On an open problem of Liu, Ouyang and Zhang

In this paper, we investigate the following
Chern-Simons-Schr\”{o}dinger system
\begin{equation*} -\Delta
u+u+\left(\int_{|x|}^{\infty}
\frac{h(s)}{s} u^{2}(s) d
s+\frac{h^{2}(|x|)}{|x|^{2}}\right)
u=f(u) \quad \text { in }
\mathbb{R}^{2}, \end{equation*}
where $h(s)=\int_{0}^{s}
\frac{t}{2} u^{2}(t) dt$ and the nonlinearity
$f\in C(\mathbb{R},
\mathbb{R})$ satisfies the Ambrosetti-Rabinowitz type
condition. By using a combination of invariant sets method and the
Ljusternik-Schnirelman type minimax method, we prove the existence of
infinitely many sign-changing solutions. It is worth mentioning that the
nonlinear term may not be 5-superlinear at infinity. In particular, it
includes the power-type nonlinearity
$|u|^{p-2}u$ with $p\in (4,
\infty)$. This work also answers the open problem raised
by Liu, Ouyang and Zhang (Nonlinearity \textbf{32}
(2019), 3082-3111).