A complete classification of bifurcation diagrams of a Dirichlet problem with concave-convex nonlinearities.
The authors study the bifurcation diagrams of positive solutions of the two-point boundary value problem \(u''(x)+\lambda{g(u(x))}+h(u(x))=0\), \(-1<x<1\), \(u(-1)=u(1)=0\), where \(g\) and \(h\) are continuous on \([0,+\infty)\) and twice continuously differentiable on \((0,+\infty)\). Moreover, \(g\) is concave and \(h\) is convex. \(\lambda\) is a positive bifurcation parameter. A complete classification of the bifurcation diagrams is given. The authors prove an exact multiplicity result for positive solutions. The paper has links to a paper by \textit{A. Ambrosetti}, \textit{H. Brézis} and \textit{G. Cerami} [J. Funct. Anal. 122, No. 2, 519--543 (1994; Zbl 0805.35028)].
scientific article; zbMATH DE number 2056500