On the exact structure of positive solutions of an Ambrosetti-Brezis-Cerami problem and its generalization in one space variable

Abstract

We study the exact structure of positive solutions of an Ambrosetti-Brezis-Cerami problem and its generalization in one space variable and in the classical Laplacian case. We prove the exact multiplicity result when $\lambda $ ranges over the whole interval $\left( 0,\infty \right) $ and get more detailed results of the solution curve. The proof of our exact multiplicity result uses the modified time-map techniques which can be adapted, and the exact multiplicity result can be extended to a more general $k$-Laplacian problem with $k>1.$