RETRACTED ARTICLE: Existence results for the general Schr\xf6dinger\t\t\tequations with a superlinear Neumann boundary value problem

Zhen Liu

doi:10.1186/s13661-019-1174-4

Abstract

The main goal of this paper is to study the general Schrödinger\t\t\t\tequations with a superlinear Neumann boundary value problem in domains with conical\t\t\t\tpoints on the boundary of the bases. First the formulation and the complex form of\t\t\t\tthe problem for the equations are given, and then the existence result of solutions\t\t\t\tfor the above problem is proved by the complex analytic method and the fixed point\t\t\t\tindex theory, where we absorb the advantages of the methods in recent works and give\t\t\t\tsome improvement and development. Finally, we are also interested in the asymptotic\t\t\t\tbehavior of solutions of the mentioned equation. These results generalize some\t\t\t\tprevious results concerning the asymptotic behavior of solutions of non-delay\t\t\t\tsystems of Schrödinger equations or of delay Schrödinger equations.

Highlights

1 Introduction This article deals with solutions of the general Schrödinger equation with a superlinear Neumann boundary value problem
From a physical point of view, such Schrödinger equations with a superlinear Neumann boundary value problem have gained a lot of interest in recent years, in particular in the context of systems for the mean field dynamics of Bose–Einstein condensates [2, 5] and in applications to fields like nonlinear and fibers optics [25]
5 Conclusions In this paper, we studied a class of Schrödinger equations with Neumann boundary condition Lεg = div(ωε(x)|∇g| (x)–2∇g) = 0 on a compact metric space S ⊂ Rn, n ≥ 2, with a positive weight ωε(x)

Summary

Introduction

This article deals with solutions of the general Schrödinger equation with a superlinear Neumann boundary value problem. G : g ∈ Wl1o,c1(T), g = g(x) d (x) ∈ L1loc(T) This set is a Sobolev space of functions, locally summable on S together with their first order generalized derivatives. It follows that there exists a good approximation of g based on a set of independent and identically distributed random samples w = {wi}mi=1 = {(si, ti)}mi=1 ∈ Zm drawn according to the measure. In the case when ωε(x) and (x) are fixed constants, there have been many results about the existence, uniqueness, blowing-up and so on; we refer to the bibliography (see [19, 29]) It follows that the hypothesis space is a Hilbert space HE induced by a Mercer kernel K which is a continuous, symmetric, and positive semi-definite function on S × S (see [24]).

Main results

Conclusions