Existence and concentration of positive ground states for Schrödinger-Poisson equations with competing potential functions

Abstract

This article concerns the Schrödinger-Poisson equation -ε2Δu + V(x)u + K(x)φu = P(x)|u|p-1 u + Q(x)|u|q-1u, x ∈ ℝ3, -ε2Δφ = K(x)u2, x ∈ ℝ3, where 3 < q < p < 5 = 2* - 1. We prove that for all ε > 0, the equation has a ground state solution. The methods used here are based on the Nehari manifold and the concentration-compactness principle. Furthermore, for ε > 0 small, these ground states concentrate at a global minimum point of the least energy function.