Ground state sign-changing solutions for a Schrödinger–Poisson system with a 3-linear growth nonlinearity

Abstract

In this paper, we investigate the existence and asymptotic behavior of ground state sign-changing solutions to a class of Schrödinger–Poisson systems{−△u+V(x)u+μϕu=λf(x)u+|u|2u,x∈R3,−△ϕ=u2,x∈R3, where V is a smooth function, f is nonnegative, μ>0, λ<λ1 and λ1 is the first eigenvalue of the problem −△u+V(x)u=λf(x)u in H. With the help of the sign-changing Nehari manifold, we obtain that the Schrödinger–Poisson system possesses at least one ground state sign-changing solution uμ for all μ>0 and each λ<λ1. Moreover, we prove that its energy is strictly larger than twice that of ground state solutions. Besides, we give a convergence property of uμ as μ↘0. This paper can be regarded as the complementary work of Shuai and Wang [23], Wang and Zhou [24].