Multi-peak Positive Solutions of a Nonlinear Schrödinger–Newton Type System

Abstract

Abstract In this paper, we study the following nonlinear Schrödinger–Newton type system: { - ϵ 2 ⁢ Δ ⁢ u + u - Φ ⁢ ( x ) ⁢ u = Q ⁢ ( x ) ⁢ | u | ⁢ u , x ∈ ℝ 3 , - ϵ 2 ⁢ Δ ⁢ Φ = u 2 , x ∈ ℝ 3 , \left\{\begin{aligned} &\displaystyle{-}\epsilon^{2}\Delta u+u-\Phi(x)u=Q(x)|u% |u,&&\displaystyle x\in\mathbb{R}^{3},\\ &\displaystyle{-}\epsilon^{2}\Delta\Phi=u^{2},&&\displaystyle x\in\mathbb{R}^{% 3},\end{aligned}\right. where ϵ > 0 {\epsilon>0} and Q ⁢ ( x ) {Q(x)} is a positive bounded continuous potential on ℝ 3 {\mathbb{R}^{3}} satisfying some suitable conditions. By applying the finite-dimensional reduction method, we prove that for any positive integer k, the system has a positive solution with k-peaks concentrating near a strict local minimum point x 0 {x_{0}} of Q ⁢ ( x ) {Q(x)} in ℝ 3 {\mathbb{R}^{3}} , provided that ϵ > 0 {\epsilon>0} is sufficiently small.