We study positive solutions to classes of steady state reaction diffusion equations that arise naturally in applications. In particular, we study models arising from population dynamics and combustion theory. The main focus of this dissertation is the mathematical analysis of a challenging new class of problems when a certain nonlinear boundary condition is satisfied. In particular, we establish existence and multiplicity results by making use of the Quadrature method, the method of sub-super solutions, and degree theory. The results in this dissertation provide a significant contribution towards the analysis of elliptic boundary value problems with nonlinear boundary conditions.

The method of dual steepest descent is used to solve ordinary differential equations with nonlinear boundary conditions. A general boundary condition is B(u) = 0 where where B is a continuous functional on the nth order Sobolev space Hn[0.1J. If F:HnCO,l] —• L2[0,1] represents a 2 differential equation, define *(u) = 1/2 IIF < u) li and £(u) = 1/2 l!B(u)ll2. Steepest descent is applied to the functional 2 £ a * + £. Two special cases are considered. If f:lR —• R is C^(2), a Type I boundary condition is defined by B(u) = f(u(0),u(1)). Given K: [0,1}xR—•and g: [0,1] —• R of bounded variation, a Type II boundary condition is B(u) = ƒ1/0K(x,u(x))dg(x).

In this dissertation, we study the existence and nonexistence of positive solutions to some classes of singular p-Laplacian boundary value problems with a parameter. In the first study, we discuss positive solutions for a class of sublinear Dirichlet p- Laplacian equations and systems with sign-changing coefficients on a bounded domain of Rn via Schauder Fixed Point Theorem and the method of sub- and supersolutions. Under certain conditions, we show the existence of positive solutions when the parameter is large and nonexistence when the parameter is small. In the second study, we discuss positive radial solutions for a class of superlinear p- Laplacian problems with nonlinear boundary conditions on an exterior domain via degree theory and fixed point approach. Under certain conditions, we show the existence of positive solutions when the paprameter is small and nonexistence when the paramter is large. Our results provide extensions of corresponding ones in the literature from the Laplacian to the p-Laplacian, and can be applied to the challenging infinite semipositone case