Exact multiplicity results for a one-dimensional prescribed mean curvature problem related to MEMS models

2012 ◽  
Vol 13 (5) ◽  
pp. 2432-2445 ◽  
Author(s):  
Hongjing Pan ◽  
Ruixiang Xing
Keyword(s):  
Mean Curvature ◽  
Multiplicity Results ◽  
One Dimensional ◽  
Prescribed Mean Curvature ◽  
Exact Multiplicity ◽  
Curvature Problem

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.

A VARIATIONAL APPROACH FOR ONE-DIMENSIONAL PRESCRIBED MEAN CURVATURE PROBLEMS

2014 ◽  
Vol 97 (2) ◽  
pp. 145-161 ◽  
Author(s):  
GHASEM A. AFROUZI ◽  
ARMIN HADJIAN ◽  
GIOVANNI MOLICA BISCI
Keyword(s):  
Variational Methods ◽  
Mean Curvature ◽  
Variational Approach ◽  
Regularity Result ◽  
Nontrivial Solutions ◽  
One Dimensional ◽  
Prescribed Mean Curvature ◽  
Nonnegative Solutions ◽  
Curvature Problem

AbstractWe discuss the multiplicity of nonnegative solutions of a parametric one-dimensional mean curvature problem. Our main effort here is to describe the configuration of the limits of a certain function, depending on the potential at zero, that yield, for certain values of the parameter, the existence of infinitely many weak nonnegative and nontrivial solutions. Moreover, thanks to a classical regularity result due to Lieberman, this sequence of solutions strongly converges to zero in $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}C^1([0,1])$. Our approach is based on recent variational methods.

Positive Solutions of an Indefinite Prescribed Mean Curvature Problem on a General Domain

2004 ◽  
Vol 4 (1) ◽  
pp. 1-13 ◽  
Author(s):  
Patrick Habets ◽  
Pierpaolo Omari
Keyword(s):  
Positive Solutions ◽  
Mean Curvature ◽  
General Domain ◽  
Prescribed Mean Curvature ◽  
Existence Of Positive Solutions ◽  
Curvature Problem

AbstractThe existence of positive solutions is proved for the prescribed mean curvature problemwhere Ω ⊂ℝ

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.

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