Regularity of weak solutions of elliptic and parabolic equations with some critical or supercritical potentials.

We prove Hölder continuity of weak solutions of the uniformly elliptic and parabolic equations (0.1) ∂ i ( a i j ( x ) ∂ j u ( x ) ) − A | x | 2 + β u ( x ) = 0 ( A > 0 , β ≥ 0 ) , (0.2) ∂ i ( a i j ( x , t ) ∂ j u ( x , t ) ) − A | x | 2 + β u ( x , t ) − ∂ t u ( x , t ) = 0 ( A > 0 , β ≥ 0 ) , 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 [3] treated the case when A < 0 , ( a i j ) = I and β = 0 for which they show that there does not exist any regular positive solution or singular positive solutions, depending on the size of | A | . When A > 0 , β = 0 and ( a i j ) = I , P. D. Milman and Y. A. Semenov [7,8] obtain bounds for the heat kernel. [ABSTRACT FROM AUTHOR]