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

TL;DR

This paper investigates the existence and multiplicity of positive solutions for nonlinear boundary value problems involving the $ extit{ extbf{ extit{ extphi}}}$-Laplacian, providing bifurcation diagrams and new solution patterns.

Contribution

It extends previous work by analyzing cases where $f(0)=0$, establishing solution counts and bifurcation structures for a class of $ extit{ extbf{ extphi}}$-Laplacian boundary value problems.

Findings

Exact number of positive solutions determined for various parameters.

New bifurcation patterns identified.

Global bifurcation diagrams constructed.

Abstract

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