An inhomogeneous semilinear equation in entire space

This dissertation is a study of the following partial differential equation which arises in the analysis of measure valued stochastic processes known as super-Brownian motion {dollar}{dollar}left{lcub}eqalign{lcub}&Delta u + usp{lcub}p{rcub} + f(x) = 0qquad {lcub}rm in{rcub} IRsp{lcub}n{rcub}cr&u > 0cr{rcub}right.{dollar}{dollar}where {dollar}Delta = sumsbsp{lcub}i = 1{rcub}{lcub}n{rcub}{lcub}partialsp2overpartial xsbsp{lcub}i{rcub}{lcub}2{rcub}{rcub}{dollar} is the Laplacian operator, {dollar}p > 1, n in rm I!N{dollar} with {dollar}nge 3{dollar} and {dollar}fin C(IRsp{lcub}n{rcub}){dollar} with {dollar}fge 0, fnotequiv 0.{dollar}; The question of existence of solutions to this equation is treated in two parts: Necessary conditions, and secondly sufficient conditions. Integral a priori estimates are established leading to necessary conditions on the exponent p and the inhomogeneous term f. From these results follows a nonexistence theorem involving the magnitude of the inhomogeneous term f.; In the second part, existence theorems are established using mainly the super-subsolution method. Explicit conditions on the inhomogeneous term limit its size to ensure the existence of solutions. Three existence theorems are presented yielding respectively slow decaying solutions, fast decaying solutions and solutions which lie in a neighborhood of the solutions of the corresponding homogeneous equation.; Lastly, the question of radial symmetry of this equation is treated. Given that the inhomogeneous term is radially symmetric about the origin and satisfies two monotonicity conditions, then all solutions of this equation are shown to be also radial.