On the Schrödinger–Poisson system with a general indefinite nonlinearity

We study the existence and multiplicity of positive solutions of a class of Schrödinger–Poisson system: {−Δu+u+l(x)ϕu=k(x)g(u)+μh(x)uinR3,−Δϕ=l(x)u2inR3, where k∈C(R3)k∈C(R3) changes sign in R3R3, lim∣x∣→∞k(x)=k∞<0lim∣x∣→∞k(x)=k∞<0, and the nonlinearity gg 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>μ1μ>μ1 and near μ1μ1, where μ1μ1 is the first eigenvalue of −Δ+id−Δ+id in H1(R3)H1(R3) with weight function hh, whose corresponding positive eigenfunction is denoted by e1e1. An interesting phenomenon here is that we do not need the condition ∫R3k(x)e1pdx<0, 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).