In this talk, we will discuss global higher integrability for the gradient of weak solutions to quasilinear equations of $p$-Schrödinger type

$$-\mathrm{div}\,(|Du|^{p-2}Du) + V|u|^{p-2}u =-\mathrm{div}\,(|F|^{p-2}F)\ \ \text{in }\Omega, \ \ u=0\ \ \text{on }\Omega$$.

Here, $V$ called the potential function is basically nonnegative and the domain $\Omega\subset\mathbb R^n$ is bounded and might be a non-graph domain such as fractal domain. In particular,  we present a suitable condition on $V$ to obtain higher integrability of $Du$. This work is a generalization of the result in [Z. Shen, $L^p$ estimates for Schrödinger operators with certain potentials, Ann. Inst. Fourier (Grenoble) 45 (1995), no. 2, 513-546] to nonlinear equations in non-smooth domains.

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