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

Abstract

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.