RETRACTED ARTICLE: Normal families of solutions for modified equilibrium equations and their applications

Yili Tan,Zongjing Jiang,Weiyong Ding,Yourong Wang

doi:10.1186/s13661-017-0906-6

Abstract

Using boundary behaviors of solutions for certain Laplace equation proved by Yan and Ychussie (Adv. Difference Equ. 2015:226, 2015) and applying a new method to dispose of the impulsive term with finite mass subject presented by Shi and Liao (J. Inequal. Appl. 2015:363, 2015) from another point of view, we prove that there exists a supra-open in (X,tau) for each V insigma in which the modified equilibrium equation has normal families of solutions. Moreover, we establish a new expression of a harmonic multifunction for the above equation. As applications, we not only prove the existence of normal families of solutions for modified equilibrium equations but also obtain several characterizations and fundamental properties of these new classes of superharmonic multifunctions.

Highlights

As in [ ], the modified equilibrium equations for a self-gravitating fluid rotating about theE x axis with prescribed velocity (r) can be defined as follows: T ⎧ ⎨∇P = ρ∇(– + r s(s) ds), C⎩ = πgρ. ( . )AHere ρ, g, and denote the density, gravitational constant, and gravitational potential, respectively, P is the pressure of the fluid at a point x ∈ R, and r = x + x
We proved that there exists a supra-open set in (X, τ ) for each V ∈ σ in Twhich the modified equilibrium equation has normal families of solutions
We established a new expression of harmonic multifunctions for the above equation

Summary

Introduction

As in [ ], the modified equilibrium equations for a self-gravitating fluid rotating about the. E In the study of this model, Yan and Ychussie [ ] proved the existence of the modified R Laplace solution if the angular velocity satisfies certain decay conditions. The existence and uniqueness of the generalized solutions for the boundary value. Many important physical phenomena on the engineering and science fields are frequently modeled by nonlinear differential equations. Such equations are often difficult or impossible to solve analytically. R (lower) α-continuous and upper (lower) β-continuous harmonic multifunctions defined by Wine [ ]. Characterization of these new harmonic multifunctions by many.

Preliminaries

A These imply that

Proof Let

Conclusions

12 November 2017 I References