjie Peng. Bound states for fractional Schrödinger-Poisson system with critical exponent. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021038 [4] 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 [5] 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 [6] Daniele Cassani, Luca Vilasi, Jianjun Zhang. Concentration phenomena at saddle points of potential for Schrödinger-Poisson systems. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021039 [7] Juntao Sun, Tsung-fang Wu. The number of nodal solutions for the Schrödinger–Poisson system under the effect of the weight function. Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3651-3682. doi: 10.3934/dcds.2021011 [8] Kazuhiro Kurata, Yuki Osada. Variational problems associated with a system of nonlinear Schrödinger equations with three wave interaction. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021100 [9] 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 [10] Alfonso Castro, Jorge Cossio, Sigifredo Herrón, Carlos Vélez. Infinitely many radial solutions for a $p$-Laplacian problem with indefinite weight. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021058 [11] Sishu Shankar Muni, Robert I. McLachlan, David J. W. Simpson. Homoclinic tangencies with infinitely many asymptotically stable single-round periodic solutions. Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3629-3650. doi: 10.3934/dcds.2021010 [12] Tianyu Liao. The regularity lifting methods for nonnegative solutions of Lane-Emden system. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021036 [13] 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 [14] 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 [15] 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 [16] Yingte Sun. Floquet solutions for the Schrödinger equation with fast-oscillating quasi-periodic potentials. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021047 [17] Maoding Zhen, Binlin Zhang, Xiumei Han. A new approach to get solutions for Kirchhoff-type fractional Schrödinger systems involving critical exponents. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021115 [18] Norman Noguera, Ademir Pastor. Scattering of radial solutions for quadratic-type Schrödinger systems in dimension five. Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3817-3836. doi: 10.3934/dcds.2021018 [19] Shu-Yu Hsu. Existence and properties of ancient solutions of the Yamabe flow. Discrete & Continuous Dynamical Systems, 2018, 38 (1) : 91-129. doi: 10.3934/dcds.2018005 [20] Hai-Liang Li, Tong Yang, Mingying Zhong. Diffusion limit of the Vlasov-Poisson-Boltzmann system. Kinetic & Related Models, 2021, 14 (2) : 211-255. doi: 10.3934/krm.2021003

Impact Factor: 0.263