Measure data problems, lower-order terms and interpolation effects

We deal with the solutions to nonlinear elliptic equations of the form $$ −div a(x,Du)+g(x,u)=f, $$ with $f$ being just a summable function, under standard growth conditions on $g$ and $a$. We prove general local decay estimates for level sets of the gradient of solutions in turn implying very general estimates in rearrangement and non-rearrangement function spaces, up to Lorentz–Morrey spaces. The results obtained are in clear accordance with the classical Gagliardo–Nirenberg interpolation theory.