A complete classification of bifurcation diagrams of classes of multiparameter \(p\)-Laplacian boundary value problems (Q1006102)

Earlier the second author and \textit{T.-S. Yeh} [Nonlinear Anal., Theory Methods Appl. 64, 2412--2432 (2006; Zbl 1096.34014)] gave a complete classification of bifurcation diagrams of positive solutions of the problem \[ (|u'|^{p-2}u')'+f(u)=0,\quad u(-1)=u(1)=0 \] in the case \(f(u)=\lambda g(u)+h(u)\), where \(\lambda >0\) is a bifurcation parameter. In this paper, the authors give a similar analysis in the more general case \(f(u)=g(u,\lambda )+h(u,\mu )\), where \(\lambda\) and \(\mu\) are bifurcation parameters. Under appropriate conditions, they give a classification of eight qualitatively different bifurcation diagrams. They also establish the exact multiplicity of positive solutions.