Structures and evolution of bifurcation diagrams for a multiparameter $p$-Laplacian Dirichlet problem - Tsung-Yi Hsieh; Shin-Hwa Wang (Electronic Journal of Qualitative Theory of Differential Equations)
We study the multiparameter $p$-Laplacian Dirichlet problem \begin{equation*} \begin{cases} \left( \varphi _{p}(u^{\prime }(x))\right) ^{\prime }+\lambda (ku^{p-1}+\sum _{i=1}^{m}a_{i}u^{q_{i}})-\mu \sum _{j=1}^{n}b_{j}u^{r_{j}}=0,% \text{ }-1<x<1, \\ u(-1)=u(1)=0,% \end{cases} \end{equation*} where $p>1,$ $\varphi _{p}(y)=\left \vert y\right \vert ^{p-2}y$, $\left( \varphi _{p}(u^{\prime })\right) ^{\prime }$ is the one-dimensional $p$-Laplacian, $\lambda >0$ and $\mu \geq 0$ are two bifurcation parameters. We assume that $k\geq 0,$ $0<p-1<q_{1}<q_{2}<\cdot \cdot \cdot <q_{m}<r_{1}<r_{2}<\cdot \cdot \cdot <r_{n},$ $m,n\geq 1,$ $a_{1}=1,$ $a_{i}>0$ for $i=1,2,\dots,m$ and $b_{1}=1,$ $b_{j}>0$ for $j=1,2,\dots,n$. We mainly prove that, on the $(\lambda ,\left \Vert u\right \Vert _{\infty })$-plane, the bifurcation diagram consists of a strictly decreasing curve for $\mu =0,$ and always consists of a $\subset $-shaped curve for fixed $\mu >0$. We then study the structures and evolution of the bifurcation diagrams with varying $\mu \geq 0$.