Weighted Orlicz gradient estimates for the class of singular p-Laplace system

Abstract

Let $n\in \{2,3,4,\dots \}$, $N\in \{1,2,3,\dots \}$ and $p\in (1,2-\frac{1}{n}]$. Let $\beta \in (1,\infty )$ be such that \[ \frac{np}{n-p}<β'<\frac{n}{n(2-p)-1} \] and $f\in L^{\beta }({\mathbb{R}}^n;{\mathbb{R}}^N)$. Consider the $p$-Laplace system \[ -Δ_p u=-\operatorname{div}\big(|Du|^{p-2}Du\big)=f \quad in \quad \mathbb R^n. \] We obtain a weighted gradient estimate for distributional solutions of this system.

29 pages, 0 figures