Regularizing effect of the lower-order terms in elliptic problems with Orlicz growth

Under various conditions on the data we analyse how appearence of lower order terms affects the gradient estimates on solutions to a general nonlinear elliptic equation of the form $$-\mathrm{d}\mathrm{i}\mathrm{v}a(x,Du)+b(x,u)=\mu$$ with data $\mu$ not belonging to the dual of the natural energy space but to Lorentz/Morrey-type spaces. The growth of the leading part of the operator is governed by a function of Orlicz-type, whereas the lower-order term satisfies the sign condition and is minorized with some convex function, whose speed of growth modulates the regularization of the solutions.