Positive solutions of Schrödinger-Poisson systems with Hardy potential and indefinite nonlinearity

Abstract

In this article, we study the nonlinear Schrödinger-Poisson system -Δu + u - μ u/|x|2 + l(x)φu = k(x)|u|p-2u x ∈ ℝ3, -Δφ = l(x)u2 x ∈ ℝ3, where k ∈ C(ℝ3) and 4 < p < 6, k changes sign in ℝ3 and lim sup|x|→∞ k(x) = k∞ < 0. We prove that Schrödinger-Poisson systems with Hardy potential and indefinite nonlinearity have at least one positive solution, using variational methods.