$L^q$-regularity for nonlinear elliptic equations with Schr\"odinger-type lower order terms

TL;DR

This paper establishes interior and boundary $L^q$ estimates for the gradient of solutions to nonlinear elliptic equations with Schrödinger-type lower order terms, under sharp regularity conditions, without relying on Fefferman-Phong inequalities.

Contribution

It provides new $L^q$ regularity results for nonlinear elliptic equations with potentials satisfying reverse Hölder conditions, avoiding traditional linear techniques.

Findings

Interior and boundary $L^q$ estimates obtained

Results hold under sharp regularity conditions

Proof avoids Fefferman-Phong inequalities

Abstract

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].