The structure of the solution set of a generalized Ambrosetti--Brezis--Cerami problem in one space variable
This paper deals with the structure of the set of positive solutions of the nonlinear two-point boundary value problem \[ u''(x) + f_\lambda (u(x)) = 0\,, \quad -1 < x < 1,\quad u(-1) = u(1) = 0, \] where \(\lambda > 0\) is a bifurcation parameter and \[ f_\lambda(u) = \lambda \sum_{i=1}^m a_i u^{q_i} + \sum_{j=1}^n b_j u^{p_j} \] satisfies certain conditions which do not involve sign conditions on the coefficients \(a_i, b_j\). Using classical ODE methods based on the study of an appropriate time map, the authors give a precise description of the bifurcation diagram. In particular, they prove that there exists \(\lambda^* > 0\) such that the problem has exactly two positive solutions for \(0 < \lambda < \lambda^*\), exactly one positive solution for \(\lambda = \lambda^*\), and no positive solution for \(\lambda > \lambda^*\).
scientific article; zbMATH DE number 5016112