Ground state sign-changing solutions for nonlinear Schr\"{o}dinger-Poisson system with indefinite potentials
Commun. Korean Math. Soc.
Published online May 13, 2022
Shubin Yu and Ziheng Zhang
TianGong University; TianGong University
Abstract : This paper is concerned with the following Schr\"{o}dinger-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 (Nonlinear Anal. Real World Appl., {\bf 39} (2018), 142-156).
Keywords : Schr\"{o}dinger-Poisson system; nonlocal term; sign-changing solution
MSC numbers : 35A15, 35J20, 35J50
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