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

Abstract

<p style='text-indent:20px;'>We consider the following nonlinear Schrödinger-Poisson system</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> $ \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. $ </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">$ \lambda>0 $</tex-math></inline-formula> and <inline-formula><tex-math id="M2">$ f $</tex-math></inline-formula> is continuous. By combining delicate analysis and the method of invariant subsets of descending flow, we prove the existence and asymptotic behavior of infinitely many radial sign-changing solutions for odd <inline-formula><tex-math id="M3">$ f $</tex-math></inline-formula>. The nonlinearity covers the case of pure power-type nonlinearity <inline-formula><tex-math id="M4">$ f(u) = |u|^{p-2}u $</tex-math></inline-formula> with the less studied situation <inline-formula><tex-math id="M5">$ p\in(3, 4). $</tex-math></inline-formula> 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.</p>