A note on sign-changing solutions for the Schrödinger Poisson system

  • Received: 01 January 2020 Revised: 01 February 2020
  • Primary: 35J20, 35J60

  • 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[J]. Electronic Research Archive, 2020, 28(1): 195-203. doi: 10.3934/era.2020013

    Related Papers:

  • 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.



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    [1] Multiple bound states for the Schrödinger-Poisson problem. Commun. Contemp. Math. (2008) 10: 391-404.
    [2] Solitary waves for nonlinear Klein-Gordon-Maxwell and Schrödinger-Maxwell equations. Proc. Roy. Soc. Edinburgh Sect. A (2004) 134: 893-906.
    [3] An eigenvalue problem for the Schrödinger-Maxwell equations. Topol. Methods Nonlinear Anal. (1998) 11: 283-293.
    [4] Nonexistence of least energy nodal solutions for Schrödinger-Poisson equation. Appl. Math. Lett. (2017) 68: 135-142.
    [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] Infinitely many sign-changing solutions for the nonlinear Schrödinger-Poisson system. Ann. Mat. Pura Appl. (2016) 195: 775-794.
    [9] The Schrödinger-Poisson equation under the effect of a nonlinear local term. J. Funct. Anal. (2006) 237: 655-674.
    [10] On the Schrödinger-Poisson-Slater system: Behavior of minimizers, radial and nonradial cases. Arch. Ration. Mech. Anal. (2010) 198: 349-368.
    [11] On nonlinear Schrödinger-Poisson equations with general potentials. J. Math. Anal. Appl. (2013) 401: 672-681.
    [12] Existence of solitary waves in higher dimensions. Comm. Math. Phys. (1977) 55: 149-162.
    [13] Sign-changing solutions for the nonlinear Schrödinger-Poisson system in $\Bbb{R}^3$. Calc. Var. Partial Differential Equations (2015) 52: 927-943.
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