Classification and evolution of bifurcation curves for a one-dimensional Dirichlet-Neumann problem with a specific cubic nonlinearity

Abstract

We study the classification and evolution of bifurcation curves of positive solutions of the one-dimensional Dirichlet-Neumann problem with a specific cubic nonlinearity given by \begin{document}$ \left \{ \begin{array} [c]{l}u^{\prime \prime}(x)+\lambda(-\varepsilon u^{3}+u^{2}+u+1) = 0,\;0<x<1,\\ u(0) = 0,\ u^{\prime}(1) = -c<0, \end{array} \right. $\end{document} where $ 1/10\leq \varepsilon \leq1/5 $. It is interesting to find that the evolution of bifurcation curves is not completely identical with that for the one-dimensional perturbed Gelfand equations, even though it is the same for these two problems with zero Dirichlet boundary conditions. In fact, we prove that there exist a positive number $ \varepsilon^{\ast}\,(\approx0.178) $ and three nonnegative numbers $ c_{0}(\varepsilon)<c_{1}(\varepsilon)<c_{2}(\varepsilon) $ defined on $ [1/10,1/5] $ with $ c_{0} = 0 $ if $ 1/10<\varepsilon \leq \varepsilon^{\ast} $ and $ c_{0}>0 $ if $ \varepsilon^{\ast}<\varepsilon \leq1/5 $, such that, on the $ (\lambda,\Vert u\Vert_{\infty}) $-plane, (ⅰ) when $ 0<c\leq c_{0}(\varepsilon) $ and $ c\geq c_{2}(\varepsilon) $, the bifurcation curve is strictly increasing; (ⅱ) when $ c_{0}(\varepsilon)<c<c_{1}(\varepsilon) $, the bifurcation curve is $ S $-shaped; (ⅲ) when $ c_{1}(\varepsilon)\leq c<c_{2}(\varepsilon) $, the bifurcation curve is $ \subset $-shaped.