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.

中文翻译：

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内部和边界 W1,-Schrödinger 型拟线性椭圆方程的估计

我们在变分法的设置中考虑由椭圆Schr\"odinger方程$-\Delta u +Vu=0$自然得到的非线性椭圆方程，得到弱梯度的$L^q$-估计解决方案。特别是，我们通过使用不同的方法在非线性设置中推广了 Shen 在 [Ann. Inst. Fourier 45 (1995), no. 2, 513--546] 中的结果。这允许我们考虑不连续系数一个小的 BMO 半范数和非光滑边界，可能不是 Lipschitz 连续的。