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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

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##### 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
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