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.

We study positive solutions to nonlinear elliptic systems of the form: \begin{eqnarray*} -\Delta u =\lambda f(v) \mbox{ in }\Omega\\-\Delta v =\lambda g(u) \mbox{ in }\Omega\\\quad~~ u=0=v \mbox{ on }\partial\Omega \end{eqnarray*} where $\Delta u$ is the Laplacian of $u$, $\lambda$ is a positive parameter and $\Omega$ is a bounded domain in $R^n$ with smooth boundary $\partial\Omega$. In particular, we will analyze the combined effects of the nonlinearities on the existence and multiplicity of positive solutions. We also study systems with multiparameters and stronger coupling. We extend our results to $p$-$q$-Laplacian systems and to $n\times n$ systems. We mainly use sub- and super-solutions to prove our results.

We study positive solutions to classes of nonlinear elliptic singular problems of the form: -Δpu = λ g(u) uα in Ω u = 0 on δΩ where Ω is a bounded domain in ℝN, N ≥ 1 with smooth boundary δΩ, &lambda¸ is a positive parameter, α ∈(0; 1), Δpu := div(⌊∇u⌋p-2 ∇u); p > 1 is the p-Laplacian operator, and g is a smooth function. Such elliptic problems naturally arise in the study of steady state reaction diffusion processes. In particular, we will be interested in the challenging new class of problems when g(0) < 0 (hence lims→0+g(s) sα = - ∞ which we refer to as infinite semipositone problems. Our focus is on existence results. We obtain results for the single equation case as well as to the case of systems. We use the method of sub-super solutions to prove our results. The results in this dissertation provide a solid foundation for the analysis of such infinite semipositone problems.

In this thesis we study two reaction-diffusion models that have been used to analyze the existence of alternate stable states in ecosystems. The first model describes the steady states of a logistic growth model with grazing in a spatially homogeneous ecosystem. It also describes the dynamics of the fish population with natural predation. The second model describes phosphorus cycling in stratified lakes. The same equation has also been used to describe the colonization of barren soils in drylands by vegetation. In this study we discuss the existence of multiple positive solutions, leading to the occurrence of S-shaped bifurcation curves. We were able to show that both the models have alternate stable states for certain ranges of parameter values. We also introduce a constant yield harvesting term to the first model and discuss the existence of positive solutions including the occurrence of a Sigma-shaped bifurcation curve in the case of a one-dimensional model. Again we were able to establish that for certain ranges of parameter values the model has alternate stable states. Thus we establish analytically that the above models are capable of describing the phenomena of alternate stable states in ecological systems. We prove our results by the method of sub-super solutions and quadrature method.