Analysis of positive solutions for classes of nonlinear boundary value problems

Abstract

In this dissertation, we study the existence and uniqueness of positive solutions for classes of nonlinear boundary value problems. In the first study, we the 𝑝-superlinear case, we prove the existence of a large positive solution when a parameter is small and if, in addition, the reaction term satisfies a concavity-like condition at the origin, the existence of two positive solutions for a certain range of the parameter. In the 𝑝-sublinear case, we establish the existence of a large positive solution when a parameter is large. We also investigate the number of positive solutions for the general 𝜙-Laplacian with nonlinear boundary conditions when the reaction termis positive. Our results can be applied to the challenging infinite semipositone case and complement or extend previous work in the literature. Our approach depends on the Krasnoselskii’s fixed point in a Banach space, degree theory, and comparison principles.