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

1. 关于退化椭圆方程奇异解的结构：(1)研究了各种类型的Hardy-Sobolev-Rellich不等式，并通过找出明显的缺失项对其进行了改进。我们将它们应用于爆破溶液的研究。(2)研究了加权Hardy-Sobolev不等式。对于Caffarelli-Kohn-Nirenberg型权函数，通过添加尖锐的余数项，不等式得到了改善。特别是对于临界权值情况和超临界情况，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况。这不仅给出了新的嵌入定理，而且为PDE问题提供了新的框架。我们还建立了极值的存在性和不存在性结果、最佳常数的渐近性以及极值的一些定性性质。在非平凡泛函框架下，应用我们的不等式研究了以最佳常数为系数的奇异势边值问题。拟线性椭圆偏微分方程的线性化问题。系统地研究了具有强正非线性项的p -调和方程，建立了最小解的唯一存在性结果。此外，我们对最小解下的线性化算子进行了很好的研究，明确了算子的矫顽性和正性，构造了分岔理论。

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.