| The focus of this work is the analysis of semi-linear elliptic POE related to prescrbed- curvature problems or having critical Sobolev nonlinearities. Three main theorems will be given, one in each chapter. In the first chapter, a precise asymptotic profile for a sequence of blow-up solutions to a prescribed-Gaussian-curvature equation will be obtained. In the second chapter, a Harnack-type inequality will be obtained for a prescribed-scalar-curvature equation on a domain with boundary. As a corollary to the Harnack-type inequality, an energy estimate for solutions to the prescribed-scalar curvature POE is obtained. In the final chapter, a classification theorem is given for positive solutions of a semi-linear elliptic system of POE on the upper half of Euclidean space. |
