Classification and evolution of bifurcation curves for a one-dimensional Neumann–Robin problem and its applications

Abstract

We study the classification and evolution of bifurcation curves of positive solutions for the one-dimensional Neumann–Robin boundary value problem \begin{equation*} \begin{cases} u^{\prime \prime }(x)+\lambda f(u(x))=0,\quad 00$ is a bifurcation parameter, $\alpha >0$ is an evolution parameter, and nonlinearity $f$ satisfies $f(0)\geq 0$ and $f(u)>0$ for $u>0$. We obtain the multiplicity of positive solutions for $\alpha >0$ and $\lambda >0$. Applications are given.

Keywords

• bifurcation
• multiplicity
• positive solution
• neumann–robin problem
• $s$-shaped bifurcation curve
• $\subset$-shaped bifurcation curve
• time map