Regularity of Weak Solutions of Elliptic and Parabolic Equations with Some Critical or Supercritical Potentials

We prove H"older continuity of weak solutions of the uniformly elliptic and parabolic equations %$\mathrm{\Delta }u-\frac{A}{\mathrm{\mid }x{\mathrm{\mid }}^{2+\beta }}u=0,(\beta \ge 0)$, and variable second order term coefficients case $%% \begin{equation}\label{01} \partial_{i} (a_{ij}(x) \partial_{j}u(x)) - \frac{A}{|x|^{2+\beta}} u(x) =0\quad (A>0,\quad\beta\geq 0), \end{equation} \begin{equation}\label{02} \partial_{i} (a_{ij}(x,t) \partial_{j}u(x,t)) - \frac{A}{|x|^{2+\beta}} u(x,t)-\partial_{t}u(x,t) =0\quad (A>0,\quad\beta\geq 0), \end{equation} with critical or supercritical 0-order term coefficients which are beyond De Giorgi-Nash-Moser’s Theory. We also prove, in some special cases, weak solutions are even differentiable. Previously P. Baras and J. A. Goldstein \cite{Baras1984} treated the case when $A<0$, $(a_{ij})=I$ and $\beta =0$ for which they show that there does not exist any regular positive solution or singular positive solutions, depending on the size of $\mathrm{\mid }A\mathrm{\mid }$. When $A>0$, $\beta =0$ and $(a_{ij})=I$, P. D. Milman and Y. A. Semenov \cite{Milman2003}\cite{Milman2004} obtain bounds for the heat kernel.