Communications of the
Korean Mathematical Society CKMS

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Commun. Korean Math. Soc. 2022; 37(4): 1269-1284

Online first article May 13, 2022      Printed October 31, 2022

https://doi.org/10.4134/CKMS.c210337

Copyright © The Korean Mathematical Society.

Ground state sign-changing solutions for nonlinear Schr\"{o}dinger-Poisson system with indefinite potentials

Shubin Yu, Ziheng Zhang

TianGong University; TianGong University

Abstract

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This paper is concerned with the following Schr\"{o}dinger-\linebreak Poisson system
$$
\left\{\begin{array}{ll}
 -{\Delta}u+V(x)u+K(x){\phi}u=a(x)|u|^{p-2}u  &\mbox{in}\ \mathbb{R}^3, \\[0.1cm]
 -{\Delta}{\phi}=K(x)u^{2}&\mbox{in}\ \mathbb{R}^3, \\[0.1cm]
\end{array}
\right.
$$
where $4<p<6$. For the case that $K$ is nonnegative, $V$ and $a$ are indefinite, we prove the above problem possesses one ground state sign-changing solution
with exactly two nodal domains by constraint variational method and quantitative deformation lemma. Moreover, we show that the energy of sign-changing solutions is
larger than that of the ground state solutions. The novelty of this paper is that the potential $a$ is indefinite and allowed to vanish at infinity. In this sense, we complement
the existing results obtained by Batista and Furtado \cite{BF18}.

Keywords: Schr\"{o}dinger-Poisson system, nonlocal term, sign-changing solution

MSC numbers: Primary 35A15, 35J20, 35J50

Supported by: This work was financially supported by the National Natural Science Foundation of China (Grant No.11771044 and No.12171039).

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