Global gradient estimates for very singular quasilinear elliptic equations with non-divergence data

Abstract

This paper continues the development of regularity results for quasilinear elliptic equations −div(A(x,∇u))=μinΩ,andu=0on∂Ω,in Lorentz and Lorentz–Morrey spaces, where Ω⊂Rn (n≥2); A is a monotone Carathéodory vector valued operator acting between W01,p(Ω) and its dual W−1,p′(Ω); and μ is a datum in some Lebesgue space Lm(Ω), for m<p′. It emphasizes that in this paper, we restrict our study to the case of ‘very singular’ when 1<p≤3n−22n−1, and under mild assumption that the p-capacity uniform thickness condition is imposed on the complement of domain Ω. There are two main results obtained in our study pertaining to the global gradient estimates of solutions in Lorentz and Lorentz–Morrey spaces involving the use of maximal and fractional maximal operators. The idea for writing this working paper comes directly from the recent results by others in the same research topic, where global estimates for gradient of solutions for the ‘very singular’ case still remains a challenge, specifically related to Lorentz and Lorentz–Morrey spaces.