A complete classification of bifurcation diagrams of classes of a multiparameter Dirichlet problem with concave-convex nonlinearities (Q952152)

The authors study the bifurcation diagrams for positive solutions of the multi-parameter Dirichlet problem \[ u^{''}(x)+f_{\lambda,\mu} (u(x))=0, \quad-1<t<1, \] \[ u(-1)=u(1)=0, \] where \(f_{\lambda,\mu}(u)=g(u,\lambda)+h(u,\mu)\), \(\lambda>\lambda_0\), \(\mu>\mu_0\) are two bifurcation parameters, \(\lambda_0\) and \(\mu_0\) are two given real numbers. Under some reasonable conditions on \(g\) and \(f\), they give a classification of totally \textbf{eight} qualitatively different bifurcation diagrams, and prove the exact multiplicity of positive solutions. In addition, they give interesting examples which show complete evolution of bifurcation diagrams as \(\mu\) (resp. \(\lambda\)) varies. For earlier work, see \textit{A. Ambrosetti} et al. [J. Funct. Anal. 122, No. 2, 519--543 (1994; Zbl 0805.35028)].