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

Abstract

In this paper, we study the following nonlinear Schrödinger–Newton type system:\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 {\epsilon>0} and {Q(x)} is a positive bounded continuous potential on {\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}} of {Q(x)} in {\mathbb{R}^{3}}, provided that {\epsilon>0} is sufficiently small.

Existence and local uniqueness of normalized peak solutions for a Schrodinger-Newton system

“…Very recently, in [23] Luo, Peng and Wang proved the uniqueness of the concentrated solutions of (1.4) obtained in [34] by some local Pohozaev identities, blowup analysis and the maximum principle. For more other results of the existence of solutions with concentration, one can refer to [5,8,30,32] and the references therein. For a very similar nonlocal problem i.e.…”

Existence of Multispike Positive Solutions for a Nonlocal Problem in ℝ3

“…Proof. The proof of this Lemma can be obtained arguing as Lemma B.1 of [24] exactly. We omit the details here.…”