Least energy radial sign-changing solution for the Schrödinger–Poisson system in R3 under an asymptotically cubic nonlinearity

Abstract

In this paper we consider the following Schrödinger–Poisson system in the whole R3,{−Δu+u+λϕu=f(u) in R3,−Δϕ=u2 in R3, where λ>0 and the nonlinearity f is “asymptotically cubic” at infinity. This implies that the nonlocal term ϕu 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.