Existence of solutions to the nonlinear, singular second order Bohr boundary value problems

• Published in: Nonlinear analysis: real world applications Vol. 36; pp. 183 - 202
• Main Author: Fewster–Young, Nicholas
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier Ltd 01.08.2017; Elsevier BV
• Subjects: Bohr; Boundary conditions; Boundary value problem; Boundary value problems; Differential equations; Existence of solutions; Liapunov functions; Nonlinear systems; Resonant BVPs; Singular; Thomas–Fermi; Singular; Bohr; Resonant BVPs; Boundary value problem; Thomas–Fermi; Existence of solutions
• ISSN: 1468-1218; 1878-5719
• DOI: 10.1016/j.nonrwa.2017.01.009

This paper investigates the existence of solutions for nonlinear systems of second order, singular boundary value problems (BVPs) with Bohr boundary conditions. A key application that arises from this theory is the famous Thomas–Fermi equations for the model of the atom when it is in a neutral state. The methodology in this paper uses an alternative and equivalent BVP, which is in the class of resonant singular BVPs, and thus this paper obtains novel results by implementing an innovative differential inequality, Lyapunov functions and topological techniques. This approach furnishes new results in the area of singular BVPs for a priori bounds and existence of solutions, where the BVP has unrestricted growth conditions and subject to the Bohr boundary conditions. In addition, the results can be relaxed and hold for the non-singular case too.