A Constructive Approach About the Existence of Positive Solutions for Minkowski Curvature Problems

Abstract

In this paper, we give an existence theorem about positive solutions for the Dirichlet boundary value problem of one dimensional Minkowski curvature equations. We apply the theorem to one parameter family of problems to investigate a constructive method for numerical range of parameters where positive solutions exist. Moreover, we establish a nonexistence theorem of positive solutions for the corresponding one parameter family of problems. The coefficient function may be singular at the boundary and nonlinear term satisfies a sublinear growth condition. Main argument for the proof of existence theorem is employed by Krasnoselskii’s theorem of cone expansion and compression. We give a numerical algorithm and various examples to illustrate numerical information about ranges of the existence and nonexistence parameters which have been given only in a theoretical manner so far.