On the Schrödinger–Poisson system with a general indefinite nonlinearity

Abstract

We study the existence and multiplicity of positive solutions of a class of Schrödinger–Poisson system: [View the MathML source Turn MathJax on] where k∈C(R3) changes sign in R3, lim∣x∣→∞k(x)=k∞<0, and the nonlinearity g behaves like a power at zero and at infinity. We mainly prove the existence of at least two positive solutions in the case that μ>μ1 and near μ1, where μ1 is the first eigenvalue of −Δ+id in H1(R3) with weight function h, whose corresponding positive eigenfunction is denoted by e1. An interesting phenomenon here is that we do not need the condition View the MathML source, which has been shown to be a sufficient condition to the existence of positive solutions for semilinear elliptic equations with indefinite nonlinearity (see e.g. Costa and Tehrani, 2001).