Uniform estimates for some degenerating quasilinear elliptic equations and a bound on the Harnack constant for linear equations

Abstract

Let u be any smooth solution of −εa_iju_xixj+a_iju_xiu_xj+b_iu_xi=εc in B₁={x:|x|<1}, where ε is a small parameter. We prove some uniform estimates on u in W^1,q under weak assumptions on the coefficients a_ij, b_i and c. This is motivated by the study of exponential behaviors for solutions of linear elliptic equations −εa_ijv_xixj+b_iv_xi+cv=0, as ε goes to 0. For this equation, our result provides an estimate of Harnack type sup _Brv≤Cinf _Brv, with C=e^−krα/ε for some α≤1.