Abstract
In this paper, chapter 1 is introduction. Chapter 2, we study the exact structure of positive solutions of the Laplacian Dirichlet problem which is a generalized Ambrosetti-Brezis-Cerami problem in one space variable. We choose the nonlinearities are the polynomial nonlinearities with positive coefficients. We are applying a modified time-map method to study this problem. Chapter 3, we study above problem with class of general nonlinearities. We assume that nonlinearities satisfy hypotheses (A1)-(A3). Under hypotheses (A1)-(A3), we give a complete classification of bifurcation diagrams, each bifurcation diagram consists of exactly one curve which is either a monotone curve or has exactly one turning point where the curve turns to the left. More precisely, we prove the exact multiplicity and ordering properties of positive solutions. Chapter 4, we study above problem with another general nonlinearities satisfying (B1)-(B4). Under hypotheses (B1)-(B4), we prove the exact structure of positive solutions. Chapter 5, we study the exact multiplicity and ordering properties of positive solutions of the p-Laplacian Dirichlet problem. By these results, we can extend the results in chapter 2 from p=2 to any p>1. Another application is to study a stationary singular diffusion problem.