On the exact structure of positive solutions of an Ambrosetti-Brezis-Cerami problem and its generalization in one space variable.
The classical Ambrosetti-Brezis-Cerami problem deals with the existence of positive solutions to the Dirichlet BVP \[ \Delta u+\lambda u^q+u^p,\;x\in \Omega ,\quad u=0,\;x\in \partial \Omega , \] where \(\lambda >0\), \(0<q<1<p\), and \(\Omega \subset \mathbb R^N\) is a bounded domain with smooth boundary, and includes (among others) several open questions for the one-dimensional case \(N=1\) and \(\Omega =(-1,1)\). Motivated by the paper of \textit{Z.\,Liu} [Nonlinear Anal., Theory Methods Appl.\ 46, 181--197 (2001; Zbl 1003.34020)], the authors study in detail the structure of positive solutions in this one-dimensional case, extending the obtained results to the more general differential equation \[ u''+\lambda \sum _{i=1}^m a_iu^{q_i}+\sum _{j=1}^n b_ju^{p_j}=0 \] where \(\lambda >0\), \(0<q_i<1 \), \(a_i>0\), \(i=1,\dots ,m\), \(1<p_j\), \(b_j>0\), \(j=1,\dots ,n\). The proofs of the results of the paper are based on a modified time-map technique.
scientific article; zbMATH DE number 5138394