On the structure of singular solutions for degenerate elliptic equations

1.关于退化椭圆型方程奇异解的结构：(1)系统地研究了右端具强正非线性项的p-调和方程，建立了极小解的唯一存在性结果。此外，我们还研究了极小解上的线性化算子。证明了线性化算子的基本性质，明确了算子的矫顽性和正性，并将其应用于构造分支理论。(2)研究了各种类型的广义Hardy-Sobolev-Rellich不等式，并通过找出尖锐的缺失项对其进行了改进。2.关于真退化椭圆方程解的正则性：研究了退化椭圆型方程有界解的存在性。为了进一步研究正则性，建立了带权重的乘性Sobolev不等式。3.关于具有临界非线性项和奇异解的变分问题：利用上述结果，研究了奇异变分问题。4.研究了退化椭圆算子的位势理论：多重抛物算子，建立了广义中值性质。

1.On the structue of singular solutions for degenerate ellptic equations :(1)We studied p-harmomic equation with strong positive nonlinear terms in the right-hand side systematically, and established the unique existence results of minimal solutions. Moreover we studied very well linearized operators at the minimal solutions. Proving fundamental properties of the linearized operators, we made clear the coercivity and positivity of the operator and applied them to construct the theory of Bifurcation.(2)We studied various type of generalized Hardy-Sobolev-Rellich inequalities, and improved them by finding out sharp missing terms. We applied them to the study of Blow-up solutions.2.On the regularity of solutions for genuinely degenerated elliptic equations :The existence of bounded solutions for degenerate elliptic equations were studied. To study further regularities, multiplicative Sobolev inequalities with weghts were established.3.On the variational problems with critical nonlinear terms and singular solutions :Using the results in the above, singular variational problems were investigated.4.The potential theory for degenerate elliptic operators :Multi-parabolic operator was studied and generalized mean-value property was established.