Two positive solutions of a class of Schrödinger–Poisson system with indefinite nonlinearity

Abstract

We study the existence and multiplicity of positive solutions of a class of Schrödinger–Poisson system:{−Δu+u+l(x)ϕu=k(x)|u|p−2u+μh(x)uin R3,−Δϕ=l(x)u2in R3, where 4<p<6, k∈C(R3), k changes sign in R3 and lim|x|→∞k(x)=k∞<0. 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 eigenfunction is denoted by e1. An interesting phenomenon is that we do not need the condition ∫R3k(x)e1pdx<0, which has been shown to be a necessary condition to the existence of positive solutions for semilinear elliptic equations with indefinite nonlinearity (see e.g. S. Alama, G. Tarantello, On semilinear elliptic equations with indefinite nonlinearities, Calc. Var. Partial Differential Equations 1 (1993) 439–475).