Abstract
We investigate the one-dimensional prescribed mean curvature equation with concave-convex nonlinearities in the form of − ( u ′ 1 + u ′ 2 ) ′ =λ( u p + u q ), u(x)>0, 0<x<1, u(0)=u(1)=0, where λ>0 is a parameter and p, q satisfy −1<p<q<+∞, and we obtain new exact results of positive solutions. Our methods are based on a detailed analysis of time maps.
Highlights
Mean curvature equations arise in differential geometry, physics, and other applied subjects
We investigate the one-dimensional prescribed mean curvature equation with concave-convex nonlinearities in the form of
Classical existence theorems for this problem are presented in [ ] with references to the original papers by Bombieri, Finn, Miranda, etc
Summary
Mean curvature equations arise in differential geometry, physics, and other applied subjects. The existence theorems established in most of those papers are concerned with solutions of the prescribed mean curvature problem as global minimizers of the corresponding energy functionals. There are some papers considering the exact number of positive solutions of In [ ], Habets and Omari considered the nonlinear boundary value problem of the one-dimensional prescribed mean curvature equation with f (t, u) =. In [ ], the authors considered a general class of f (t, u) involving a singularity, and they obtained the result that there exists a positive solution for a small parameter, and they pointed out f (t, u) = u–p, (R – u)–q, u–p(R – u)–q (where p, q > ). Consider the following boundary value problem of the one-dimensional prescribed mean curvature equation:.
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