Sufficient and necessary conditions for ground state sign-changing solutions to the Schrödinger–Poisson system with cubic nonlinearity on bounded domains

Abstract

In this paper we are interested in the existence of ground state sign-changing solutions to the following Schrödinger–Poisson system −Δu+ϕu=λu+μ|u|2uinΩ−Δϕ=u2inΩu=ϕ=0on∂Ω,where Ω is a bounded smooth domain of R3, μ>0, λ<λ1 and λ1 is the first eigenvalue of −Δ,H01(Ω). Using construction technique, we show that the above system possesses one ground state sign-changing solution if and only if μ>0, which improves the recent results by Khoutir (2021).