  We consider nonlinear elliptic equations that are naturally obtained from the
elliptic Schr\"odinger equation $-\Delta u +Vu=0$ in the setting of the
calculus of variations, and obtain $L^q$-estimates for the gradient of weak
solutions. In particular, we generalize a result of Shen in [Ann. Inst. Fourier
45 (1995), no. 2, 513--546] in the nonlinear setting by using a different
approach. This allows us to consider discontinuous coefficients with a small
BMO semi-norm and non-smooth boundaries which might not be Lipschitz
continuous.
