Establishing existence and uniqueness of solutions to the boundary value problem involving a generalized Emden equation, embracing Thomas–Fermi-like theories

Abstract

The purpose of this article is to construct a firm mathematical foundation for the boundary value problem associated with a generalized Emden equation that embraces Thomas–Fermi-like theories. Boundary value problems for the relativistic and non-relativistic Thomas–Fermi equations are included as special cases. Questions of existence and uniqueness of solutions to these boundary value problems form a fundamental and important area of investigation regarding whether these mathematical models for physical phenomena are actually well-posed. However, these questions have remained open for the generalized Emden problem and the relativistic Thomas–Fermi problem. Herein we advance current understanding of existence and uniqueness of solutions by proving that these boundary value problems each admit a unique solution. Our methods involve an analysis of the problems through arguments that apply differential inequalities and fixed-point theory. The new results guarantee the existence of a unique solution, ensuring the generalized Emden equation that embraces Thomas–Fermi theory sits on a firm mathematical foundation.