Least energy radial sign-changing solution for the Schr\"oinger-Poisson system in r3 under an asymptotically cubic nonlinearity

In this paper we consider the following Schr"odinger-Poisson system in the whole ${\mathbb{R}}^3$, \begin{equation*} \left{ \begin{array}{ll} -\Delta u+u+ \lambda \phi u=f(u) &$\text{ in }$ \mathbb R^3, -\Delta \phi= u^2 &$\text{ in }$ \mathbb R^3, \end{array} \right. \end{equation*} where $\lambda >0$ and the nonlinearity $f$ is “asymptotically cubic” at infinity. This implies that the nonlocal term $\phi u$andthenonlinearterm$f(u)$ are, in some sense, in a strict competition. We show that the system admits a least energy sign-changing and radial solution obtained by minimizing the energy functional on the so-called {nodal Nehari set).