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

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We prove Hölder continuity of weak solutions of the uniformly elliptic and parabolic equations %Δu-A|x|²⁺βu=0,(β≥ 0), and variable second order term coefficients case %% equation01 ∂ᵢ (aij(x) ∂ⱼu(x)) - A|x|²⁺β u(x) =0 (A>0,β≥ 0), equation equation02 ∂ᵢ (aij(x,t) ∂ⱼu(x,t)) - A|x|²⁺β u(x,t)-∂ₜu(x,t) =0 (A>0,β≥ 0), 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 Baras1984 treated the case when A<0, (aij)=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 (aij)=I, P. D. Milman and Y. A. Semenov Milman2003Milman2004 obtain bounds for the heat kernel.