Executive Summary : | Semilinear elliptic boundary value problems with Dirichlet boundary condition are commonly used in nonlinear heat generation, combustion theory, chemical reactor theory, and population dynamics. Positone problems are a class of such problems where the reaction term is positive, and solutions can be obtained using topological or variational methods. However, non-positone problems often require additional work to obtain positivity. In recent years, authors have focused on semipositone problems on smooth bounded domains in Rn with negative reaction terms at the origin. They have also studied the extension of this problem to exterior domains in Rn, assuming that the weight function is radial. The Kelvin transformation reduces the problem to a two-point boundary value problem when the dimension is greater than 2. Some authors have allowed the weight function to be non-radial and established the existence of positive solutions for various classes of reaction terms by studying the PDE in its original setting without reducing it to a two-point boundary value problem. However, restrictions on the dimension continue to appear. The proposed project aims to study semipositone problems on exterior domains in R2 for various classes of reaction terms, allowing the weight function to be non-radial whenever possible. The researchers aim to investigate whether this result continues to hold when the reaction term is sign changing, particularly for semipositone problems. The project will also explore positone problems remaining open in exterior domains in R2 and both positone and non-positone problems on more general unbounded domains in R2. |