  We consider nonlinear elliptic equations of the $p$-Laplacian type with lower
order terms which involve nonnegative potentials satisfying a reverse H\"older
type condition. Then we obtain interior and boundary $L^q$ estimates for the
gradient of weak solutions and the lower order terms, independently, under
sharp regularity conditions on the coefficients and the boundaries. In
particular, the proof in this paper does not employ Fefferman-Phong type
inequalities which are essential tools in the linear cases in [3,47].
