Interior and boundary $W^{1,q}$-estimates for quasi-linear elliptic equations of Schr\"odinger type

TL;DR

This paper establishes interior and boundary gradient estimates in $W^{1,q}$ for nonlinear elliptic equations derived from Schr"odinger equations, accommodating discontinuous coefficients and irregular boundaries.

Contribution

It generalizes Shen's linear results to nonlinear equations, introducing a new approach to handle discontinuous coefficients and non-smooth boundaries.

Findings

Established $L^q$-estimates for weak solutions' gradients.

Extended results to equations with discontinuous coefficients and irregular boundaries.

Provided a novel method for nonlinear elliptic equations of Schr"odinger type.

Abstract

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.