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

1.关于退化椭圆型方程奇异解的结构：(1)研究了各种类型的Hardy-Sobolv-Rellich不等式，并通过找出尖锐的缺失项对它们进行了改进。我们将它们应用于爆破解的研究。(2)研究加权Hardy-Sobolev不等式。对于Caffarelli-Kohn-Nirenberg型权函数，通过增加尖锐的余项，改进了不等式。特别地，对于临界权和超临界情形，得到了新的CKN-不等式。(3)研究了CKN-不等式，并将其推广到临界和超临界情形。这不仅给出了新的嵌入定理，而且为偏微分方程组问题提供了新的框架。我们还建立了关于极值的存在和不存在的结果，最佳常数的渐近行为和极值的一些定性性质。在非平凡函数框架下，应用我们的不等式研究了系数为最佳常数的奇异势的边值问题。在拟线性椭圆型偏微分方程线性化问题上，系统地研究了右端具有强正非线性项的P-调和方程，建立了其极小解的唯一存在性结果。此外，我们还研究了极小解的线性化算子，明确了算子的矫顽力和正性，并建立了分支理论。

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.