In this article we investigate the existence of a solution to a semilinear,elliptic, partial differential equation with distributional coefficients anddata. The problem we consider is a generalization of the Lichnerowicz equationthat one encounters in studying the constraint equations in general relativity.Our method for solving this problem consists of solving a net of regularized,semilinear problems with data obtained by smoothing the original,distributional coefficients. In order to solve these regularized problems, wedevelop a priori pointwise bounds and sub- and super-solutions and then apply afixed-point argument for order-preserving maps. We then show that the net ofsolutions obtained through this process satisfies certain decay estimates bydetermining estimates for the sub- and super-solutions and by utilizingclassical, a priori elliptic estimates. The estimates for this net of solutionsallow us to regard this collection of functions as a solution in aColombeau-type algebra. We motivate this Colombeau algebra framework by firstsolving an ill-posed critical exponent problem. To solve this ill-posedproblem, we use a collection of smooth, "approximating" problems and then usethe resulting sequence of solutions and a compactness argument to obtain asolution to the original problem. This approach is modeled after the moregeneral Colombeau framework that we develop, and it conveys the potential thatsolutions in these abstract spaces have for obtaining classical solutions toill-posed nonlinear problems with irregular data.
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