Good-[formula omitted] type bounds of quasilinear elliptic equations for the singular case

Abstract

In this paper, we study the good-λ type bounds for renormalized solutions to nonlinear elliptic problem: −div(A(x,∇u))=μinΩ,u=0on∂Ω.where Ω⊂Rn, μ is a finite Radon measure and A is a monotone Carathédory vector valued function defined on W01,p(Ω). The operator A satisfies growth and monotonicity conditions, and the p-capacity uniform thickness condition is imposed on Rn∖Ω, for the singular case 3n−22n−1<p≤2−1n. In fact, the same good-λ type estimates were also studied by Quoc-Hung Nguyen and Nguyen Cong Phuc. For instance, in Nguyen and Phuc (0000) and Nguyen (0000) authors’ method was also confined to the case of 3n−22n−1<p≤2−1n but under the assumption of Ω is the Reifenberg flat domain and the coefficients of A have small BMO (bounded mean oscillation) semi-norms. Otherwise, the same problem was considered in Phuc (2014) in the regular case of p>2−1n. In this paper, we extend their results, taking into account the case 3n−22n−1<p≤2−1n and without the hypothesis of Reifenberg flat domain on Ω and small BMO semi-norms of A. Moreover, in rest of this paper, we also give the proof of the boundedness property of maximal function on Lorentz spaces and also the global gradient estimates of solution.