Least energy radial sign-changing solution for the Schröinger-Poisson system in r3 under an asymptotically cubic nonlinearity

Abstract

In this paper we consider the following Schrödinger-Poisson system in the whole ${\mathbb{R}}^3$, \begin{equation*} \left\{ \begin{array}{ll} -Δu+u+ λϕu=f(u) &\text{ in } \mathbb R^3, -Δϕ= 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 $\varphi u$ 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).

References in corpus (1)

• A sign-changing solution for the Schrödinger-Poisson equation