Multiple positive solutions of a class of non autonomous Schrödinger–Poisson systems

For the nonlinear Schrödinger equation coupled with Poisson equation of the version −Δu+u+ϕu=a(x)∣u∣p−2u+λk(x)u−Δu+u+ϕu=a(x)∣u∣p−2u+λk(x)u in R3R3 and −Δϕ=u2−Δϕ=u2 in R3R3, we prove the existence of two positive solutions in H1(R3)H1(R3) when a(x)a(x) is sign changing and the linear part is not coercive. We show that the coupled term ϕuϕu is helpful to find multiple positive solutions when a(x)a(x) is sign changing, which gives striking contrast to the known result where ϕuϕu is proven to be an obstacle to get the existence of nontrivial solutions. Surprisingly we show that the term ϕuϕu can play the role similar to a sign condition ∫a(x)e1pdx<0, which has turned out to be a necessary condition to the existence of positive solutions for semilinear elliptic equations with indefinite nonlinearity (see e.g. Alama et al. (1993)).