Abstract

The aim of this paper is to present the global estimate for gradient of renormalized solutions to the following quasilinear elliptic problem: {−div(A(x,∇u))=μu=0inΩ,on∂Ω, in Lorentz–Morrey spaces, where Ω⊂Rn (n≥2), μ is a finite Radon measure, A is a monotone Carathéodory vector-valued function defined on W1,p0(Ω) and the p-capacity uniform thickness condition is imposed on the complement of our domain Ω. It is remarkable that the local gradient estimates have been proved first by Mingione in [Gradient estimates below the duality exponent, Math. Ann.346 (2010) 571–627] at least for the case 2≤p≤n, where the idea for extending such result to global ones was also proposed in the same paper. Later, the global Lorentz–Morrey and Morrey regularities were obtained by Phuc in [Morrey global bounds and quasilinear Riccati type equations below the natural exponent, J. Math. Pures Appl.102 (2014) 99–123] for regular case p>2−1n. Here in this study, we particularly restrict ourselves to the singular case 3n−22n−1<p≤2−1n. The results are central to generalize our technique of good-λ type bounds in the previous work [M.-P. Tran, Good-λ type bounds of quasilinear elliptic equations for the singular case, Nonlinear Anal.178 (2019) 266–281], where the local gradient estimates of solution to this type of equation were obtained in the Lorentz spaces. Moreover, the proofs of most results in this paper are formulated globally up to the boundary results.