The research on degenerate elliptic partial differential equations by the method of real analysis

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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 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 weights were established.3. On the quasilinear elliptic equations with critical nonlinear terms: 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.4. Missing terms in the generalized Hardy inequalities were investigated in connection with 3.