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

TL;DR

This paper establishes the existence and local uniqueness of normalized single-peak solutions for a Schr"odinger-Newton system with degenerate trapping potential, highlighting differences from the classical Schr"odinger equation due to nonlocal effects.

Contribution

It provides the first analysis of existence and local uniqueness of prescribed $L^{2}$-norm solutions for the Schr"odinger-Newton system, including nonexistence of multi-peak solutions.

Findings

Existence of single-peak solutions under degenerate potential.

Local uniqueness of these solutions proved.

Nonexistence of multi-peak solutions due to nonlocal term.

Abstract

In this paper, we investigate the existence and local uniqueness of normalized peak solutions for a Schr\"odinger-Newton system under the assumption that the trapping potential is degenerate and has non-isolated critical points. First we investigate the existence and local uniqueness of normalized single-peak solutions for the Schr\"odinger-Newton system. Precisely, we give the precise description of the chemical potential $\mu$ and the attractive interaction $a$. Then we apply the finite dimensional reduction method to obtain the existence of single-peak solutions. Furthermore, using various local Pohozaev identities, blow-up analysis and the maximum principle, we prove the local uniqueness of single-peak solutions by precise analysis of the concentrated points and the Lagrange multiplier. Finally, we also prove the nonexistence of multi-peak solutions for the Schr\"odinger-Newton system, which is markedly different from the corresponding Schr\"odinger equation. The nonlocal term results in this difference. The main difficulties come from the estimates on Lagrange multiplier, the different degenerate rates along different directions at the critical point of $P(x)$ and some complicated estimates involved by the nonlocal term. To our best knowledge, it may be the first time to study the existence and local uniqueness of solutions with prescribed $L^{2}$-norm for the Schr\"odinger-Newton system.