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

We study a class of boundary value problems with $φ$-Laplacian (e.g., the prescribed mean curvature equation, in which $φ(s)=\frac{s}{\sqrt{1+s^2}}$) \begin{center} $-\left(φ(u')\right)'=λf(u)\; \text{ on }(-L, L),\quad u(-L)=u(L)=0,$ \end{center} where $λ$ 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.

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