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We consider second order linear elliptic equations −div(A(x)∇u) + b(x) • ∇u = 0 with a singular vector field b. We prove a refined subsolution estimate, which contains a precise dependence of the quantities of b, for weak subsolutions and a weak Harnack inequality for weak supersolutions under certain assumptions on b.

We study the existence of positive solutions to quasilinear elliptic equations of the typein the sub-natural growth case 0 < q < p − 1, where ∆ p u = ∇ · (|∇u| p−2 ∇u) is the p-Laplacian with 1 < p < n, and σ and µ are nonnegative Radon measures on R n . We construct minimal generalized solutions under certain generalized energy conditions on σ and µ. To prove this, we give new estimates for interaction between measures. We also construct solutions to equations with several sub-natural growth terms using the same methods.✩ This is a pre-print of an article published in Nonlinear Analysis. The final authenticated version is available online at: https://doi.

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