Elliptic equations in Sobolev spaces with Morrey drift and the zeroth-order coefficients

Abstract

We consider elliptic equations with operators $L=a^{ij}D_{ij}+b^{i}D_{i}-c$ with $a$ being almost in VMO, $b$ in a Morrey class containing $ L_{d}$, and $c\geq0$ in a Morrey class containing $L_{d/2}$. We prove the solvability in Sobolev spaces of $Lu=f\in L_{p}$ in bounded $C^{1,1}$-domains, and of $\lambda u-Lu=f$ in the whole space for any $\lambda>0$. Weak uniqueness of the martingale problem associated with such operators is also discussed.