Analysis of classes of nonlinear eigenvalue problems on exterior domains.

摘要

In this dissertation, we establish new existence, multiplicity, and uniqueness results on positive radial solutions for classes of steady state reaction diffusion equations on the exterior of a ball. In particular, for the first time in the literature, this thesis focuses on the study of solutions that satisfy a general class of nonlinear boundary conditions on the interior boundary while they approach zero at infinity (far away from the interior boundary). Such nonlinear boundary conditions occur naturally in various applications including models in the study of combustion theory. We restrict our analysis to reactions terms that grow slower than a linear function for large arguments. However, we allow all types of behavior of the reaction terms at the origin (cases when it is positive, zero, as well as negative). New results are also added to ecological systems with Dirichlet boundary conditions on the interior boundary (this is the case when the boundary is cold). We establish our existence and multiplicity results by the method of sub and super solutions and our uniqueness results via deriving a priori estimates for solutions.