The Existence of Normalized Solutions for a Nonlocal Problem in ℝ3

Jing Yang

doi:10.1155/2020/3186135

Abstract

In this paper, we study the following fractional Schrödinger equation in ℝ3−Δσu−λu=up−2u, in ℝ3 with σ∈0,1,λ∈ℝ and p∈2+σ,2+4/3σ. By using the constrained variational method, we show the existence of solutions with prescribed L2 norm for this problem.

Highlights

This paper concerns with the following fractional Schrödinger problem: ð−ΔÞσu − λu = jujp−2u, in R3, ð1Þ where σ ∈ ð0, 1Þ, p ∈ ð2 + σ, 2 + ð4/3ÞσÞ, and λ ∈ R
The motivation for studying such equations comes from mathematical physics: searching for the form of standing wave ψ = e−ihtu of the evolution equation i jψjp−2ψ in R+
Mainly by variational methods, many researches have been devoted to the study of the existence, multiplicity, uniqueness, regularity, and asymptotic decay properties of the solutions to fractional Schrödinger equation (1)

Summary

Introduction

Part of the motivation is to consider h ∈ R as a fixed parameter and to search for a solution u ∈ HσðR3Þ solving (1) In this direction, mainly by variational methods, many researches have been devoted to the study of the existence, multiplicity, uniqueness, regularity, and asymptotic decay properties of the solutions to fractional Schrödinger equation (1). In the present paper, inspired by the fact that physicists are often interested in normalized solutions, we look for solutions in HσðR3Þ having a prescribed L2 norm to equation (1). Such types of problems were studied extensively in recent years for the classical Schrödinger equations with the standard Laplacian operator.

Preliminaries

Proof of the Main Result