Exact multiplicity for degenerate two-point boundary value problems with \(p\)-convex nonlinearity
The authors investigate the exact number of positive solutions to degenerate two-point boundary value problems \[ (|u'|^{p-2}u')'+f(u)=0,\;-R<x<R,\;u(-R)=u(R)=0, \] under the assumption that \(f\) is p-convex, which is a generalization of convexity. They obtain the following result: Suppose \(f\) is \(C^2,\) strictly p-convex and \(f(u)/u^{p-1}\) is infinite when u is infinite. If \(f(0)\leq 0\), then the problem has a unique solution. If \(f(0)>0\), then there are \(0,1\) or \(2\) solutions provided that the number of solutions is finite. The proofs are based on the shooting method for the characterization of the shape of the time-map.
degenerate two-point boundary value problems
bifurcation diagram of positive solutions
scientific article; zbMATH DE number 1880112
scientific article