On the regularity of solutions for degenerate elliptic equations

1.关于退化椭圆型方程解的可去性：(1)当算子的主体为线性时，我们成功地建立了一个充分条件来消除带吸收项的退化椭圆型方程解的可去性。此外，如果算子退化的集合足够光滑，那么这一条件也是必要的。(2)我们把我们关于线性算子的结果推广到了主部是非线性的情况。为了克服非线性带来的困难，我们使用了吸收项和修正电容的共轭函数(勒让德变换)。关于具临界非线性项的拟线性椭圆型方程：(1)成功地研究了具临界非线性项的拟线性椭圆型方程，并作为应用，确定了各类带权的加权Sobolev不等式的最佳常数。我们还证明了最佳常数和极值函数的存在性本质上依赖于算子的退化程度。(2)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，建立了带权的乘性Sobolev不等式。

1. On the Removable singularities of solutions for degenerate ellptic equations :(1) When the principal part of the operator is linear, we have succeeded in establishing a sufficient condition in order to remove singularities of solutions of degenerate elliptic equations with absorption terms. Moreover, if the set on which the operator may be degenerate is smooth enough, then this condition is also necessary.(2) We have extended our results on the linear operators to the case when the principal part is nonlinear. In order to overcome the difficulties caused from nonlinearlity, we used the conjugate function (Legendre transform) of the absorption term and modified capacities.2. On the quasilinear elliptic equations with critical nonlinear terms :(1) Quasilinear elliptic equations with critical nonlinear terms were successfully investigated, and as an application, the best constants for various types of weighted Sobolev inequalities with weights were determined. We also showed that the best constants and the existence of extremal functions essentially depend on the degeneracy of the operators in a very subtle way.(2) The existence of bounded solutions for degenerate elliptic equations were studied. To study further regularities, multiplicative Sobolev inequalities with weights were established.