Semilinear elliptic boundary value problems on unbounded domains

Abstract

Study of positive solutions to problems of the form (0.1) for various classes of newlinereaction terms has been of interest to many in the recent past. There are several newlineexistence, uniqueness and multiplicity results available in the recent literature. In many of these works, authors assume the weight function K to be radial and study newlinepositive radial solutions to (0.1) by reducing the problem to a two point boundary value problem via the Kelvin transformation. In this thesis, we extend some of these studies by considering weights which are nonradial. We study the problem (0.1) in its original setting for various classes of reaction terms and prove some existence, nonexistence and multiplicity of positive solutions. We allow the reaction term to be negative at the origin. When f(0) lt 0, problems of the form (0.1) are known as semipositone problems and they are known for the difficulty in establishing the positivity of the solution. When f has a superlinear growth at 1, we prove the existence of a positive solution for small values of using variational methods. A nonexistence result is also established for large values of . In order to study (0.1) with asymptotically linear reaction terms, we first prove the compactness of the solution operator for a class of singular semilinear elliptic problems on the exterior of a ball in Rn, ngt2. Compactness of solution operators for similar problems in Rn, n gt 2 are also established. Using these compactness results and employing Tychonoff fixed point theorem, we prove the existence of a positive solution to (0.1) and similar problems in Rn, for a range of , when f has an asymptotically linear growth. newline