Ground state sign-changing solutions for a Schrödinger–Poisson system with a critical nonlinearity in [formula omitted

Abstract

In this paper, we investigate the existence of ground state sign-changing solutions to a class of Schrödinger–Poissonsystems −△u+u+k(x)ϕu=λf(x)u+|u|4u,x∈R3,−△ϕ=k(x)u2,x∈R3,where k and f are nonnegative functions, 0<λ<λ1 and λ1 is the first eigenvalue of the problem −△u+u=λf(x)u in H1(R3). With the help of the constraint variational method, we obtain that the Schrödinger–Poisson system possesses at least one ground state sign-changing solution for each 0<λ<λ1. Moreover, we prove that its energy is strictly larger than twice that of ground state solutions. This paper can be regarded as the complementary work of Huang et al. (2013), Shuai and Wang (2015), Wang and Zhou (2015) and Zhang (2015).