Elliptic Boundary-Value Problems

Abstract

The first two sections of this chapter are devoted to the development of the monotone sequences for elliptic boundary-value problems and to the justification that the limits of the monotone sequences are classical solutions. Various sufficient conditions are given in later sections to ensure the uniqueness and the multiplicity of positive solutions. When the reaction function involves a parameter the spectrum of positive solutions and the dependence of the solutions on the parameter are analyzed using the method of upper and lower solutions. An extension of the monotone method to a class of integroelliptic boundary-value problems is included in the discussion. The results for elliptic boundary-value problems are applied to a number of specific models as applications of the theory, including a discussion of the existence of multiple positive solutions.