The investigation of positive steady states to reaction diffusion models in bounded domains with Dirichlet boundary conditions has been of great interest since the 1960’s. We study reaction diffusion models where the reaction term is negative at the origin. In the literature, such problems are referred to as semipositone problems and have been studied for the last 30 years. In this dissertation, we extend the theory of semipositone problems to classes of singular semipositone problems where the reaction term has singularities at certain locations in the domain. In particular, we consider problems where the reaction term approaches negative infinity at these locations. We establish several existence results when the domain is a smooth bounded region or an exterior domain. Some uniqueness results are also obtained. Our existence results are achieved by the method of sub and super solutions, while our uniqueness results are proved by establishing a priori estimates and analyzing structural properties of the solution. We also extend many of our results to systems.

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.

Submitted by Erika Demachki (erikademachki@gmail.com) on 2017-03-09T21:07:51Z No. of bitstreams: 2 Dissertação - Steffânio Moreno de Sousa - 2017.pdf: 1402439 bytes, checksum: 6a4985baeae0454d7088588c3c87e295 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Approved for entry into archive by Luciana Ferreira (lucgeral@gmail.com) on 2017-03-10T10:48:27Z (GMT) No. of bitstreams: 2 Dissertação - Steffânio Moreno de Sousa - 2017.pdf: 1402439 bytes, checksum: 6a4985baeae0454d7088588c3c87e295 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Made available in DSpace on 2017-03-10T10:48:28Z (GMT). No. of bitstreams: 2 Dissertação - Steffânio Moreno de Sousa - 2017.pdf: 1402439 bytes, checksum: 6a4985baeae0454d7088588c3c87e295 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Previous issue date: 2017-03-03 / Conselho Nacional de Pesquisa e Desenvolvimento Científico e Tecnológico - CNPq / In this work we study existence and regularity of non-negative solution of elliptical systems of the type 8>< >: 􀀀Dpu = f (x;u;v); 􀀀Dqv = g(x;u;v) in W; u;v > 0 in W; where 1 < p;q <N, W IRN is a bounded domain with smooth boundary ¶W, and f ;g are of the type singular-convex or W=IRN and f ;g are concave-convex. In case W we will find solutions that cancel out ¶W while in the case W = IRN solutions in C1(IRN)\L¥(IRN).We will use the Galerkin method and the comparison principle. In case W = IRN we will use the method of sub and super solutions, variational methods and principles of maximum. / Neste trabalho estudaremos existência e regularidade de soluções positivas de sistemas elípticos do tipo 8>< >: 􀀀Dpu = f (x;u;v); 􀀀Dqv = g(x;u;v) in W u;v > 0 in W; onde 1 < p;q < N, W IRN é um domínio limitado com fronteira ¶W regular, e f , g são do tipo convexo-singular ou W = IRN e f , g são do tipo côncavo-convexo. No caso W limitado encontraremos soluções que se anulam em ¶W, enquanto que, no caso W = IRN as soluções em C1(IRN) \ L¥(IRN). No caso f , g singulares utilizaremos o método de Galerkin e princípio de comparação. No caso W = IRN utilizaremos o método de sub e super-soluções, métodos variacionais e princípios de máximo.