On the linearization of quasilinear degenerate elliptic equations and the structure of singular solutions

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1. On the structure of singular solutions for degenerate elliptic equations:(1) We studied various types of Hardy-Sobolev-Rellich inequalities, and improved them by finding out sharp missing terms. We applied them to the study of Blow-up solutions.(2) We studied the weighted Hardy-Sobolev inequalities. For the Caffarelli-Kohn-Nirenberg type weight functions, the inequalities have been improved by adding sharp remainder terms. In particular for the critical weight case and the supercritical case new CKN-inequalities were obtained.(3) We studied CKN-inequalities and extended them to the critical and supercritical case. This gives not only new imbedding theorems but also new framework for the PDE problems. We also established Existence and Nonexistence results on the extremals, asymptotic behaviors of the best constant and some qualitative properties of extremals. We applied our inequalities to study the boundary value problems having singular potential with the best constant as its coefficient in nontrivial functional framework.2. On the linearization of quasi-linear elliptic PDE.P-harmonic equation with strong positive nonlinear terms in the right-hand side has been studied systematically and established the unique existence results of minimal solutions. Moreover we studied very well linearized operators at the minimal solutions to made clear the coercivity and positivity of the operator and to construct the theory of Bifurcation.