scholarly journals Existence of Positive Solutions of One-Dimensional Prescribed Mean Curvature Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-10
Author(s):  
Ruyun Ma ◽  
Lingfang Jiang
Keyword(s):  
Positive Solutions ◽  
Mean Curvature ◽  
Quadrature Method ◽  
One Dimensional ◽  
Curvature Equation ◽  
Exact Number ◽  
Prescribed Mean Curvature ◽  
Mean Curvature Equation ◽  
Prescribed Mean Curvature Equation ◽  
Existence Of Positive Solutions

We consider the existence of positive solutions of one-dimensional prescribed mean curvature equation−(u′/1+u′2)′=λf(u),0<t<1,u(t)>0,t∈(0,1),u(0)=u(1)=0whereλ>0is a parameter, andf:[0,∞)→[0,∞)is continuous. Further, whenfsatisfiesmax{up,uq}≤f(u)≤up+uq,0<p≤q<+∞, we obtain the exact number of positive solutions. The main results are based upon quadrature method.

Bifurcation diagrams and exact multiplicity of positive solutions of one-dimensional prescribed mean curvature equation in Minkowski space

2019 ◽  
Vol 21 (03) ◽  
pp. 1850003 ◽  
Author(s):  
Xuemei Zhang ◽  
Meiqiang Feng
Keyword(s):  
Minkowski Space ◽  
Mean Curvature ◽  
Bifurcation Diagrams ◽  
One Dimensional ◽  
Curvature Equation ◽  
Prescribed Mean Curvature ◽  
Mean Curvature Equation ◽  
Prescribed Mean Curvature Equation ◽  
The One ◽  
Exact Multiplicity

In this paper, bifurcation diagrams and exact multiplicity of positive solution are obtained for the one-dimensional prescribed mean curvature equation in Minkowski space in the form of [Formula: see text] where [Formula: see text] is a bifurcation parameter, [Formula: see text], the radius of the one-dimensional ball [Formula: see text], is an evolution parameter. Moreover, we make a comparison between the bifurcation diagram of one-dimensional prescribed mean curvature equation in Euclid space and Minkowski space. Our methods are based on a detailed analysis of time maps.

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