On divergence form second-order PDEs with growing coefficients in $W^{1}_{p}$ spaces without weights

Abstract

We consider second-order divergence form uniformly parabolic and elliptic PDEs with bounded and $VMO_{x}$ leading coefficients and possibly linearly growing lower-order coefficients. We look for solutions which are summable to the $p$th power with respect to the usual Lebesgue measure along with their first derivatives with respect to the spatial variables.