Existence of positive solutions for a Schrödinger-Poisson system with bounded potential and weighted functions in R 3 $\mathbb{R}^{3}

Abstract In this paper we study the Schrödinger-Poisson system 0.1 { − Δ u + V ( x ) u + K ( x ) ϕ u = a ( x ) | u | m − 2 u + λ b ( x ) | u | q − 2 u , in R 3 − Δ ϕ = K ( x ) u 2 , lim | x | → ∞ ϕ ( x ) = 0 , in R 3 , $$ \textstyle\begin{cases} -\Delta u+V(x)u+K(x)\phi u=a(x)|u|^{m-2}u+\lambda b(x)|u|^{q-2}u, &\mbox{in } \mathbb {R}^{3} \\ -\Delta\phi=K(x)u^{2},\qquad \lim_{|x|\to\infty}\phi(x)=0, & \mbox{in } \mathbb {R}^{3}, \end{cases} $$ where the potential V ( x ) $V(x)$ and the weighted functions a ( x ) , b ( x ) $a(x), b(x)$ are positive and bounded in R 3 $\mathbb {R}^{3}$ , K ( x ) ∈ L 2 ( R 3 ) ∪ L ∞ ( R 3 ) $K(x)\in L^{2}(\mathbb {R}^{3})\cup L^{\infty}(\mathbb {R}^{3})$ and K ( x ) ≥ 0 $K(x)\ge0$ in R 3 $\mathbb {R}^{3}$ . We prove the existence of a positive solution ( u , ϕ ) ∈ W 1 , 2 ( R 3 ) × D 1 , 2 ( R 3 ) $(u,\phi)\in W^{1,2}(\mathbb {R}^{3})\times\mathcal{D}^{1,2}(\mathbb {R}^{3})$ for 4 < q < m < 2 ∗ = 6 $4< q< m