  We establish the existence and uniqueness of solutions of fully nonlinear
elliptic second-order equations like $H(v,Dv,D^{2}v,x)=0$ in smooth domains
without requiring $H$ to be convex or concave with respect to the second-order
derivatives. Apart from ellipticity nothing is required of $H$ at points at
which $|D^{2}v|\leq K$, where $K$ is any given constant. For large $|D^{2}v|$
some kind of relaxed convexity assumption with respect to $D^{2}v$ mixed with a
VMO condition with respect to $x$ are still imposed. The solutions are sought
in Sobolev classes.
