On the Existence of Positive Solutions for Some Nonlinear Boundary Value Problems II

Abstract

We study a class of boundary value problems with $\phi$-Laplacian (e.g., the prescribed mean curvature equation, in which $\phi (s)=\frac{s}{\sqrt{1+s^2}}$) \begin{center} $-{\left(\phi (u^{\prime })\right)}^{\prime }=\lambda f(u)\text{ on }(-L,L),u(-L)=u(L)=0,$ \end{center} where $\lambda$ and $L$ are positive parameters. For convex $f$ with $f(0)=0$, we establish various results on the exact number of positive solutions as well as global bifurcation diagrams. Some new bifurcation patterns are shown. This paper is a continuation of Pan and Xing [13], where the case $f(0)>0$ has been investigated.

43 pages, 22 figures