Positive solutions for a nonhomogeneous Schrödinger-Poisson system

Abstract

Abstract In this article, we consider the following Schrödinger-Poisson system: − Δ u + u + k ( x ) ϕ ( x ) u = f ( x ) ∣ u ∣ p − 1 u + g ( x ) , x ∈ R 3 , − Δ ϕ = k ( x ) u 2 , x ∈ R 3 , \left\{\begin{array}{ll}-\Delta u+u+k\left(x)\phi \left(x)u=f\left(x)| u{| }^{p-1}u+g\left(x),& x\in {{\mathbb{R}}}^{3},\\ -\Delta \phi =k\left(x){u}^{2},& x\in {{\mathbb{R}}}^{3},\end{array}\right. with p ∈ ( 3 , 5 ) p\in \left(3,5) . Under suitable assumptions on potentials f ( x ) f\left(x) , g ( x ) g\left(x) and k ( x ) k\left(x) , then at least four positive solutions for the above system can be obtained for sufficiently small ‖ g ‖ H − 1 ( R 3 ) \Vert g{\Vert }_{{H}^{-1}\left({{\mathbb{R}}}^{3})} by taking advantage of variational methods and Lusternik-Schnirelman category.