*Mathematical Methods in the Applied Sciences*Multiple nodal solutions for the Schr\”odinger-Poisson
system with an asymptotically cubic term

This paper deals with the following
Schr\“odinger-Poisson system
\begin{equation}\label{zhaiyaofc}\left\{\begin{aligned}
&-\Delta u+u+ \lambda\phi
u=f(u)\quad\mbox{in
}\mathbb{R}^3,\\
&-\Delta
\phi=u^{2}\quad\mbox{in
}\mathbb{R}^3,
\end{aligned}\right.\end{equation}
where $\lambda>0$ and $f(u)$ is a
nonlinear term asymptotically cubic at the infinity. Taking advantage of
the Miranda theorem and deformation lemma, we combine some new analytic
techniques to prove that for each positive integer $k,$ system
\eqref{zhaiyaofc} admits a radial nodal solution
$U_k^{\lambda}$, which has exactly $k+1$ nodal
domains and the corresponding energy is strictly increasing in $k$.
Moreover, for any sequence
$\{\lambda_n\}\to
0_+$ as $n\to\infty,$ up to a
subsequence, $U_k^{\lambda_n}$ converges to some
$U_k^0\in
H_r^1(\mathbb{R}^3)$, which is a radial nodal
solution with exactly $k+1$ nodal domains of
\eqref{zhaiyaofc} for $\lambda=0 $.
These results give an affirmative answer to the open problem proposed in
[Kim S, Seok J. Commun. Contemp. Math., 2012] for the
Schr\”odinger-Poisson system with an asymptotically
cubic term.

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