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

TL;DR

This paper proves the existence of a least energy, sign-changing, radial solution for a Schrödinger-Poisson system in three dimensions with an asymptotically cubic nonlinearity, using variational methods on the nodal Nehari set.

Contribution

It introduces a novel approach to find least energy sign-changing solutions for the Schrödinger-Poisson system with asymptotically cubic nonlinearities.

Findings

Existence of a least energy sign-changing radial solution.

Solution obtained via minimization on the nodal Nehari set.

Addresses the competition between nonlocal and nonlinear terms.

Abstract

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 $\phi u$ and the nonlinear term $f(u)$ 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).