Linear and fully nonlinear elliptic equations with $L_d$-drift

In subdomains of ${\mathbb{R}}^d$ we consider uniformly elliptic equations $H(v(x),Dv(x),D^{2}v(x),x)=0$ with the growth of $H$ with respect to $\mathrm{\mid }Dv\mathrm{\mid }$ controlled by the product of a function from $L_d$ times $\mathrm{\mid }Dv\mathrm{\mid }$. The dependence of $H$ on $x$ is assumed to be of BMO type. Among other things we prove that there exists $d_0\in (d\mathrm{/}2,d)$ such that for any $p\in (d_0,d)$ the equation with prescribed continuous boundary data has a solution in class $W_{p,\text{loc}}^2$. Our results are new even if $H$ is linear.